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motion, and consequently reduce the number of degrees of freedom in the .... the nodes of both meshes are superposed along their common boundary. .... during the Eulerian step, and the calculation of the volume of material ...... K23. K33. ⎤. ⎦ with Kij = K1 ij + K2 ij + K3 ij. In the fluid part. K1 ii = ∫. V. ∂N. ∂xj μ. ∂NT. ∂xj.
Arbitrary Lagrangian-Eulerian and Fluid –Structure Interaction

Arbitrary Lagrangian-Eulerian and Fluid–structure Interaction Numerical Simulation

Edited by Mhamed Souli David J. Benson

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd 2010 The rights of Mhamed Souli and David J. Benson to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Arbitrary Lagrangian-Eulerian and fluid–structure interaction : numerical simulation / edited by Mhamed Souli, David J. Benson. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-131-5 1. Fluid-structure interaction. I. Souli, M. II. Benson, D. J. (David J.), 1955TA357.5.F58A44 2009 624.1'71--dc22 2009041738 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-131-5 Editorial services provided by Aptara Corporation, New Delhi, India Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mhamed S OULI Chapter 1. Introduction to Arbitrary Lagrangian–Eulerian in Finite Element Methods David J. B ENSON 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . 1.2. Governing equations . . . . . . . . . . . . . . . 1.3. Operator splitting . . . . . . . . . . . . . . . . . 1.4. The Lagrangian step . . . . . . . . . . . . . . . 1.4.1. Governing equations . . . . . . . . . . . . 1.4.2. The central difference method . . . . . 1.4.3. Element formulation . . . . . . . . . . . . 1.4.4. Hourglass modes . . . . . . . . . . . . . . 1.4.5. Stress rates . . . . . . . . . . . . . . . . . . 1.4.6. Shock viscosity . . . . . . . . . . . . . . . . 1.4.6.1. von Neumann–Richtmyer . . . . . 1.4.6.2. Standard quadratic and linear formulation . . . . . . . . . . . . . . . 1.4.6.3. Effect on time step size . . . . . . . 1.4.7. Mixture theories . . . . . . . . . . . . . . . 1.4.7.1. Mean strain rate mixture theory 1.4.7.2. Mean stress mixture theory . . . 1.5. Mesh relaxation . . . . . . . . . . . . . . . . . .

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1.6. The Eulerian step . . . . . . . . . . . . . . . . . . . 1.6.1. Transport in one dimension . . . . . . . . . 1.6.2. Multidimensional transport by operator splitting . . . . . . . . . . . . . . . . . . . . . . 1.6.3. Multidimensional transport on unstructured meshes . . . . . . . . . . . . . 1.6.4. Momentum transport . . . . . . . . . . . . . 1.6.4.1. Momentum transport using a dual mesh in one dimension . . . . . . . . . 1.6.4.2. Element-centered transport in one dimension . . . . . . . . . . . . . . . . . . 1.6.5. Interface reconstruction . . . . . . . . . . . 1.6.5.1. Lagrangian methods . . . . . . . . . . 1.6.5.2. Level set methods . . . . . . . . . . . . 1.6.5.3. Volume of fluid methods . . . . . . . . 1.7. Future research directions . . . . . . . . . . . . . 1.8. Bibliography . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Fluid–Structure Interaction: Application to Dynamic Problems . . . . . . . . . . . Mhamed S OULI

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2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. General ALE description of Navier–Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Fluid–structure interaction . . . . . . . . . . . . . . 2.3.1. Contact algorithms for fluid–structure interaction problems . . . . . . . . . . . . . . . . 2.3.2. Euler–Lagrange coupling . . . . . . . . . . . . 2.3.3. Damping in the coupling . . . . . . . . . . . . . 2.4. Numerical applications . . . . . . . . . . . . . . . . . 2.4.1. Piston problem . . . . . . . . . . . . . . . . . . . . 2.4.2. Two-dimensional slamming modeling . . . . 2.4.2.1. Numerical approach of a two-dimensional slamming problem . . 2.4.2.2. Numerical approach for rigid structure 2.4.3. Airbag deployment . . . . . . . . . . . . . . . . . 2.4.4. Sloshing tank problem . . . . . . . . . . . . . .

51 54 57 59 62 69 72 72 74 76 76 81 84

Table of Contents

2.4.4.1. Analytical treatment of the sloshing problem . . . . . . . . . . . . . 2.4.4.2. Sloshing in a rigid tank . . . . . . . . 2.4.4.3. Frequency analysis for sloshing . . . 2.4.4.4. Application to a cylindrical flexible tank subjected to seismic loading . . 2.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Acknowledgments . . . . . . . . . . . . . . . . . . . 2.7. Bibliography . . . . . . . . . . . . . . . . . . . . . .

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87 89 96

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Chapter 3. Implicit Partitioned Coupling in Fluid–Structure Interaction . . . . . . . . . . . . . . . Michael S CHÄFER

109

3.1. Introduction . . . . . . . . . . . . . . . . . . . 3.2. Computational fluid mechanics . . . . . 3.2.1. Governing equations . . . . . . . . . . 3.2.1.1. Incompressible flows . . . . . . 3.2.1.2. Inviscid flows . . . . . . . . . . . 3.2.2. Finite volume discretization . . . . . 3.2.2.1. Solution algorithms . . . . . . . 3.3. Computational structural mechanics . . 3.3.1. Governing equations . . . . . . . . . . 3.3.1.1. Linear elasticity . . . . . . . . . 3.3.1.2. Plane stress problems . . . . . 3.3.1.3. Hyperelasticity . . . . . . . . . . 3.3.2. Finite element methods . . . . . . . . 3.4. Fluid–structure interaction algorithms 3.4.1. ALE formulation . . . . . . . . . . . . 3.4.2. Mesh dynamics . . . . . . . . . . . . . 3.4.2.1. Algebraic approaches . . . . . . 3.4.2.2. Elliptic approaches . . . . . . . 3.4.3. Coupling methods . . . . . . . . . . . . 3.4.3.1. Implicit partitioned coupling . 3.5. Results and applications . . . . . . . . . . 3.5.1. Verification results . . . . . . . . . . . 3.5.2. Validation results . . . . . . . . . . . . 3.5.3. Flow induced by solid deformation

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3.5.4. Interaction of flow and solid deformation . . 3.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. Avoiding Instabilities Caused by Added Mass Effects in Fluid–Structure Interaction Problems . . . . . . . . . . . . . . . . . . . . . 165 Sergio I DELSOHN, Facundo D EL P IN and Riccardo R OSSI 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The discretized equations to be solved in a FSI problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Monolithic solution of the FSI equations by pressure segregation . . . . . . . . . . . . . . . . . . 4.4. Static condensation of the pressure . . . . . . . . 4.4.1. Approximation to the static condensation 4.4.2. Definition of the stabilizing matrix . . . . . 4.5. Evaluation of the Laplace matrix for FSI problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. The partitioned (or staggered) scheme . . . . . . 4.7. Numerical examples . . . . . . . . . . . . . . . . . . 4.7.1. Fluid column interacting with an elastic solid bottom . . . . . . . . . . . . . . . . . . . . . 4.7.2. Mesh sensitivity analysis . . . . . . . . . . . 4.7.3. 3D flexible flap in a converging channel . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4. Experimental validation with aortic valve 4.7.5. Flexible valve in pulsatile flow . . . . . . . . 4.8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 4.9. Acknowledgments . . . . . . . . . . . . . . . . . . . . 4.10. Bibliography . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Multidomain Finite Element Computations: Application to Multiphasic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Thierry C OUPEZ, Hugues D IGONNET, Elie H ACHEM, Patrice L AURE, Luisa S ILVA, Rudy VALETTE 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1.1. A classification of multidomain approaches . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Notations . . . . . . . . . . . . . . . . . . . . . . . 5.1.2.1. Functional and discrete spaces . . . . . 5.1.2.2. Bubble space . . . . . . . . . . . . . . . . . . 5.1.2.3. Element length definition . . . . . . . . . 5.2. Characterization of different phases . . . . . . . . 5.2.1. The level set or signed distance functions . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Mesh adaptation . . . . . . . . . . . . . . . . . . 5.2.3. Displacement of the level set function . . . . 5.2.3.1. The reinitialization . . . . . . . . . . . . . 5.2.3.2. Convective reinitialization . . . . . . . . 5.2.3.3. The local distance function . . . . . . . . 5.2.3.4. Parameters of reinitialized and truncated convective equations . . . . . 5.3. Stabilized finite element formulations . . . . . . . 5.3.1. Multidomain Navier–Stokes equations . . . 5.3.2. Weak formulation of incompressible Navier–Stokes equations . . . . . . . . . . . . . 5.3.3. Stable multiscale variational approach for Navier–Stokes equations . . . . . . . . . . . . . 5.3.3.1. The fine scale subproblem . . . . . . . . . 5.3.3.2. The coarse scale subproblem . . . . . . . 5.3.3.3. Time advancing . . . . . . . . . . . . . . . . 5.3.3.4. Matrix formulation of the problem . . . 5.3.4. Stabilized weak formulation for convection equation . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Multiphasic problems with fluid–air and fluid–fluid interface . . . . . . . . . . . . . . . . . . . 5.4.1. Two-dimensional Zalesak’s problem . . . . . 5.4.2. Gravitational flow . . . . . . . . . . . . . . . . . 5.4.3. The surface tension . . . . . . . . . . . . . . . . 5.5. Immersion of solid bodies in fluid . . . . . . . . . . 5.5.1. Immersed body having an imposed velocity 5.5.2. Velocity-pressure formulation for solid bodies in a fluid . . . . . . . . . . . . . . . . . . .

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5.5.3. Weak formulation of fluid/solid system . . . 5.5.4. Discrete formulation and Uzawa algorithm 5.5.5. Lagrangian particle displacement . . . . . . . 5.5.6. A sphere in a shear flow . . . . . . . . . . . . . 5.5.7. Two rigid spheres in stokes flow . . . . . . . . 5.5.7.1. Influence of mesh . . . . . . . . . . . . . . . 5.5.7.2. Particle displacements . . . . . . . . . . . 5.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7. Acknowledgements . . . . . . . . . . . . . . . . . . . 5.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . .

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List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . .

291

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293

Introduction

Fluid–structure interaction (FSI) and, in a broader sense, multiphysics have become more and more the focus of computational engineering in the recent years. Fluid structure coupling can occur in many fields of engineering, and is a crucial consideration in the design of many engineering systems; for example, stability and response of aircraft wings in aerospace industry, flow of blood through arteries in biomedical applications, response of bridges and tall buildings to winds in civil engineering, and oscillation of heat exchangers in nuclear industry. These problems are often too complex to solve analytically and so they have to be analyzed by means of numerical simulations. Numerical simulations of coupled problems have been increasing for applications where experimental studies are very expensive and time consuming. These problems are computer time consuming and require new stable and accurate coupling algorithms to solve these problems. For the past decades, new development in coupling algorithms and the increased performance of computer have allowed solving some of these problems, besides solving some physical applications that were not accessible in the past. In the future this trend might continue to address more realistic problems. The use of numerical simulation can reduce the amount of time spent using experimental methods to assess a large number of design alternatives. A better understanding of the problem is obtained through a computational approach because

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of increased amount of information gathered during computation. There is continuous research in the fields of computational fluid dynamics and computational solid mechanics, and it has reached a maturity level to help in solving large industrial and academic problems that were not accessible in the past. However, this level of achievement has not been reached to solve numerical simulation of fluid–structure interaction problems, and there are still several key questions in FSI regarding robustness, accuracy, stability, and computation time to be resolved satisfactorily. Various approaches have been investigated to model fluid–structure interaction problems. The added mass techniques have been extensively developed and used for problems involving small displacement and small deformation of structures coupled to potential flow. The objective of the added mass techniques is to investigate the dynamic behavior of the structure without computing the fluid motion, and consequently reduce the number of degrees of freedom in the problem and save computational time. Since the CFD techniques that solve the full Navier–Stokes equations to obtain the hydrodynamic force on the structure are extremely time consuming, for practical use the added mass techniques have been used by designers for predicting vibration levels and fatigue damage of the structure in a coupling. This approach is also used by different authors in solving flow-induced vibration problems, assuming small amplitude vibrations of the structure and potential flow for the fluid. For general hydrodynamic problems involving large deformation or moving structure, the full Navier–Stokes equations need to be solved in a moving mesh, since the fluid is described in a moving domain because of the structure motion. The most natural approach to solve these coupling problems is the ALE (Arbitrary Lagrangian–Eulerian) formulation for the fluid domain and the Lagrangian formulation for the structure domain. The ALE method has been intensively used for problems involving small and large structure displacements with no topological changes in the structure. In the presence of large structure displacements and complex geometries, the actual ALE remeshing algorithms are not general

Introduction

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and robust enough to handle complex meshes and topological changes in geometry, and fail for most complex three dimensional problems. For most three dimensional applications, automatic mesh generators are called on internally to create a new mesh with a new topology given the boundary segments. This method refers to a rezoning method; the dependent variables: velocity, pressure, internal energy, stress components, and plastic strain are updated on the new mesh by using a remap algorithm. Unlike a rezoning method, the topology of the mesh is fixed in an ALE method. The accuracy of an ALE calculation is often superior to the accuracy of a rezoned calculation because the algorithms used to remap the solution from distorted to undistorted mesh is second-order accurate for the ALE formulation when using second-order advection algorithms, while the algorithm for the remap in the rezoning is only first-order accurate. The purpose of this book is to address the challenges encountered in solving coupling problems and describe different algorithms and numerical methods being used to solve problems where fluid and structure can be weakly coupled using partitioned methods or strongly coupled using monolitical approach. In many applications, mainly industrial applications, partitioned methods are the most used techniques for solving coupled problems, because they allow use of specific designated fluid and structure codes and offer significant benefits in term of efficiency. Using the partitioned method, small and better conditioned systems are solved instead of a single solver. Chapter 1 presents an introduction to ALE formulation, which is crucial to solve most of the fluid–structure interaction problems. For most FSI problems, one of the numerical challenges is the mesh motion and the ALE formulations that follow. Special care is needed while using ALE formulations, because classical time integration methods may lose their stability when used on moving mesh. The Lagrangian formulation where the mesh moves with the material, which is commonly used to solve problems in solid mechanics. This choice is very economical and resolves the material boundaries and

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free surfaces very accurately. The main limitation of the Lagrangian formulation is that the deformations must be limited otherwise the distortion in the mesh will result in inaccuracies and numerical instability. The Eulerian formulation, where the mesh is fixed in space, is commonly used to solve problems in fluid mechanics; using a fixed mesh eliminates the concern of mesh distortion, but introduces additional complexity of the convective terms associated with the transport of the material through the mesh. Additional computation time and concerns need to be considered to take into account the advection in the Navier–Stokes equations. For problems involving fluid and structure, such as fluid–structure interaction, neither the Lagrangian nor the Eulerian formulation can be used for the entire domain; the ALE formulation permits the mesh to transition from being Eulerian for modeling the fluid flow to a nearly Lagrangian that follows the deformation of the structure. For general 3D applications, the remesh of the fluid domain is more complex due to the complexity of the 3D geometry of the computational domain. Chapter 2 describes a survey related to numerical simulation of fluid–structure interaction problems for dynamic explicit problems. Dynamic fluid–structure interaction analysis is useful for a wide range of applications such as hydrodynamic impact, airbag inflation, fuel tank sloshing, and many other industrial applications. In recent years these problems have been considered to be of increased importance due to new development in industry to analyze the safety and integrity of the structure as a result of dynamic loading; in particular in the naval industry for high hydrodynamic impact analysis of deformable structure, in the automotive industry for airbag deployment and fuel tank sloshing, and in the aerospace industry for bird impact analysis and helicopter ditching. A demand for better design of the structure requires improved procedure analysis, and in many cases the safety and integrity of the structure cannot be estimated unless hydrodynamic forces for the structure loading are computed accurately by solving Navier–Stokes equations. In order to describe the algorithms

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used for the ALE formulation and coupling, a detailed presentation of the ALE and coupling are given in this chapter. The coupling algorithm based on penalty coupling, spring damping system, is described in detail in this chapter. These algorithms allow us to solve coupling problems, where the structure is described in a Lagrangian formulation and the fluid in a Eulerian formulation. The mesh of the Lagrangian structure is allowed to deform with no mesh interface constraints inside a Eulerian fluid mesh. Coupling algorithms evaluate interaction forces connecting fluid and structure. To illustrate these methods, two academic problems, a piston problem, and a two-dimensional water impact problem are illustrated. Both applications carry analytical solutions that can be used for comparison and to evaluate the accuracy of the coupling algorithm. To extend the coupling algorithm to more complex problems, we consider two industrial examples. The simulation of the deployment of an airbag due to a high velocity gas flowing out of an inflator is described first. This problem is of high interest in the automotive industry and presents several complexities involving a contact algorithm between different airbag layers and the large deformation of flexible membrane elements of the airbag. The second example consists of a sloshing fluid tank under seismic loading. This application, used for earthquake response of liquid storage tanks, is of great interest in civil engineering. Chapter 3 covers the second numerical challenge that concerns the numerical realization of the coupling mechanisms between structure and fluid. These mechanisms can be invoked at different levels within the numerical schemes, resulting either in more weakly or more strongly coupled procedures. The first and by far the simplest is the fully explicit partitioned coupling involving an alternating solution of solid and fluid problems with exchange of boundary conditions. Moreover, the partitioned approach allows us to solve the flow equations and the structural equations with different, possibly more efficient techniques that have been developed specifically for either flow equations or structural equations. This

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approach is very flexible concerning the choice of the solvers for the individual fluid and solid subtasks, but it often suffers from poor stability and convergence properties, time-step restriction, and energy conservation at the fluid structure interface. In the past years, important improvements have been achieved in the partitioned coupling, which goes beyond data mapping, communication between different codes, and interpolation and projection techniques. In order to preserve the flexibility and modularity that are inherent in the partitioned coupling, strongly or implicit partitioned technique, using an iterative process, have been intensively developed and used to improve the properties of the partitioned coupling. After reviewing the basics of computational fluid and structural mechanics, the main aspects of fluid–structure interaction algorithms are developed, and numerical results for a variety of academic and industrial applications are described in this chapter. Chapter 4 consists of the description of the monolithic approach involving the simultaneous solution of all unknowns. The equations governing the flow velocity and displacement structure are solved simultaneously with a single solver. In this case the convergence rate with respect to the coupling is usually optimal, but the full system is hard to solve and extensive modifications of the individual fluid and solid solvers are necessary. The iterative process used in the implicit partitioned coupling may be difficult to converge for problems where the fluid density is very close to the structure density. For such problems the monolithic approach is necessary to ensure numerical stability of the coupling problem. This difficulty does not occur for problems such as aeroelasticity where the fluid density is orders of magnitude smaller than the material structure density, but it becomes very important in biomechanical application where the arteries material and blood densities are of the same order. Chapter 5 describes a monolithic approach to explain multiphase problems. In this approach different materials

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described by different domains can be involved in the simulation, fluid, rigid or deformable solids. All materials are solved using a Eulerian mesh. The interaction between several phases involving liquid, gas, rigid solid and elastic material are considered in this chapter. In this approach, a monolithic coupling is performed to solve problems with different materials and their interaction, gas-solid or liquid-solid, by tracking the interface between materials using a level set function, which solves the material boundary and the free surface accurately. Interfaces between different domains are known implicitly through the values of a characteristic function defined on the whole computational domain and a specific solver associated with each domain. The main advantage of the multiphase formulation is that the mesh can be fixed. Since the coupling interface is a material interface and it is not defined geometrically through the mesh for accuracy of the multiphase problem, the mesh has to be refined at the interface. Level set technique used for interface tracking is described in this chapter. The numerical method used is based on the use of stable mixed formulation, which consists of continuous piecewise linear functions enriched with a bubble function for the velocity and piecewise linear functions for the pressure. Mhamed S OULI

Chapter 1

Introduction to Arbitrary Lagrangian–Eulerian in Finite Element Methods

1.1. Introduction The choice of the coordinate system for the numerical solution of a partial differential equation is both the first and arguably the most important decision. An inappropriate choice will lead to a numerical method that is both expensive and inaccurate. Traditionally, problems in structural engineering and solid mechanics have used a Lagrangian coordinate system, with the computational mesh moving with the material. This choice is very economical and resolves the material boundaries very accurately. Its primary limitation is that the deformations must be limited otherwise the distortion in the mesh will result in inaccuracies and numerical instability. The traditional choice for fluid mechanics is a Eulerian coordinate system having the mesh fixed in space. Using a fixed

Chapter written by David J. B ENSON.

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mesh eliminates the limitation on the degree of deformation permitted in the material, but introduces the additional complexity of the convective terms associated with the transport of the material through the mesh. For problems with both fluids and solids, such as fluid– structure interaction, neither the Lagrangian nor the Eulerian formulations are optimal for the entire domain. A Lagrangian formulation cannot handle the large deformations of the fluid and a Eulerian formulation sacrifices some accuracy when applied to solids. Most fluid–structure interaction formulations use a Eulerian formulation for the fluid and a Lagrangian formulation for the structure, and introduce a coupling algorithm between them. The coupling algorithm is quite complex if it is required to handle an arbitrary Lagrangian mesh overlapping a Eulerian mesh. It can be simplified to a larger extent if the boundary of the fluid mesh conforms to the boundary of the Lagrangian mesh, and the nodes of both meshes are superposed along their common boundary. Arbitrary Lagrangian–Eulerian (ALE) methods allow the mesh to move in an arbitrary manner, with the two limiting cases reducing to the Lagrangian and Eulerian formulations. An ALE mesh that conforms to the Lagrangian mesh for the structure along part of its boundary while the rest remains fixed providing a convenient transition between the fluid and the structure. Since the ALE mesh moves relative to the material, it has transport terms similar to those found in the Eulerian formulation, and therefore it has many algorithms in common with computational fluid dynamics (CFD). This chapter provides an introduction to the basic ideas behind the ALE formulation with an emphasis on methods appropriate for problems in solid mechanics and introduces some of the numerical methods most commonly used in ALE calculations to solve them.

Introduction to ALE in FEM

3

1.2. Governing equations In this section, the governing equations are derived for the case when the reference coordinates move at an arbitrary velocity [NOH 64, HUG 81, DON 83]. This formulation is referred to as the Arbitrary Lagrangian–Eulerian formulation as it contains both the Lagrangian and Eulerian equations as subsets. The velocity of the material is u, the velocity of the reference coordinates is v , and their difference, u − v , is denoted w. The Jacobian, J  , is the relative differential volume between the reference and the spatial coordinates, ∂J  ∂vi = J . ∂t ∂xi

(1.1)

The material time derivative can be expressed in terms of both the spatial and reference coordinates, where f r means that f is expressed as a function of the reference coordinates, f˙ = f˙ =

∂f ∂f + ui ∂t ∂xi r ∂f ∂f + (ui − vi ) . ∂t ∂xi

(1.2) (1.3)

The ALE equations are derived by substituting equation (1.2) into the usual Lagrangian equations, but the results are not in conservation form, ∂ρr ∂t ∂ur ρ i ∂t ρ

∂er ∂t

= −ρ

∂ui ∂ρ − wi ∂xi ∂xi

= (σij ,j +ρbi ) − ρwj

(1.4) ∂ui ∂xj

= (σij ui ,j +ρbi ui ) − ρwj

(1.5) ∂e . ∂xj

(1.6)

To put them into conservation form, an additional identity is derived by multiplying equation (1.4) by J  , multiplying

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equation (1.1) by ρ, and adding them: ∂J  ρ ∂ρwi = −J  . ∂t ∂xi

(1.7)

After multiplying equation (1.7) by f , equation (1.3) by ρJ  , and arranging terms, the ALE equation for f is written in its general form. The first term on the right-hand side of equation (1.8) is the source term for f , and the second term, the transport of f : ∂(J  ρf ) ∂ρf wi = J  ρf˙ − J  . ∂t ∂xi

(1.8)

The conservation form of the ALE equations is obtained by substituting the Lagrangian equations into equation (1.8): ∂ρJ  ∂t ∂ρJ  ui ∂t ∂ρJ  e ∂t

= −J 

∂ρwi ∂xi

= J  (σij ,j +ρbi ) − J 

(1.9) ∂ρui wj ∂xj

= J  (σij ui ,j +ρbi ui ) − J 

∂ρewj . ∂xj

(1.10) (1.11)

When w is zero, J  is 1 and the Lagrangian equations are recovered from equation (1.9). If the reference coordinates are the current spatial coordinates, w is v , J  is again 1, and the Eulerian equations are recovered.

1.3. Operator splitting ALE methods advance the solution in time using operator splitting, which breaks the governing partial differential equation into a series of simpler ones that are solved sequentially. The basic concept is easily illustrated using a simple

Introduction to ALE in FEM

5

linear differential equation [CHO 78], du = (A + B)u, dt

u(0) = u0 ,

(1.12)

where u is a vector and A and B are matrices. The solution for an interval t, letting eC denote the matrix exponential of the matrix C , is u(t) = e(A+B)t u0 . (1.13) Suppose that instead of solving equation (1.12) in a single step, the equation is rewritten as du dt du dt

= Au

0 < t < t

(1.14)

= Bu

0 < t < t,

(1.15)

which leads to the solution u(t) = eBt eAt u0 .

(1.16)

If A and B are scalars, the solutions in equations (1.13) and (1.16) are identical. For certain special matrices, such as the spin in two dimensions,     0 −ωA 0 −ωB B= , A= (1.17) ωA 0 ωB 0 the solution (the rotation of a vector u in the x − y plane about the z axis by an angle (ωA + ωB )t) will also be identical. However, the general case for arbitrary matrices A and B will not give the exact solution defined by equation (1.13). This is easily demonstrated by expressing the matrix exponential as a Taylor series and comparing the leading terms in equations (1.13) and (1.16) [CHO 78]. A careful analysis [CHO 78] proves that for arbitrary A and B the solution given by equation (1.16) is at best second-order accurate. In other words, the coefficients in the Taylor expansion can be

6

ALE and Fluid–Structure Interaction

made to agree up to terms in (t)2 , but there will always be an error in coefficient for (t)3 . Operator splitting shows up in several places within a typical ALE formulation for problems in solids mechanics. The first split occurs in stress update, where the deviatoric stress is updated separately and independent of the pressure, allowing the effect of the work performed by the deviatoric stress during plastic deformation to be more easily accounted for in the internal energy used to evaluate the pressure. Furthermore, the deviatoric stress is a function both of the spin (rigid body rotation) and the strain rate (deformation), and their effects are typically accounted for in two steps by first updating the stress to account for the rigid body rotation then, in a second step, for the increment in strain. Many of the transport algorithms that were originally derived for one dimension are extended to multiple dimensions by performing a sequence of one-dimensional sweeps along the mesh lines. To illustrate how splitting is applied to the ALE formulation, consider the model PDE in one dimension, ∂φ ∂φ +w = S(x, t) ∂t ∂x

(1.18)

with φ as the solution variable and S a source term. Using operator splitting, the equation is re-written as two equations, ∂φ ∂t ∂φ ∂t

= S(x, t) = −w

∂φ . ∂x

(1.19) (1.20)

The first equation has the form of the Lagrangian formulation (w = 0) and its solution is advanced with the Lagrangian step. The second equation includes only the term associated with the convection of the solution variable, and its solution is commonly referred to as the Eulerian step.

Introduction to ALE in FEM

7

According to the previous operator splitting analysis, this approach should be second-order accurate at best, but the solutions obtained in practice are often better than the theory would predict. Another way of looking at the solution process [JOH 81] is as a Lagrangian solution followed by a projection of the solution from one mesh onto another. If the projection is perfect, i.e. the new mesh is capable of exactly representing the solution on the old mesh, and the mapping from one mesh to the other does not introduce any error, then the solution should be every bit as accurate as the Lagrangian solution, and the method is no longer limited to second-order accuracy. Operator splitting has historically been the starting point for deriving the ALE formulation but viewing the Eulerian step as a projection operation is actually closer to how the ALE formulation is actually implemented. Time does not evolve during the Eulerian step, and the calculation of the volume of material transported between elements is typically a geometrical calculation based on how the new mesh overlaps the old one.

1.4. The Lagrangian step The Lagrangian step for ALE advances is virtually identical to the formulation used in explicit finite element codes, e.g. LS-DYNA [HAL 98]. The primary difference is multimaterial ALE formulations permit multiple materials in a single element. The interaction between the materials in a multimaterial (or mixed) element is handled by a mixture theory.

1.4.1. Governing equations The three fundamental conservation equations are (1) the conservation of mass, (2) the conservation of momentum, and (3) the conservation of energy. The Lagrangian formulation

8

ALE and Fluid–Structure Interaction

trivially enforces the conservation of mass by defining the density as the ratio of the element mass to the current volume. The conservation of energy, in the absence of heat conduction, is enforced locally at selected points within each element or control volume. Different finite element and finite difference methods are therefore defined in terms of how they handle the conservation of momentum. Finite difference methods, such as the integral difference method [WIL 64, NOH 64], work directly with the conservation of momentum in its differential form; equation (1.9). Finite element methods use the weak form of the equation, which is also known as the principle of virtual work or the principle of virtual power within the solid mechanics community, 

 [ρ¨ xδx + σ : δ]dV = V

 ρbδxdV +

V

τ δxdS,

(1.21)



where V is the volume of the domain, S is the surface, divided into Sτ and Su , where traction and displacement boundary conditions are imposed respectively. The virtual displacement δx is an arbitrary bounded vector satisfying the constraint δx = 0 on Su .

1.4.2. The central difference method The solution is advanced in time using the central difference method, which is also called the Verlet or leap frog method. It is based on the central difference approximation for evaluating the derivative of a function f (t) at t, f (t + 12 t) − f (t − 12 ) df (t) = . dt t

(1.22)

Introduction to ALE in FEM

9

The approximation is derived by subtracting the Taylor series for f (t + 12 t) from f (t − 12 t),  f

t+

1 t 2

 = +

 f

1 t − t 2

 = −

 f

   1 1 t + t − f t −  2 2

=

   2 df (t) t t 1 d2 f (t) · · + dt 2 2 dt2 2   3   t 1 d3 f (t) (1.23) · + O (t)4 3 6 dt 2  2   df (t) t 1 d2 f (t) t + · f (t) − · dt 2 2 dt2 2   3   t 1 d3 f (t) (1.24) · + O (t)4 3 6 dt 2  3 df (t) t 1 d3 f (t) · + .... · t + + dt 3 dt3 2 f (t) +

(1.25)

The derivation itself shows that the central difference approximation is second-order accurate through its truncation error, which is proportional to (t)3 . The approximation of the first derivative may be inverted to provide a method for integrating f (t) forward in time,     1 1 df (t) = f t − t + t f t + t (1.26) 2 2 dt     3 1 df (t + t) = f t + t + t f t + t (1.27) 2 2 dt ... where t is commonly referred to as the time step size. To simplify the notation, the time step number n is defined as tn =

n

tn .

(1.28)

i=1

This notation is extended to the midpoint values by tn+1/2 = tn + t/2. All variables evaluated at tn are also superscripted with n. Note that the time step size can change as the solution advances in time. Using this new notation and indicating

10

ALE and Fluid–Structure Interaction

a time derivative by a “·” above the variable, the central difference integration method is written as f n+1/2 = f n−1/2 + tf˙n .

(1.29)

Applying this method to the equations of motion gives x ¨n = F n /M

(1.30)

x˙ n+1/2 = x˙ n−1/2 + t¨ xn n+1

x

= x + tx˙ n

n+1/2

(1.31) ,

(1.32)

where F is the force and M is the mass. The central difference method has its step size limited by stability considerations like all explicit integration methods. In one dimension, for systems without damping, von Neumann found in his pioneering work (see [ROA 76] for the history) that the stable time step size was limited to t ≤ , c

(1.33)

where is the length of the shortest

element and c is the speed of sound. For linear elasticity, c = E/ρ. Notice that the form of this stability limit is similar to the one in transport algorithms, where c is the velocity of the fluid. The stability limit may be interpreted physically in either of two ways. The first is the time step size must be less than the length of time required for a wave to propagate across the element. For larger time steps, waves would have to jump across an element without the element directly experiencing their effect. The second interpretation is based on the highest natural frequency of the element. A simple eigenvalue analysis for a single linear element gives two natural frequencies, E ω1 = 0 and ω2 = 2 . (1.34) ρ 2

Introduction to ALE in FEM

11

Substituting in the highest frequency into the stability limit gives 2 t ≤ , (1.35) ω2 a result that may be derived directly using standard methods for difference equations. This result may be interpreted as requiring a minimum of three or four time steps to resolve a single oscillation cycle of the element at its highest frequency of vibration. The stable time step size decreases as the amount of damping in the system increases,

2 2 t ≤ ξ +1−ξ , (1.36) ω where ξ is the damping coefficient. A means of estimating ξ is discussed in the section on the shock viscosity. 1.4.3. Element formulation The finite element method is used here to derive the Lagrangian step. Other methods are equally applicable, but the finite element method permits a reasonably simple and straightforward derivation. An expression for the acceleration is required to use the central difference method to advance the solution in time. Rearranging the weak form of the momentum equation, equation (1.21), gives  ρ¨ xδxdV = F ext − F int (1.37) V

 F

ext

=

 ρbδxdV +

V

 F

int

=

τ δxdS

(1.38)



σ : δdV.

(1.39)

V

The coordinates, velocity, acceleration and virtual displacement are interpolated from their values at the nodes using

12

ALE and Fluid–Structure Interaction

shape functions, NA , where A is the node number, x= NA xA x˙ = NA x˙ A A

¨= x



A

¨A NA x

δx =

A



(1.40) NA δxA .

A

The shape functions are expressed in terms of an isoparametric coordinate system, and their functional form may be found in any standard finite element textbook [BEL 00, HUG 87, BAT 82, COO 89]. For a four-node quadrilateral with the nodes numbered counterclockwise starting at the lower left, the shape functions are defined in terms of the parametric coordinates s = (s1 , s2 ), N1 (s) = N2 (s) = N3 (s) = N4 (s) =

1 (1 − s1 ) (1 − s2 ) 4 1 (1 + s1 ) (1 − s2 ) 4 1 (1 + s1 ) (1 + s2 ) 4 1 (1 − s1 ) (1 + s2 ) . 4

(1.41) (1.42) (1.43) (1.44)

The shape functions are defined identically for each element and exist only within a single element. A node is therefore associated with as many shape functions as there are elements attached to it, and the collection of them may be thought of as a single function that is differentiable within each element and continuous across the element boundaries. Substituting the relations from equation (1.40) into the inertial term in the weak form gives   ¨A ρ¨ xδx = ρ NA x NB δxB dV (1.45) V

V

=

A

B

 A

B



¨ A δxB ρNA NB dV x V

(1.46)

Introduction to ALE in FEM

 c MAB

13



=

(1.47)

ρNA NB dV , V

where M c is called the consistent mass matrix because its derivation is consistent with the finite element method. Using the consistent mass matrix would require the solution of a system of linear equations for the accelerations at each time step, which would be costly, therefore a lumped diagonal mass matrix is commonly used,   MAB = δAB ρNB dV . (1.48) V

Since the lumped mass matrix is diagonal, the diagonal term may be thought of as the mass of the node, and the matrix is inverted by simply inverting the terms on the diagonal,  M −1 = [1/MB ] , MB = ρNB dV. (1.49) V

The external force vector is also derived by a direct substitution of the appropriate relations from equation (1.40),   F ext = ρbδxdV + τ δxdS (1.50) V



 =

ρb



V

=



τ NB δxB dS (1.51)

NB δxB dV + Sτ

B

V

B

 



τ NB dS δxB . (1.52)

ρbNB dV + Sτ

The internal force vector requires the evaluation of the virtual strain, δ. Again using the relations from equation (1.40),   1 ∂δxi ∂δxj δ ij = + (1.53) 2 ∂xj ∂xi  1  ∂NB ∂NB = δxBi + δxBj . (1.54) 2 ∂xj ∂xi B

14

ALE and Fluid–Structure Interaction

This relation may be put in matrix form, which defines the B matrix, B , δ = Bδx (1.55) and the internal force may be written as  int F = σ : δdV V



=

B

(1.56)

 B B σdV

δxB ,

(1.57)

V

where B B is the section of the B matrix associated with node B . All the terms in the weak form have now been written in terms of the unknown coordinates and the virtual displacements. Collecting the terms gives      1 ¨B − x ρbNB dV + τ NB dS − B B σdV MB V Sτ V B

(1.58)

δxB = 0.

Since the virtual displacements are arbitrary, this equation can only be satisfied if the individual terms enclosed by {. . .} are individually zero,     1 ¨B = x ρbNB dV + τ NB dS − B B σdV . (1.59) MB V Sτ V 1.4.4. Hourglass modes The integrals in equation (1.59) are evaluated using numerical quadrature, with one-point Gauss quadrature [DAH 74] the most commonly used rule,  f (x)dV ≈ f (0)V, (1.60) V

Introduction to ALE in FEM

15

letting the origin, x = 0, be the centroid of the element. A four node quadrilateral element in two dimensions has eight degrees of freedom, three of them consisting of rigid body modes, leaving five modes of deformation that must be resisted by the stress tensor. The Cauchy stress tensor in two dimensions has three independent components acting in opposition to the corresponding strains, leaving two modes of deformation that are not resisted by the element. In a similar manner, a hexahedral element in three dimensions has eight nodes, giving it 24 degrees of freedom, six of which are rigid body modes. The six independent components of the stress tensor act against the six homogenous strain deformations, leaving 12 modes that are not resisted by the element. These modes are commonly called hourglass, keystone, or zero energy modes. Using more integration points will eliminate these modes, but at the expense of additional computational cost and the response of the element will be too stiff. Triangular and tetrahedral elements have exactly the right number of degrees of freedom to avoid hourglass modes, but their response is also too stiff. A means of hourglass control is therefore necessary in ALE calculations. The mode shapes can be determined by inspection for a square element by noting that the lengths of the two lines bisecting the element must not change (corresponding to the strains 11 and 22 ), and they must remain perpendicular (no shear strain); see Figure 1.1. As can be seen, a single element has the shape of a keystone in an archway, and a pair of them form an hourglass. The general condition on the modes, denoted hi here, for quadrilaterals of arbitrary shape is they must be orthogonal to the rigid body modes, T 1 · hi = 0

T 1 = {1, 0, 1, 0, 1, 0, 1, 0}T

(1.61)

T 2 · hi = 0

T 2 = {0, 1, 0, 1, 0, 1, 0, 1}

(1.62)

S 3 · hi = 0

S 3 = {−y1 , x1 , −y2 , x2 , −y3 , x3 , −y4 , x4 } (1.63)

T

16

ALE and Fluid–Structure Interaction

Figure 1.1. A quadrilateral element deformed in a hourglass mode. Note the deformation at the four Gauss point used in full integration and the absence of deformation at the element centroid, the Gauss point for reduced integration

with T i being the rigid body translation mode in direction i, and S i the rigid body spin about axis i, and the nodal coordinates are measured relative to the centroid of the element. The modes also be orthogonal to the strains, Bhi = 0.

(1.64)

Gram-Schmidt orthogonalization can therefore be used to calculate the shapes of the hourglass modes for arbitrary elements [FLA 81]. The simplest means of controlling the hourglass modes is to add a force that acts opposite to them and which is proportional to their magnitude, hi · u, or rate, hi · v , F hi = −hi ⊗ hi {ku + cv}

(1.65)

with k and c being appropriate stiffness and damping constants [BEL 00, GOU 82, WIL 64].

1.4.5. Stress rates Suppose that a bar is stretched in the x1 direction so that the stress is σ in the x1 direction and zero in the others. Now suppose the bar is rigidly rotated 90o so that it is aligned with the x2 direction. The stress should now be in the x2 direction

Introduction to ALE in FEM

17

and zero in the others. Most material models are expressed in a rate form, ˙ σ˙ = C . (1.66) During the initial stretching of the bar, it is subjected to a strain rate ˙11 over a period of time to generate the stress σ11 . During the rigid-body rotation, the strain rate is zero (by definition, a rigid-body rotation is one that does not deform the body), making σ˙ = 0 and the final stress is still in the x1 direction. Clearly, the simple expression in equation (1.66) is inadequate when large deformations and rotations occur. Additional terms must be added to equation (1.66) to account for the large rotations, giving rise to stress rates, the most popular being the Jaumann rate [BEL 00, HAL 98]. It can be understood by considering two coordinate systems, one being a local system that rotates with time and the second is the global coordinate system. The components of a vector and tensor in the two systems (superscripted with L and G respectively) are connected by a rotation matrix, aG = Rik aLk i

aG = RaL

G L σij = Rik Rj σk

(1.67)

σ G = Rσ L RT

(1.68)

Differentiating equation (1.68) with time gives ˙ L RT + Rσ˙ L RT + Rσ L R ˙T σ˙ G = Rσ

(1.69) T

T ˙ ˙ (1.70) = R(R R)σ L RT + Rσ˙ L RT + Rσ L (RT R)R

˙ T )σ G + Rσ˙ L RT + σ G (RR ˙ T) = (RR ∇

T

˙ T )σ G + σ G (RR ˙ ). = σ + (RR

(1.71) (1.72)



The term σ is the objective stress rate. Different stress rates ˙ T. are defined by choosing different representations for RR In terms of the implementation in a finite element code, the stress rate may be thought of as the call to the material model library to update the stress due to an increment in strain, e.g. ∇

˙ σ = C .

(1.73)

18

ALE and Fluid–Structure Interaction

Differentiating the identity RRT = I with time, ˙ T + RR ˙T =0 RR

(1.74)

˙ T = −RR ˙ T = −(RR ˙ T )T , RR

(1.75)

˙ T is skew symmetric. The Jaumann demonstrates that RR ˙ T where rate uses the spin, W , for RR W =

 1 L + LT 2

L=

∂ x˙ . ∂x

(1.76)

This rate was used exclusively in the early finite difference hydrocodes [ALD 64] because it is inexpensive to compute and also works well, and it is still the most popular rate in modern explicit codes. 1.4.6. Shock viscosity The shock viscosity is only necessary for problems with shocks, which are simply jump discontinuities in the stress. Typical shock viscosity formulations resolve a shock in three to six elements. The width is influenced by the strength of the shock and its direction through the mesh. Acoustic waves, which are weak shocks, tend to have the greatest width as they do not have the converging characteristics that constantly attempt to steepen the shock front. Shocks propagating diagonal to the mesh direction are wider than those aligned with the mesh. Oscillations were observed behind the shock in the earliest numerical shock wave calculations [VON 50]. Von Neumann and Richtmyer [VON 50] introduced the artificial viscosity as a means of resolving shocks without resorting to moving internal boundary conditions. It has remained the predominant strategy for resolving shocks in solid mechanics problems. Most shock viscosities are turned on only in compression since rarefaction waves are generally smooth and do not give any problem.

Introduction to ALE in FEM

19

The underlying reasons for needing a shock viscosity can be stated in different ways: – Physically, shocks are irreversible processes and a mechanism to introduce the proper amount of irreversibility in the calculation is necessary for a correct solution. The shock viscosity is an approximation to the viscosity in the shock front. – From the mathematics of partial differential equations, it is known that the weak form of a hyperbolic system of equations may have more than one solution [LAX 54]. The physically correct solution that satisfies the entropy condition is the one obtained by adding viscous terms to the system of equations and obtaining the limit as the viscous terms go to zero [HAR 76]. – In approximation theory the oscillations are regarded as Gibb’s phenomena, the consequence of trying to approximate a discontinuous function by a series of continuous functions. The viscosity acts to smooth the solution so that the continuous interpolation functions can approximate it without oscillations. Just as many existence and uniqueness proofs in mathematics fail to give insight into constructing a solution, an understanding of the origin of the oscillations behind the shock does not lead to a unique, optimal form of the shock viscosity. 1.4.6.1. von Neumann–Richtmyer Von Neumann and Richtmyer [VON 50] introduced the concept of a shock viscosity. Their viscosity, q , is quadratic in the strain rate of the element and it is added to the pressure throughout their finite difference formulation. In one dimension, for an element with a length x, their viscosity is     2 ∂v  ∂v  q = −ρ(cx) (1.77) , ∂x  ∂x  where c is a constant close to 1.0. For a two-node element, the viscosity can be expressed in terms of the difference in the

20

ALE and Fluid–Structure Interaction

velocity, v , of the two nodes, q = −ρc2 v |v| .

(1.78)

In their landmark paper [VON 50], von Neumann and Richtmyer analytically solved the steady-state shock solution for an ideal gas with their viscosity to prove that the Rankine– Hugoniot equations were satisfied. The shock profile was sinusoidal and approximately six elements wide. 1.4.6.2. Standard quadratic and linear formulation Later formulations of the shock viscosity have added a small linear term to the quadratic formulation of von Neumann and Richtmyer. Kuropatenko [KUR 67] noted the similarities between the jump in pressure across a shock for an ideal gas and an elastic solid. When the velocity jump, v , is much greater than the speed of sound, the pressure jumps are P+ − P− =

γ+1 ρ− (v)2 2

P+ − P− = ρ− (v)2

for an ideal gas

for an elastic fluid

(1.79) (1.80)

and when it is small, P+ − P− = ρ− c|v|

(1.81)

for both materials. Combining the two relations and adding the scale factors CQ and CL for the quadratic and linear terms respectively gives the standard form of the shock viscosity in popular use, q = CQ ρ(v)2 + CL ρc|v|.

(1.82)

Based on equation (1.81), the expected value of CL is 1.0, but in practice, a value of 0.05 is commonly used. A typical value for the quadratic scale factor is 1.5, which is closer to the value expected for an ideal gas than for an elastic material.

Introduction to ALE in FEM

21

1.4.6.3. Effect on time step size The shock viscosity is a damping term, and therefore reduces the stable time step size. From the stability analysis for the central difference method, the stable time step size (equation (1.36)) is

2 2 2

. t = ξ + 1 − ξ =

(1.83) ω 2 ω ξ +1+ξ Hicks [HIC 78] analyzed an infinite one-dimensional mesh of elements with a length L. Rather than a formal finite difference stability analysis, Rayleigh quotients are used here to motivate his results. The mode of the highest frequency is assumed to be of the form {. . . , +δ, −δ, +δ, −δ, . . .}T . For a linear system with the nodes having a mass M = ρAL, and the elements having stiffness and damping coefficients of K = EA/L and C = q  (v) = ∂q/∂v , the natural frequency and damping coefficients are K(2δ)2 uT Ku 4E = = uT M u M δ2 ρL2 2c L

ω2 = ω = 2ξω = ξ =

(1.84) (1.85)

C(2δ)2 uT Cu 4q  = = uT M u M δ2 ρL

(1.86)

q . ρc

(1.87)

Letting q  /ρ = s, substituting in the previous results gives t = c =

L   s 2 c

 +1+

L √ , s + s2 + c2

the form given by Hicks [HIC 78].

(1.88)

s c

(1.89)

22

ALE and Fluid–Structure Interaction

1.4.7. Mixture theories Mixture theories [BEN 92a, BEN 97] are required to handle elements containing more than one material, which are commonly called mixed or multimaterial elements. Given the stresses in each material m, σ m , the material volumes, Vm , and the mean strain rate of the element, ¯˙ , the mixture theory performs the following operations to calculate the element ¯ , used to calculate the accelerations: stress, σ 1) Partition the strain rate between the materials, subject to the constraint ˙ m Vm = ¯˙ Vm . (1.90) m

m

2) Independently update the stress in each material the strain rate for that material. 3) Calculate the mean stress of the element,  m m σ Vm ¯=  σ . (1.91) m Vm Since the last two steps are identical for all mixture theories, their differences lie in how they partition the strain rate between the materials. Only the simplest, most commonly used mixture theories are examined here; additional information is available in [BEN 92a]. 1.4.7.1. Mean strain rate mixture theory The simplest mixture theory gives each material the mean strain rate of the element, thereby trivially satisfying the constraint equation (equation (1.90)). In addition to its simplicity, the mean strain rate mixture theory is robust and conserves energy exactly. The limitation of this mixture theory is the error it introduces when an element contains a very soft material and a very hard material. To make the situation clear physically,

Introduction to ALE in FEM

23

consider an element containing a vacuum in between two steels. If the element is being compressed, the vacuum should be compressed preferentially until it is gone, and similarly when the element expands. However, since all the materials have the same strain rate, the vacuum can never be compressed out completely. An additional error is that the steel materials are compressed before they are in contact. Under tensile loading, there are two possible responses. First, if the element is being pulled apart in the direction normal to the steel–vacuum interfaces, no force should be generated because the vacuum cannot resist expansion. Secondly, if the element is pulled apart in the direction parallel to the interfaces, the steel should carry the load, making the response very stiff. This mixture theory does not account for the orientation of the interfaces, and therefore gives the same response regardless of how the element is pulled apart. Although these two situations introduce an error into the solution, there is a third problem which may terminate the calculation. Consider the situation where the element contains only a small fragment of steel, say 1% of the volume, and the remainder is vacuum. Given the volume weighting of the stresses, the element behaves as if it contains a material with only 1% of the stiffness of the steel, making it very easy to compress. If it is adjacent to an element completely filled with steel, the mixed element has to compress 100 times as much as the steel element to maintain pressure equilibrium. In the presence of strong shocks, this can lead to an error termination. Most equations of state for metals go to infinity at a finite density, typically somewhere around a volume strain of 1/3. During a single time step, the small fragment may go from having no pressure to an infinite one, immediately blowing up the calculation. There are two strategies for handling these problems with the mean strain rate mixture theory. The first strategy avoids the overcompression of the fragments by simply deleting the

24

ALE and Fluid–Structure Interaction

small fragments when they are detected. This is also done to improve the speed of the calculation because mixed elements are several times costlier than an element with only one material. The second strategy expands or compresses the vacuum preferentially to give a more accurate response. If an element is compressed to the degree where the volume of the vacuum goes to zero, the remainder of the compression is applied to other materials. 1.4.7.2. Mean stress mixture theory The mean stress mixture theory partitions the strain rate among the M materials so that all the materials have the same stress, σ m = σ m+1 , m = 1, M − 1, (1.92) subject to equation (1.90). Compared to the mean strain rate theory, this one is attractive because it automatically compresses soft materials preferentially, guaranteeing that the vacuum will be compressed out between two impacting solids. Additionally, the issue of over-compressing fragments to a density resulting in an infinite pressure is also automatically eliminated. The mixture theory is not without its problems. One of them is that the response is extremely soft. For example, in an element containing steel and vacuum, the stress will always be zero in steel because the vacuum must have zero stress. A second problem is that energy conservation is no longer guaranteed. Consider two materials in an element undergoing a compression of V , with one of the materials, m, initially at a much higher pressure than the other. During the  time step, the high pressure material will expand, with − P dV m < 0, while the work the material performs on the nodes will be −P V · (V m /V ) > 0. To eliminate possible problems this inconsistency may result in, some codes partition the work done by the element among the materials based on a mass weighting.

Introduction to ALE in FEM

25

This mixture theory requires the solution of 3M (two dimensions) or 6M (three dimensions) nonlinear equations, which are usually costlier. A more important problem is that there is no guarantee that a physically meaningful equilibrium solution exists for arbitrary material models at arbitrary initial stress states (which may be introduced by transport from adjacent elements). Identifying this situation a priori is impossible, so codes typically use the mean strain rate theory when the nonlinear iteration scheme fails to find a feasible solution. One approach to reducing the cost of the solution is to perform one Newton iteration per time step with a heuristic procedure applied to bound the changes in the state of the materials during a single time step. Materials therefore relax toward equilibrium over several time steps, a situation that is more physically realistic than instantaneous equilibration. 1.5. Mesh relaxation Most ALE codes have adopted mesh relaxation procedures based on elliptic mesh generation methods. The coordinates of the nodes are calculated by minimizing a functional, I , which usually has elliptic Euler–Lagrange equations. Jacobi iteration is used instead of a direct equation solver, resulting in a difference stencil that can be applied in a vectorized manner over the interior mesh. A mesh generator typically iterates hundreds of times to calculate the mesh, while ALE codes make only a few sweeps through the mesh at each time step. The displacement increment calculated by the Jacobi iteration is limited by Courant restrictions associated with the advection algorithms. For simplicity, the relaxation stencils are developed here on a logically regular two-dimensional mesh, where nodes can be labeled by the the intersections of the vertical mesh line j with the horizontal mesh line k as (j, k). The coordinates x = (x1 , x2 ) are replaced with (x, y) in this section to eliminate any confusion between the mesh lines and the coordinate directions in the subscripts.

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One of the oldest relaxation stencils was developed by Winslow [WIN 63], [WIN 82]. He inverted Laplace’s equation and derived a stencil so that the mesh forms lines of equal potential on a logically regular mesh. The mesh directions are φ and ϕ instead of ξi to avoid putting subscripts on the subscripts in the following equations. The inverted form of Laplace’s equation is given by equation (1.93), with the coefficients given by equation (1.95). The differencing for quadrilateral elements is given by equation (1.102) and stencils for a wide variety of triangular mesh configurations have been derived [WIN 81], αxφφ − 2βxφϕ + γxϕϕ = 0

(1.93)

αyφφ − 2βyφϕ + γyϕϕ = 0

(1.94)

α = x2ϕ + yϕ2

(1.95)

β = xφ xϕ + yφ yϕ

(1.96)

γ = x2φ + yφ2 .

(1.97)

The difference stencils for y are derived by substituting y for x in equations (1.98) through (1.102):

xφ =

1 (x − x(i,j−1) ) 2 (i,j+1)

(1.98)

xϕ =

1 (x − x(i−1,j) ) 2 (i+1,j)

(1.99)

xφφ = x(i,j+1) − 2x(i,j) + x(i,j−1)

(1.100)

xϕϕ = x(i+1,j) − 2x(i,j) + x(i−1,j)

(1.101)

xφϕ =

1 (x − x(i−1,j+1) + x(i−1,j−1) − x(i+1,j−1) ). 4 (i+1,j+1) (1.102)

Introduction to ALE in FEM

27

After substituting equation (1.95) and equation (1.102) into equation (1.93), the coordinates of the center node are calculated, x(i,j) = −

y(i,j) = −

1 (α(x(i,j+1) + x(i,j−1) ) + γ(x(i+1,j) + x(i−1,j) ) 2(α + γ) β (x − x(i−1,j+1) + x(i−1,j−1) − x(i+1,j−1) )) 2 (i+1,j+1) (1.103) 1 (α(y(i,j+1) + y(i,j−1) ) + γ(y(i+1,j) + y(i−1,j) ) 2(α + γ) β (y − y(i−1,j+1) + y(i−1,j−1) − y(i+1,j−1) )). 2 (i+1,j+1) (1.104)

The basic idea behind the Winslow algorithm has been extended in many ways, including by Winslow [WIN 82], Brackbill and Saltzman [BRA 82], and Giannakopoulos and Engel [GIA 88]. Winslow’s variable diffusion method is used in CAVEAT [ADD 90], where w(x) is a positive function,    ∂ϕ ∂ϕ 1 ∂φ ∂φ I= + dΩ. (1.105) ∂xi ∂xi ∂xi ∂xi Ω w i

When w is constant, the original Winslow algorithm is recovered. The mesh is concentrated in areas where the logarithmic gradient, ∇w/w, is largest. To concentrate the mesh in a region, a large value of w can be assigned in the neighborhood of the feature. A smoothing filter is applied to w to eliminate any high-frequency components before it is used in equation (1.105). 1.6. The Eulerian step The Eulerian step accounts for the transport of the material due to its motion relative to the mesh. In practice, the

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Eulerian step usually amounts to a projection of the solution from one mesh on to another rather than the solution of convective partial differential equation, ∂φ ∂φ + wi = 0. ∂t ∂xi

(1.106)

i

In the following discussion, the transport algorithms are developed from a largely geometrical viewpoint in keeping with the idea that the solution is being projected from one mesh to another. There are other approaches to developing transport algorithms that lead to ones that are different from those presented here, and the interested reader is referred to [BEN 92a]. The two most desirable qualities in a transport algorithm are conservation and monotonicity. Conservation requires that the integral of the transport variable over the domain remain unchanged by the transport. Applying this requirement to the density, for example, is equivalent to requiring the conservation of mass. Not all the solution variables, however, are governed by physical conservation laws, e.g. stress is not conserved, but they are transported as if they are. The requirement of monotonicity is that the transport should not introduce any new maxima or minima in the solution or amplify the existing ones. This means that no numerical oscillations will be introduced into the solution by the transport algorithm, which was a serious problem with the first method. From a practical standpoint, a minimum of second-order accuracy is a third requirement. First-order accuracy excessively diffuses the solution unless a very fine mesh is used in the calculation. Since the Lagrangian step is generally no better than the second-order accurate, third-order or higher accuracy in the transport does not improve the overall accuracy of the solution.

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29

1.6.1. Transport in one dimension Most transport algorithms are developed in one dimension and extended to two and three dimensions by operator splitting. The elements are numbered sequentially from left to right, and element i+1/2 is defined by nodes i and i+1. Following the usual convention for structured meshes (meshes with a regular grid pattern), the transport volume, Vi is positive if it is transporting material from element i − 1/2 to i + 1/2. The simplest stable transport method is donor cell. The values of the variable being transported, φ, are piecewise constant with one-point integration. New values are calculated by directly invoking conservation: the new value of φ in element i+1/2 is the sum of the products of the individual values of φ and the element volumes from the old mesh that overlap the new element. Taking into account that the material may move in either the positive or negative direction, the donor cell algorithm may be summarized as  φi−1/2 if Vi > 0 φi = (1.107) φi+1/2 otherwise φi+1/2 =

φi+1/2 Vi+1/2 + φi V i − φi+1 V i+1 (1.108) Vi+1/2 + V i − V i+1

The element chosen for evaluation φi is the upwind element. Upwinding is crucial for stable transport, and all modern transport algorithms are careful to use upwind data (perhaps from more than a single element) to evaluate φi . Unfortunately, the donor cell algorithm is only first-order accurate and it diffuses the solution very rapidly throughout the mesh. For pure Eulerian calculations, it is never acceptable, but in ALE calculations that are run in a nearly Lagrangian manner (the mesh is moved relative to the material as little as possible), it may provide an acceptable answer. The inaccuracy of the donor cell algorithm stems from the piecewise constant representation of φ within the elements.

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Better accuracy is obtained by replacing the constant by a linear function (MUSCL [VAN 77]), a quadratic function (PPM [COL 84]), or a more general polynomial (e.g. ENO [HAR 89, HAR 97, SHU 88]). Any function that is introduced must satisfy two important conditions: (1) its integral over the element must equal the mean value located at the integration point times the element volume, φi+1/2 Vi+1/2 , and (2) its maximum and minimum values must be bounded by values in the surrounding elements to avoid introducing or amplifying maxima and minima. As an illustration of how higher order accurate methods are constructed, the MUSCL algorithm [VAN 77] is developed here. MUSCL is probably the most commonly used transport algorithm in ALE codes today because it is simple, efficient, and accurate enough to prevent the Eulerian step from being the dominant contributor to the error in the solution. The piecewise constant approximation of φ within an element is replaced by a linear approximation, φ(x) = φ− i+1/2 +

∂φ (x − xi+1/2 ). ∂x

(1.109)

The value of the slope is calculated using additional values of φ from the adjacent elements. Conservation is guaranteed by choosing the midpoint of the element, xi+1/2 , as the origin for the linear function,  xi + 1 φ(x)dx = φ− (1.110) i+1/2 (xi+1 − xi ). xi

To avoid creating new extreme values, the slope must be limited so that extreme values of φ(x) lie within the range defined − − by φ− i−1/2 , φi+1/2 , and φi+3/2 . Accuracy is obtained by using a second-order accurate difference approximation of the slope, which can be calculated by fitting a parabola through three successive elements and

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31

evaluating the derivative at the center of the element,   VR 2 VL sC = φL + φR (1.111) VL + VR VL VR φL = φi+1/2 − φi−1/2

(1.112)

φR = φi+3/2 − φi+1/2

(1.113)

where VL is the distance between φi+1/2 and φi−1/2 and VR is the distance between φi+3/2 and φi+1/2 . Two bounding firstorder accurate slopes can be evaluated by using the values to the left and right, sL =

φL VL

sR =

φR . VR

(1.114)

The MUSCL algorithm chooses the slope to be zero if φi+1/2 is either a local minimum or maximum, or equivalently, φL · φR ≤ 0, making it equivalent to the donor cell method at extremums. Second-order accuracy is obtained by using sC , but to maintain monotonicity, the smallest slope must be used, s=

1 (sign(φL ) + sign(φR )) min (|sC |, |sL |, |sR |) . 2

(1.115)

The transport is performed using equation (1.108), but with φi and φi+1 evaluated at the center of the transport volumes using appropriate linear approximation, equation (1.109), from the upwind element. Essentially non-oscillatory methods [HAR 89], [HAR 97], [SHU 88] are derived by integrating the solution along the mesh line, with the values of the integral evaluated at the nodes,  xi Φi = φ(x)dx. (1.116) x0

A smooth, higher order accurate polynomial Φ(x) that interpolates Φi is constructed using divided differences, with the ENO methods differing largely in how they select the “smoothest” interpolating polynomial. The derivative of Φ(x)

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is the value of φ(x), and the method is conservative by construction since the integral of φ over element i + 1/2 is Φ(xi+1 )−Φ(xi ). In practice, the derivative of Φ(x) is not needed during the transport because it is the integral of φ over the transport volume and not φ itself that is required in equation (1.108), φi Vi = Φ(xn+1 ) − Φ(xni ). (1.117) i 1.6.2. Multidimensional transport by operator splitting The most obvious way of extending one-dimensional transport methods to two and three dimensions is to apply them in an uncoupled manner along each element direction [BEN 89]. This strategy works well for logically regular meshes, and it is adequate for ALE problems where the amount of material transported is small or where the transport is roughly aligned with the mesh directions. However, if the amount of transport is large and it is along the mesh diagonal, the solution may be poor. An example of a square pulse transported along the mesh diagonal in this manner with the MUSCL algorithm is shown in Figure 1.2. The initially square shape diffuses in the direction perpendicular to the transport due to a lack of corner coupling. Corner coupling accounts for the transport between elements that share only a node instead of a common face. Letting the velocity be (vx , vy ), the amount of material transported between the diagonal elements is (vx t)(vy t). For small velocities or small time step sizes, this term is insignificant. For logically regular meshes, the one-dimensional transport algorithm is first applied to each row in the x direction, then it is applied in the y direction on each column of elements. On the next step, the y direction transport is performed first, and each subsequent step alternates the sequence of onedimensional transport directions. The solution obtained using

Introduction to ALE in FEM

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Figure 1.3. The contours of the final state for logically regular mesh using operator splitting in combination with the one-dimensional donor cell and MUSCL algorithms

operator splitting is shown in Figure 1.3. Unfortunately, operator splitting is not viable for unstructured meshes. 1.6.3. Multidimensional meshes

transport

on

unstructured

The transport algorithms that preserve the geometric shape of the solution during its transport on an unstructured mesh are more complicated than their one-dimensional counterparts, e.g. [VAN 84, THU 96, DEN 02]. It is beyond the scope of this chapters to describe any of them in detail. Perhaps the simplest method [CHR 87] for including corner coupling on unstructured meshes uses a two stage RungeKutta integration scheme [DAH 74]. For an ordinary differential equation y = y  (y, t), the solution is advanced in two steps, y n+1/2 = y n +

t  n n y (y , t ) 2

y n+1 = y n + ty  (y n+1/2 , tn + t/2).

(1.118) (1.119)

Introduction to ALE in FEM

35

The first stage is performed using a donor cell (since only firstorder accuracy is required to maintain second-order accuracy overall), and the second using MUSCL. In terms of the actual implementation, the first stage is performed using transport volumes that are half of the full-time step values, while the second stage uses the full values. The results, labeled “RK MUSCL” in Figure 1.2, show the shape preservation of this method for both logically regular and unstructured meshes. 1.6.4. Momentum transport Ideally, both momentum and kinetic energy should be conserved by the transport. Unfortunately this is not possible while maintaining the monotonicity of the solution. As a consequence, the usual choice is to conserve momentum and accept some loss in the kinetic energy. There are codes that calculate the loss of kinetic energy and add the lost energy to the internal energy, but some stability and accuracy issues remain. Formulations that have the velocity field at the same points because the other state variables [PON 98, DUK 89] can use their standard transport methods to update the momentum in the same manner as the other state variables. Most finite element and finite difference methods for problems in solid mechanics, however, use a staggered mesh with the velocities located at the nodes and the other state variables located at the centroids of the elements. The momentum, the product of the velocity and the density, is therefore defined in terms of variables that do not share a common location in space. Two different approaches have been developed to modify element-centered transport algorithms for transporting the momentum. The first approach constructs a dual mesh that has the nodes of the original mesh at the centroid of the dual mesh elements, and the dual mesh nodes are defined as the

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ALE and Fluid–Structure Interaction

centroids of the original mesh. The second constructs additional solution variables that are transported using a standard element-centered transport method (e.g. MUSCL), then reconstructs the velocity field after the transport. The two different approaches are comparable in terms of computational cost, with the most efficient one depending on the particular structure of the ALE code. Both these methods give virtually identical answers when implemented properly. The two approaches are illustrated in one dimension for conceptual clarity; the multidimensional extensions may be found in [AMD 80, BEN 92b, BEN 08]. 1.6.4.1. Momentum transport using a dual mesh in one dimension The dual mesh method, pioneered in YAQUI [AMS 73], constructs a mesh with the nodes at the centroids of the dual mesh elements, and the element centroids of the original mesh are the dual mesh nodes; see Figure 1.4. The same transport algorithm used on the original mesh is applied to the dual mesh, giving the dual mesh analog to equation (1.108), vi =

Mi vi + M i−1/2 v i−1/2 − M i+1/2 v i+1/2 . Mi + M i−1/2 − M i+1/2 Original Mesh

i−1

i i–1/2

ΔM i–1

i–1

ΔM i+1 ΔM i+1/2

i i–1/2

Nodes Elements

i+1/2 ΔM i

ΔM i–1/2 Dual Mesh

i+1

(1.120)

i+1 i+1/2

Elements Nodes

Figure 1.4. The top mesh is used for the HIS momentum transport and the bottom, the mesh for the dual mesh method

Introduction to ALE in FEM

37

The new mass of the node is given by the denominator of equation (1.120), Mi = Mi + M i−1/2 − M i+1/2

(1.121)

and it can also be expressed in terms of the new element masses,

+ + Mi = 12 Mi−1/2 + Mi+1/2   = 12 Mi−1/2 + M i−1 − M i   (1.122) + Mi+1/2 + M i − M i+1   = Mi + 12 M i−1 − M i + M i − M i+1     = Mi + 12 M i−1 + M i − 12 M i + M i+1 . A consistency condition, first discussed by DeBar [DEB 74], imposes a constraint on the transport: If a body has a uniform velocity and a spatially varying density before the transport, then the velocity should be uniform and unchanged after the transport. This condition is satisfied if the new mass for a node in equation (1.120) is the same as in equation (1.122). Comparing the terms in these equations gives the consistent definition of the transport masses on the dual mesh, M i+1/2 =

 1 M i + M i+1 . 2

(1.123)

1.6.4.2. Element-centered transport in one dimension SALE [AMD 80] introduced the first element-centered transport method. In one dimension, the momentum density and element momentum are defined as 1 pi+1/2 = ρi+1/2 (vi + vi+1 ) , 2

1 Pi+1/2 = Mi+1/2 (vi + vi+1 ) , 2 (1.124)

and they are updated with the element-centered transport method. The nodal velocities are updated using the increment

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in the momentum from the adjacent elements, vi =

M i vi +

1 2

  Pi−1/2 + Pi+1/2

Mi + M i−1/2 − M i+1/2

+ − Pi+1/2 = Pi+1/2 − Pi+1/2 .

(1.125) (1.126)

This method is easy to implement but it is subject to dispersion errors and a loss of monotonicity near discontinuities in the velocity, e.g. shock fronts. It is, however, adequate for many engineering applications [BEN 92b]. Benson [BEN 92b] introduced the half index shift (HIS) algorithm to eliminate the dispersion and monotonicity errors at the expense of additional computations. Two new elementcentered variables are defined, 1 ψi+1/2 = vi ,

2 ψi+1/2 = vi+1 ,

(1.127)

then transported using the element-centered transport algorithm with the transport masses. The velocity at a node is calculated directly from the updated values, vi+1/2 =

 1 1 /(Mi+1/2 + M i − M i+1 ). Mi ψi2 + Mi+1 ψi+1 2 (1.128)

For unstructured meshes, the HIS algorithm has become popular for research codes [PAN 04, FRE 07], production codes developed under the Department of Energy Advanced Strategic Computing Initiative [PEE 00] and in generalpurpose commercial codes ls970,abaqus. It is attractive for parallel calculations because it does not require a second domain decomposition for the dual mesh or any of the additional data structures associated with the typical implementations of the dual mesh methods. Recent research by Benson [BEN 08] has demonstrated that the HIS algorithm

Introduction to ALE in FEM

39

is a close approximation to the dual mesh method, and the results in two dimensions are virtually indistinguishable. Motivated by the analysis of the two methods, Benson [BEN 08] showed that the dual mesh algorithm can be restructured to be as simple to implement HIS algorithm. 1.6.5. Interface reconstruction The interfaces between materials are not required to follow the mesh lines in multimaterial ALE (MM-ALE) formulations. Instead, the material interfaces run through the elements just as they do in multiphase CFD calculations. There are three basic strategies for handling the interfaces: 1) Lagrangian methods use particles that are connected by line segments. The positions of the particles is updated by interpolating the velocity from the nodes. This interface method was used in the earliest ALE codes. While it is still used in some research codes, it is rarely used in modern production codes. 2) Level set methods define a level set function that has a value of zero on the interface. The value of the function is evaluated at the nodes and the location of the interface is defined as the zero contour level. Level set methods are popular in CFD, but are less popular in ALE codes for solid mechanics because they typically do not conserve mass exactly. 3) Volume of fluid (VOF) methods use the volume of each material in an element and its surrounding neighbors to construct the interfaces within the element. The interfaces are discontinuous between elements, because they are constructed element-by-element. This approach is the one most commonly used in production codes because mass is exactly conserved by construction. Reviews of early tracking methods are given in [HYM 84] and [ORA 87]. Most of the reviewed methods were based on

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Lagrangian representations of the surface with marker particles or derived from the surface definitions from the volume fractions of different materials. A review of explicit methods for hydrocodes, including interface reconstruction methods, is presented in [BEN 92a]. Several different algorithms were implemented and evaluated in the papers by Rider and Kothe [RID 95, RID 98]. The latest published reviews appear to be by Scardovelli and Zaleski [SCA 99] and Benson [BEN 02]. 1.6.5.1. Lagrangian methods Lagrangian methods are conceptually simple: Before the simulation, particles are distributed along the material interfaces and connected with line segments. During each time step, the positions of the particles are updated by interpolating the velocity from the nodes and integrating with respect to time. For simple problems, this approach can work well. In comparison to volume of fluid methods, it represents triple points well. There are, however, several problems that have eroded its popularity over time. First, Lagrangian interface methods, by their nature, assume a fixed connectivity between materials similar to Lagrangian finite element methods. Materials that come into contact, or separate, change the topology of the system. A Lagrangian representation of the boundary must, therefore, have its connectivity dynamically updated. When a material fails by fracture, it generates a new free surface, which must be created. Similarly, new materials may be produced by chemical reactions, and their interfaces must also be dynamically created. However, it is quite difficult to correctly recognize and change the connectivity of the Lagrangian interfaces. Second, errors in the motion of the particles can cause the boundaries to become tangled. This problem is generally severe in high-velocity impact simulations.

Introduction to ALE in FEM

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Third, interpolating the particle velocity from the mesh can introduce significant errors. Consider the case when an element containing steel and air is compressed. The particle resides on the boundary between the steel and the air. Clearly, the correct particle velocity should account for the preferential compression of the air. An interpolation of the velocity from the mesh to the current position of particle, however, corresponds to compressing the air and steel equally. Lastly, the resolution of the interface changes as the boundary is stretched or compressed. Particles can be added or deleted to successfully address this problem. 1.6.5.2. Level set methods Level set methods are popular in the CFD community because of their simplicity and speed. They were originally developed by Osher and his students, and their books [SET 99, OSH 02] are highly recommended. A level set function is defined for the interface between each pair of contacting materials. It is usually the signed distance function, d(x), d(x) = n · d(x),

(1.129)

where n is the normal to the boundary and d(x) is a vector from the closest point on the boundary to x. The location of the interface at d = 0 is determined by interpolating from the values of d at the nodes. In the normal direction to the boundary, the level set function is linear, which most transport algorithms can transport exactly. Specialized fast-marching schemes [SET 99, OSH 02] have been developed to update the values of the level set functions at the nodes of each time step more efficiently and accurately than the standard transport algorithms. To maintain accuracy, the level set function must be periodically reinitialized. For applications in solid mechanics, the primary difficulty with using level set methods is that they are not exactly conservative on the element level. Methods have been developed

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that are globally conservative in terms of volume or mass, but for solid mechanics, it is critical that conservation is also guaranteed at the element level. Experience has shown that small conservation errors will lead to instabilities in the solution due to the high bulk modulus of metals relative to typical fluids. 1.6.5.3. Volume of fluid methods Volume of fluid methods are currently the most popular interface reconstruction methods because they are conservative by construction. The interface is calculated at each time step from the volume of the material in each element and its surrounding neighbors instead of the other way around. Each interface within an element is determined independent of its neighbors in most VOF implementations, and therefore, unlike the other methods, the interface is discontinuous between elements. Recently developed VOF methods [WIL 02, GAR 05, RID 95, RID 98] have minimized the discontinuity between elements to a large degree. The most commonly used class of VOF methods are the piecewise linear interface construction (PLIC) methods. Within each element, the interface is a line or a plane. Parabolic functions have been used successfully in two dimensions [PRI 98], but most of the current research still focuses on PLIC methods. Sharp corners, or other types of geometric discontinuities, are therefore not represented exactly. After propagating across several element, a sharp corner is typically rounded to a radius of two to five elements. The first PLIC method to gain widespread popularity was developed by Youngs [YOU 82]. Each interface is constructed in a two step process. First, the normal direction of the interface, n, is determined by applying a finite difference stencil to the volume fraction of the material in the element and its surrounding neighbors to calculate the gradient of the volume fraction. The gradient is normalized to give the normal, and

Introduction to ALE in FEM

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the function defining the interface may be written as n · x − d = 0,

(1.130)

where d is the shortest distance from the origin to the interface. The value of d is determined by requiring the volume behind the interface to equal the volume of the material in the element. On unstructured meshes, a fixed finite difference stencil cannot be used. A typical approach for evaluating the normal is to average the volume fractions to the nodes and then evaluate the gradient using the element’s B matrix [PEE 00]. Multiple material interfaces within an element are handled via the onion skin model. Consider a sequence of materials numbered 1 through n, with n − 1 interfaces. The first interface, which is between materials 1 and 2, is calculated using the volume of material 1. The second interface, which is between materials 2 and 3, is calculated by using the sum of the volumes of materials 1 and 2. Each successive interface k is calculated by summing the volumes of materials 1 through k , building up layers of material like the layers of an onion. As a consequence of this model, other topologies, such at Tintersections, are not handled well. A fixed ordering of the materials does not work well for many problems. For example, consider the system of circular particles packed in Figure 1.5. A background material should be between every pair of materials in the system, but there is no fixed sequence of material numbers for this system that will keep the background material between the others. For simple problems of this type, a simple rule stating the background material is always between the other materials is sufficient but a dynamic scheme for ordering the materials is required for more complicated cases [BEN 98, MOS 94]. The coordinates centroid of each material in the element are added as solution variables, and updated as the materials are transported. A least squares solution for a line through the centroids is calculated, then the centroids are projected on to the

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a

b

Figure 1.5. System of packed circular particles packed

line. Their ordering along the line provides the ordering used in the onion skin model. Other methods have been equally successful with this type of problem [BEL 92, GAR 05]. 1.7. Future research directions There are several areas where current ALE methods can be improved. The most obvious one, probably, is in interface tracking. Great advances have been made recently, but the underlying approximation of the interface as a piecewise linear polynomial remains. Although quadratic interface reconstruction has been successful in two dimensions [PRI 98], an extension to three dimensions has not been achieved. The increasing accuracy of the current generation of interface reconstruction methods has also increased their cost. For problems with a large number of mixed elements, the interface reconstruction can be the largest single cost of the calculation. Unfortunately, the problems with a large number of mixed elements are the ones that need the high-quality interface reconstruction the most. Transport methods on unstructured meshes are another area of research. One area that is critically lacking is the development of monotonic transport methods for higher order

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elements, e.g. those with quadratic basis functions. Virtually all the current research is devoted to elements that use onepoint integration, restricting their applicability to linear basis functions. The methods for controlling the motion of the mesh have improved over the years but there are still substantial research opportunities. Given a mesh, there are algorithms that are very good at smoothing out the distortions. The control of these algorithms, e.g. which areas should be smoothed and which should have more elements, is still very difficult for users. In the Lagrangian step, the contact methods for interfaces that run through the elements still need improvement. Until recently [VIT 06, BEN 04], all interfaces were treated as bonded. This is not a severe problem for fluid–structure interaction but it is a critical one for modeling the relative slip between metals. Additionally, there are areas where the Lagrangian and Eulerian steps interact. For example, material may fail during the Lagrangian step but the transport during the Eulerian step may “heal” the failed material by dropping the damage parameter within an element from 1.0 to something much less. Ad hoc tracer particle methods have been used in the past to indicate regions where failure must be maintained; however a better and more robust method is desirable. 1.8. Bibliography [ADD 90] A DDESSIO F. L., B AUMGARDNER J. R., D UKOWICZ J. K., J OHN SON N. L., K ASHIWA B. A., R AUENZAHN R. M., Z EMACH C., CAVEAT: A Computer Code for Fluid Dynamics Problems with Large Distortion and Internal Slip (revised edition, 1990), Los Alamos National Laboratory, 1990. [ALD 64] A LDER B., F ERNBACH S., R OTENBERG M., Methods in Computational Physics, Volume 3, Fundamental Methods in Hydrodynamics, Academic Press, New York, 1964.

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[AMD 80] A MDSDEN A. A., R UPPEL H. M., H IRT C. W., SALE: A Simplified ALE Computer Program for Fluid Flow at All Speeds, Report , Los Alamos Scientific Laboratory, 1980. [AMS 73] A MSDEN A. A., H IRT C. W., YAQUI: An Arbitrary LargrangianEulerian Computer Program for Fluid Flow at All Speeds, Report num. LA-5100, Los Alamos Scientific Laboratory, 1973. [BAT 82] B ATHE K. J., Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982. [BEL 92] B ELL R. L., J R . E. S. H., An Improved Material Interface Reconstruction Algorithm for Eulerian Codes, Report num. SAND 92-1716, Sandia National Laboratories, Albuquerque, NM, September 1992. [BEL 00] B ELYTSCHKO T., L IU W. K., M ORAN B., Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons, New York, 2000. [BEN 89] B ENSON D., “An efficient, accurate, simple ALE method for nonlinear finite element programs”, Computer Methods in Applied Mechanics and Engineering, vol. 72, p. 305–350, 1989. [BEN 92a] B ENSON D. J., “Computational methods in Lagrangian and Eulerian hydrocodes”, Computer Methods in Applied Mechanics and Engineering, vol. 99, p. 2356–394, 1992. [BEN 92b] B ENSON D. J., “Momentum advection on a staggered mesh”, Journal of Computational Physics, vol. 100, num. 1, p. 143–162, 1992. [BEN 97] B ENSON D., “A mixture theory for contact in multi-material Eulerian formulations”, Computer Methods in Applied Mechanics and Engineering, vol. 140, p. 59–86, 1997. [BEN 98] B ENSON D. J., “Eulerian finite element methods for the micromechanics of heterogeneous materials: dynamic prioritization of material interfaces”, Computer Methods in Applied Mechanics and Engineering, vol. 151, p. 343–360, 1998. [BEN 02] B ENSON D. J., “Volume of fluid interface reconstruction methods for multi-material problems”, Applied Mechanics Reviews, vol. 55, num. 2, p. 151–165, 2002. [BEN 04] B ENSON D., O KAZAWA S., “Contact in the multi-material Eulerian formulation”, Computer Methods in Applied Mechanics and Engineering, vol. 193, p. 4277–4298, 2004. [BEN 08] B ENSON D. J., “Momentum advection on an unstructured quadrilateral mesh”, International Journal for Numerical Methods in Engineering, vol. 75, num. 13, p. 1549–1580, 2008.

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[BRA 82] B RACKBILL J. U., S ALTZMAN J. S., “Adaptive zoning for singular problems in two dimensions”, Journal of Computational Physics, vol. 46, num. 3, p. 342–368, 1982, 1982. [CHO 78] C HORIN A., H UGHES T. J. R., M C C RACKEN M. F., M ARSDEN J. E., “Product formulas and numerical algorithms”, Communications on Pure and Applied Mathematics, vol. 31, p. 205–256, 1978. [CHR 87] C HRISTENSEN R., Personal communication, Lawrence Livermore National Laboratory, 1987. [COL 84] C OLELLA P., W OODWARD P., “The piecewise parabolic method (PPM) for gas dynamical simulations”, Journal of Computational Physics, vol. 54, p. 174–201, 1984. [COO 89] C OOK R. D., M ALKUS D. S., P LESHA M. E., Concepts and Applications of Finite Element Analysis, 3rd edition, John Wiley & Sons, New York, N. Y., 1989. [DAH 74] D AHLQUIST G., B JORCK A., Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974. [DEB 74] D E B AR R. B., Fundamentals of the KRAKEN Code, Report num. UCIR-760, Lawrence Livermore Laboratory, 1974. [DEN 02] D ENDY E. D., PADIAL -C OLLINS N. T., VANDER H EYDEN W. B., “A general-purpose finite-volume advection scheme for continuous and discontinuous fields on unstructured grids”, Journal of Computational Physics, vol. 180, p. 559–583, 2002. [DON 83] D ONEA J., Arbitrary Lagrangian-Eulerian Finite Element Methods, p. 473–516, Elsevier Science Publishers, B. V., 1983. [DUK 89] D UKOWICZ J. K., C LINE M. C., A DDESSIO F. L., “A general topology godunov method”, Journal of Computational Physics, vol. 82, p. 29–63, 1989. [FLA 81] F LANAGAN D. P., B ELYTSCHKO T., “A uniform strain hexahedron and quadrilateral with orthogonal hourglass control”, International Journal for Numerical Methods in Engineering, vol. 17, p. 679–706, 1981. [FRE 07] F RESSMANN D., W RIGGERS P., “Advection approaches for singleand multi-material arbitrary Lagrangian-Eulerian finite element procedures”, Computational Mechanics, vol. 39, p. 153–190, 2007. [GAR 05] G ARIMELLA R. V., D YADECHKO V., S WARTZ B. K., S HASKOV M. J., “Interface reconstructions in multi-fluid, multi-phase flow simulations”, Proceedings of the 14th International Meshing Roundtable, San Diego, CA, 2005.

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[GIA 88] G IANNAKOPOULOS A. E., E NGEL A. J., “Directional control in grid generation”, Journal of Computational Physics, vol. 74, p. 422–439, 1988. [GOU 82] G OUDREAU G. L., H ALLQUIST J. O., “Recent developments in large-scale finite element lagrangian hydrocode technology”, Computer Methods in Applied Mechanics and Engineering, vol. 33, p. 725–757, 1982. [HAL 98] H ALLQUIST J. O., LS-DYNA Theoretical Manual, Report, Livermore Software Technology Corporation, 1998. [HAR 76] H ARTEN A., H YMAN J. M., L AX P. D., “On finite-difference approximations and entropy conditions for shocks”, Communications on Pure and Applied Mathematics, vol. 29, p. 297–322, 1976. [HAR 89] H ARTEN A., “ENO schemes with subcell resolution”, Journal of Computational Physics, vol. 83, p. 148–184, 1989. [HAR 97] H ARTEN A., E NGQUIST B., O SHER S., “Uniformly high order accurate essentially non-oscillatory schemes”, Journal of Computational Physics, vol. 131, num. 1, p. 3–47, 1997. [HIC 78] H ICKS D. L., “Stability analysis of WONDY for a special case of Maxwell’s law”, Mathematics of Computation, vol. 32, p. 1123, 1978. [HUG 81] H UGHES T. J. R., L IU W. K., Z IMMERMAN T. K., “LagrangianEulerian finite element formulation for incompressible viscous flows”, Computer Methods in Applied Mechanics and Engineering, vol. 29, p. 329–349, 1981. [HUG 87] H UGHES T. J. R., The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987. [HYM 84] H YMAN J. M., “Numerical methods for Tracking Interfaces”, Physica 12D, p. 396–407, 1984. [JOH 81] J OHNSON J. N., “Dynamic fracture and spallation in ductile solids”, Journal of Applied Physics, vol. 52, num. 4, p. 2812–2825, 1981. [KUR 67] K UROPATENKO V. F., “On difference methods for equations of hydrodynamics”, J ANENKO N. N., Ed., Difference Methods for Solutions of Problems of Mathematical Physics, I, American Mathematical Society, Providence, RI, 1967. [LAX 54] L AX P. D., “Weak solutions of nonlinear hyperbolic equations and their numerical computation”, Communications on Pure and Applied Mathematics, vol. 7, p. 159–193, 1954.

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[MOS 94] M OSSO S., C LANCY S., Geometrically Derived Priority System for Youngs’ Interface Reconstruction, Report , Los Alamos National Laboratory, 1994. [NOH 64] N OH W. F., “CEL: A time-dependent, two-space-dimensional, coupled Eulerian-Lagrange code”, Methods in Computational Physics, p. 117–179, Academic Press, New York, 1964. [ORA 87] O RAN E. S., B ORIS J. P., Numerical Simulation of Reactive Flow, Elsevier Sciences Publishing, Inc., 1987. [OSH 02] O SHER S. J., F EDKIW R. P., Level Set Methods and Dynamic Implicit Surfaces, Springer, Berlin, 2002. [PAN 04] PANTALE O., B ACARIA J.-L., D ALVERNY O., R AKOTOMALALA R., CAPERAA S., “2D and 3D numerical models of metal cutting with damage effects”, Computer Methods in Applied Mechanics and Engineering, vol. 193, p. 4383–4399, 2004. [PEE 00] P EERY J. S., C ARROLL D. E., “Multi-material ALE methods in unstructured grids”, Computer Methods in Applied Mechanics and Engineering, vol. 187, p. 591–619, 2000. [PON 98] P ONTHOT J.-P., B ELYTSCHKO T., “Arbitrary LagrangianEulerian formulation for element-free Galerkin method”, Computational Methods for Applied Mechanics and Engineering, vol. 152, p. 19–46, 1998. [PRI 98] P RICE G. R., R EADER G. T., R OWE R. D., B UGG J. D., “A piecewise parabolic interface calculation for volume tracking”, Proceedings of the Sixth Annual Conference of the Computational Fluid Dynamics Society of Canada, University of Victoria, Victoria, British Columbia, 1998. [RID 95] R IDER W. J., K OTHE D. B., Stretching and Tearing Interface Tracking Methods, Report num. AIAA 95-1717, AIAA, 1995. [RID 98] R IDER W. J., K OTHE D. B., “Reconstructing volume tracking”, Journal of Computational Physics, vol. 141, p. 112–152, 1998. [ROA 76] R OACHE P. J., Computational Fluid Dynamics, Hermosa Publishers, 1976. [SCA 99] S CARDOVELLI R., Z ALESKI S., “Direct numerical simulation of free-surface and interfacial flow”, Annual Review of Fluid Mechanics, vol. 31, p. 567–603, 1999. [SET 99] S ETHIAN J. A., Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge, 1999.

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[SHU 88] S HU C. W., O SHER S., “Efficient implementation of essentially non-oscillatory shock-capturing schemes”, Journal of Computational Physics, vol. 77, num. 2, p. 439–471, 1988. [THU 96] T HUBURN J., “Multidimensional flux-limited advection schemes”, Journal of Computational Physics, vol. 123, p. 74–83, 1996. [VAN 77] VAN L EER B., “Towards the ultimate conservative difference scheme IV: A new approach to numerical convection”, Journal of Computational Physics, vol. 23, p. 276–299, 1977. [VAN 84] VAN L EER B., Multidimensional Explicit Difference Schemes for Hyperbolic Conservation Laws, p. 493, Elsevier Science, New York, 1984. [VIT 06] V ITALI E., B ENSON D. J., “An extended finite element formulation for contact in multi-material arbitrary Lagrangian-Eulerian calculations”, International Journal for Numerical Methods in Engineering, vol. 67, p. 1420–1444, 2006. [VON 50] V ON N EUMANN J., R ICHTMYER R. D., “A method for the numerical calculation of hydrodynamic shocks”, Journal of Applied Physics, vol. 21, p. 232–237, 1950. [WIL 64] W ILKINS M., Calculation of Elastic – Plastic Flow, vol. 3, p. 211– 263, Academic Press, New York, 1964. [WIL 02] W ILLIAMS M. W., K OTHE D., K ORZEKWA D., T UBESING P., “Numerical methods for tracking interfaces with surface tension in 3-D mold filling processes”, Proceedings of FEDSM’02 ASME Fluid Engineering Division Summer Meeting, 2002. [WIN 63] W INSLOW A. M., Equipotential Zoning of Two-Dimensional Meshes, Report num. UCRL-7312, Lawrence Radiation Laboratory, 1963. [WIN 81] W INSLOW A. M., Adaptive Mesh Zoning by the Equipotential Method, Report num. UCID-19062, Lawrence Livermore National Laboratory, 1981. [WIN 82] W INSLOW A. M., B ARTON R. T., Rescaling of Equipotential Smoothing, Report num. UCID-19486, Lawrence Livermore National Laboratory, 1982. [YOU 82] Y OUNGS D. L., “Time dependent multi-material flow with large fluid distortion”, M ORTON K. W., B AINES M. J., Eds., Numerical Methods for Fluid Dynamics, p. 273–285, Academic Press, New York, 1982.

Chapter 2

Fluid–Structure Interaction: Application to Dynamic Problems

2.1. Introduction This chapter gives a survey related to numerical simulation of fluid–structure interaction applications for dynamic explicit problems. In recent years, these problems have assumed increased importance because of new developments in industry to analyze the safety and integrity of the structure against dynamic loading; in particular, in the naval industry for high hydrodynamic impact analysis of deformable structure, in the automotive industry for airbag deployment and fuel tank sloshing, and in the aerospace industry for bird impact analysis and helicopter ditching. A demand for better design of the structure requires improved procedure analysis, and in many cases the safety and integrity of the structure cannot be estimated unless hydrodynamic forces for the structure loading are computed accurately by solving Navier–Stokes equations.

Chapter written by Mhamed S OULI.

51

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In order to describe the algorithms used for the ALE formulation and coupling, a detailed presentation of the ALE and coupling are given. To illustrate these methods, a simple academic piston problem and two industrial application are described. There are several ways of solving fluid–structure interaction problems. The classical one is a Lagrangian formulation for the structure and ALE formulation for the fluid; at the fluid structure interface, a contact or interface algorithm is used, this algorithm can be explicit or implicit. For most of the dynamic problems, fluid mesh undergoes high mesh distortion, the existing ALE mesh smoothing, equipotential algorithm and elasticity like problems for mesh motion, are not robust and stable enough to solve many dynamic problems including bird strike and airbag deployment. The main concern in fluid–structure interaction problems is the computation of the fluid forces that act on a rigid or deformable structure. These problems are mainly encountered by the structure community, thus several simplified methods have been used to solve hydrodynamic equations, for instance in shipbuilding industry, and empirical solutions and handbook calculations have been used to estimate forces due to hydrodynamic effects. The hydrodynamic forces are applied as external loads on the structural dynamic model to predict failure, stress fatigue, and creep damage to the structure. Since force calculations rely on an accurate knowledge of the flow field, an accurate fluid–structure analysis requires skilled engineering talent and careful modeling. The application of fluid–structure interaction technology allows us to move beyond handbook-design techniques, which use generic experimental correlation, to correctly predict the hydrodynamic forces by solving the hydrodynamic equations and using an appropriate coupling algorithm to communicate forces between fluid and structure for dynamic equilibrium.

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In the fluid–structure coupling algorithm, two superimposed meshes are considered, a fixed Eulerian or ALE mesh for the fluid and a deformable Lagrangian mesh for the structure. Unlike existing algorithms that couple two separate codes, a CFD and a structure code, the fluid–structure interaction algorithm developed in the next section is a fully coupled algorithm. In the first section, the governing equations for the fluid and structure are presented together with boundary conditions. In section 2.2, a detailed description of the Euler– Lagrange coupling algorithm is presented together with the description of a regular penalty contact algorithm. The damping formulation implemented in the coupling is described in the last part of this section. In section 2.3, the improvements to the Euler–Lagrange coupling are validated in a simple test problem where a column of fluid compressed by an elastic piston. The fluid–structure interaction in this example are modeled using two different methods, an Euler–Lagrange coupling algorithm and a classical Lagrangian formulation using a tied contact at the fluid–structure interface. Since the piston problem is a simple one, fluid mesh distortions are not severe and thus the approach of the problem by a Lagrangian formulation is reliable. The solution of the coupling problem is compared to the Lagrangian solution, which is considered to be a reference solution. In the second part of section 2.3, the analytical pressure from Zhao et al. [ZHA 93] is compared to the numerical results obtained by the Euler–Lagrange coupling with damping algorithm. Finally, the slamming modeling is described and the numerical results are compared to theoretical results. The coupling method can be applied in many fields of engineering, for example in the automotive industry, where a simulation of an airbag deployment is described. In civil engineering, the coupling technique is applied to the response of sloshing tank under seismic loading.

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2.2. General equations

ALE

description

of

Navier–Stokes

In the ALE description, an arbitrary referential coordinate is introduced in addition to the Lagrangian and ALE coordinates. The material time derivative of a variable f with respect to a reference coordinate can be described as → − → −−→ → df ( X , t) ∂f (− x , t) → → = + (− v −− w ).gradf (− x , t), dt ∂t

(2.1)

→ − → x the ALE coordinate, where X is the Lagrangian coordinate, − → − → − v the particle velocity, and w the velocity of the reference coordinate, which will represent the grid velocity for the numerical simulation, and the system of reference will be later the ALE grid.

The equations of mass, momentum, and energy conservation for a Newtonian fluid in ALE formulation in the reference domain, are given by

ρ

ρ

∂ρ → → → + ρ div(− v ) + (− v −− w )grad(ρ) = 0 ∂t

(2.2)

→ −→ → − ∂− v → → → + ρ(− v −− w ).grad(− v ) = div(σ) + f ∂t

(2.3)

−−→ →→ − ∂e → → → + ρ(− v −− w ).grad(e) = σ : grad(− v ) + f .− v, ∂t

(2.4)

where ρ is the density and σ is the total Cauchy stress given by → → σ = −p.Id + μ(grad(− v ) + grad(− v )T ), (2.5) where p is the pressure and μ is the dynamic viscosity. Equations (2.2)–(2.4) are completed with appropriate boundary conditions.

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One of the major difficulties in time integration of the ALE Navier–Stokes equations is because of the nonlinear term in the general ALE Navier–Stokes equations (2.2)–(2.4), with the → → v −− w ), for some ALE formulations, the relative velocity (− mesh velocity can be solved using a remeshing and smoothing process. → − → w = 0, In the Eulerian formulation, the mesh velocity − this assumption eliminates and simplifies the remeshing and smoothing process, but does not simplify the Navier–Stokes equations (2.2)–(2.4).

Basically, there are two approaches to implement the Navier–Stokes equations. The first approach is to solve the fully coupled equations (2.2)–(2.4), this approach is used by Hughes et al. [HUG 81]. Kennedy et al. [KEN 81] adopted this approach to solve fluid–structure interaction problems. An alternative approach is the split method adopted in this chapter, and used by Souli et al. [SOU 00] and Lee et al. [LEE 02], the split method is used in most hydrocodes. Operator splitting is a convenient method for breaking complicated problems into series of less complicated problems. In this approach, first a Lagrangian phase is performed, using an explicit finite element method, in which the mesh moves with the fluid particle. In CFD community, this phase is referred to as a linear Stokes problem. In this phase, the changes in velocity, pressure, and internal energy because of external and internal forces are computed. The equilibrium equations for the Lagrangian phase are → −→ → − v d− = div(σ) + f ρ (2.6) dt →→ − de → = σ : grad(− v ) + f .− v. (2.7) dt For Lagrangian formulation, there is no need for continuity equation, since in Lagrangian meshes, to compute the density, mass conservation equation is used in its integrated form ρ

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rather than as a partial differential equation. This eliminates the need to include a continuity equation. In the second phase, called advection or transport phase, the transportation of mass, momentum, and energy across element boundaries are computed. This may be thought of as remapping the displaced mesh at the Lagrangian phase back to its initial position. The transport equations for the advection phase are ∂φ → − ∂t + u .grad(φ) = 0 (2.8) → φ(− x , 0) = φ0 (x), → → → where − u =− v −− w is the convected velocity.

Equation (2.8) is solved successively for the conservative variables: mass, momentum, and energy with initial condition φ0 (x) taken at the current time step is the solution from the Lagrangian calculation (equations (2.6)–(2.7)). In equation (2.8), the time t is a fictitious time: in this chapter, the time step is not updated when solving the transport equation. There are different ways of splitting the Navier–Stokes problems; in some split methods, the Stokes problem and transport equation are solved successively for half a time step. The hyperbolic equation system (2.8) is solved for mass, momentum, and energy by using a finite volume method. Either a first-order upwind method or a second-order Van Leer advection algorithm [LEE 77] can be used to solve equation (2.8). To solve the fluid problem for one time step, we proceed in two phases: In the first phase, we use an explicit time integration method to solve equations (2.6)–(2.7) for mass, momentum, and energy conservation. A finite element formulation is used for the first phase. The density, momentum, and energy from the first phase are used as initial conditions for the transport

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equation in the second phase. A second-order time integration explicit method is used for the transport equation. The following section compares two fluid–structure interaction problems: the contact problem where a contact interface is used to separate the fluid mesh from the structure mesh and the Euler–Lagrange coupling problem where the Eulerian grid overlaps the structure mesh.

2.3. Fluid–structure interaction In this section, we analyze and compare two fluid– structure interaction problems: the contact problem where a contact interface is used to separate the fluid mesh from the structure mesh and the Euler–Lagrange coupling problem where the Eulerian grid overlaps the structure mesh. The fluid–structure interaction problem considered in this chapter can be treated using two finite element formulations. The first one is the contact formulation, where a contact interface is used to separate the fluid mesh from the structure mesh, for these problems, contact algorithms, described in detail in [ZHO 93], are used to compute the contact forces applied from the fluid to the structure and vice versa. For explicit methods, nodal forces at the contact interface are updated at each time step to take into account contact forces. Since the fluid nodes at the contact interface move in order to remain in contact with the Lagrangian structure, an ALE method is required to remesh the fluid domain. For small fluid mesh deformations, classical ALE methods described in [SOU 01] can be used, but for large mesh distortion, ALE methods cannot be used for the mesh motion. To solve the problem, a rezoning or automatic remesh method is required for the fluid domain, which is CPU time consuming. This method is based on an interpolation algorithm, which is first-order accurate, non-conservative and numerically dissipative. An alternative

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algorithm to avoid fluid mesh distortion is to use an Euler– Lagrange coupling, a Eulerian formulation for the fluid and a Lagrangian formulation for the structure. Using Euler–Lagrange coupling allows us to treat the impact problems involving fluids, because this coupling treats the interactions between a Lagrangian formulation by modeling the structure and a Eulerian formulation by modeling the fluid. The Lagrangian finite element formulation uses a computational mesh that follows the material deformation. This approach is efficient and accurate for problems involving moderate deformations such as structure motions or flows that are essentially smooth. When the latter departs from this kind of smoothness, the ALE formulation must be used because the finite element mesh is allowed to move independently from the material flow. In fact, ALE codes allow material to flow through the mesh and therefore the remap step in the ALE algorithm needs an advection algorithm like upwind method for first-order advection, or Van Leer or Godunov methods for higher order advection. The convection schemes available in the explicit finite element code used in this chapter are the donor-cell algorithm, or upwind method for first order, and the Van Leer algorithm for second order. In this chapter, we use second-order advection to minimize dissipation and dispersion effects. If the fluid is subjected to large deformations, Lagrangian or ALE meshes are strongly distorted, which jeopardizes the simulation because distorted elements have low accuracy and their stable time step sizes are small for explicit time integration algorithms. For single material or multimaterial Eulerian formulation, the mesh is fixed in space and materials flow through the mesh using an advection scheme to update fluid velocity and history variables, thereby eliminating all problems associated with distorted mesh that are commonly encountered with a Lagrangian or ALE formulation. Therefore, the Euler–Lagrange coupling using Eulerian multimaterial formulation for the fluid is more suitable for solving slamming problems and, more generally, fluid–structure interaction problems. First, the multimaterial Eulerian formulation is able to simulate large deformations of the free

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surface, and second, the coupling treats fluid–structure impacts. From a mechanical point of view, the coupling algorithm introduced in this chapter is similar to penalty contact algorithm; it is mainly based on force equilibrium and energy conservation, and can be described as Eulerian contact. We first present interface conditions for contact and coupling formulations as well as the contact and coupling algorithm description. A classical impact problem is used in order to highlight the limitation of contact algorithms when the fluid mesh is highly distorted. The end of this section is devoted to the damping formulation introduced in the coupling algorithm.

2.3.1. Contact algorithms for fluid–structure interaction problems The detailed description of finite element contact algorithms is not the goal of this chapter. However, since the coupling method described in this chapter is based on the penalty method for contact algorithms, the contact approach is a good introduction to this method. In contact algorithms, a contact force is computed proportional to the penetration vector, the amount the constraint is violated. In an explicit FEM method, contact algorithms compute interface forces because of impact of the structure on the fluid (Figure 2.1), these forces are applied to the fluid and structure nodes in contact in order to

Slave node Master node Structure

Ff = -k.d

k

d Fluid

Slave node

Master node Fs = k.d

t=

t0

t=

t0

+ Δt

Figure 2.1. Sketch of contact algorithm

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ALE and Fluid–Structure Interaction

prevent a node from passing through contact interface. An ALE or Lagrangian mesh is used for the fluid. The literature on contact algorithms is extensive, but most of it is devoted to static problems. The literature devoted to contact for dynamic fluid–structure interaction problems is very limited: one of the problems encountered in these applications is the high mesh distortion at the contact interface as a result of high fluid nodal displacement and velocity. This problem is still unsolved since most of the ALE remeshing algorithms including the equipotential methods, simple and volume average, are not efficient to maintain a regular mesh for the calculation to continue, see [SOU 00], [BEN 89]. In contact algorithms, one surface is designated as a slave surface, and the second as a master surface. The nodes lying on both surfaces are also called slave and master nodes, respectively. For a fluid–structure coupling problem, for example fluid with an initial velocity impacting a structure, as described in Figure 2.2, the fluid nodes at the interface are considered as slave, and the structure elements as master. The first approach for contact is the kinematic contact, constraining fluid and structure nodes to the same velocity. Kinematic contact conserves total momentum, but not total energy. The second approach, the penalty contact, is different from the previous one. The penalty method imposes a resisting force to the slave node, proportional to its penetration through the master segment. This force is applied to both the slave node and the nodes of

Fluid

V Structure

Figure 2.2. Contact between two distinct Lagrangian meshes, time = 0 ms

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61

the master segment in opposite directions to satisfy equilibrium: the force applied to the nodes of the master segment are scaled by the shape functions: Fs = −k · d

(2.9)

i Fm = Ni kd,

(2.10)

where N i is the shape function at node i (i = 1, 2 in two dimensions, and i = 1, . . . , 4 in three dimensions) of the master segment taken at slave node location and d the penetration vector. In case the slave node coincides exactly with one of the master node (see Figure 2.1) , node 1 for instance, we will have Fs = −k · d 1 2 3 4 Fm = k · d and Fm = Fm = Fm = 0.

The coefficient k represents the stiffness of a spring. In fact, this method consists in placing normal interface springs between all penetrating nodes and the contact surface. The spring stiffness is given by equation (2.11) in terms of the bulk modulus K of the master material, V the volume of the master element and A the area of the master segment: k = pf

KA2 , V

(2.11)

where pf is scale factor for the interface stiffness, which satisfies 0 ≤ pf ≤ 1. Nevertheless, the contact algorithms have a drawback: for large deformations of materials, the mesh distortions are important. To illustrate the contact algorithm in fluid–structure interaction problem, we consider a volume of fluid with ini , impacting an elastic structure, modeled with tial velocity V shell elements, Belytschko-type shell, described in detail in [BEL 84], [HAL 98]. Mesh distortion appears in Figures 2.3 and 2.4 at t = 0.15 ms and t = 0.5 ms, respectively. At the earlier time t = 0.15 ms, we have small mesh deformations

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ALE and Fluid–Structure Interaction

Figure 2.3. Contact between two distinct Lagrangian meshes, time = 0.15 ms

and the time step controlling the explicit calculation does not decrease significantly. But later in time, at 0.5 ms, large mesh distortion causes the time step to decrease, and the accuracy decreases as a result of highly distorted elements at the contact interface. To prevent high mesh distortion, an explicit Euler–Lagrange coupling method has been developed to compute coupling forces at the fluid–structure interface nodes. 2.3.2. Euler–Lagrange coupling For simplicity, we assume the fluid is solved in a Eulerian fixed mesh, although there is no restriction in the coupling formulation to a Eulerian mesh. The method has been extended and used for general ALE problems. In the Euler–Lagrange coupling, the structure is embedded in a Eulerian fixed mesh,

Figure 2.4. Large deformations of the material involving mesh distortion, time = 0.5 ms

FSI: Application to Dynamic Problems

Fluid 2

n=

63

grad f grad f

Material interface

Fluid 1

Multimaterial Eulerian cell

Figure 2.5. VOF method

as shown in Figures 2.6 and 2.7. Since the mesh is fixed and the fluid material flows through the mesh, a larger Eulerian mesh is required instead of the physical fluid mesh. When fluid material leaves the mesh through the element faces, it will occupy adjacent elements that are initially voided or they are air material elements. During material transport or advection, the interface elements are partially voided, thus the volume fraction f of these elements satisfies 0 < f < 1. To track the material interface in elements, the Young method is used. In this method, the material layout is described solely by the volume fraction of the fluid material in the element. Specifically, a straight line using the SLIC technique (simple linear interface calculation) of Woodward and Collela [WOO 82] approximates, the interface in the cell described in Figure 2.5. A detailed multimaterial and VOF (volume of fluid) formulation is described in [AQU 03]. This chapter describes in detail the split method used for ALE multimaterial and single-material formulations, but coupling with structure is not involved. It is the aim of this chapter, however, to describe the coupling and damping algorithms to solve fluid–structure interaction problems.

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In an explicit time integration problem, the main part of the procedure in the time step is the calculation of the nodal forces. After computation of fluid and structure nodal forces, we compute the forces due to the coupling, these will only affect nodes that are on the fluid structure → interface. For each − structure node, a depth penetration d is incrementally up→ → vs − − vf ) dated at each time step, using the relative velocity (− at the structure node, which is considered as a slave node, and the master node within the Eulerian element. The location of the master node is computed using the isoparametric coordi→ − nates of the fluid element. At time t = 0, we consider d 0 = 0: the master and slave node coincide. At time tn = tn−1 +Δt, the penetration vector is updated as follows: −n − → → → → d = d n−1 + (− vs − − vf ) · Δt.

(2.12)

→ vf is the velocity at the master node locaThe fluid velocity − tion, interpolated from the nodes of the fluid element at the current time step. For this coupling, the slave node is a structure mesh node, whereas the master node is not a fluid mesh node, it can be viewed as a fluid particle within a fluid element, with mass and velocity interpolated form the fluid element nodes using finite element shape functions. The vector →n − d represents the penetration depth of the structure inside the fluid during the time step, which is the amount the constraint is violated. The coupling force acts only if penetration →n →·− → is built up by averaging nord < 0 , where − n n occurs, − s s mals of structure elements connected to the structure node. For clarity, the superscript of the penetration has been omit→ − ted, we will use d instead of d n , for the penetration vector.

Penalty coupling behaves like a spring system and penalty forces are calculated proportionally to the penetration depth and spring stiffness. The head of the spring is attached to the structure or slave node and the tail of the spring is attached to the master node within a fluid element that is intercepted by the structure, as illustrated in Figure 2.6.

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Void

65

Void

Structure Ff = -k.d Slave node Fluid

Fluid element

Fluid Fs = k.d

Fluid node Master node (fluid particle)

t=

t0

d

t = t0 + Δt

Figure 2.6. Description of the coupling scheme

Similarly to penalty contact algorithm, the coupling force is described by F = k · d, (2.13) where k represents the spring stiffness and d is the penetration. The force F in equation (2.13) is applied to both master and slave nodes in opposite directions to satisfy force equilibrium at the interface coupling, and thus the coupling is consistent with the fluid–structure interface conditions, namely the action–reaction principle. The main difficulty in the coupling problem comes from the evaluation of the stiffness coefficient k in equation (2.13). The stiffness value is problem dependent, a good value for the stiffness should reduce energy interface in order to satisfy total energy conservation, and prevent fluid leakage through the structure. The value of the stiffness k is still a research topic for explicit contact-impact algorithms in structural mechanics. For fluid–structure coupling, the spring stiffness is deduced from explicit contact with the penalty method. This value is difficult to obtain unless numerical experiments are run systematically. For some industrial problems, the automotive industry for instance, experimental tests are done prior to analysis to evaluate the stiffness value for penalty contact algorithm.

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In this chapter, the stiffness k is based on stiffness used in explicit contact algorithms in [ZHO 93]. Here k is given in term of the bulk modulus K of the fluid element in the coupling containing the slave structure node, the volume V of the fluid element that contains the master fluid node, and the average area A of the structure elements connected to the structure node. KA2 . k = pf (2.14) V However, to avoid numerical instabilities, a scalar factor pf , 0 ≤ pf ≤ 1, is introduced as in the contact method. For impact problems, we always have to examine the influence of this parameter on the solution. In impact problems, after contact, the master and slave nodes may separate: this phenomenon is known as release. If the penalty force is very large, the impact and the release may occur in the same time step involving a numerical instability. To avoid this anomaly, the force F in equation (2.13) can be bounded by the contact force between two spheres, defined by Belytschko and Neal [BEL 89], which is given by → → vs − − vf ) Ms Mm (− , F ≤ (2.15) Δt(Ms + Mm ) where Ms is the mass of the slave or structure node, Mm is the mass of the master fluid node, interpolated from the fluid → → vs − − vf ) is the relative velocity defined in element nodes, and (− equation (2.12), Δt is the current time step. As mentioned in the previous section, the coupling algorithm can be used for problems involving large mesh distortion that contact algorithm cannot handle. To illustrate the capability of the coupling for solving problems where contact algorithms fail because of high mesh distortion, we consider an impact problem described in the previous section (Figures 2.2–2.4) that we run long enough in time for the fluid mesh to become highly distorted. In this case, the problem is treated using a multimaterial or VOF formulation for the fluid, and by applying the coupling forces defined by equation (2.13) at

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Fluid

Void

V Structure

Figure 2.7. Coupling between Lagrangian and Eulerian mesh, time = 0 ms

the fluid structure interface. The problem is illustrated in Figures 2.7–2.9. As shown in Figures 2.8 and 2.9, the fluid material flows through a fixed mesh, and the coupling interface between the fluid and the structure is no longer a fluid boundary interface defined by the mesh as in a contact algorithm, but it is a material surface defined by fluid elements that

Figure 2.8. Coupling between Lagrangian and Eulerian mesh, time = 0.15 ms

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Figure 2.9. Coupling between Lagrangian and Eulerian mesh, time = 0.50 ms

are partially filled. The impact at time t = 0.5 ms showed in Figures 2.4 and 2.9 is modeled by two algorithms, coupling and the contact algorithms, respectively. In the contact algorithm in Figure 2.4, the fluid undergoes high mesh distortion, which makes it difficult for the problem to continue, since time step decreases. For the coupling algorithm in Figure 2.9, fluid mesh distortion is no longer a problem. However, problems related to fluid leakage through the structure may occur for high-velocity impact problems. In such cases, the fluid particle penetrates so deep within the structure that the coupling force in equation (2.13) is not large enough to return it to the coupling interface. For general problems, one solution to this problem is to reduce the time step. But for some problems involving highly compressible gas, expression (2.13) needs to be modified: the spring system is nonlinear and the stiffness k should be penetration dependent: F = k(d) · d.

(2.16)

Fluid leakage through the structure is a difficult problem to solve, it is not the aim of this chapter to describe different

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numerical algorithms to prevent leakage. Most of the research in fluid–structure interaction has focused on developing numerical algorithms that prevent leakage and conserve energy. 2.3.3. Damping in the coupling In general, the numerical dissipation in the numerical scheme is a problem and its limitation is mandatory to respect the physical phenomenon. For instance, Piperno et al. [PIP 96] studied the numerical damping errors resulted due to discrete fluid structure coupling in order to enhance the numerical scheme by compensating these errors. Several methods can be considered to obtain a filtered curve. Some codes of numerical simulation use numerical implicit schemes with integration parameters, which can be adjusted to achieve numerical accuracy and stability. The classical Newmark scheme, second-order and stable scheme can filter out numerical noise. Nevertheless, the numerical damping introduced in the Newmark method brings the order of accuracy to a lower level. The HHT method developed by Hilber, Hughes, and Taylor improves the Newmark approach by introducing the a-method [HIL 77]. In this chapter, a viscous coupling is added to the coupling to damp out high frequency oscillations, but a better way can be found to stabilize the numerical scheme. For the penalty coupling algorithm, since the finite element software used in this chapter is an explicit code, a damping force is added to the penalty force to have a smoother response and consequently better convergence characteristics. In order to prevent damping force from altering the physics of the problem and give rise to an inconsistent scheme, an optimal damping is used. In fact, the process is a classical one: the spring system described for the contact and coupling force is completed by a damper as shown in Figure 2.10 for contact and in Figure 2.11 for coupling. Let us note k is the spring stiffness, C is the damping coefficient, mstructure is the structure nodal mass, and mf luid is the

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Slave node Master node

.

Structure

Ff = -k.d - C.d k

d Fluid

C

.Slave node Fs = -k.d + C.d

Master node

t = t0 + Δt

t = t0

Figure 2.10. Contact algorithm with a dashpot

mass of the fluid node or particle. If physical stiffnesses are small compared to k, the coupling and contact algorithm with damping enhancement can be represented by the scheme 1 of Figure 2.12. This scheme is equivalent to the scheme 2 with M, the equivalent mass. 2

The equilibrium of the inertia force M · ddt2 d, the damping force C · ddt d, and the stiffness force k · d satisfy the following equation: d2 d M · 2 d + C · d + k · d = 0, (2.17) dt dt

Void

Void

Structure

.

C

Ff = -k.d - C.d C

Slave node

k Fluid

k

Fluid element

Fluid

.

Fs = k.d - C.d

Fluid node Master node (fluid particle)

t=

t0

d

t = t0 + Δt

Figure 2.11. Coupling algorithm with a dashpot

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Sketch 1

71

Sketch 2

mfluid

k

k

C

C M

mstructure

mstructuremfluid

M=

mstructure + mfluid

Figure 2.12. Schemes of the penalty method with damping



·m f l u i d where M = mmssttrruuccttuurree+m and C = ξ · f lu id damping factor.

k · M with ξ , the

Equation (2.17) can be written again as d d2 d + ξω · d + ω 2 · d = 0 2 dt dt

(2.18)

with the pulsation ω=

 k·

m s t r u c t u r e +m f l u i d m s t r u c t u r e ·m f l u i d

.

(2.19)

The damping is optimal when ξ is critical. For ξ = 2, the numerical oscillations must be completely damped out. The main drawback of this approach is a possible alteration of the physical aspects if the numerical stiffness k is close to the highest physical stiffness: physical oscillations may be damped out. However, the numerical stiffness is generally large compared to the physical stiffness as shown in the following applications.

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2.4. Numerical applications 2.4.1. Piston problem In order to highlight numerical applications of the algorithms evoked in the previous section, we consider the following simple problem: a column of water 0.5 m high is compressed by a piston moving with constant velocity of 10 m/s. Sliding conditions are applied along the walls of the cavity. A Newtonian material model with constant viscosity is used for the water and the equation of state is •

ΔP = −ρc2 tr(ε)Δt,

(2.20)

where ΔP is the incremental pressure, Δt the is time step, • ρ is the density, c is the sound velocity and ε is the rate of deformation. The bulk modulus, K = ρc2 = 2.25 GPa, is important enough to consider water as slightly compressible. A rigid material model is used for the structure of the piston. The simple problem can be solved using a classical Lagrangian formulation for fluid and structure, and shared nodes at the interface with no contact involvement. The fluid mesh remains regular during fluid compression, pressure, and forces at the fluid interface can be considered as reference data. The problem is illustrated in Figure 2.13, which shows the configuration at initial time. At time t = 2.5 ms, 5 ms, and 7.5 ms, we keep integrity and regularity of the fluid mesh, the mesh is linearly compressed and mesh regularity is satisfied. Structure V Water t = 0 ms

t = 2.5 ms

t = 7.5 ms

Figure 2.13. Piston problem with shared nodes

t = 10 ms

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Void Structure V Water t = 0 ms

t = 2.5 ms

t = 7.5 ms

t = 10 ms

Figure 2.14. Piston problem with coupling

Pressure (Pa) E+06)

In Figure 2.14, a Euler–Lagrange coupling method is used for the same problem. Because of fluid incompressibility and the nature of the penalty coupling, the coupling method generates high oscillations for interface forces and pressure time history as shown in Figure 2.15. The damping force, proportional to relative velocity of the fluid and the structure, is added to the coupling force for the evaluation of nodal forces. The damping force developed in the previous section is used to damp out high frequencies. The high frequency phenomena are related to fluid incompressibility, and are significant for incompressible fluid. For fluid–structure problems where the fluid

10

5

0 A Coupling B Coupling with damping C Lagrangian curve

–5 0

0.5

1

1.5

Time (s) E-03)

Figure 2.15. Fluid pressure histories: (a) coupling without damping, (b) coupling with damping, (c) reference curve

2

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is highly compressible, e.g. airbag inflation, pressure or interface force time history are not affected by high oscillations, and no damping force is required. The equivalent stiffness of the fluid in the piston is determined by equation (2.20): •

ΔP = −K tr(ε)Δt = −K

ΔΩf ΔD , = −K Ωf D

(2.21)

where Ωf is the fluid cavity volume, D = 0.5 m is the depth, ΔD is the incremental displacement of the piston. The equivalent stiffness by area unit is ρc2 K = = 4.5 GPa.m−1 . D D

(2.22)

The numerical stiffness by area unit is given by equation (2.14) for pf = 1 and the mesh size dx = VA = D8 according to Figure 2.14 : KA K K k = pf = = 8 = 36 GPa.m−1 . A V dx D

(2.23)

Thus the numerical stiffness is sufficiently large compared to the physical stiffness. 2.4.2. Two-dimensional slamming modeling In order to highlight the capability of the coupling algorithm described in this chapter, we consider the twodimensional impact problem described in Figure 2.16. A rigid wedge impacting water with a deadrise angle, the angle between the water-free surface and the wedge, of 10o . Unlike the piston problem, the contact algorithm fails when solving this problem, because of high mesh distortion for the fluid. This problem has been selected for the following three reasons.

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Wedge V

Free surface

y x 10°

Figure 2.16. Rigid wedge impacting water free surface with a deadrise of 10 degrees

First, there are several papers published earlier followed the theoretical approach for solving the pressure applied from the fluid on the structure. For details, we can refer to the work done by Zhao et al. [ZHA 93], Wagner [WAG 32], Dyment [DYM 86]. A development of Wagner’s theory is used for a two-dimensional rigid wedge with small deadrise angle. A detailed aspect of the theoretical approach of the problem is described later. The numerical results for the slamming problem are compared to the theoretical approach described by Wagner [WAG 32] using a potential flow theory for the fluid. Second, unlike the piston problem, this problem cannot be solved using a Lagrangian formulation for the fluid, and a sideline at the fluid structure interface, or a simulation similar to the one used for the piston problem in Figure 2.13. Using a Lagrangian method for the fluid, the fluid mesh will undergo high distortion after few time steps, which causes the time step to decrease. To solve the problem, the coupling method where the fluid is solved with a Eulerian formulation and the structure with a Lagrangian formulation is required. Third, the fluid is incompressible, the penalty coupling using the

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ALE and Fluid–Structure Interaction

coupling force based on the spring system will generate high oscillations, and thus damping forces are required to smooth out high frequencies and reduce the noise in the solution. 2.4.2.1. Numerical approach of a two-dimensional slamming problem The model represented by a Eulerian formulation with eight-node brick elements (see Figure 2.18) is considered. The grid is sandwiched between both planes of symmetry orthogonal to z-axis to obtain a two-dimensional model. A third vertical plane of symmetry orthogonal to x-axis goes through the apex of the wedge to reduce the number of finite elements. Transmitting boundary conditions are applied to the three other Eulerian boundaries. The wedge is modeled by three jointed strips consisting of Lagrangian shell elements. The position of the three flat plates is presented in Figure 2.16. The first one is inclined by an angle of 10o , the second one is vertical and the last one is horizontal. The dimensions of the structure are shown in Figure 2.16. The sizes of Eulerian mesh are roughly about dx = 2.5 mm and are twice as large as the Lagrangian meshes. The structure penetrates into the water with a constant velocity of Vo = 5.425 m/sec. 2.4.2.2. Numerical approach for rigid structure The two-dimensional water impact problem with rigid structure has an analytic solution developed by Wagner; Figure 2.17 describes the fluid–structure coupling problem, where both materials air and water need to be modeled. The mesh of both fluid materials and the mesh of the shell structure are shown in Figures 2.18 and 2.19. For the structure mesh to move and deform without affecting the fluid mesh, both meshes are superimposed. The numerical results presented in this section are obtained with the Euler–Lagrange penalty coupling. Because of the nature of the penalty coupling, numerical high-frequency oscillations can be generated which may pollute the computed pressure. To damp out these high

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Figure 2.17. Lagrangian structure in a Eulerian fluid domain

oscillations, the damping formulation described above is performed. In the simulation, the local peak pressure on the structure is sensitive to the scalar factor pf used in equation (2.14) for the coupling force, although the impulse and the resultant force on the structure are not sensitive to the variation of the scalar factor as they are average values on the structure and the pressure we are looking for is a local value. In order to

Figure 2.18. Mesh of the fluid, water, and air domains

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Figure 2.19. Mesh of the Lagrangian structure with Shell elements

get a good correlation for the local peak pressure with theoretical results, we set pf = 0.1. For this value and the mesh size dx = 2.5 mm, the numerical stiffness by unit area computed by the following equation: KA K k = pf = pf = 90 GPa.m−1 . A V dx

(2.24)

The maximal local physical stiffness by unit area can be evaluated when the pressure peak shown in Figure 2.23 is reached. The theoretical curve on this graph shows the peak

Figure 2.20. Free surface at time = 0.03 s

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Figure 2.21. Free surface at time = 0.04 s

pressure value, which is close to 1.2 MPa and occurs 24 ms after impact. Thus, during this time interval, the wedge penetrates 0.13 mm into the free surface. The numerical results are the histories of local pressures at 40 mm from the apex and vertical force on the wedge. These curves are shown on 18, Figures 2.23 and 2.25, respectively.

Figure 2.22. Free surface at time = 0.05 s

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ALE and Fluid–Structure Interaction 1.4 Theoretical Numerical

Pressure (MPa)

1.2 1 0.8 0.6 0.4 0.2 0

0

1

2

3

4

Time (s)

Figure 2.23. Pressure at 40 mm from the apex: (o) theoretical, (x) numerical

On these graphs, the numerical pressure curves are superimposed with the corresponding theoretical results. For the local pressure at 40 mm, the penalty coupling is used with and without damping force. The superimposition of both cases is presented on Figure 2.24, which illustrates damping effect on the solution. The presence of the damping force reduces the 1.4 without damping

Pressure (MPa)

1.2

with damping

1 0.8 0.6 0.4 0.2 0

0

1

2

3

4

Time (ms)

Figure 2.24. Comparisons of numerical pressure at 40 mm from the apex: (a) without damping, (b) with damping

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Force per unit length (N/m) (E+3)

40 Theoretical Numerical 30

20

10

0

0

0.5

1

1.5

4

Time (ms)

Figure 2.25. The vertical force per unit length on the wedge: (o) theoretical, (x) numerical

numerical oscillations. The curves plotted in Figure 2.23 and Figure 2.25 show good correlations between analytical and numerical results for the vertical force on the wetted wedge. 2.4.3. Airbag deployment To present the efficiency of the coupling algorithm presented in this chapter, when applied to problems involving large deformation of the structure; the method is applied to the deployment of an airbag. The problem consists of a high-velocity gas flowing out of an inflator into an airbag. The airbag material consists of flexible membrane elements that deform under tensile stress and cannot carry any compressive loads. The gas is modeled using compressible Navier–Stokes equations with ideal gas equation of state for pressure. In the automotive industry, the simulation of the deployment process of an airbag was first performed using uniform pressure technique. Assuming an ideal gas law and an adiabatic process, and using scalar thermodynamic equations, the pressure can be determined as a function of the density and the specific gas constant. This pressure is applied uniformly on the membrane elements of the structure. By using this technique, fluid

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ALE and Fluid–Structure Interaction

flow velocity and pressure are not computed. The accuracy of this simplified method depends largely on the position of the passenger with respect to the deploying airbag. When the passenger is close to the airbag during inflation, experiments have shown that the gas pressure inside the airbag and the resulting force applied to the passenger are not accurate to obtain validated results. This situation, called Out Of Position, requires us to solve the compressible Navier–Stokes equation for gas dynamics and a coupling algorithm as described in this chapter. Because of large deformation of the airbag membrane elements, a fixed Eulerian mesh is used for the fluid as described in Figure 2.26. Velocity boundary conditions are prescribed

Figure 2.26. Eulerian mesh for gas and air

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83

at the inflator, simulated by few nodes inside the airbag. The fluid domain needs to be large enough to cover the deformed structure during the simulation. To reduce the size of the fluid domain, and thus the number of fluid elements, an ALE moving mesh that moves and extends in the three directions following the motion of the structure can be performed. To properly solve the airbag problem, the air material outside the airbag needs to be modeled, since it applies an external load, the atmospheric pressure, on the airbag. The external pressure inside the outside air material needs to be taken into account to obtain validated results. Figures 2.27 to 2.31 show the gas inside the airbag and the deformed membrane

Figure 2.27. Airbag deployment and gas expansion at time = 0.0004 sec

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ALE and Fluid–Structure Interaction

Figure 2.28. Airbag deployment and gas expansion at time = 0.0008 sec

elements of the airbag at times t = 0.4 ms, 0.8 ms, 1.2 ms, 1.6 ms, and 2 ms.

2.4.4. Sloshing tank problem The disturbance of a partially filled container by external forces causes movement of the fluid-free surface. Any motion of the free surface inside its container is called sloshing. Depending on the type of disturbance, the free liquid surface can experience different types of motion including breaking of the surface waves and droplet formations resulting due to abrupt changes in acceleration. This phenomenon is of great concern in many engineering fields especially in the aerospace, civil, and automobile industries since adverse

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Figure 2.29. Airbag deployment and gas expansion at time = 0.0012 sec

effects of sloshing can affect stability of the overall fluid– structure system. Fluid-free surface, especially in liquid containment tanks when subjected to earthquake motion, undergoes large amplitude sloshing waves. The method employed to clarify sloshing problem has to cope with large deformations of fluid-free surface in order to accurately predict the hydrodynamic forces acting on the structure generated by high-speed impacts of sloshing liquid. However, this problem is generally treated with analytical methods which are based on the potential flow theory. In these methods, the wave amplitude of the free surface is considered to be very small in comparison with the wavelengths and depths. The container is considered as rigid. Moreover, these methods are limited to regular geometric tank shapes such as cylindrical and rectangular. Therefore,

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Figure 2.30. Airbag deployment and gas expansion at time = 0.0016 sec

analytical formulations derived for the analysis of sloshing in tanks may not be capable of capturing the complex fluid– structure effects when large amplitude the free-surface displacements take place. However, appropriate numerical methods using fluid–structure interaction techniques can be used to predict the hydrodynamic forces due to the sloshing liquid in a tank. The coupling algorithm detailed in the previous sections provides an efficient way for the solution of sloshing problems. In order to verify the accuracy of the developed method for analysis of the sloshing phenomenon, rectangular and cylindrical tank models, which are subjected to harmonic and earthquake motions, respectively, are analyzed using the coupling method presented in this section.

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Figure 2.31. Airbag deployment and gas expansion at time = 0.0020 sec

2.4.4.1. Analytical treatment of the sloshing problem A tank problem with sloshing liquid is generally regarded as an initial boundary-value problem. The small amplitude sloshing of irrotational flow of an incompressible and inviscid fluid within a rigid container is treated by potential flow theory in which the motion of the fluid inside a tank is governed by the Laplace equation: ∇2 Φ = 0,

(2.25)

where Φ represents velocity potential function of fluid. Solution of the Laplace equation under zero-velocity boundary condition at the tank base and wall along with the combination

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ALE and Fluid–Structure Interaction

h

b

Figure 2.32. 2D tank model

of dynamic and kinematic free surface boundary conditions provides circular frequencies ωn of the sloshing modes:   (2n + 1)π (2n + 1)π 2 tanh h ωn = g (2.26) b b where b represents the length of the tank, h is the fluid depth and n is the sloshing mode numbers (Figure 2.32). If the container is forced by a horizontal harmonic displacement, de , at its base, it may be represented as de = D sin(ωt),

(2.27)

the total velocity potential function can be split into a tank potential function, Φ1 , and a fluid disturbance potential function, Φ2 . In equation (2.27), D represents the harmonic external force amplitude and ω is the circular frequency of the applied motion. The tank potential function defines the motion of the tank and computed by the following equation: Φ1 = −ωD cos(ωt)x.

(2.28)

For the forced excitation case, the combination of kinematic and dynamic boundary conditions at the free surface are presented in terms of fluid disturbance potential function, Φ2 : ∂Φ2 ∂ 2 Φ2 = −ω 3 D cos(ω t) x. +g 2 ∂t ∂y

(2.29)

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89

It should be noted that the r.h.s. of equation (2.29) is zero for free vibration case. In combination with zero-velocity boundary condition on the wetted boundary and equation (2.29), the solution of Laplace equation (equation (2.25)) gives the following expression for velocity potential of fluid domain inside the tank for a Cartesian coordinate system (x, y), which is defined at the center of the fluid-free surface [FAL 78]:      (2n+1)π (2n+1)π Φ(x, y, t) = ∞ sin x cosh (y + h) n=0 b b (2.30) ∗ {An cos(ωn t) + C n cos(ωt)} − ωD cos(ωt)x, where An = −Cn − Kn ω ωKn Cn = 2 ωn − ω 2  2 b 4 ω2 D   Kn = − (−1)n . (2n+1)rmπ b (2n + 1)π cosh h b

The free-surface displacement, η , measured from the undisturbed liquid surface at equilibrium, and pressure, p, at any point inside the tank can be defined in terms of velocity potential function as follows: p (x, y, t) = −ρ

∂Φ ∂t

(2.31)

1 ∂Φ , (2.32) g ∂t where ρ is the mass density of the liquid and g is the gravitational acceleration. η (x, t) =

2.4.4.2. Sloshing in a rigid tank The sloshing event inside a 3D rigid tank subjected to resonant and non-resonant harmonic motions is investigated with a fully nonlinear fluid–structure interaction method based on the ALE coupling algorithm implemented in the previous sections. Time histories of the free-surface wave height and

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ALE and Fluid–Structure Interaction

pressure obtained at specific locations by the numerical method are compared with results of an existing experimental test and analytical solution. The experimental study carried out by Liu and Lin [LIU 08] is regarded as reference solution to evaluate the numerical findings. The tank dimensions and material properties are determined in accordance with this reference model. The analytical method given in section 2.4.4.1, which is based on the linear potential flow theory, is used for comparing the nonlinear effects of fluid sloshing with that of linear. In the numerical model, a three-dimensional rigid rectangular tank with a width of 0.57 m, breath of 0.31 m, and total height of 0.30 m is filled with water (ρ = 1000 kg/m3 ) up to a height of 0.15 m. In the numerical analyses, the fluid region is treated on a moving mesh using an ALE formulation whereas the structure is characterized with a deformable mesh using a Lagrangian formulation. The interior liquid is discretized with uniform mesh (Figure 2.33). The sizes of the Lagrangian Time = 6.68

Tank

Fluid Mesh

Y X

Figure 2.33. Finite element model of the rigid tank with fluid

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shell and ALE solid elements are 0.01 m. The model is fixed at its base and harmonic motion is applied as displacement. Two loading cases as nonresonance and resonance are considered. In view of the first fundamental sloshing frequency, which can be obtained from equation (2.26) as ω0 = 6.0578 rad/s, the excitation frequency of the first case is considered as ω = 0.583 ω0 . The second loading is intended to simulate sloshing phenomenon under resonant frequency, therefore the excitation frequency is taken as the same as first fundamental frequency. The amplitudes of the horizontal harmonic excitations are 0.005 m for both cases. The hydrostatic pressure field generated gradually increases for 1 sec. In order to optimize the fluid mesh, a moving ALE mesh that follows the structure motion is employed in the numerical analyses. To perform the coupling as described in section the structure mesh is embedded inside the fluid mesh during the analyses. This technique allows the high distortion of the structure mesh since both meshes are not connected at their contact interface. The time step size is 1.0 × 10−4 s throughout the simulation. In the experimental study, the time history response of free-surface elevation is measured at three locations, which were near left (i.e. x = −0.265) and right (i.e. x = 0.265) ends of tank and at the middle of the free surface (i.e. x = 0). Figure 2.34 presents the time history response of free-surface elevations at three measurement location (i.e. x = −0.265, 0 and 0.265 m) that is extended to 20 s. For nonresonant frequency motion, the numerical solution of sloshing by the proposed method is in a quite acceptable agreement with the reference solution and analytical formulation in terms of elevation of free surface. In both cases, the numerical results are quite close to the experimental ones. For negative (downward) wave amplitudes, numerical results are more consistent than those of analytical, since numerical method takes into

Surface Displacement (m)

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ALE and Fluid–Structure Interaction 0.01 ALE Coupling Analytical Solution Experimental

x=0 0.005 0 −0.005 −0.01 0

2

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Figure 2.34. Comparisons of the time histories of free-surface elevation for the present numerical method, the analytical solution, and experimental data (non-resonant case)

account nonlinear sloshing behavior. As expected, the wave height is almost zero at the middle of the free surface for each method. For the resonant frequency case, the wave height increases continuously over time for all solution types at the near left and right end of the tank. The comparison of three solution

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0.15 0.1

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x=0

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0.05 .0 −0.05 −0.1 −0.15 0

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0.15 0.1

x = 0.265

0.05 .0 −0.05 −0.1 −0.15 0

Figure 2.35. Comparisons of the time histories of free-surface elevation for the present numerical method, the analytical solution, and experimental data (resonant case)

methods reveals that analytical study overestimates negative surface amplitudes, whereas it underestimates the positive ones (Figure 2.35). Numerical and experimental results are highly consistent in terms of peak-level timing, shape, and amplitude of sloshing wave. The free-surface displacement time histories obtained from numerical and experimental studies show that the positive (upward) sloshing wave

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amplitudes are always larger than the negative (downward) ones. This phenomenon is a classical indication of a nonlinear behavior of sloshing and caused by suppression effect of the tank base on the waves with negative amplitude. Although the gravity effects exist for both upward and downward fluid motion, the downward motion of fluid is blocked by the tank bottom. The ratio of positive amplitude to absolute negative amplitude increases as the fluid depth decreases. This phenomenon cannot be observed from analytical solution because it is derived under linearized assumptions. Since the pressures were not measured during the experimental study, the analytical results are considered as reference solution. Yet, it should be borne in mind that the analytical method does not include the nonlinear effects. Pressure (including hydrostatic pressure) time histories obtained at three specific locations, which are at the two edges of the tank wall, 0.01 m above the base and at the middle of the tank base plate, are given in Figures 2.36 and 2.37. In these figures, there is a high-frequency oscillation region at the beginning of the pressure plots but they disappear after 1 second. For both the resonance and nonresonance cases, the presented numerical method predicts the pressure slightly higher than the analytical method at the bottom of the tank wall (x = −0.265 and 0.265 m). However, the pressure measured at the middle of the tank base (x = 0.0 m) is exactly the same for both methods. At this location, the analytical method gives a single value throughout the analysis. For resonant frequency loading case, pressures obtained from analytical study at the right and left walls of the tank increase continuously over time. However, pressure observed from numerical models fluctuates between the same negative and positive values after 8 seconds. It is obviously verified that the free-surface profiles obtained from experimental and numerical studies perfectly match each other for both resonance and non-resonance loading conditions, although analytical results highly deviate from experimental results for loading with resonant frequency. The

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x = 0.0; y=0.0 2000 15000 1000 500 0

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Figure 2.36. Comparisons of the time histories of pressure for the present numerical method and analytical data (non-resonant case)

pressures computed at specific locations by the analytical method continuously amplify over the time for resonance loading case, whereas numerical method gives bounded results after 8 seconds. This verifies that the analytical method is not reliable for resonant frequencies where nonlinear sloshing behavior is extremely dominant. On the other hand, the present numerical algorithm can be used for the analysis of sloshing problems in practice for every frequency range of external excitation.

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Figure 2.37. Comparisons of the time histories of pressure for the present numerical method and analytical data (non-resonant case)

2.4.4.3. Frequency analysis for sloshing The fluid motions of a partially filled tank can be studied using a two-dimensional mathematical model as mentioned in section 2.4.4.1. The liquid in an open tank can flow back and forth across the basin in standing waves at discrete natural frequencies. The purpose of this example is to find the natural frequency and mode shape of a two-dimensional, partially filled tank by using Fourier series expansion. To expand

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the analytical solution, the liquid is assumed homogenous, non-viscous, irrotational, and incompressible. The boundaries are rigid: the fluctuation in pressure on the walls as result of sloshing exclude flexing of the tank wall that can have a significant influence on the natural frequencies and mode shapes of sloshing in extreme case. Nonlinear effects are neglected: the wave amplitudes are very small in comparison with the wavelengths and depths. In the analytical study, the pressure at the free surface is set to zero, and surface tension is negligible. Referring to equation (2.26) the natural frequencies of sloshing in a rigid 2D tank can be defined by the following formulae:   (2 n + 1) π 1 (2 n + 1) tanh h . fn = g (2.33) 2 bπ b Two extreme cases can be studied in order to understand the physical meaning of this formula, the basin depth h is smaller than the breadth b or, conversely, the depth of the tank is bigger than the breadth. In the shallow liquid case, h is smaller than b. Therefore, the fundamental natural frequency is estimated by the following equation: √ 1 gh h < . f0 = if (2.34) 2b b 10 The resonance frequency of surface waves in a harmonic rolling tank with large lateral dimension appears as the wavelength λ equals twice the breadth b and the √ celerity equals the gravity wave velocity: λ0 = 2b and c0 = g · h. For a lake whose breadth is of the order of kilometers, the natural period is of the order of minutes or even hours. In the deep liquid case, h is higher than b. The fundamental frequency is approximated by the following expression:   gb/ 1 gb h π = > 1. f0 = if (2.35) 2 π 2b b

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The resonance frequency of surface waves in a harmonic rolling tank with large lateral dimension appears as the wavelength λ equals twice the breadth b and the celerityequals a particular gravity wave velocity: λ0 = 2b and c0 = g.b/π . In both particular cases, the fundamental natural frequency of a basin decreases with increasing tank breadth. To validate the numerical model, equation (2.33) is used to predict sloshing frequencies for a simple tank with various fuel amounts. The tank chosen was approximately 100 inches (254 cm) by 50 inches (127 cm). Three different fuel heights were analyzed: 25%, 50%, and 75%. Next, this specific tank was modeled by a mesh composed of 17,988 solid elements and a sloshing event was introduced. Sloshing event is generated by initially assigning velocities to both the tank and the fluid, then stopping the tank motion at 0.00001 seconds. The velocity boundary condition applied to the tank and the fluid is given in Figure 2.38. Rigid material was used to represent tank walls. The finite element model and the induced sloshing event are shown in Figure 2.39.

40

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30 20 10 0 −10 −0.2

0

0.2

0.4 Time (s)

0.6

0.8

Figure 2.38. Velocity boundary conditions for tank

1

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t=0s

t = 0.15 s

t=2s

t = 10.6 s

Figure 2.39. Finite element model and analysis depicting a sloshing event

Comparing the predicted sloshing frequencies against calculated values for three different fuel levels reveals a maximum error below 3%, Figure 2.40. This analysis demonstrates the numerical model can accurately predict sloshing frequencies.

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Fluid Depth / Tank Height

1 Numerical Method Analtytical Solution 0.75

0.5

0.25

0 0.3

0.35

0.4 0.45 Frequency (Hz)

0.5

0.55

Figure 2.40. Plot comparing calculated and predicted sloshing frequencies

2.4.4.4. Application to a cylindrical flexible tank subjected to seismic loading The applicability of coupling algorithm presented in this chapter to seismic analysis of tank problems is verified with the results of an existing experimental study carried out by Manos and Clough [MAN 82]. The analyses are performed for the same tank dimensions and material properties which are used in the experimental study. The tank is made of aluminum with a density of 2700 kg/m3 , elastic modulus of 71.0 GPa and yield stress of 100 MPa. The model has a radius of 1.83 m and a total height of 1.83 m. Water (ρ = 1000 kg/m3 ) is filled up to a height of 1.53 m. The thicknesses of the aluminum bottom plate and the shell section nearest to the bottom is 0.002 m. The second tank shell course has a thickness of 0.0013 m. In numerical simulations, fluid is represented by Navier–Stokes equations in ALE formulation while the structure is discretized by a Lagrangian approach. The tank base is constrained in five degrees-of-freedom using single-point constraints at all nodes. The resulting discretized model of the fluid–tank system is given in Figure 2.41. Both material

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Fluid Mesh

Tank

Figure 2.41. Finite element model of the cylindrical tank

and geometric nonlinearities are considered in the numerical model in order to accurately determine stress, strain, and strain rate distributions throughout the tank and fluid. A vertical acceleration field of a 1 g is applied to give the correct hydrostatic pressure in the fluid. Nonlinear dynamic time history analyses are performed under unidirectional horizontal earthquake motion recorded during the 1940 El Centro earthquake with 0.50 g√peak acceleration, which is scaled with regard to time by 1/ 3 (Figure 2.42). Figure 2.43 shows the stresses developed on the tank.

Acceleration (g)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6

0

1

2

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6

Time (s)

Figure 2.42. Input motion used for the analysis of cylindrical tank

7

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t = 1.6 s

t = 2.5 s

t = 2.72 s

Figure 2.43. Stress developed on the tank

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Free Surface Displacement (m)

0.1 0.08 0.06 0.04 0.02 0

−0.02

ALE Coupling Experimental (EERC-82/07)

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−0.08

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−0.1 0

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3

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7

Time (s)

Figure 2.44. Comparison of time histories of free-surface elevation obtained by the coupling algorithm and experimental data (left: x = −1.72 m; right: x = 1.72 m)

Figure 2.44 presents the time history response of freesurface elevations at two measurement locations (i.e. r = −1.72 and 1.72 m) that is extended to 7 seconds. These locations are situated on the loading axis. The maximum sloshing amplitudes at r = −1.72 m are measured as 0.08 m for both downward and upward directions from the experimental study, whereas it is obtained as 0.08 and 0.045 m from the numerical solution for positive and negative amplitudes, respectively. At the second measurement location, numerical method gives 0.07 m for both positive and negative of sloshing amplitudes while negative and positive sloshing amplitudes from experimental study are 0.1 and 0.05 m, respectively. As can be noticed from the graphs given in Figure 2.44, the numerical and experimental models lead to a quite accurate description of the water sloshing in terms of the displacement of free surface for the earthquake motion. Therefore, it can be justified that the presented method is reliable for the analysis of sloshing inside a cylindrical tank when subjected to earthquake excitation.

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2.5. Conclusion In the slamming modeling, the model has been restricted to a rigid body, since for water impact problem, a theoretical solution is available only for rigid structure. In the description of the method, there is no restriction to rigid bodies, the method can be applied to general structures. The slamming modeling with the Euler–Lagrange coupling showed numerical oscillation problems. More specifically, for times close to the instant of pressure peak, “numerical noise” produced by the penalty spring perturbed the investigated curve after the peak instant. To treat this problem, a damping algorithm has been added to the penalty algorithm in order to obtain smooth pressures. In the sample modeling of the piston, the new penalty coupling has given good results by damping high oscillations. The comparison between the results obtained for the Euler–Lagrange coupling with and without damping after penalty stiffness calibration has highlighted the positive effect of damping: the added damping option in the coupling code permits to find the theoretical results given by Wagner’s approach without numerical dissipative errors. The future investigations will focus on the determination of the coupling stiffness. The challenging problem, which remains open, is to predict the value of this numerical parameter for a large range of fluid–structure interaction problems. The coupling method presented in this chapter has been successfully used for different applications where classical contact methods fail or cannot be used, including airbag deployment in automotive industry and bird impact in aerospace industry. There are, however, still class of problems where neither method, contact nor coupling, appears adequate.

2.6. Acknowledgments The author would like to thank Nicolas Aquelet from LSTC, Livermore Software Technology Corp., and Zuhal Ozdemir

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from Bogazici university in Istanbul, for their contributions. Ozdemir Zuhal was sponsored by the Eiffel Excellence Scholarship Program, during her stay at Lille university in France.

2.7. Bibliography [AQU 03] A QUELET N., S OULI M., G ABRYS J., O LOVSSON L., “A new ALE formulation for sloshing analysis”, Structural Engineering & Mechanics, vol. 16, p. 423–440, 2003. [BEL 84] B ELYTSCHKO T., L IN J., T SAY C., “Explicit algorithms for nonlinear dynamics of shells”, Computational Methods in Applied Mechanical Engineering, vol. 42, p. 225–251, 1984. [BEL 89] B ELYTSCHKO T., N EAL M., “Contact-impact by the pinball algorithm with penalty, projection, and Lagrangian methods”, Proceedings Symposium on Computational Techniques for Impact, Penetration, and Performation of Solids, ASME, New York, NY, vol. 103, p. 97–140, 1989. [BEN 89] B ENSON D., “An efficient accurate simple ALE method for nonlinear finite element programs”, Computational Methods in Applied Mechanical Engineering, vol. 72, p. 305–350, 1989. [DYM 86] D YMENT A., “Self similar unsteady Boundary layers”, Acta Mechanica, vol. 59, p. 91–102, 1986. [FAL 78] FALTINSEN O., “A numerical nonlinear method of sloshing in tanks with two-dimensional flow”, Journal of Ship Research, vol. 22, p. 193–202, 1978. [HAL 98] H ALLQUIST J., LS-DYNA – Theoretical Manual, Livermore Software Technology Corporation, Livermore, 1998. [HIL 77] H ILBER H., H UGHES T., T AYLOR R., “Improved numerical dissipation for time integration algorithms in structural dynamics”, Earthquake Engineering and Structural Dynamics, vol. 5, p. 283–292, 1977. [HUG 81] H UGHES T., L IU W., Z IMMERMAN T., “Lagrangian-Eulerian finite element formulation for incompressible viscous flows”, Computational Methods in Applied Mechanical Engineering, vol. 29, p. 329–349, 1981.

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[KEN 81] K ENNEDY J., B ELYTSCHKO T., “Theory and application of finite element method for arbitrary Lagrangian-Eulerian fluids and structures”, Nuclear Engineering Designs, vol. 68, p. 129–146, 1981. [KáR 29] K ÁRMÁN T. V., The impact on seaplane floats during landing, Report , N.A.C.A. TN321- Washington, 1929. [LEE 77] L EER B. V., “Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection”, Journal Computational Physics, vol. 167, p. 276–299, 1977. [LEE 02] L EE S., C HO J., PARK T., L EE W., “Baffled fuel-storage container:parametric study on transient dynamic characteristics”, Structural Engineering and Mechanics, vol. 13, 2002. [LIU 08] L IU D., L IN P., “A numerical study of three-dimensional liquid sloshing in tanks”, Journal of Computational Physics, vol. 227, p. 3921– 3939, 2008. [MAN 82] M ANOS G., C LOUGH R., Further study of the earthquake response of a broad cylindrical liquid-storage tank model, Report UCB/EERC 82-07, Earthquake Engineering Research Center University of California Berkeley, July 1982. [PIP 96] P IPERNO S., L ARROUTUROU B., L ESOINNE M., “Analysis and compensation of numerical damping in one-dimensional acoustic piston simulations”, International Journal of Computational Fluid Dynamics, vol. 6, p. 157–174, 1996. [SOU 00] S OULI M., O UAHSINE A., L EWIN L., “ALE and fluid-structure interaction problems”, Computational Methods in Applied Mechanical Engineering, vol. 190, p. 659–675, 2000. [SOU 01] S OULI M., Z OLESIO J., “Arbitrary Lagrangian-Eulerian and free surface methods in fluid mechanics”, Computer Methods in Applied Mechanics and Engineering, vol. 191, p. 451–466, 2001. [WAG 32] WAGNER H., “Über Stoß- und Gleitvogänge an der Oberfläche von Flüssigkeiten”, Zeitschrift f. Angew. Math. Und Mech., Vol. 12, p. 193–235, 1932. [WAT 86] WATANABE I., Analytical Expression of Hydrodynamic Impact Pressure by Matched Asymptotic Expansion Technique, T. West-Japan Soc. Nav. Arch., 71, 1986. [WOO 82] W OODWARD P., C OLLELA P., The Numerical Simulation of Twodimensional Fluid Flow with Strong Shocks, Lawrence Livermore National Laboratory, UCRL-86952, 1982.

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[ZHA 93] Z HAO R., FALTINSEN O., “Water entry of two-dimensional bodies”, Journal of Fluid Mechanics, vol. 246, p. 593–612, 1993. [ZHO 93] Z HONG Z., Finite Element Procedures for Contact-Impact Problems, Oxford University Press, Oxford, 1993.

Chapter 3

Implicit Partitioned Coupling in Fluid–Structure Interaction

3.1. Introduction The numerical simulation of engineering applications in many cases requires the coupled solution of problems from structural mechanics and fluid mechanics. Examples of such mechanically coupled fluid–solid problems can be found, for instance, in machine and plant building, engine manufacturing, turbomachinery, heat exchangers, offshore structures, chemical engineering processes, microsystem techniques, biology, or medicine to mention only a few of them. Usually, already the numerical solution of the individual fluid problems is a non-trivial task, requiring the use of complex (nonlinear) models, i.e. for turbulence, and high numerical resolution in order to achieve the necessary modeling quality and discretization accuracy. Therefore, a reliable simulation of processes involving interacting fluid and solid

Chapter written by Michael S CHÄFER.

109

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phenomena is a matter of high complexity, which is very demanding with respect to the underlying numerical methods and computational resources. Here, in particular, it is very essential to consider advanced numerical techniques for the treatment of the underlying problems. One of the crucial issues is the numerical realization of the coupling mechanisms, which can be invoked at different levels within the numerical scheme resulting either in a more weakly or more strongly coupled procedure. In the following, after reviewing the basics of computational fluid and structural mechanics, we will discuss important aspects of fluid–structure interaction algorithms and give results for a variety of academic and industrial applications. 3.2. Computational fluid mechanics In continuum mechanics, two descriptions are considered for the motion in a continuum media: The first is the Eulerian description, where we focus attention on a particular volume in space. The volume is fixed with respect to a laboratory frame, and we study the fluid as it passes through the fixed volume. The description is one in which the fluid is continuously renewed inside the volume, the Eulerian description is not the simplest in which the basic equations of fluid motion can be formulated. A convective term is introduced to express the material time derivative in the reference configuration. The convective term gives a non-symmetrical form of the Galerkin formulation. Since the computational domain is fixed, the Eulerian description has the advantage of preserving the mesh regularity. The second is the Lagrangian description, in which we identify and follow a particular region of fluid. The volume of fluid changes in shape, while the total mass remains constant. In the Lagrangian description, the mesh of the computational domain moves with the particle fluid velocity. In the Lagrangian description, the motion of the mesh may lead to an element entanglement; this description is preferred for problems with small motion.

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3.2.1. Governing equations For the description of fluid flows usually the Eulerian formulation is employed, because we are usually interested in the properties of the flow at certain locations in the flow domain. We restrict ourselves to the case of linear viscous isotropic fluids known as Newtonian fluids, which are by far the most important ones for practical applications. Newtonian fluids are characterized by the following material law for the Cauchy stress tensor T (unless otherwise stated, here and throughout the chapter Einsteins summation convention is used):   ∂vj ∂vi 2 ∂vk Tij = μ + − δij − pδij (3.1) ∂xj ∂xi 3 ∂xk with the velocity vector vi with respect to Cartesian coordinate xi , the pressure p, the dynamic viscosity μ, and the Kronecker symbol δij . With this the conservation laws for mass, momentum, and energy take the form ∂ρ ∂(ρvi ) + ∂t ∂xi ∂(ρvi ) ∂(ρvi vj ) + ∂t ∂xj

= 0, =

(3.2)

   ∂vj ∂vi ∂ 2 ∂vk μ + − δij (3.3) ∂xj ∂xj ∂xi 3 ∂xk

∂p + ρfi , ∂xi      2 ∂vi 2 ∂vi ∂vi ∂vj − = μ + (3.4) ∂xj ∂xj ∂xi 3 ∂xi   ∂T ∂vi ∂ κ + ρq , − p + ∂xi ∂xi ∂xi −

∂(ρe) ∂(ρvi e) + ∂t ∂xi

where ρ is the fluid density, e is the specific internal energy, and fi and q are external forces and heat sources, respectively. In the energy balance (3.4) for the heat flux vector hi Fourier’s law ∂T hi = −κf (3.5) ∂xi

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with the thermal conductivity κf is used, i.e. a constant specific heat capacity is assumed and the work done by pressure and friction forces is neglected. The system (3.2)–(3.4) has to be completed by two equations of state of the form p = p(ρ, T )

and e = e(ρ, T ) ,

which define the thermodynamic properties of the fluid, i.e. the thermal and caloric equation of states, respectively. When the fluid can be considered as an ideal gas the thermal equation of state reads p = ρRT

(3.6)

with the specific gas constant R of the fluid. The internal energy in this case is only a function of the temperature, such that one has a caloric equation of state of the form e = e(T ). For a caloric ideal gas, for instance, one has e = cv T

with a constant specific heat capacity cv (at constant volume). Frequently, it is not necessary to solve the equation system in the above most general form, i.e. it is possible to make additional assumptions to further simplify the system. The most relevant assumptions for practical applications are the incompressibility and the inviscidity. 3.2.1.1. Incompressible flows In many applications, the fluid can be considered as approximatively incompressible. Because of the conservation of mass, this is tantamount to a divergence-free velocity vector, i.e. ∂vi /∂xi = 0. For a criterion for the validity of this assumption, the Mach number Ma =

v¯ a

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is taken into account, where v¯ is a characteristic flow velocity of the problem and a is the speed of sound in the corresponding fluid (at the corresponding temperature). Incompressibility usually is assumed if Ma < 0.3. Flows of liquids in nearly all applications can be considered as incompressible, but this assumption is also valid for many flows of gases, which occur in practice. For incompressible flows the stress tensor becomes [HOL 00]   ∂vj ∂vi − pδij . Tij = μ + (3.7) ∂xj ∂xi The conservation equations for mass, momentum, and energy then read ∂vi = 0, ∂xi

(3.8)

   ∂vj ∂vi ∂p ∂ ∂(ρvi ) ∂(ρvi vj ) μ − + = + + ρfi , (3.9) ∂t ∂xj ∂xj ∂xj ∂xi ∂xi     ∂ ∂T ∂vi ∂vi ∂vj ∂(ρe) ∂(ρvi e) + κ + ρq . + = μ + ∂t ∂xi ∂xj ∂xj ∂xi ∂xi ∂xi

(3.10) We can observe that for isothermal processes in the incompressible case the energy equation does not need to be taken into account. The equation system (3.8)–(3.10) has to be completed by boundary conditions, and in the unsteady case by initial conditions. As boundary conditions for the velocity, for instance, the velocity components can be explicitly prescribed: vi = vbi .

Here, vb can be a known velocity profile at an inflow boundary or, in the case of an impermeable wall where a no-slip condition has to be fulfilled, a prescribed wall velocity (vi = 0 for a

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fixed wall). Attention has to be paid to the fact that the velocities cannot be prescribed completely arbitrarily on the whole boundary Γ of the problem domain, since the equation system (3.8)–(3.10) only admits a solution, if the integral balance  vbi ni dΓ = 0 Γ

is fulfilled. This means that there flows as much mass into the problem domain as it flows out, which, of course, is physically evident for a “reasonably” formulated problem. At an outflow boundary, where usually the velocity is not known, a vanishing normal derivative for all velocity components can be prescribed. In general, for incompressible flows, the pressure is uniquely determined only up to an additive constant (in the equations there appear only derivatives of the pressure). This can be fixed by one additional condition, e.g. by prescribing the pressure in a certain point of the problem domain or by an integral relation. 3.2.1.2. Inviscid flows As one of the most important quantity in fluid mechanics the ratio of inertia and viscous forces in a flow is expressed by the Reynolds number Re =

v¯Lρ , μ

where v¯ is again a characteristic flow velocity and L is a characteristic length of the problem (e.g. the pipe diameter for a pipe flow or the cross-sectional dimension of a body for the flow around it). The assumption of an inviscid flow, i.e. μ ≈ 0 can be made for “large” Reynolds numbers (e.g. Re > 107 ). Compressible flows at high Mach numbers (e.g. flows around airplanes or flows in turbomachines) are often treated as inviscid.

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The neglection of the viscosity automatically entails the neglection of heat conduction (no molecular diffusivity). Also heat sources are usually neglected. Thus, in the inviscid case the conservation equations for mass, momentum, and energy read ∂ρ ∂(ρvi ) + ∂t ∂xi

= 0,

∂(ρvi ) ∂(ρvi vj ) + ∂t ∂xj

= −

∂(ρe) ∂(ρvi e) + ∂t ∂xi

(3.11)

∂p + ρfi , ∂xi

= −p

∂vi . ∂xi

(3.12) (3.13)

This system of equations is called Euler equations. To complete the problem formulation one equation of state has to be added. For an ideal gas, for instance, we have p = Rρe/cv .

It should be noted that the neglection of the viscous terms results in a drastic change in the nature of the mathematical formulation, since all second derivatives in the equations disappear and, therefore, the equation system is of another type. This also causes changes in the admissible boundary conditions, since for a first-order system fewer conditions are necessary. For instance, at a wall only the normal component of the velocity can be prescribed and the tangential components are then determined automatically. For details concerning these aspects we refer to [HIR 88]. 3.2.2. Finite volume discretization For the spatial discretization of fluid flow problems frequently a finite volume method with a colocated arrangement of variables is employed. The solution domain is discretized finite volume cells (see Figure 3.1 for a two-dimensional example) and the transport equations are integrated over these control volumes (CVs), leading, after the application of the

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nn nw

nw

S n

sw

ne

P

w

e s

V se

ne

ns

Figure 3.1. Quadrilateral control volume (CV)

Gauss theorem, to a balance equation for the fluxes through the CV faces and the volumetric sources. The convection and diffusion contribution to the fluxes are evaluated by numerical integration (e.g. the midpoint rule) and proper interpolation schemes. For the convective part usually some flux blending technique is used, which can be implemented using the deferred correction approach proposed by Khosla and Rubin [KHO 74]. For the time discretization, for instance, the so-called θmethod (e.g. Cuvelier et al. [CUV 86]) can be employed. Then, approximations vhn and pnh to the solution of (3.8)–(3.9) at the time level tn = nΔt (n = 1, 2, . . .) are defined as solutions of nonlinear algebraic systems of the form Lh vhn = 0 ,

(3.14)

vhn + θΔt [Ah (vhn )vhn + Gh pnh ] = vhn−1 + (θ − 1) Δt   . + Ah (vhn−1 )vhn−1 Gh pn−1 h

(3.15) The discrete operators Ah , Gh , Fh , and Lh are defined according to the spatial discretization including the corresponding discretization of the boundary conditions. The

Implicit Partitioned Coupling

117

parameter h is a measure for the spatial resolution (e.g. the maximum CV diameter), and Δt > 0 is the time step. The parameter θ ∈ (0, 1] is a blending factor for the explicit and implicit contribution of the time discretization. We remark that the cases θ = 1 and θ = 0.5 correspond to the first-order fully implicit Euler scheme and the second-order implicit Crank–Nicolson scheme, respectively. Both methods are unconditionally stable, but for spatially non-smooth solutions the Crank–Nicolson scheme may cause numerical oscillations (e.g. Cuvelier et al. [CUV 86]), while the implicit Euler scheme does not show such a behavior (strong A-stability). The time stepping process defined by (3.14)–(3.15) is started from the initial values vh0 = v0h ,

p0h = 0,

(3.16)

where v0h is a suitable spatial approximation of the initial are already value given for v . Assuming that vhn−1 and pn−1 h computed, one is faced with the problem to solve the nonlinear system (3.14)–(3.15) for vhn and pnh . 3.2.2.1. Solution algorithms For incompressible flows, the computation of the pressure constitutes a particular problem. The pressure only appears in the momentum equations, but not in the continuity equation, which in a sense would be available for this purpose. There are several techniques for dealing with this problem. One possibility is offered by what is called pressure-correction methods, which can be derived in different variants and are mostly used in actual flow simulation programs. Alternatively, there are artificial compressibility methods, which are based on the addition of a time derivative of the pressure in the continuity equation: 1 ∂p ∂vi + =0 ρβc ∂t ∂xi with an arbitrary parameter βc > 0, with which the portion of the artificial compressibility can be controlled. The “proper”

118

ALE and Fluid–Structure Interaction

choice of this parameter is crucial for the efficiency of the method. However, obvious criteria for this are not available. The solution techniques employed together with artificial compressibility are derived from schemes developed for compressible flows (which usually fail in the borderline cases of incompressibility). It should be mentioned that pressurecorrection methods, which originally were developed for incompressible flows, also can be generalized for the computation of compressible flows (see, e.g. [FER 01]). The general idea of a pressure-correction method is to first compute preliminary velocity components from the momentum equations and then to correct this together with the pressure, such that the continuity equation is fulfilled. This proceeding is integrated into an iterative solution process, at the end of which both the momentum and continuity equations are approximately fulfilled. In the following one iteration step (k − 1) → k is described, assuming that vhn,k−1 and pn,k−1 are h n,k n,k already computed. The determination of vh and ph is done in several steps leading to a decoupling with respect to the different variables. We first introduce a splitting of Ah of the form Ah (vhn,k−1 ) = An,k−1 + An,k−1 + An,k−1 Dh Oh Nh

(3.17)

n,k−1 into a diagonal part An,k−1 Dh , and off-diagonal parts AOh and An,k−1 corresponding to orthogonal and non-orthogonal Nh (cross-derivatives) contributions of the space discretization, respectively. The splitting is illustrated in Figure 3.2 showing the assignment for a CV with its eight neighbors involved in the local discretization. n,k−1/2

to In a first step, an intermediate approximation vh n,k vh is obtained by solving the momentum equation (3.15) with the pressure term, the temperature term, and the matrix coefficients formed with values of the preceding iteration k − 1,

Implicit Partitioned Coupling

AN h

AOh

AN h

AOh

ADh

AOh

119

AN h

AOh

AN h

Figure 3.2. Splitting of the operator Ah into orthogonal and non-orthogonal off-diagonal parts and a diagonal part

treating the term with An,k−1 explicitly, and introducing an Nh underrelaxation: n,k−1/2

vh

  n,k−1/2 n,k−1 n−1 + θΔt An,k−1 + α A = αv Svh vh v Dh Oh

  n,k−1 + (1 − αv ) vhn,k−1 + θΔtAn,k−1 v Dh h   n,k−1 n,k−1 n,k−1 , v + G p + F T − αv θΔt An,k−1 h h h h Nh h

(3.18) where the underrelaxation factor αv is in the interval (0, 1], n−1 all terms of (3.15) containing and for abbreviation, in Svh only values from the preceding time level n − 1, which are not affected by the iteration process, are summarized:   Shn−1 = vhn−1 + (θ − 1) Δt Ah (vhn−1 )vhn−1 + Gh pn−1 + Fh Thn−1 . h

(3.19) Following the spirit of a pressure-correction approach, in a second step we are now looking for corrections p˜n,k ˜hn,k h and v to obtain the new pressure pn,k = pn,k−1 + p˜n,k and the new h h h n,k−1/2 n,k n,k velocity vh = vh + v˜h exactly fulfilling the discrete

120

ALE and Fluid–Structure Interaction

continuity equation (3.14). For this a momentum equation for vhn,k of the form   n,k n,k−1 n,k−1 n,k−1 vhn,k + θΔtAn,k−1 v = T (1 − α ) v + θΔtA v v Dh h h Dh h n−1 + αv Svh −   n,k−1/2 n,k−1 n,k−1 n,k n,k−1 v + A v + G p + F T αv θΔt An,k−1 h h h Oh Nh h h h

(3.20) n,k−1/2

is considered, which differs from the one for vh , i.e. (3.18), n,k−1 by the treatment of the term with AOh , and by taking the pressure term at the new iteration level k . Now, substracting (3.18) from (3.20) yields ¯ n,k−1 v˜n,k = −αv θΔtG ¯ h p˜n,k , v˜hn,k + θΔtA Dh h h

(3.21)

where the overbar on the operators indicates the selective interpolation technique used for making the cell face velocities dependent on the nodal pressures, which is necessary to avoid oscillatory solutions that may occur due to the non-staggered grid arrangement (Rhie and Chow [RHI 83]). Equation (3.21) represents an explicit expression for the new velocity in terms of the pressure-correction equation. By applying the operator Lh to both sides of (3.21), and taking into account the validity of the continuity equation (3.14) for vhn,k an equation for p˜n,k h is obtained, ¯ h p˜n,k = −Lh G h

1 1 ¯ n,k−1 n,k−1/2 n,k−1/2 Lh vh ADh Lh vh + . (3.22) αv θΔt αv

Equation (3.22) corresponds to a discrete Poisson equation with homogenous Neumann boundary conditions for p˜n,k h . To keep the structure of the pressure-correction equation the same as for the discrete momentum equation (3.18) and to improve its diagonal dominance the non-orthogonal part of ¯ h is neglected (see, e.g. Peri´c [HOR 90]), and p˜n,k Ch := Lh G h is computed from (3.22) with Ch replaced by CDh + COh using

Implicit Partitioned Coupling

121

a splitting analogous to the one in (3.17). If the grid is highly non-orthogonal the influence of the neglected pressure crossderivatives can be accounted for by the solution of a second pressure-correcton equation similar as in the PISO algorithm to account for the influence of neglected velocity corrections (Issa [ISS 86]). is computed, vhn,k can easily be obtained from Once p˜n,k h (3.21). The validity of the continuity equation for vhn,k directly follows from (3.21) and (3.22). For the new pressure an underrelaxation with 0 < αp < 1 is also employed: n,k−1 pn,k + αp p˜n,k h = ph h .

(3.23)

Theoretical results concerning the convergence properties of the considered pressure-correction approach are discussed by Wittum [WIT 90] in the more general setting of transforming smoothers. 3.3. Computational structural mechanics In structural mechanics problems, in general, the task is to determine deformations of solid bodies, which arise because of the action of various kinds of forces. From this, for instance, stresses in the body can be determined, which are of great importance for many applications. For the different material properties there exist a large number of material laws, which together with the balance equations lead to diversified complex equation systems for the determination of deformations (or displacements). 3.3.1. Governing equations In principle, for structural mechanics problems we distinguish between linear and nonlinear models, where the nonlinearity can be of a geometrical and/or physical nature. Geometrically, linear problems are characterized by the linear

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strain-displacements relation for the strain tensor εij   ∂uj 1 ∂ui , εij = + 2 ∂xj ∂xi

(3.24)

with the displacement vector ui , whereas physically linear problems are based on a material law involving a linear relation between strains and stresses. We restrict ourselves to the formulation of the equations for the linear elasticity theory, which can be used for many typical engineering applications. Furthermore, we briefly address hyperelasticity as an example of a nonlinear model class. For other classes, i.e. elastoplastic, viscoelastic, or viscoplastic materials, we refer to the corresponding literature (e.g. [HOL 00]). 3.3.1.1. Linear elasticity The theory of linear elasticity is a geometrically and physically linear one. As already outlined in section 3.3.2, there is no need to distinguish between Eulerian and Lagrangian description for a geometrically linear theory. In the following the spatial coordinates are denoted by xi . The equations of the linear elasticity theory are obtained from the linearized strain-displacement relations (3.24), the momentum conservation law (3.25) formulated for the displacements (in the framework of structural mechanics this often also is denoted as equation of motion) ρ

∂Tij D 2 ui = + ρfi , 2 Dt ∂xj

(3.25)

and the assumption of a linear elastic material behavior, which is characterized by the constitutive equation Tij = λεkk δij + 2μεij .

(3.26)

Equation (3.26) is known as Hooke’s law. λ and μ are the Lamé constants, which depend on the corresponding material (μ is also known as bulk modulus). The elasticity modulus (or Young modulus) E and the Poisson ratio ν are often employed

Implicit Partitioned Coupling

Partially plastic

Stress

Elastic

123

Strain

Figure 3.3. Qualitative strain–stress relation of real materials with linear elastic range

instead of the Lamé constants. The relations between these quantities are λ=

Eν (1 + ν)(1 − 2ν)

and

μ=

E . 2(1 + ν)

(3.27)

Hooke’s material law (3.26) is applicable for a large number of applications for different materials (e.g. steel, glass, stone, wood, etc.). Necessary prerequisites are that the stresses are not “too big”, and that the deformation happens within the elastic range of the material (see Figure 3.3). The material law for the stress tensor frequently is also given in the following notation: ⎡ ⎤ T11 ⎢ T22 ⎥ ⎢ ⎥ ⎢ T33 ⎥ ⎢ ⎥ ⎢ T12 ⎥ = ⎢ ⎥ ⎣ T13 ⎦ T23 ⎤ ⎡ ⎤⎡ ε11 1−ν ν ν 0 0 0 ⎢ ⎥ ⎢ ν 1−ν ν 0 0 0 ⎥ ⎢ ⎥⎢ ε22 ⎥ ⎢ ⎢ ⎥ E ν 1−ν 0 0 0 ⎥⎢ ε33 ⎥ ⎥. ⎢ ν ⎢ ⎥ 0 0 1−2ν 0 0 ⎥ (1+ν)(1−2ν) ⎢ ⎢ 0 ⎥⎢ ε12 ⎥ ⎣ ⎣ 0 ⎦ ε13 ⎦ 0 0 0 1−2ν 0 0 0 0 0 0 1−2ν ε23 &' ( % C

124

ALE and Fluid–Structure Interaction

Because of the principle of balance of moment of momentum, T has to be symmetric, such that only the given six components are necessary in order to fully describe T. The matrix C is called material matrix. Putting the material law in the general form: Tij = Eijkl εkl , the fourth-order tensor E with the components Eijkl is called the elasticity tensor (of course, the entries in the matrix C and the corresponding components of E match). Finally, one obtains from (3.24), (3.25), and (3.26) by eliminating εij and Tij , the following system of differential equations for the displacements ui : ρ

∂ 2 uj D 2 ui ∂ 2 ui = (λ + μ) + μ + ρfi . Dt2 ∂xi ∂xj ∂xj ∂xj

(3.28)

These equations are called (unsteady) Navier–Cauchy equations of linear elasticity theory. For steady problems correspondingly we have (λ + μ)

∂ 2 uj ∂ 2 ui +μ + ρfi = 0 . ∂xi ∂xj ∂xj ∂xj

(3.29)

Possible boundary conditions for linear elasticity problems are − Prescribed displacements: ui = ubi on Γ1 , − Prescribed stresses:

Tij nj = tbi on Γ2 .

The boundary parts Γ1 and Γ2 should be disjoint and should cover the full problem domain boundary Γ, i.e. Γ1 ∩ Γ2 = ∅ and Γ1 ∪ Γ2 = Γ. Besides the formulation given by (3.28) or (3.29) as a system of partial differential equations, there are other equivalent formulations for linear elasticity problems. We will give here two other ones that are important in connection with different numerical methods. We restrict ourselves to the steady case.

Implicit Partitioned Coupling

125

(Scalar) multiplication of the differential equation system (3.29) with a test function ϕ = ϕi ei , which vanishes at the boundary part Γ1 , and integration over the problem domain Ω yields     ∂ 2 uj ∂ 2 ui (λ+μ) ϕi dΩ + ρfi ϕi dΩ = 0 . (3.30) +μ ∂xi ∂xj ∂xj ∂xj Ω

Ω

By integration by parts of the first integral in (3.30) we obtain      ∂uj ∂ui ∂ϕi (λ+μ) +μ dΩ = Tij nj ϕi dΓ + ρfi ϕi dΩ . ∂xi ∂xj ∂xj Ω

Γ

Ω

(3.31) Since ϕi = 0 on Γ1 , in (3.31) the corresponding part in the surface integral vanishes and in the remaining part over Γ2 for Tij nj the prescribed stress tbi can be inserted. Thus, one obtains      ∂uj ∂ui ∂ϕi (λ+μ) +μ dΩ = tbi ϕi dΓ + ρfi ϕi dΩ . ∂xi ∂xj ∂xj Ω

Γ2

Ω

(3.32) The requirement that the relation (3.32) is fulfilled for a suitable class of test functions (let this be denoted by H) results in a formulation of the linear elasticity problem as a variational problem: Find u = ui ei with ui = ubi on Γ1 , such that      ∂uj ∂ui ∂ϕi (λ+μ) +μ dΩ = ρfi ϕi dΩ + tbi ϕi dΓ ∂xi ∂xj ∂xj Ω

Ω

Γ2

for all ϕ = ϕi ei in H. (3.33) The question remains of which functions should be contained in the function space H. Since this is not essential for

126

ALE and Fluid–Structure Interaction

the following, we will not provide an exact definition (this can be found, for instance, in [BRA 01]). It is important that the test functions ϕ vanish on the boundary part Γ1 . Further requirements mainly concern the integrability and differentiability properties of the functions (all appearing terms must be defined). Formulation (3.33) is called a weak formulation, where the term “weak” relates to the differentiability of the functions involved (there are only first derivatives, in contrast to the second derivatives in the differential formulation (3.29)). Frequently, in the engineering literature, the formulation (3.33) is also called principle of virtual work (or principle of virtual displacements). The test functions in this context are called virtual displacements. So far, we have considered the general linear elasticity equations for three-dimensional problems. In practice, very often these can be simplified by suitable problem-specific assumptions, in particular with respect to the spatial dimension. This way equations for bars, beams, disks, plates, or shells can be derived. As an example, which will serve later as exemplary equations for the discussion of the finite element method, we will consider disks in plane stress state. 3.3.1.2. Plane stress problems To save indices we denote the two spatial coordinates by x and y and the two unknown displacements by u and v (see Figure 3.4). Furthermore, since different index ranges occur, for clarity all occurring summations will be given explicitly (i.e. no Einstein summation convention). The underlying linear strain-displacement relations are ε11

∂u , = ∂x

ε22

∂v , = ∂y

and

ε12

1 = 2



∂u ∂v + ∂y ∂x

 ,

(3.34)

Implicit Partitioned Coupling

127

tb Γ1

n

Ω y, v ub x, u Γ2

Figure 3.4. Disk in plane stress state with notations

and the (linear) elastic material law for the plane stress state will be used in the form ⎡ ⎤ ⎡ ⎤ ⎤⎡ T11 1 ν 0 ε11 ⎣ T22 ⎦ = E ⎣ ν 1 0 ⎦ ⎣ ε22 ⎦ . (3.35) 2 1 − ν T12 ε12 0 0 1−ν &' ( % &' ( % &' ( % ε T C Here, exploiting the symmetry properties, we summarize the relevant components of the strain and stress tensors into the vectors ε = (ε1 , ε2 , ε3 ) and T = (T1 , T2 , T3 ). As boundary conditions at the boundary part Γ1 the displacements u = ub

and

v = vb

and on the boundary part Γ2 the stresses T1 n1 + T3 n2 = tb1

and

T3 n1 + T2 n2 = tb2

are prescribed (see Figure 3.4). As a basis for the finite element approximation, the weak formulation of the problem is employed. We denote the test functions (virtual displacements) by ϕ1 and ϕ2 and define   ∂ϕ1 ∂ϕ2 1 ∂ϕ1 ∂ϕ2 , ψ2 = , and ψ3 = + . ψ1 = ∂x ∂y 2 ∂y ∂x

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ALE and Fluid–Structure Interaction

The weak form of the equilibrium condition (momentum conservation) then can be formulated as follows: Find (u, v) with (u, v) = (ub , vb ) on Γ1 , such that ⎛ ⎞    3 2 ⎝ ρfj ϕj dΩ + tbj ϕj dΓ⎠ Ckj εk ψj dΩ = k,j=1 Ω

j=1

Ω

Γ2

for all test functions (ϕ1 , ϕ2 ) with ϕ1 = ϕ2 = 0 on Γ1 . (3.36) Employing the problem formulation in this form, further considerations are largely independent of the special choices of the material law and the strain–stress relation. Only the correspondingly modified definitions for the material matrix C and the strain tensor ε have to be taken into account. In this way also an extension to nonlinear material laws (e.g. plasticity) or large deformations (e.g. for rubber-like materials, see section 3.3.1.3) is quite straightforward (see, e.g. [BAT 95]). 3.3.1.3. Hyperelasticity As an example for a geometrically and physically nonlinear theory, we will give the equations for large deformations of hyperelastic materials. This may serve as an illustration of the high complexity that structural mechanics equations may take if nonlinear effects appear. The practical relevance of hyperelasticity is because it allows for a good description of large deformations of rubber-like materials. A hyperelastic material is characterized by the fact that the stresses can be expressed as derivatives of a strain energy density function W with respect to the components Fij = ∂xi /∂aj of the deformation gradient tensor F: Tij = Tij (F) =

∂W (F) . ∂Fij

In this case, we have a constitutive equation for the second Piola–Kirchhoff stress tensor of the form Pij = ρ0 (γ1 δij + γ2 Gij + γ3 Gik Gkj )

(3.37)

Implicit Partitioned Coupling

with the Green–Lagrange strain tensor   ∂uj 1 ∂ui ∂uk ∂uk . Gij = + + 2 ∂aj ∂ai ∂ai ∂aj

129

(3.38)

The coefficients γ1 , γ2 , and γ3 are functions of the invariants of G (see, for instance [SAL 01]) i.e. they depend in a complex (nonlinear) way on the derivatives of the displacements. Relations (3.37) and (3.38), together with the momentum conservation equation in Lagrange formulation ∂ D2 xi ρ0 = 2 Dt ∂aj



∂xi Pkj ∂ak

 + ρ0 fi

give the system of differential equations for the unknown deformations xi or the displacements ui = xi −ai . The displacement boundary conditions are ui = ubi , as usual, and stress boundary conditions take the form ∂xi Pkj nj = tbi . ∂ak

As can be seen from the above equations, in the case of hyperelasticity we are faced with a rather complex nonlinear system of partial differential equations together with usually also nonlinear boundary conditions

3.3.2. Finite element methods In practice, nowadays the numerical study of structural mecahnics problems involves almost exclusively finite element methods. In this section we will address some the particularities and the treatment of corresponding problems. We will do this exemplarily for a 4-node quadrilateral element for elastic plane stress problems. However, the formulations employed allow in a very simple way an understanding of the

130

ALE and Fluid–Structure Interaction y P4

η

Qi

P1

1

P3

P˜3

P˜4

Q0 P2

P˜1

x 0

P˜2

ξ

1

Figure 3.5. Transformation of arbitrary quadrilateral to unit square

necessary modifications if other material laws, other strainstress relations, and/or other types of elements are used. A practically important element class for structural mechanics applications are the isoparametric elements, which we will introduce by means of an example. The basic idea of the isoparametric concepts is to employ the same (isoparametric) mapping to represent the displacements as well as the geometry with local coordinates (ξ, η) in a reference unit area. The mapping to the unit area (triangle or square) is accomplished by a variable transformation, which corresponds to the ansatz for the unknown function. As an example, we consider an isoparametric quadrilatral four-node element, which usually provides a good compromise between accuracy requirements and computational effort. However, it should be noted that the considerations are largely independent from the element employed (triangles or quadrilaterals, ansatz functions). A coordinate transformation of a general quadrilateral Qi to the unit square Q0 (see Figure 3.5) is given by x=

4 j=1

Nje (ξ, η)xj

and y =

4 j=1

Nje (ξ, η)yj ,

(3.39)

Implicit Partitioned Coupling

131

where Pj = (xj , yj ) are the vertices of the quadrilateral (here and in the following we omit for simplicity the index i on the element quantities). The bilinear isoparametric ansatz functions N1e (ξ, η) = (1 − ξ)(1 − η) , N2e (ξ, η) = ξ(1 − η) , N4e (ξ, η) = (1 − ξ)η

N3e (ξ, η) = ξη ,

correspond to the local shape functions, which were already used for the bilinear parallelogram element. For the displacements, one has the local shape functions representation

u(ξ, η) =

4

Nje (ξ, η)uj

and v(ξ, η) =

j=1

4

Nje (ξ, η)vj (3.40)

j=1

with the displacements uj and vj at the vertices of the quadrilateral as nodal variables. By considering relations (3.39) and (3.40) the principal idea of the isoparametric concept becomes apparent, i.e. for the coordinate transformation and the displacements the same shape functions are employed. Next, the element stiffness matrix and the element load vector are determined. As a basis, we employ a weak form of the equilibrium condition within an element with the test functions (Nje , 0) and (0, Nje ) for j = 1, . . . , 4. In order to allow a compact notation, it is helpful to introduce the nodal displacement vector as φ = [u1 , v1 , u2 , v2 , u3 , v3 , u4 , v4 ]T

and write the test functions in the following matrix form:  N=

N1e

0

N2e

0

N3e

0

N4e

0

0

N1e

0

N2e

0

N3e

0

N4e

 .

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ALE and Fluid–Structure Interaction

As analogon to ψ = (ψ1 , ψ2 , ψ3 ) within the element we further define the matrix A= ⎡ ∂N e 1 ⎢ ∂x ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎣ 1 ∂N e 1

2 ∂y

∂N2e ∂x

0 ∂N1e ∂y 1 ∂N1e 2 ∂x

∂N3e ∂x

0 ∂N2e

0

∂y

1 ∂N2e 2 ∂y

1 ∂N2e 2 ∂x

0 1 ∂N3e 2 ∂y

∂N4e ∂x

0 ∂N3e ∂y 1 ∂N3e 2 ∂x

0 1 ∂N4e 2 ∂y

⎤ 0

⎥ ⎥ ⎥ ⎥ ⎥. ∂y ⎥ ⎥ ⎥ e⎦ 1 ∂N ∂N4e

4

2 ∂x

The equilibrium relation corresponding to (3.36) for the element Qi now reads 3



 Ckl εk Alj dΩ =

k,l=1 Q

2 k=1

i

⎜ ⎝





Qi

ρfk Nkj dΩ +

⎞ ⎟ tbk Nkj dΓ⎠

Γ2 i

(3.41) for all j = 1, . . . , 8. Γ2i denotes the edges of the element Qi located on the boundary part Γ2 with a stress boundary condition. If there are no such edges the corresponding term is just zero. Inserting expressions (3.40) into the strain–stress relations (3.34) we get the strains εi dependent on the ansatz functions: ε1 =

4 ∂Nje j=1

∂x

uj , ε2 =

 4  ∂Nje 1 ∂Nje vj , ε3 = uj + vj . ∂y 2 ∂y ∂x

4 ∂Nje j=1

j=1

With the matrix A these relations can be written in compact form as 8 εj = Ajk φk for j = 1, 2, 3 . (3.42) k=1

Implicit Partitioned Coupling

133

Inserting this into the weak element formulation (3.41) we finally obtain ⎞ ⎛    8 2 3 ⎜ ⎟ φk Cnl Anj Alk dΩ = ⎝ ρfk Nkj dΩ+ tbk Nkj dΓ⎠ k=1

n,l=1 Q

k=1

i

Qi

Γ2 i

for j = 1, . . . , 8. For the components of the element stiffness matrix Si and the element load vector bi we thus have the following expressions: i Sjk

=

 3 n,l=1 Q

bij

Cnl Anj Alk dΩ ,

i

⎛ ⎞   2 ⎜ ⎟ = ⎝ ρfk Nkj dΩ + tbk Nkj dΓ⎠ k=1

(3.43)

Qi

(3.44)

Γ2 i

for k, j = 1, . . . , 8. In the above formulas, for the computation of the element contributions the derivatives of the shape functions with respect to x and y appear, which cannot be computed directly because the shape functions are given as functions depending on ξ and η . The relation between the derivatives in the two coordinate systems is obtained by employing the chain rule as ⎤⎡ ⎡ ⎤ ⎤ ⎡ ∂Nke ∂Nke ∂y ∂y − ⎥ ⎢ ∂x ⎥ ⎢ ∂ξ ⎥ 1 ⎢ ⎢ ∂η ⎥ ⎢ ∂ξ ⎥ ⎥= ⎢ e e ⎣ ∂N ⎦ det(J) ⎣ ∂x ∂x ⎦ ⎣ ∂Nk ⎦ k − ∂η ∂ξ ∂y ∂η &' ( % J−1 with the Jacobi matrix J. The derivatives of x and y with respect to ξ and η can be expressed by using the transformation rules (3.39) by derivatives of the shape function with respect

134

ALE and Fluid–Structure Interaction

to ξ and η as well. In this way we obtain for the derivatives of the shape functions with respect to x and y : / 0 4 ∂Nje ∂Nke ∂Nke 1 ∂Nje ∂Nke = − yj , ∂x det(J) ∂η ∂ξ ∂ξ ∂η j=1

∂Nke

1 det(J)

=

∂y

4 j=1

/

(3.45) 0 ∂Nje ∂Nke ∂Nje ∂Nke − xj , ∂ξ ∂η ∂η ∂ξ

where ⎛ det(J) = ⎝

⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ 4 4 4 ∂Nje ∂Nje ∂Nje xj⎠⎝ yj⎠ − ⎝ yj⎠⎝ xj⎠ . ∂ξ ∂η ∂ξ ∂η j =1 j =1 j =1

4 ∂Nje j=1

Thus, all quantities required for the computations of the element contributions are computable directly from the shape functions and the coordinates of the nodal variables. For the unification of the computation over the elements the integrals are transformed to the unit square Q0 . For example, for the element stiffness matrix we obtain i Sjk

=

 3 n,l=1 Q

Cnl Anj (ξ, η)Alk (ξ, η) det(J(ξ, η)) dξdη . (3.46)

0

The computation of the element contributions – different from the bilinear parallelogram element – in general can no longer be performed exactly because the factor 1/ det(J) in relations (3.45) rational functions appear in the matrix A (transformed in the coordinates ξ and η ). Thus, numerical integration is required, for which Gauss quadrature should be advantageously employed. Here, the order of the numerical integration formula has to be compatible with the order of the finite element ansatz. We will not address the issue in detail (see, e.g. [BAT 95]), and mention only that for the considered quadrilateral four-node element a second-order Gauss

Implicit Partitioned Coupling

135

quadrature is sufficient. For instance, the contributions to the element stiffness matrix are computed with this according to Sijk

=

3 4 1 Cnl Anj (ξp , ηp )Alk (ξp , ηp ) det(J(ξp , ηp )) (3.47) 4 n,l=1 p=1

√ √ with the nodal points (ξp , ηp ) = (3 ± 3/6, 3 ± 3/6). The computation of the element load vector can be performed in a similar way (see, e.g. [BAT 95]).

Having computed the element stiffness matrices and element load vectors for all elements, the assembling of the global stiffness matrix and the global load vector can be done according to the procedure described in section 3.3.2. We will illustrate this in the next section by means of an example. The spatially discretized equation (3.36) can briefly be written as a discrete system of the form: u ¨h + Dh u˙ h + Bh uh = Fh (vh , ph ) ,

(3.48)

with M, D, and K denoting the mass, damping, and stiffness matrices. Any external forces are gathered in the source term F. For the time discretization of the fluid part, the secondorder Crank–Nicolson method is employed (e.g. Cuvelier et al. [CUV 86]) and for the solid part structural dynamical equations the approved Nemarks scheme (e.g. [BEL 00]) is applied. With this, approximations vhn , pnh , and unh of vh , ph , and uh of to the solution of (3.48) at the time level tn = nΔt (n = 1, 2, . . .) are defined as solutions of nonlinear algebraic systems of the form (2 Ih − (1 − 2γ)Δt)unh = C1h un−1 + C2h un−2 + Snh h h

(3.49)

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ALE and Fluid–Structure Interaction

with C1h = 2 Ih − (1 − 2γ)Δt Dh − (0.5 + γ − 2β)Δt2 Kh , C2h = −Ih − (γ − 1)Δt Dh − (0.5 − γ + β)Δt2 Kh ,   Snh = Δt2 βFh (vhn , pnh ) + (0.5+γ −2β)Fh (vhn−1 , pn−1 h )   + Δt2 (0.5−γ +β)Fh (vhn−2 , pn−2 h ) ,

where γ and β are the Newmark parameters, which for stability reasons should satisfy the condition (see [BEL 00]) 2β =

(γ + 0.5)2 ≥γ. 2

(3.50)

One is faced with the problem to solve the discrete system (3.49) unh . For this, in particular, multigrid methods or preconditioned Krylov subspace methods have turned out to be efficient approaches.

3.4. Fluid–structure interaction algorithms The basis of the mathematical problem formulation of FSI problems are the equations in the individual fluid and solid parts of the problem domain, as they are exemplarily given in the previous sections. Concerning the boundary conditions on solid and fluid boundaries Γs and Γf , standard conditions as for individual solid and fluid problems can be employed. On a fluid–solid interface Γi the velocities and the stresses have to fulfill the conditions Dubi vi = and σij nj = Tij nj , (3.51) Dt where ubi and Dubi /Dt are the displacement and velocity of the interface, respectively.

Implicit Partitioned Coupling

137

3.4.1. ALE formulation For the fluid part it can be convenient to consider the conservation equations in the so-called arbitrary Lagrangian– Eulerian (ALE) formulation (see, e.g. [DON 04]) in order to combine the advantages of the Lagrangian and Eulerian formulations as they are usually exploited for individual structural and fluid mechanics approaches, respectively. The principal idea of the ALE approach is that an observer is neither located at a fixed position in space nor moves with the material point, but can move “arbitrarily”. Mathematically this can be expressed by employing a relative velocity in the convective terms of the conservation equations. For instance, for an incompressible fluid the conservation equations governing transport of mass and momentum for a (moving) control volume V with surface S in the (integral) ALE formulation read   D ρ dV + ρ(vi −vig )ni dS = 0 , (3.52) Dt D Dt

V





[ρvj (vi −vig )

ρvi dV + V

S

 − Tij ] nj dS =

S

ρfi dV , (3.53) V

where vig is the velocity with which S moves due to displacements of solid parts in the problem domain. In the context of a numerical scheme, vig is also called grid velocity. Note that with vig = 0 and vig = vi the pure Euler and Lagrange formulations are recovered, respectively. In the framework of a numerical scheme, it is common practice to take into account a discrete form of the so-called space conservation law (e.g. [FER 01])   d dV = vjg nj dS (3.54) dt Vf

Sf

when computing the additional convective fluxes in equations (3.52)–(3.53). This is done via the swept volumes δVc of

138

ALE and Fluid–Structure Interaction

the control volume faces for which one has the relation (see [FER 01]): δV n c

c

Δtn

=

g Vfn − Vfn−1 = (vj nj Sf )nc , Δtn c

(3.55)

where the summation index c runs over the faces of the control volume, the index n denotes the time level tn and Δtn is the time-step size. This way, interface displacements enters the fluid problem part in a manner strictly ensuring mass conservation. 3.4.2. Mesh dynamics An important issue when the problem domain dynamically changes due to movements or deformations of solid parts is the corresponding mesh dynamics. The methods to handle this can be classified into fixed and moving grid approaches (see Figure 3.6). Fixed-grid methods can be employed together with fictitious domain, level set, or Chimera techniques. Moving mesh approaches can be realized either by simple interpolation techniques or by more sophisticated pseudo-structure techniques of varying complexity. Besides the requirements that no grid folding occurs and that the mesh exactly fits the moving boundaries we have to take care that distortions of control volumes are kept to a minimum in order not to deteriorate the discretization accuracy and the efficiency of the

Figure 3.6. Examples of moving (left) and fixed (right) grids

Implicit Partitioned Coupling

139

II II Structured block I

fsi−state n

III

Deformation process

Structured block

III

fsi−state n+1

y

I

y x

IV

x

IV

Figure 3.7. Deformation of structured two-dimensional block

solver. Here, also strategies for possible remeshings have to be taken into account in order to ensure grids of reasonable quality. For a more detailed discussion of these aspects we refer to [LOH 01] and the literature given there. As an example, we consider here algebraic and elliptic mesh generation techniques for the grid movement involving boundary orthogonality and grid spacing control. To simplify the presentation we describe the approaches for a single twodimensional structured block surrounded by 4 boundary curves I to IV (see Figure 3.7). The generalization to the threedimensional case and to multiple blocks is straightforward. For a structured two-dimensional block there is a one-toone mapping x(ξ, η) = (x(ξ, η), y(ξ, η)) of the physical coordinates x = (x, y) to computational coordinates (ξ, η) where without loss of generality 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ 1. In each FSI iteration the coordinates of the interior grid points have to be computed from the given boundary point distribution. 3.4.2.1. Algebraic approaches First, we consider algebraic approaches. A very simple method is obtained by linear interpolation between opposite boundaries, e.g. boundary I to III. Let the distances between neighboring grid points be di,j = xi,j − xi−1,j 

i = 1, . . . , N, j = 0, . . . , M ,

(3.56)

140

ALE and Fluid–Structure Interaction

with the overall lengths Lj =

N

di,j

j = 0, . . . , M .

(3.57)

i=1

The normalized lengths for the parametrization are i ¯ i,j = 1 dm,j, L Lj

¯ 0,j = 0 . i = 1, . . . , N, j = 0, . . . , M, L

m=1

(3.58) The grid-point coordinates in the domain are then computed by ¯ i,j + x 0,j xi,j = (x N,j − x 0,j ) · L

i = 0, . . . , N, j = 0, . . . , M .

(3.59) Another algebraic method is the linear transfinite interpolation (TFI), where the interior grid points are computed by x(ξ, η) = (1 − η)x(ξ, 0) + ηx(ξ, 1) + (1 − ξ)x(0, η) + ξx(1, η) −ξ [ηx(1, 1) + (1 − η)x(1, 0)] −(1 − ξ) [ηx(0, 1) + (1 − η)x(0, 0)] .

(3.60)

For boundary movement a simple, but efficient algebraic method can be employed. The cubic polynomial interpolation ¯ , see (3.58), combines two given with normalized parameter L ¯ = 0) and x(L ¯ = 1) also involving their boundary points x(L ¯ = 0) and xt (L ¯ = 1): corresponding tangents xt (L ¯ 3 + a2 L ¯ 2 + a1 L ¯ + a0 , ¯ = a3 L x(L)

¯ ≤ 1, 0≤L

(3.61)

with the coefficients a3 = 2x(0) − 2x(1) + xt (0) + xt (1) ,

(3.62)

a2 = −3x(0) + 3x(1) − 2xt (0) − xt (1) ,

(3.63)

a1 = xt (0) ,

(3.64)

a0 = x(0) .

(3.65)

Implicit Partitioned Coupling

141

The tangents xt are chosen to be perpendicular to boundary surfaces. 3.4.2.2. Elliptic approaches We now turn our attention to the elliptic methods. We describe an approach following [SPE 95] which is based on an (elliptic) Poisson equation for the physical coordinates:  1  1 1 xξ axξξ − 2bxξη + cxηη + aP11 − 2bP12 + cP13  2  2 2 + aP11 − 2bP12 + cP13 xη = 0 (3.66) with the control functions P 11 =

1 sη tξ − sξ tη

P 12 =

1 sη tξ − sξ tη

P 13 =

1 sη tξ − sξ tη

  

tη −tξ

−sη sξ

tη −tξ

−sη sξ

tη −tξ

−sη sξ

  



sξξ tξξ

(3.67)

sξη tξη sηη tηη



(3.68) 

(3.69)

and the abbreviations a = xη xη + yη yη ,

b = xξ xη + yξ yη ,

c = xξ xξ + yξ yξ . (3.70)

The indices ξ and η denote the corresponding derivatives. Figure 3.8 shows how the mapping between the computational space and the physical domain is performed through the parameter space (s, t) that can be used to control the quality of the mesh. n

t

1

1

II

I

II

I

III

II

y IV 0 1 0 Computational space

ξ

00

IV 1 Parameter space

III

I

III

s

IV x Physical domain

Figure 3.8. Mapping strategy for elliptic grid movement

142

ALE and Fluid–Structure Interaction

We consider a parameter space with s(I) = 0, s(III) = 1, t(IV) = 0, and t(II) = 1. The point distribution along s(II), s(IV), t(I), and t(III) arises from linear interpolation along these boundaries involving the normalized arc length. The inner parameter domain is adapted by solving simultaneously: s = s(IV)(1 − t) + s(II)t ,

(3.71)

t = t(I)(1 − s) + t(III)s .

(3.72)

i , P i , P i , i = 1, 2 can be comNext, the control functions P11 12 13 puted according to (3.67) to (3.69) and remain unchanged during the solution of (3.66). The derivatives in equation (3.66) are approximated by central differences (CDS) and a Picard iteration process is used for linearization:

ak−1 xkξξ − 2bk−1 xkξη + ck−1 xkηη 1 1 1 +(ak−1 P11 − 2bk−1 P12 + ck−1 P13 )xkξ 2 2 2 +(ak−1 P11 − 2bk−1 P12 + ck−1 P13 )xkη

= 0.

(3.73)

In each step, this equation system can be solved, for instance, by the Gauss–Seidel algorithm for the unknown grid coordik , i = 1, . . . , N − 1, j = 1, . . . , M − 1. The Picard nates xki,j and yi,j iteration process is repeated as long as a convergence criterion εGRID is satisfied: N M act old i=0 j=0 xi,j − xi,j ∞ < εGRID . (3.74) N M act act act act i=0 xi,0 − xi,M ∞ · j=0 x0,j − xN,j ∞ Within an FSI iteration process the coordinates of the previous iteration can be applied as initial values. The whole solution algorithm, which is summarized schematically in Figure 3.9, operates like a mesh smoother. In any case a boundary conforming mesh without grid folding results for which the interior grid point distribution is a good reflection of the prescribed boundary grid point distribution. The method described above can be extended to ensure boundary orthogonality. First, a boundary conforming grid

Implicit Partitioned Coupling

143

Calculate boundaries by interpolation, update moved coupled edges, fixed edges remain unchanged Calculate appropriate parameter space Determine control functions P11,P12,P13 with CDS approximation k=1

Solve linear equations for x-coordinates k=k+1

Solve elliptic equations

Assemble discrete equations with CDS, boundary points remain unchanged

Solve linear equations for y-coordinates

Global convergence satisfied ?

no

yes Solution of the flow field

Figure 3.9. Flow chart of the elliptic mesh movement method

without grid folding is computed applying the elliptic method explained above. For this mesh the following Laplace equations are considered:     ∂2s ∂2s 1 1 1 1 asξ − bsη + − bsξ + csη s = + 2 = = 0, ∂x2 ∂y J J J J ξ η

t =

∂2t ∂2t + = ∂x2 ∂y 2



1 1 atξ − btη J J

 ξ

(3.75)   1 1 + − btξ + ctη = 0, J J η (3.76)

with the abbreviations a = xη xη + yη yη , J = xξ yη − xη yξ ,

b = xξ xη + yξ yη ,

c = xξ xξ + yξ yξ ,

144

ALE and Fluid–Structure Interaction

together with the Neumann boundary conditions ∂s = 0, ∂n

∂t = 0, ∂n

(3.77)

where n = (n1 , n2 ) is the outward unit normal vector. Equations (3.75) and (3.76) involve a divergence expression that allows for applying the finite volume method:   Ω

1 (asξ − bsη ) J





+ ξ

= =

  Ω

1 (atξ − btη ) J



 1 (−bsξ + csη ) dξ dη J η    1 (sξ (a n1 − b n2 ) + sη (−b n1 + c n2 )) dσ J ∂Ω 0

(3.78)



+ ξ

= =

 1 (−btξ + ctη ) dξ dη J η    1 (tξ (a n1 − b n2 ) + tη (−b n1 + c n2 )) dσ J ∂Ω 0,

(3.79)

where the integration is done for a control volume Ω and its boundary ∂Ω with the line element dσ , respectively. The computational domain is discretized by unit control volumes for inner points and half control volumes for boundary points, leading to a system of linear equations for s and t, respectively. Since the boundary conditions (3.77) transform to 1 J

(sξ (a n1 − b n2 ) + sη (−b n1 + c n2 )) = 0,

(3.80)

1 J

(tξ (a n1 − b n2 ) + tη (−b n1 + c n2 )) = 0,

(3.81)

these terms have to be set to zero for the desired orthogonal grid lines at boundaries. During the solution procedure,

Implicit Partitioned Coupling

145

boundary points are moved along edges (s(IV), s(II)) until convergence is reached. These points are combined by cubic Hermite interpolation: s = s(IV)(1 + 2t)(1 − t)2 + s(II)(3 − 2t)t2 ,

0 ≤ t ≤ 1,

(3.82) t = t(I)(1 + 2s)(1 − s)2 + t(III)(3 − 2s)s2 ,

0 ≤ s ≤ 1.

(3.83) Since the interpolation is given analytically, the Jacobian matrix and its inverse can easily be calculated and solved simultaneously for s and t by the Newton algorithm. As initial condition the parameter space values from the first elliptic solution are applied, leading to convergence after one or two Newton iterations. This way a new parameter space is obtained to compute the desired control functions for boundary orthogonalization. Finally, the desired grid is computed by solving the elliptic equations once again. The main steps of the elliptic-orthogonal method can be summarized as follows: 1. Calculate boundaries by interpolation, update moved coupled edges, fixed edges remain unchanged. 2. Calculate parameter space by normalized arc length. 3. Determine control functions P11 , P12 , P13 with CDS approximation. 4. Solve elliptic equations, according to Figure 3.9. 5. Solve Laplace equations for s and t to get new boundary point distribution. 6. Perform cubic Hermite interpolation for s and t by Newtons method. 7. Determine control functions P11 , P12 , P13 with CDS approximation. 8. Solve elliptic equations once again, according to Figure 3.9.

146

ALE and Fluid–Structure Interaction Fluid

Solid

Robust

Flexible

Weak coupling

Strong coupling Figure 3.10. Schematic view of possible numerical coupling strategies

The advantages of the above approach can be seen in its high flexibility allowing independent orthogonalization of all boundaries while keeping the edge point distribution unchanged. Furthermore, the method is very robust and may even work in case of rather large deformations. 3.4.3. Coupling methods A key component within any solution procedure for coupled fluid–solid problems is the numerical realization of the coupling mechanisms among the fluid and solid parts. These mechanisms can be invoked at different levels within the numerical scheme, resulting either in more weakly or more strongly coupled procedures (see Figure 3.10). The one extreme is a fully explicit partitioned coupling involving an alternating solution of solid and fluid problems with simple interchange of boundary conditions. This approach is very flexible concerning the choice of the solvers for the individual fluid and solid subtasks, but it is often suffering from poor convergence properties. The other extreme consists of a fully implicit monolithic approach involving the simultaneous solution for all unknowns. In this case usually the convergence

Implicit Partitioned Coupling

Fluid

Fluid

Fluid

Fluid

Solid

Solid

Solid

Solid

tn

tn+1

tn

147

tn+1

Figure 3.11. Schematic view of explicit partitioned solution method (left) and stabilized implicit version with predictor-corrector technique (right)

rate with respect to the coupling is optimal, but the full system is hard to solve and much modifications of the individual fluid and solid solvers are necessary. In practice, in many cases an intermediate strategy trying to combine the advantages (or to avoid disadvantages) of the two extreme cases is applied. For instance, an explicit partitioned strategy can be made more robust and less restrictive concerning time-step limitations by making it more implicit via a predictor-corrector iteration technique (see Figure 3.11). Another possibility is to combine some global (monolithic) solver, e.g. multigrid approaches or Krylov subspace methods (e.g. conjugate gradient type methods), with a partitioned scheme acting as a smoother or a preconditioner. For various variants we refer to [CAS 95, CAS 01, FEL 01, LES 98, LET 01, SIE 01, TES 01]. 3.4.3.1. Implicit partitioned coupling We consider an implicit partitioned coupling approach according to [SCH 06a]. In Figure 3.12, a schematic view of the iteration process is given. After the initializations the flow field is determined in the actual flow geometry. From this the friction and pressure forces on the interacting walls are computed. These are passed to the structural solver as boundary conditions. The structural solver computes the deformations, with which then the fluid mesh is modified. Afterwards the flow solver is started again.

148

ALE and Fluid–Structure Interaction

End

Start

yes no last time step?

yes

next time step no next coupling step

FSI converged?

Flow solver

Compute Flow Field − FVM − SIMPLE

Mesh Movement − linear, elliptic, trans− finite grid generation − moving grids

− multigrid vi , p

u

Comp. Wall Forces

Underrelaxation static and adaptive underrelaxation of structural displacement

− shear forces − normal forces

Structure solver forces Interface

Comp. Deformation − FEM

displacements Interface

Figure 3.12. Flow chart of coupled solution procedure

The fluid–structure interaction (FSI) iteration loop is repeated until a convergence criterion ε is reached, which is defined by the change of the mean displacements FSI =

N 1 uk,m−1 − uk,m ∞ < ε, N uk,m ∞

(3.84)

k=1

where m is the FSI iteration counter, N is the number of interface nodes, and  · ∞ denotes the infinite norm. Note that an explicit coupling method would be obtained, if only one FSI iteration is performed. The data transfer between the flow and solid solvers within the partitioned solution procedure is performed via an interface, e.g. realized by the coupling library MpCCI (see

Implicit Partitioned Coupling

149

[MPC 04]), that controls the data communication and also carries out the interpolations of the data from the fluid and solid grids. Various test computations have shown that the coupling scheme is rather sensitive with respect to the deformations especially in the first FSI iterations. Here, situations that are far away from the physical equilibrium can arise, which may lead to instabilities or even the divergence of the FSI iterations. In order to counteract this effect, an adaptive underrelaxation is employed. m the actually computed By using an relaxation factor αFSI m ˜ are linearly weighted with the values um−1 displacements u from the preceding iteration to give the new displacements um+1 : m m ˜ m + (1 − αFSI u um+1 = αFSI )um−1 , (3.85) m ≤ 1. Note that the underrelaxation does not where 0 < αFSI change the final converged solution. The basic effects of the underrelaxation has already been shown in [SCH 06b]. m , different methods For the adaptive determination of αFSI are known. We employ here an approach based on the Aitken method for vectorial equations, which is an extrapolation approach frequently applied in the context of Newton–Raphson iterations. The basis of this approach was proposed by Aitken in 1937. It was identified as very efficient for computations in the field of fluid–structure interaction by Mok [MOK 01].

Employing the values from two preceding iterations the socalled Aitken factor γ m is extrapolated by  m−1 T Δu − Δum · Δum m m−1 m−1 γ =γ + (γ − 1) (3.86) (Δum−1 − Δum )2 ˜ m−1 and Δum = um−1 − u ˜ m . The actual with Δum−1 = um−2 − u m underrelaxation factor αFSI is then defined by m αFSI = 1 − γm .

(3.87)

150

ALE and Fluid–Structure Interaction

The last Aitken factor from the preceding time step can be taken as the first Aitken factor in each time step γ 0 . For the first time step some reasonable value can be chosen, e.g. γ 0 = 0. 3.5. Results and applications 3.5.1. Verification results For verifying the functionality of the subproblem solvers and the coupled solution procedure, we consider the benchmark configuration proposed in [HRO 06] for comparing pure fluid, pure structural, and fluid–structure interaction computations for both steady and unsteady cases. For details of the geometry and the physical parameters, we refer to [HRO 06]. Although the benchmarks are two-dimensional, in order to test the full functionality of the present approaches, we consider all cases in a fully three-dimensional setting with symmetry boundary conditions in the third (z -)direction (see Figure 3.13). For the spatial discretizations of the flow domain 19, 968 control volumes for the x-y -plane and two control volumes for the z -direction are employed. On the structure side only the flexible part (flag) is taken into account and discretized with 21 × 4 × 2 trilinear eight-node hexahedral solid elements for x-, y -, and z -directions respectively, i.e. the thickness of the flag is resolved by four elements. The rigid cylinder is not discretized. Remark that the fluid and structure meshes do not match, which deliberately are selected in order to verify this functionality.

Inlet

Outlet

Figure 3.13. Geometric configuration of benchmark test case

Implicit Partitioned Coupling

151

The results for the fluid–structure interaction test cases FSI1 and FSI3 are summarized in Table 3.1. For the transient case FSI3, that is characterized by periodic oscillations of the flag, a time step size of Δt = 0.01 s is used. In Figure 3.14 the temporal development of the drag force for FSI3 is shown. Already with the relatively coarse spatial grid the results are in good agreement. FSI1 (steady) FSI3 (unsteady) Drag (kg/ms2 ) present from [HRO 06]

14.277 14.295

467.0 ± 24.20 458.2 ± 24.57

Lift (kg/ms2 ) present from [HRO 06]

0.772 0.767

8.8 ± 163.9 2.5 ± 149.5

Frequency (1/s) present from [HRO 06]

– –

10.8 10.7

Table 3.1. Results for fluid–structure interaction benchmark cases

3.5.2. Validation results For model validation a corresponding reference configuration has been set up in cooperation with the Institute of Fluid Mechanics at University of Erlangen–Nürnberg (see [GOM 06]). A sketch of the test channel is shown in Figure 3.15. For the (highly viscous) fluid polyethylene glycol is chosen with density ρf = 1050 kg/m3 and kinematic viscosity νf = 1.9 · 10−4 m2 /s. According to the experiment, a uniform inlet velocity of v¯ = 1.37 m/s is applied. The structural configuration, which was set up after a variety of complementary numerical-experimental tests, is shown in Figure 3.16. It consists of a cylindrical aluminum front body (ρs = 2828 kg/m3 , E = 7 × 1010 N/m2 ), a thin membrane

152

ALE and Fluid–Structure Interaction FSI3 Δt =0.01 530

drag

520 510 Drag (kgm/s2)

500 490 480 470 460 450 440 430 1

1.5

2

2.5

3

3.5

Time (s)

Figure 3.14. Temporal behavior of drag force for fluid–structure interaction case FSI3

of stainless steel (ρs = 7855 kg/m3 , E = 2 × 1011 N/m2 ), and a rectangular rear mass (ρs = 7800 kg/m3 , E = 2 × 1011 N/m2 ). The configuration is fixed in the channel with one rotational degree of freedom positioned at the center of the cylinder, where no friction is assumed at the fixing point. The gravity force is aligned with the x-axis. The physical problem can be considered in very good approximation as two-dimensional,

U=V=W=0

Inlet

Outlet,0−Gradient

250

240

y

x 55

U=V=W=0

338

310

Figure 3.15. Geometric properties of test channel (units in mm)

Implicit Partitioned Coupling

153

0.04 4

11

R=

10 71

Figure 3.16. Sketch of structural configuration

which was confirmed by several measurements. In the simulations this is represented by symmetry boundary conditions in z -direction. For the structural discretizations linear solid hexahedrons (1 840 elements) are applied in combination with enhanced strain formulations allowing for very large aspect ratios. The fluid domain is discretized by 53,776 control volumes. For mesh adaption the elliptic method is applied. For a better understanding of the the fluid–structure interaction mechanisms first an eigenfrequency analysis for the structure is carried out by a pure structural simulation. The first two eigenmodes are shown in Figure 3.17 (the trivial rigid-body motion with 0.048 Hz is not considered). In the first mode with frequency 6.12 Hz the front body moves in the same direction as the rear mass, while in the second mode with frequency 30.07 Hz the moving direction of the front and the end mass is opposite. For the fluid–structure interaction case in the computation as well as in the experiment the structure starts vibrating by itself until it swivels. A snapshot of the velocity component in First mode (6.12 Hz)

Second mode (30.07 Hz)

Figure 3.17. Eigenmodes of structural configuration

154

ALE and Fluid–Structure Interaction

VELx 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 -0.2

Figure 3.18. Snapshot of velocity in x-direction for swiveling fluid–structure interaction motion

x-direction is given in Figure 3.18. In both cases the structure vibrates in a state corresponding to the second mode, i.e. the front body moves in the opposite direction as the rear mass. The oscillation is periodic but not harmonic what is expected for nonlinear vibrations. The experimental and numerical frequencies are 13 Hz ± 1.5% and 12.5 Hz, i.e. there is a very good agreement. The spectral representation of the rear mass and front body oscillations obtained from the simulation results is shown in Figure 3.19 indicating that other occurring

Amplitude (m)

0.02 Rear mass Front body

0.015

0.01

0.005

0

0

10

20

30 40 Frequencies in Hz

50

60

Figure 3.19. Spectral representation of rear mass and front body oscillations (from simulation results) for swiveling fluid–structure interaction motion

70

40

40

30

30

20

20

10

10

y [mm]

y [mm]

Implicit Partitioned Coupling

0 –10

–40 0

0 –10

–20 –30

155

–20

Computation 10

20

30 40 x [mm]

50

60

70

–30

Experiment

–40 0

10

20

30 40 x [mm]

50

60

70

Figure 3.20. Comparison of trailing edge displacements for swiveling fluid-structure interaction motion

frequencies are negligibly small. We can also observe that the fluid–structure interaction swiveling motion frequencies do not coincide with the corresponding eigenfrequencies of the structure. In Figure 3.20 a comparison between experiment and simulation for the displacement of the trailing edge in the x-y -plane is shown. Again the experimental and numerical results are in good agreement. The maximum y -amplitude in both cases is between 18 and 20 mm. The reason for the slight difference of the x-y -displacements of the trailing edge, besides the usual disturbances in experiments (no perfect materials, measurement tolerances, etc.) seems to be because of the assumption of a fully two-dimensional physical problem. In the experiment the side walls and the small gap between them and the structure have a damping effect which is not represented in the simulation with symmetric boundary conditions. The influence of this effect will be investigated in future studies. 3.5.3. Flow induced by solid deformation As an example for a problem, where a flow is induced by deformation of a solid, we consider the flow in an elastic pipe,

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F

1

1

v0

F

F

2

2

2d

10d

7d

d 1.75d

z

0.05 0 -0.05 0.2

0.3

0.4

0.5

0.6

x

0.7

0.8

0.9

z

0.05 0 -0.05 0.2

0.3

0.4

0.5

0.6

x

0.7

0.8

0.9

1

1

1.1

1.1

Figure 3.21. Elastic pipe problem configuration (upper) and initial (t = 0 s) and most deformed (t = 100 s) numerical grids (lower)

which is subjected to two pinching forces F1 and F2 pointing to the center of the cross-section at two opposite points on the surface (see Figure 3.21). The pipe is fixed along a certain area at the inlet and the outlet. The forces are taken to be time-periodic with a period of T = 200 s and an amplitude of 50 N. The inlet velocity is assumed to have a constant parabolic profile with a maximal value v0 = 0.002 m/s. The pipe diameter is d = 0.1 m and the Reynolds number is Re = 100. The pressure at the outflow is assumed to be zero. The material has an elasticity modulus of E = 107 kN/m2 and a Poisson ratio of ν = 0.4. With these material properties

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the deformation of the pipe is mainly affected by the external forces then by the pressure and shear stresses induced by the flow. As it could be expected, the pipe is the most deformed when the pinching forces are the biggest (for t = 100 s). The corresponding grid deformation is presented in Figure 3.21 (right) in comparison with the undeformed state (scaled with a factor 5 for better visualization). The variations of the applied forces and the corresponding outlet mass flux are shown in Figure 3.22 (left). For comparison the inlet mass flux, which is constant during the time, is also given. As can be seen, the outflow mass flux does not strictly follow the applied forces magnitude, since the change of the pipe volume is not linearly related to the displacements. While at the beginning and at the end of the pinching the change of the volume is much smaller than the change of the displacements, this is opposite near t = 100 s. At time t = 100 s the increment displacements are zero, as well as the change of the volume. Therefore, at this moment the outlet mass flux is equal to the inlet mass flux. In Figure 3.22 (left) the change of the z -component of the fluid velocity is given at different time steps in the intersection of the pipe with the plane y = 0 (only the part of the fluid domain containing the elastic part of the pipe is shown). It can be seen how the fluid flow follows the grid movement. In the beginning the z -velocity component is nearly zero. Owing to the elastic walls movement two areas with opposite signs develop, which later exchange their signs, because the pipe shape starts returning to its initial state (after t = 100 s).

3.5.4. Interaction of flow and solid deformation As an example for a fluid–solid interaction we consider the laminar flow around an elastic thin-walled cylinder. A schematic view of the configuration, which is similar to the one considered in [WAL 99], is shown in Figure 3.23. The flow

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t = 70 s

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

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X

t = 100 s

0.1

0.2

0.3

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0.7

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0.9

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X

t = 130 s

0.1

0.2

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X 8.00 force outlet mass flux inlet mass flux

Force(N)

90

7.99 7.98

80

7.97

70

7.96

60

7.95 7.94

50

7.93

40

7.92

Mass flux (g/s)

100

30 7.91 20

7.90

10

7.89

0

7.88 50

100

150

200

Time(s) Figure 3.22. Applied force, inlet and outlet mass fluxes (lower) and z-component of fluid velocity in the plane y = 0 (upper)

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symmetry 6d

B

6d 6d

A

d = 0.006 m

symmetry 25d

Figure 3.23. Problem configuration for flow around cylinder

parameters are chosen to yield a Reynolds number of Re = 100 based on the (undeformed) cylinder diameter d. The length of the channel is chosen so that the flow behavior is not affected by the outlet boundary, where zero pressure is assumed. The cylinder, which is assumed to be fixed at point A and to move only horizontally at point B, is made of an elastic isotropic material with Young modulus E = 15000 N/m2 , Poisson ratio ν = 0.4 and density ρs = 8000 kg/m3 . At the start of the fluid– structure interaction the flow is assumed to be the developed periodic laminar flow around the rigid cylinder (Karman vortex street). The deformation of the cylinder due to the fluid pressure and shear stress is computed with the FEAP code using finite beam elements. In Figure 3.24 the temporal variation of the drag and lift coefficients are shown. At t = 0 s, the moment when the interaction starts, the drag coefficient and its oscillation amplitudes significantly increase, while the oscillation period nearly remains the same. Because of the oscillating fluid forces, it moves at a half of the cylinder diameter distance and fluctuates around this new position. In Figure 3.25, the streamlines and the horizontal velocity are shown for different consecutive time steps. We can see that the Karman vortex street is also affected by the reaction of the elastic cylinder.

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drag coeff. lift coeff.

4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -1 -1.5

0

0.5

1

1.5

2

time [s] Figure 3.24. Time history of drag and lift coefficients for flow around cylinder with fluid–structure interaction

Figure 3.25. Streamlines and horizontal fluid velocity at different time steps for flow around cylinder with fluid–structure interaction

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3.6. Bibliography [BAT 95] B ATHE K., Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1995. [BEL 00] B ELYTSCHKO T., L IU W., M ORAN B., Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Ltd., New York, 2000. [BRA 01] B RAESS D., Finite Elements, 2nd edition, Cambridge University Press, Cambridge, 2001. [CAS 95] C ASADEI M., “An algorithm for permanent fluid structure interaction in explicit transient dynamics”, Computer Methods in Applied Mechanics and Engineering, vol. 128, num. 3–4, p. 231–289, 1995. [CAS 01] C ASADEI F., H ALLEUX J., S ALA A., C HILLE F., “Transient fluid-structure interaction algorithms for large industrial applications”, Computational Methods for Applied Mechanical Engineering, vol. 190, p. 3081–3110, 2001. [CUV 86] C UVELIER C., S EGAL A., S TEENHOVEN A. V., “Finite element methods and Navier-Stokes equations”, Mathematics and Its Applications, vol. 22, p. 150–450, 1986. [DON 04] D ONEA J., H UERTA A., P ONTHOT J.-P., R ODRIGUEZ -F ERRAN A., “Arbitrary Lagrangian-Eulerian methods”, Encyclopedia of Computational Mechanics, vol. 1, E. Stein and R.D. Borst and T.J.R. Hughes and editors, John Wiley & Sons Ltd., New York, 2004. [FEL 01] F ELLIPA C., PARK K., FARHAT C., “Partitioned analysis of coupled mechanical systems”, Computational Methods for Applied Mechanical Engineering, vol. 190, p. 3247–3270, 2001. [FER 01] F ERZIGER J., P ERIC M., Computational Methods for Fluid Dynamics, 3rd edition, Springer, Berlin, 2001. [GOM 06] G OMES J., L IENHART H., “Experimental study on a fluid structure interaction reference test case”, Computational Science and Engineering, vol. 53, p. 356–370, 2006. [HIR 88] H IRSCH C., Numerical Computation of Internal and External Flows: Fundamentals of Numerical Discretization, John Wiley & Sons, New York, 1988. [HOL 00] H OLZAPFEL G., Nonlinear Solid Mechanics: A Continuum Approach for Engineering, John Wiley & Sons, New York, 2000. [HOR 90] H ORTMANN M., P ERICH M., “Finite volume multigrid prediction of solutions”, International Journal for Numerical Methods in Fluids, vol. 11, p. 189–207, 1990.

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[HRO 06] H RON J., T UREK S., “Proposal for numerical benchmarking of fluid–structure interation between elastic object and laminar incompressible flow”, B UNGARTZ H.-J., S CHAFER M., Eds., Fluid–Structure Interaction, Springer, Berlin, 2006, LNCSE 53, pp. 146–170. [ISS 86] I SSA R., “Solution of the implicity discretized fluid flow equations by operator splitting”, Journal of Computational Physics, vol. 62, p. 40– 65, 1986. [KHO 74] K HOSLA P., R UBIN S., “A diagonal dominant second order accurate implicit scheme”, Computers and Fluids, vol. 2, p. 207–209, 1974. [LES 98] L ESOINNE M., FARHAT C., “Improved staggered algorithms for the serial and parallel solution of three-dimensional non linear transient aeroelastic problems”, Computational Mechanics – New Trends and Applications, p. 96–138, 1998. [LET 01] L E T ALLEC P., M OURO J., “Fluid structure interaction with large structural displacements”, Computational Methods in Applied Mechanical Engineering, vol. 190, p. 3039–3067, 2001. [LOH 01] L OHNER R., Applied CFD Techniques, Wiley, Chichester, 2001. [MOK 01] M OK D., WALL W., “Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures”, W. Wall et al., Ed., Trends in Computational Structural Mechanics, CIMNE, Barcelona, 2001, pp. 689–698. [MPC 04] M P CCI, Mesh-Based Parallel Code Coupling Interface, User Guide V2.0 Fraunhofer SCAI, 2004. [RHI 83] R HIE C., C HOW W., “Numerical study of turbulent flow past an airfoil with trailing adge separation”, American Institute of Aeronautics and Astronautics Journal, vol. 21, p. 1525–1532, 1983. [SAL 01] S ALENCON J., Handbook of Continuum Mechanics, Springer, Berlin, 2001. [SCH 06a] S CHAFER M., Computational Engineering Introduction to Numerical Methods, Springer, Berlin, 2006. [SCH 06b] S CHAFER M., H ECK M., Y IGIT S., “An implicit partitioned method for the numerical simulation of fluid-structure interaction”, B UNGARTZ H.-J., S CHAFER M., Eds., Fluid-Structure Interaction, Springer, Berlin, 2006, LNCSE 53, pp. 171–194. [SIE 01] S IEBER G., Numerical Simulation of Fluid-Structure Interaction Using Loose Coupling Methods, PhD Thesis, Fachgebiet Numerische Berechnungsverfahren im Maschinenbau TU Darmstadt, 2001.

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[SPE 95] S PEKREIJSE S., “Elliptic grid generation based on Laplace equations and algebraic transformations”, Journal of Computational Physics, vol. 118, p. 38–61, 1995. [TES 01] T ESCHAUER I., Numerische Simulation gekoppelter FluidStruktur Probleme mittels der Finiten-Volumen-Methode, PhD Thesis Fachgebiet Numerische Berechnungsverfahren im Maschinenbau TU Darmstadt, 2001. [WIT 90] W ITTUM G., “R-transforming smoothers for the incompressible Navier-Stokes equations”, Numerical Treatment of Navier-Stokes Equations, p. 153–162, 1990. [WAL 99] WALL W.A., M OK D.P., R AMM E., “Partitioned analysis approach for the transient, coupled response of viscous fluids and flexible structures”. In ECCM’99-Proc. European Conference on Computational Mechanics, Munich, Germany, August 31–September 3, 1999.

Chapter 4

Avoiding Instabilities Caused by Added Mass Effects in Fluid–Structure Interaction Problems

4.1. Introduction Fluid–structure interaction problems involving an incompressible viscous flow and elastic nonlinear structure have been solved in the past using different methods. Partitioned (or staggered) [FEL 80, FAR 98, PIP 01, NEU 06] schemes are probably the most popular solution techniques for the simulation of coupled problems as they allow using specifically designed codes on different domains and offer significant benefits in terms of efficiency: smaller and better conditioned subsystems are solved instead of a single problem. Loosely (or weakly) [LOH 95] and strongly coupled [RUG 00, RUG 01,

Chapter written by Sergio I DELSOHN, Facundo D EL P IN and Riccardo R OSSI.

165

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Figure 4.1. Summary of the numerical troubles induced by the added-mass effect when solving FSI problems

SOU 00, LET 01, FEL 01, WAL 05, DET 06] schemes are distinguished in the partitioned case: loosely coupled schemes require only one solution of either field per time step in a sequentially staggered manner and are thus particularly appealing in terms of efficiency. Strongly coupled schemes give the same results as non-partitioned (also named monolithic) algorithms, after an iterative process. Both, the strongly coupled and the monolithic schemes, lead to expensive simulation, since at each time step a subiteration algorithm including the fluid and the structure domains has to be performed in the partitioned strongly coupled scheme. Alternatively, fully coupled systems including the equations for the fluid and the structure must be solved for the monolithic procedure. For a graphical representation of different schemes see Figure 4.1. There is a key difference between the strongly coupled scheme and the monolithic scheme: the iterative process in the strongly coupled scheme may be difficult (even nonconvergent) when the so-called “added-mass effect” is important [CAU 05, BAD 08b]. Indeed, in such a situation, a monolithic scheme seems to be necessary to avoid numerical instabilities. In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move some volume of surrounding fluid as it moves through it, since the object and fluid cannot occupy the same physical space simultaneously.

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However the name “added-mass effect” has been used in the literature to indicate the numerical instabilities that typically occur in the internal flow of an incompressible fluid whose density is close to the structure density. We will use the same terminology to be consistent with previous chapters, but as will be shown later, the instabilities are not necessarily caused by a fluid density close to the structure density. There are other reasons, as the elasticity coefficients and time step size that must be taken into account to avoid unstable solutions. The added-mass effect does not occur in aero elasticity problems, but it becomes very important in biomechanics applications where the materials are normally muscles and arteries and the fluid is blood. Weakly coupled schemes are also affected by the addedmass effect: they become unstable when this effect is significant. There is a third situation for which the added-mass effect produces complications. It concerns the monolithic solution of the fluid–structure interaction problem when the pressure is segregated from the displacement or the velocity field. In this case, even if we are solving the fluid and the solid equations together, the iterative scheme to obtain the pressure may be difficult and even non-convergent. 1) Instabilities in the loosely coupled scheme. 2) Non-convergent iterative solutions in the strongly coupled scheme. 3) Non-convergent iterative solutions in the monolithic scheme with pressure segregation. Such monolithic schemes with pressure segregation are first addressed in [FER 07]. A further development of the idea is presented in [BAD 08b]. Exploiting the unified formulation some mathematical explanation for the so-called added mass

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effect was presented in [CAU 05]. The role played by the stabilization in the field was highlighted in [FOR 07]. An alternative coupling paradigm based on the so-called Robin–Robin condition is presented in [BAD 08a], while a different formulation based on the use of a modified projection at the interface is presented in [DET 06]. The main idea of this chapter is totally different from the previously reported literature. First, we put in evidence that the numerical instabilities that appear with the added-mass effect is a consequence of the pressure segregation (case 3 in the previous list) and that a correct understanding of the pressure segregation effect yields different solutions to the addedmass problem, which may be successfully applied to cases 1 and 2. Starting from the monolithic approach, the exact way to segregate the pressure (in fact, it is only exact for the linear case) is described. This exact way is represented by a tangent matrix that, for the linear case, gives the result in only one iteration. An approximation to this tangent matrix is then proposed. Several other approximations may also be thought with the same idea, but the one proposed here shows excellent convergence rates for strong added-mass effect problems. The achieved formulation is independent of the time-integration scheme and may be applied to fully implicit, semiimplicit, or explicit time integrations. In the following section we will study the numerical solution of the monolithic scheme with pressure segregation. At this point we can ask ourselves why we are trying to segregate the pressure from the rest of the unknowns. The answer is: 1) It is too expensive from the computational point of view to solve the pressure together with the rest of the unknowns (typically velocities and/or displacements): the nonlinear system to be solved is large and ill-conditioned with regular nondefined positive matrices.

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2) Because partitioned schemes can be classified as pressure segregated solutions. Effectively, when the structural domain is solved, the pressure of the previous iteration is used. Then, if a procedure is unstable using a monolithic algorithm with pressure segregation, it will also be unstable with a partitioned method. The segregation of the pressure can be conveniently performed using a Chorin–Temam projection scheme [CHO 68, TEM 69, COD 01]. This splitting procedure works conveniently for incompressible flows. Nevertheless, we will introduce pressure segregation via a simple static condensation procedure. This static condensation will explain the Chorin– Teman projection as a particular case and will allow generalizing the Chorin–Temam scheme for fluid–structure interaction problems. 4.2. The discretized equations to be solved in a FSI problem The equations to be solved for both the incompressible fluid and the elastic solid domains are the momentum equations, ρai = ρ

∂σij DVi = + ρ fi , Dt ∂xj

(4.1)

where σij is the Cauchy stress tensor, ρ the density, ai the acceleration vector equal to the total derivative of the velocity Vi , and fi a body force vector. In the incompressible part of the domain, mass conservation equation must be enforced, εV =

∂Vi = 0. ∂xi

(4.2)

The boundary conditions for both domains are σni = σni

in Γσ

and

Ui = Ui

in

ΓU .

(4.3)

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On the fluid domain, it is sometimes useful to use a moving framework different from the particle displacement. In this case, the acceleration vector may be written as a function of the framework velocity VM j as DF Vi ∂Vi DVi = + (Vj − VM j ) , Dt Dt ∂xj

where

DF Vi Dt

represents the framework acceleration.

Apart from the incompressible condition, the only difference between the fluid and the solid is the constitutive equations. Classical Newtonian constitutive equations for the fluid are expressed as a function of the rate of deformations and the pressure:   ∂Vj 1 ∂Vi 1 ∂Vl   σij = 2μ dij − pδij with dij = + δij , − 2 ∂xj ∂xi 3 ∂xl where μ is the viscosity parameter and p the pressure. On the other hand, the constitutive equations for an elastic solid are written as a function of the strain,   1 ∂Uk ∂Ul ∂Uk ∂Ul kl σij = Cij εkl with εkl = + + 2 ∂xj ∂xk ∂xl ∂xk Nevertheless, once the time integration scheme has been chosen, both constitutive equations may be written as a function of the displacement rates or the velocities rates (adding always the pressure in the incompressible part). Assuming for simplicity that an implicit Euler time integration has been chosen, then   Uin+1 = Δt θ Vin+1 + (1 − θ)Vin , where the upper index indicates the time position, Δt is the time step and θ is an integration parameter between 0 and 1.

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To simplify the notation, in the following the upper index n + 1 will be omitted. For instance, for a hypoelastic solid, the constitutive equation becomes   2G 2ΔtG  Δt n dij + λ+ εV δij , σij = σ ˆij + J J 3 where λ and G are the Lamé parameters, J the Jacobian man the initial stress tensor. ˆij trix, and σ In the following and without loss of generality, we will consider that the constitutive equations for the solid and the fluid domains are expressed as a function of the velocity field (plus the pressure in the incompressible region). The same results and conclusions may be obtained using the displacement field as the main unknown. Finally, even though the momentum equations for both domains (the solid and the fluid) are geometrically nonlinear, we will consider, for simplicity in the theoretical aspect of this development, only the linear term, assuming that the matrices obtained are an approximation of the exact nonlinear ones. The weighted residual form of the momentum and mass conservations equations are     DVi ∂σij − Wl ρ − ρ fi dV + Wl (σni − σni )dΓ = 0 Dt ∂xj V Γσ  Wp (−εV )dV = 0. V

Replacing the stress tensor from the corresponding constitutive equation and discretizing the velocity and the pressure fields with standard shape functions: Vi = N T Vi p = NpT P

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and using Galerkin weighting functions ⎡ ⎤ N 0 0 [W1 , W2 , W3 ] = ⎣ 0 N 0 ⎦ with Wp = Np 0 0 N the global fluid–structure interaction problem may be written in a compact monolithic form as

     Mρ ρ + K −B V Vn F+M Δt Δt , (4.4) = P 0 −B T 0 where Mρ is the mass matrix, which is a function of the fluid density ρf or the solid density ρs and the shape functions ⎡ ⎤ 0 0 M11 0 ⎦ M22 Mρ = ⎣ 0 0 0 M33 with Mii =

 V

N ρN T dV .

Here K is the stiffness matrix function depending on the viscosity μ in the fluid part of the Lamé parameters (G Δt) and (λ Δt) in the solid part, and the derivatives of the shape functions ⎡ ⎤ K11 K21 K31 K = ⎣ K12 K22 K32 ⎦ K13 K23 K33 with Kij = Kij1 + Kij2 + Kij3 . In the fluid part  Kii1

= V



∂N ∂N T μ dV ∂xj ∂xj

∂N ∂N T μ dV ∂xi V ∂xj    2μ ∂N T ∂N 3 − Kij = dV, 3 ∂xj V ∂xi Kij2

=

(4.5) (4.6) (4.7)

Avoiding Instabilities Caused by Added Mass Effects

and in the solid domain    ∂N ΔtG ∂N T 1 Kii = dV J ∂xj V ∂xj    ∂N ΔtG ∂N T 2 Kij = dV J ∂xi V ∂xj    ∂N Δtλ ∂N T 3 Kij = dV. J ∂xj V ∂xi

173

(4.8) (4.9) (4.10)

If the displacements of the moving framework are different from the particle displacements, matrix K includes the convective terms Kij4 :  ∂N T 4 Kii = N (Vj − VM j ) dV ∂xj V Matrix B affects the incompressible part of the domain. This means that B has non-zero terms only in the degrees of freedom related with the fluid including the solid–fluid interfaces. The form of matrix B is B T = [B1T , B2T , B3T ]



with BiT =

Np V

∂N T dV. ∂xi

Equations (4.4) represent the coupled monolithic fluid– structure interaction problem that must be solved. It is well known that this system of equations must be stabilized for some class of equal order interpolations (e.g. when Np = N ) [E.O 06, ONA 04, TEZ 92]. Independent of the method chosen to stabilize the problem, we will assume that the problem has been conveniently stabilized by a matrix S (to be defined later in section 4.4.2) in such way that the problem reads

     Mρ ρ + K −B V Vn F+M Δt Δt = . (4.11) P 0 −B T S

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The stabilization terms introduce an error in the original system (4.4). In particular, the incompressibility condition originally discretized by (see equations (4.4)) BiT Vi = 0,

which becomes −BiT Vi + S P = 0.

In spite of the fact that the stabilization procedure is beyond the scope of this chapter, it is interesting to note that the matrix S must be singular so that the product SP is equal to zero with P different from zero. Nevertheless, several stabilization methods presented in the literature are based on non-singular S matrices, therefore introducing compressibility in the solution that sometimes is unacceptable.

4.3. Monolithic solution of the FSI equations by pressure segregation Solution of equations (4.4) or their analogous stabilized version equation (4.11) as a fully coupled system is sometimes expensive due to ill-conditioning problems. A more convenient way to solve that system is by segregating the pressure from the remaining unknowns (in our examples the velocity field). Segregation means to separate during the solution process the pressure from the velocity variables in a staggered way: first, the velocities (or the pressure) are evaluated independent of the pressure (or the velocities) and then the solution for the pressure (or the velocities) is found using the previous results. Segregation of the pressure has several advantages, for example: 1) decreases the number of degrees of freedom to be solved simultaneously; 2) avoids ill-conditioned matrices;

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3) allows using unified formulations for the fluid and solid domains; 4) it separates the velocity components; 5) it separates the nonlinearity; 6) allows us to draw some conclusions to be used in partitioned schemes (for which the pressure is always segregated from the solid part). There are several ways to segregate the pressure from the velocity. The simplest one is to assume an initial value for the pressure, compute the velocities using this initial value and then evaluate the pressure iteratively. A more sophisticated scheme to segregate the pressure is the Chorin–Temam projection scheme [CHO 68, TEM 69, COD 01], which will be discussed later. In order to simplify the discussion, the following change of variable will be introduced: P0 being any arbitrary vector of the same dimension of the pressure, we define the following new unknown: δP = P − P0 .

(4.12)

Note that P0 is not necessarily the initial pressure vector at time t = 0. It is any arbitrary vector. The system of equations to be solved becomes 

Mρ Δt

+K

−B T

−B S



V δP



 =

F+

Mρ Δt

V n + BP0 −SP0

 . (4.13)

We emphasize that solving this problem with pressure segregation for any of the methods previously proposed, as well as using a simple iteratively procedure or a projection method, leads to stable solutions when the added-mass effect does not occur. Nevertheless, in several cases the pressure segregation

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will produce non-convergent solutions or inefficient algorithms requiring large number of iterations. These disadvantages could be found when: 1) the density of the fluid is similar or lower than the solid density; 2) the time steps become too small; 3) the stiffness of the solid is small. In the following we will propose an algorithm that allows the pressure segregation and eliminates the above disadvantages. 4.4. Static condensation of the pressure The more precise way to segregate the pressure from the velocity is via static condensation (it is exact in case of linear system of equations). Static condensation is a procedure to solve a system of equations in a partitioned way. It consists of inverting a part of the initial matrix. For instance, in the system of equation (4.13) we can condensate the pressure by inverting matrix

S , or condensate the velocity by inverting Mρ matrix Δt + K . As stated before, matrix S must be singular and then the only possibility is to condense the velocity field. From the first row of equation (4.13), the velocity field may be obtained as −1    Mρ n Mρ +K V + BP0 + BδP . F+ V = Δt Δt Inserting this into the second line of equations (4.13) gives −1    Mρ n Mρ T +K V + BP0 + BδP +SδP = −SP0 . F+ −B Δt Δt

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This means that the static condensation of the velocity allows the problem to be solved in two steps: 

T

Mρ +K Δt

−1



−1 Mρ +K −B B + S δP = B Δt   Mρ n F+ V + BP0 − SP0 (4.14) Δt     Mρ Mρ n +K V = F + V + BP0 + BδP . (4.15) Δt Δt 



T

Defining the vector V˜ as  V˜ =

Mρ +K Δt

−1   Mρ n V + BP0 . F+ Δt

Static condensation is implemented in the following three steps:    Mρ Mρ n ˜ +K V = F + V + BP0 Δt Δt   −1  M ρ +K −B T B + S δP = B T V˜ − SP0 Δt   Mρ + K (V − V˜ ) = BδP. Δt 

(4.16) (4.17) (4.18)

Equations (4.16)–(4.18) represent the way to segregate the pressure from the velocity in a exact way. It is a very expensive procedure from the computational point of view, but if enough resources are available, it is the correct method to apply. On the other hand, equations (4.16)–(4.18) suggest a procedure to approximate the exact algorithm and to obtain a more efficient way to segregate the pressure.

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It is interesting to note the analogy between equations (4.16) and (4.18) with the classical Chorin-Teman projection scheme. Effectively, the first equation (4.16) is the evaluation of the compressible velocity V˜ . Using P0 = 0 the first-order scheme is obtained. Using P0 = P n the second-order approximation may be developed. Equation (4.17) is in fact the exact equation corresponding to the Laplace equation in a projection method and equation (4.18) is the exact equation to obtain the incompressible velocity as a function of the pressure gradients. Then, we can say that Chorin–Teman projection method is an approximation to the exact static condensation of the velocity field. In the following section we present other approximations that are more suitable for FSI problems. 4.4.1. Approximation to the static condensation The Chorin–Teman projection method [20] is equivalent to

−1 ρ + K B in equation (4.17) approximate the matrix B T M Δt by the Laplace matrix L: −1  Mρ Δt T +K L, B B (4.19) Δt ρ where

 L= V

∂N ∂N T dV. ∂xj ∂xj

(4.20)

This approximation is acceptable for non-viscous or nearly inviscid flows for which matrix K is negligible versus the mass −1 ρ T Mρ matrix M . The remaining matrix B B is approxiΔt Δt mately equal to Δt ρ L (see the definition of B , Mρ , and L) for a lumped mass matrix and continuous pressure shape functions. ρ A more suitable way to approximate matrix B T M Δt + K)−1 B for cases where matrix K is not negligible will be proposed next.

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  Since M = M e and K = K e , where M e and K e are the mass and stiffness matrices of an element “e”. Let us introduce the following approximation in equation (4.17): M e ≈ ρe MDe ,

where MDe is a lumped matrix. On the other hand,   Δtλ ΔtG 1 e e + K ≈ M J J h2 D in the elastic domain and K e ≈

μ h2

(4.21)

(4.22)

MDe in the fluid domain.

If the velocity of the moving framework is different from the particle velocity, the convective term is added to the fluid lumped stiffness matrix K e :   μ |V − Vm | e MDe . K ≈ + (4.23) h2 h In all previous definitions, h represents a characteristic element size (for instance, the average distance between the element nodes). Note that previous approximations are also used by other authors in related fields, see for example [TUR 98]. It must be noted that using the same idea, different possibilities for the lumped mass and stiffness matrices may be proposed. In equations (3.13) and (3.14) just the diagonal terms of each matrix are chosen, but other most sophisticated lumped matrices may be used. We emphasize that these lumped matrices are used exclusively in the pressure equation (4.17). In equations (4.16) and (4.18) the fully consistent matrices are used. With the previous approximations, the original matrix in equation (4.17) becomes  −1 −1  e

M M e τ e [M e ]−1 = MT−1 , +K +K = ≈ D Δt Δt

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where MT−1 is a diagonal matrix obtained from the assembly of the element contributions τ e [M e ]−1 D , with

 e

τ =

Δtλ ΔtG ρs + + Δt Jh2 Jh2

−1

in the elastic domain and   ρf μ |V e − Vme | −1 e + τ = + Δt h2 h

(4.24)

in the incompressible domain. The first two approximations may be written as  B

T

Mρ +K Δt

−1

B ≈ B T MT−1 B.

Finally, a third approximation, similar to the classical one introduced in projection methods, is added: B T MT−1 B ≈ Lτ ,

where Lτ = ment.



(τ e Le ), Le being the Laplace matrix for the ele-

Then, the three steps algorithm reads     Mρ Mρ n ˜ +K V = F + V + BP0 Δt Δt 

(4.25)

(−Lτ + S) δP = B T V˜ − SP0  Mρ + K (V − V˜ ) = BδP. Δt

(4.26) (4.27) (4.28)

Of course, equation (4.27) is an approximation to equa

−1 ρ tion (4.17) when B T M + K B is replaced by Lτ . This Δt

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approximation introduces an error in the evaluation of P . In order to diminish this error, an iterative procedure may be used to approximate P0 by P . Effectively, now the introduction of the arbitrary variable P0 in equation (4.12) becomes justifiable. Hence, once P is evaluated at the k iteration, then the next iteration is started with P0 = P k in equation (4.26). Note that the error introduced by the approximation to

−1 ρ BT M + K B becomes negligible when δP → 0. Δt Another way to introduce the previous approximation is by keeping P in the r.h.s. of equation (4.27). The system of equations to be solved is now     Mρ M ρ n + K V˜ = F + V + BP0 (4.29) Δt Δt  −1 Mρ T ˜ T +K (−Lτ + S) P = B V − B BP0 (4.30) Δt  

Mρ +K V − V˜ = B (P − P0 ) . (4.31) Δt In order to facilitate the solution of equation (4.30), let us introduce the following auxiliary vector T0 : −1  Mρ +K T0 = BP0 . Δt Then the four step algorithm reads     Mρ Mρ n ˜ +K V = F + V + BP0 Δt Δt   Mρ + K T0 = BP0 Δt (−Lτ + S) P = B T V˜ − B T T0  

Mρ +K V − V˜ = B (P − P0 ) Δt

(4.32) (4.33) (4.34) (4.35)

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Note that equations (4.32)–(4.35) are conceptually different from equations (4.26)–(4.28). In fact, in equations (4.26)– (4.28) the error introduced by the approximation to the static condensation disappears when δP → 0. However, this error does not disappear in equations (4.32)–(4.35), remaining a non-zero term even for δP = 0. This remaining term introduces an additional stabilization in the pressure equation that, in certain cases, is enough to obtain stable solutions. However, this additional term does not always guarantee a stable solution and for this reason equations (4.26)–(4.28) will be used in the following part of this chapter. It is also interesting to note the similitude of equations (4.32)–(4.35) to the classical projection methods (see for instance Codina [COD 01]). Vector T0 represents the projection of the pressure gradients in the velocity field (see equation (4.33)). In our approach, this method has been obtained in a more natural way as a consequence of static condensation. Equations (4.32)–(4.35) represent a generalization of the Chorin–Teman projection method to problems for which the ρ K matrix is not negligible versus the M Δt matrix.

4.4.2. Definition of the stabilizing matrix As stated before, equation (4.27) must be stabilized for equal order elements in order to overcome the BBL (BabuskaBrezzi-Ladyzhenskaya) condition. Matrix S represents in equation (4.27) this stabilizing matrix.

−1 ρ + K B The error introduced in approximating B T M Δt by Lτ introduces a stabilization during the iterative process while δP = 0. Nevertheless, in order to have a more general stabilization algorithm independent of the iterative process, a standard projection method will be used to stabilize the pressure equation.

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Projection methods are based on the difference between the pressure gradients (which are non-continuous between elements) and a continuous function of the pressure gradients (the projection of the pressure gradients in the continuous velocity space). ∂p , Let us define the pressure gradient projections Πi = τ0 ∂x i where τ0 is a parameter that will be defined later. A simple Galerkin weighting approximation gives    ∂NpT 1 1 N dV P = N Πi dV = N N T dV Π. ∂xi τ0 τ0 V V V

Thus we obtain that: Π = Mτ−1 DP, 0

where the matrix D may be obtained from its transposed as  ∂N T DT = [D1T , D2T , D3T ] DiT = N p dV V ∂xi and

⎡ 

Mτ 0 = ⎣

V

N τ10 N T dV 0 0

 V

0 1 N τ 0 N T dV 0

⎤ 0 ⎦. 0  1 T V N τ 0 N dV

The standard stabilization term [E.O 06, ONA 04] is defined by the divergence of the difference between the gradient and the projection gradient pressure,    ∂ ∂p Πi − τ0 SP = − Np dV. (4.36) ∂xi ∂xi V Parameter τ0 has been introduced in order to keep the consistence of the dimensions. Taking into account that SP will be added to the mass conservation equation, a correct value for τ0 is τ0 = Δt ρ . It must be noted that τ0 changes for each element. This must be built element by element usmeans that matrix Mτ−1 0 ing a standard assembly process.

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In a more general way, taking into account the “natural” definition of τ for the static condensation of equation (4.24), the following definition of τ0e is also useful:  τ0e

e

=τ =

ρf μ |V e − Vme | + 2 + Δt h h

−1 .

Integrating by parts, equation (4.36) becomes    ∂ ∂p Πi − τ0 SP = − Np dV ∂xi ∂xi V      1 ∂Np ∂p ∂p Πi − τ0 νi dΓ. = dV − Np Πi − τ0 ∂xi ∂xi V ∂xi (4.37) The boundary terms of equation (4.37) are normally neglected [ONA 04], remaining:    ∂Np ∂p Πi − τ0 SP = dV ∂xi V ∂xi 0 /  ∂Np ∂Np ∂NpT N dV Π − τ0 dV P = ∂xi V ∂xi V ∂xi   D − Lτ0 P. = DT Mτ−1 0 Or simply,

S = DT Mτ−1 D − Lτ0 . 0

Note that D is different from B and also from B T . Introducing matrix S into equations (4.26)–(4.28) reads     Mρ Mρ n ˜ +K V = F + V + BP0 Δt Δt D − Lτ0 ) P = B T V˜ − Lτ P0 (−Lτ + DT Mτ−1 0  

Mρ +K V − V˜ = BδP. Δt

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The stabilization terms (DT Mτ−1 D − Lτ0 ) P may be solved 0 D) P using the previous iteraexplicitly for the term (DT Mτ−1 0 k −1 tion of the pressure vector P . In other words (DT Mτ−1 D − Lτ0 ) P ≈ DT Mτ−1 D P k−1 − Lτ0 P. 0 0

The four steps, stabilized algorithm remains     Mρ Mρ n ˜ +K V = F + V + BP0 Δt Δt Mτ0 Πk−1 = DP k−1 (−Lτ − Lτ0 )P = B T V˜ − Lτ P0 − DT Πk−1  

Mρ +K V − V˜ = BδP. Δt

(4.38) (4.39) (4.40) (4.41)

Equations (4.38)–(4.41) may be solved implicitly, which requires the solution of four systems of equations: two with the ρ same matrix M Δt + K (which may be triangularized once), one with matrix Mτ0 , and the fourth equation is a Laplace system for the pressure involving matrix Lτ + Lτ0 . Equation (4.39) is normally solved explicitly with a lumped form of matrix Mτ0 . Several strategies may be used to solve equations (4.38)– (4.41) as a nonlinear problem. One is to take P0 = P k−1 in equations (4.38). In this case, the method is called predictorcorrector and the results converge to the monolithic solution. Another possibility is to take P k−1 = P0 in equations (4.39). In this case the static condensation error introduced in equation (4.25) does not disappear, introducing an additional stabilization term. The convergence is fast but the result is not equal to the monolithic one. ρ Considering the K matrix negligible in front of M Δt (which is always true for small time steps), equations (4.41) may be

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also solved explicitly with a lumped mass matrix MD , 

   Mρ Mρ n + K V˜ = F + V + BP0 Δt Δt

(4.42)

k−1 Πk−1 = [Mτ0 ]−1 D DP

(4.43)

−(Lτ + Lτ0 )P = B T V˜ − Lτ P0 − DT Πk−1

(4.44)

Δt −1 M BδP. V = V˜ + ρ D

(4.45)

It must be noted that in both cases, equation (4.40) and equation (4.44) have a Laplace form. Then, in order to be solved, essential boundary conditions must be introduced in the pressure term to avoid the singularity of the Laplace matrix. The standard approach is to impose the pressure on the free-surface boundaries or to impose an arbitrary pressure value in one node in the case of closed domains. Observing equations (4.40) and (4.44), two different parameters multiply the Laplace matrix L. One of them is τ0 , the stabilization parameter. The other one is τ . This one is not a stabilization parameter. It is a parameter that comes from the static condensation of equation (4.25). This parameter is multiplying a Laplace matrix by δP . This means that when the convergence is achieved the τ value does not affect the results. This is not the case for τ0 , which is multiplying the difference between pressure gradient and its projection. This difference is not necessary zero after convergence, which means that τ0 stabilizes the final results. Nevertheless, the terms introduced while Lτ δP is different from zero are important to achieve convergence in FSI problems with added-mass effects. Although both parameters are equal (τ = τ0 ), they are expressed separately in the Laplace equation in order to explain the different meaning of each of them.

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4.5. Evaluation of the Laplace matrix for FSI problems When solving an incompressible fluid–elastic solid interaction problem, the incompressible condition equation (4.2) is only applied to the fluid domains. This means that the discretized form BiT Vi = 0 only affects some degrees of freedom (dof). Let us call nP the total dof corresponding to the pressure, nV the total dof corresponding to the velocity, ns the velocity dof corresponding to the solid exclusively (without the interfaces), nf the velocity dof corresponding exclusively to the fluid (without the interfaces), and nsf the velocity dof of the interfaces solid–fluid. For simplicity, we will consider only conforming meshes, this means that the interface between the fluid and the solid has exactly the same number of nodes. Then matrix B T is a matrix of nP files and nV columns, but all the columns corresponding to the ns solid dof are zero. Matrix B T has non-zero columns in the dof corresponding to the fluid domain and the interfaces. On the other hand, matrix MT−1 is a diagonal matrix, with terms  −1  ρf μ |V n+1 − Vm | −1 + MT = + (4.46) MD Δt h2 h in the nf dof MT−1

 =

ρs GΔt λΔt + + Δt Jh2 Jh2



−1 MD

(4.47)

in the ns dof, and −1  ρf μ ρs GΔt λΔt |V n+1 − Vm | + + + = ( + + )MD Δt h2 h Δt Jh2 Jh2 (4.48) in the nsf dof. MT−1

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ALE and Fluid–Structure Interaction

Performing the double product B T MT−1 B , all the terms corresponding to the ns dof are zero. Matrix Lτ may be written as Lτ = B T MT−1 B ≈ Lτf + Lτs , (4.49) where Lτf is the standard Laplace matrix corresponding to the fluid domains including the interfaces (see equations (4.19), (4.20), and (4.25))   Lτf = τfe Le , defined only on the fluid domain and Lτs is a Laplace matrix corresponding only to the fluid–solid interface  e Lτs = τs L , where e



L = V

T ∂Nsf ∂Nsf dV ∂xj ∂xj

 s

is the Laplace matrix of the solid elements evaluated only with the shape functions Nsf that are different from zero on the fluid–solid interface. Equation (4.49) may also be written as Lτ = Lτf + βLτs

with τs β= = τf

ρf Δt

and Lτf =

+

μ h2

ρs Δt

+



|V n + 1 −V m | h GΔt λΔt + J h2 J h2

+

e

τf L

(4.50)

.

This means that the Laplace interface matrix Lτf may be neglected for small values of the β parameter. This is for instance the case when ρs >>ρf and the added-mass effect is not present. However, for other physical properties the β

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189

parameter may not be negligible, and the Laplace interface matrix must be evaluated in order to obtain good results. For instance, for non-viscous flows (μ ≈ 0), a Lagrangian formulation (V ≈ Vm ), and small displacements (J ≈ 1) the β parameter is ρf

. β= (G+λ)Δt2 ρs + 2 h 2

This means that β → 0 for large values of (G+λ)Δt , that is h2 for large Δt, large shear stiffness G and λ and for small mesh size h. However, β is not negligible for ρρfs ≈ 1 or if we are using small time steps or the solid material is very soft. In the examples we will show that the Lτf matrix improves the convergence in the following cases: 1) ρf ≥ ρs (classical added-mass effect); 2) small Δt (this means that instabilities do not disappear by decreasing the time step); 3) soft materials (i.e. biomedical materials). Equation (4.50) shows that the evaluation of β is rather more complicated than a simple ratio between the fluid and the solid densities. In general β has small values, but its inclusion considerable, improves the convergence rate in all cases. Also in the cases where the added-mass effect is strong, its inclusion is crucial to achieve convergence.

4.6. The partitioned (or staggered) scheme Partitioned schemes are derived from dividing the original FSI problem in two parts: the solid one and the fluid one. The division is performed independent of sub-iterations in a strongly coupled partitioned scheme or in a loosely coupled one. The idea is similar to the pressure segregation

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described above for which the system was split in the velocity and pressure unknowns. Now the same system of equations is divided in the solid unknowns (for instance, the velocity or the displacements) and the fluid unknowns (normally the velocity and the pressure). Both systems are solved separately. Some recent work on this subject may be found in [DET 06, FER 07, GER 03]. Let us call Vs the vector containing the solid unknowns, Vf and P the vectors containing the fluid unknowns, not including the common solid–fluid unknowns, and Vsf the vector including the common solid–fluid unknowns. The transfer of information occurs on the boundary ΓSF by using techniques that guarantee momentum and energy conservation [FAR 98]. For staggered algorithms the use of non-matching meshes is common practice since both systems, fluid and structure, are completely decoupled. In this chapter only conformal (matching) meshes will be analyzed. However, we consider that the main result concerning the convergence properties of the interface Laplace matrix is also valid for nonmatching meshes. The classical boundary conditions at the interface are (Vf − Vs )T = 0

on

Γf −s

(4.51)

σ f + ts = 0

on

Γf −s ,

(4.52)

where equation (4.51) represents the consistency condition. Since the interface is modelled using a fully Lagrangian frame of reference, this condition guarantees that the fluid and solid meshes will remain tightly coupled along the FSI interface. Equation (4.52) represents the equilibrium of stresses along the interfaces.

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The original FSI equation (4.4) may then be written as ⎤ ⎡

Mρ Mρ + K + K 0 0 Δt ⎥⎡ ⎤ ⎢ Δt

Ts

sf

⎥ Vs ⎢ Mρ Mρ ⎥ ⎢ Mρ + K + K + K −B Vsf ⎥ sf ⎥ ⎢ ⎢ Δt Δt Δt ⎥ sf ssf f fs ⎢ ⎥⎢

⎣ T ⎢ ⎥ Vf ⎦ M M ρ ρ ⎢ 0 −Bf ⎥ Δt + K f s Δt + K f ⎣ ⎦ P T T 0 −Bsf −Bf 0

⎡ ⎤ ρ n F+M Δt V

s⎥ ⎢ ⎥ ⎢ F + Mρ V n ⎢ ⎥ Δt = ⎢ (4.53)

sf ⎥ . ⎢ ⎥ Mρ n V F + ⎣ ⎦ Δt f

0

In the monolithic method with the pressure segregation described earlier, equation (4.53) was partitioned into two parts: the first three rows and columns and the fourth row and column. In classical staggered methods, equation (4.53) is also partitioned into two parts, the first two rows and columns and the third and fourth rows and columns. For each subiteration, the static condensation of the terms  −1 M T +K Bsf Bsf (4.54) Δt ssf f must be taken into account when the added-mass effect is strong. Remember from section 4.3 that this term is placed on the left-hand side of the pressure equation (4.27). In the partitioned scheme, the classical Neumann boundary conditions are imposed on Γf −s and thus equation (4.54) is still of great importance. Using the same conclusion reached in section 4.4, an interface Laplace matrix βLτf must be added when solving the incompressible part of the domain. Since the meshes on the

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ALE and Fluid–Structure Interaction

interface may be non-matching, special care has to be taken when equations (4.46)–(4.48) are evaluated. In the general case, equation (4.48) takes the form MT−1

  ρf μ |V n+1 − Vm | + MDf = + Δt h2 h  −1  GΔt λΔt ρs + M + + . Ds Δt Jh2 Jh2

In conclusion, matrix βLτf must be added to the fluid equation independent of the method used to solve the incompressibility condition. This means that independent of using pressure segregation, the fluid solution must include the interface Laplace matrix. This is because, when using a partitioned solution, pressure segregation is implicitly included in the procedure as explained in equation (4.53). Partitioned algorithms are in fact more complicated to approximate with a Laplace matrix than monolithic ones using pressure segregation because of the fact that there are other connecting matrices in addition to Bsf . For example, the matrix corresponding to the velocity inside the fluid and the interfaces after static condensation, this matrix yields the terms 

M +K Δt

T  sf

M +K Δt

−1  ssf f

M +K Δt

 . sf

Fortunately, these matrices are negligible for the pressure forces on the interfaces, and therefore are normally neglected. To satisfy the boundary conditions in equations (4.51–4.52) and therefore guarantee energy conservation through the interface, the staggered approach uses subiterations where P, Vf , Vs are updated at each substep. This procedure is essential when added-mass effects are present providing a stronger coupling between the fluid and the solid variables. For the

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193

case of aero-elasticity an explicit coupling via a single substep may provide enough stability. Splitting equation (4.53) for the staggered approach results in     Mρ Mρ n (4.55) + K Vs = F(Ps f ) + V Δt Δt s s     Mρ n Mρ + K Vf − Bf P(Vs f ) = F + V (4.56) Δt Δt f f T Bsf Vsf + BfT Vf = 0.

(4.57)

As mentioned above, pressure segregation for the staggered approach implies a one-step solution for the solid problem and a fractional step method for the fluid variables. Following a similar procedure to the one described in section 4.3, equations (4.55)–(4.57) become   Mρ n Mρ + K V˜f = F + V + BP0 (4.58) Δt Δt f   Mρ M n + K Vs = F(Ps f ) + V (4.59) Δt Δt s k−1 Πk−1 = [Mτ0 ]−1 D DP

(4.60)

−(Lτf + βLτf + Lτ0 )P = B T (V˜f + Vsf ) − (Lτf + βLτf )P0 −DT Πk−1 .

(4.61)

To simplify the notation, the lower index for the matrices has been removed implying that the matrices for the fluid problem are computed on the fluid domain and similarly for the solid. In the examples presented in the next section, equations (4.58)–(4.61) are solved with P k−1 = P0 in equation (4.60) and P0 = Pn in equation (4.58) at each time step. Then, the steps from equations (4.58) and (4.59) are performed only once per time step. Substepping iterations are performed between

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equations (4.58) and  (4.59) and  the solid solver. Convergence  k k−1  is achieved when Vsf − Vsf  < ε. The iterations performed p

between equations (4.58) and (4.59) involve a simple fixedpoint method. Thus, no acceleration technique has been used to improve the convergence of the method. The time integration on the solid will not be considered in the present analysis. For instance, the Newmark method may be applied in this case. It is important to notice that different Δt could be used in the fluid and solid domains. If this is the case, the time step has to be synchronized between both solvers. Matrix βLτf has to be computed in the same way as for the monolithic approach. For non-matching meshes some considerations have to be taken into account when dealing with the matrices on the interface elements. These considerations will not be addressed here. A simplified way to deal with this situation is to evaluate the interface matrix  βLτf = βτf V

T ∂Nsf ∂Nsf dV, ∂xj ∂xj

(4.62)

which must be computed with the shape functions of the solid and fluid domain using only the shape functions of the fluid domain,  βLτf = βτf V

T ∂Nsf ∂Nsf dV ∂xj ∂xj



 ≈ βτf fs

V

T ∂Nf ∂Nf dV ∂xj ∂xj

 . f

(4.63) This approach is valid when both meshes are sufficiently similar. All the validation examples approximated with the staggered approach use equation (4.63), since the fluid and solid nodes almost match at the interface to avoid introducing errors from this approximation.

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4.7. Numerical examples 4.7.1. Fluid column interacting with an elastic solid bottom The first example is a very simple 1D problem for which an analytical result can be easily obtained. Nevertheless, from the numerical point of view it has some convergence problems. The example is ideal to test different materials and time step sizes in order to check the validity of the algorithm proposed, in particular the effectiveness of the interface matrix to improve the convergence rate. The example consists of an incompressible water column over an elastic solid (Figure 4.2). Both column walls have the horizontal displacement constrained (plane strain). The upper line is a free surface and

Figure 4.2. Water column over an elastic beam

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the bottom one has the constrained displacement. Initially the fluid is at rest. For the solid part the density is set to ρs = 1.5 g/cm3 , the Young modulus Es = 2.3 × 105 dynes/cm2 and the Poisson coefficient is ν = 0.4. On the fluid, the density is ρf = 1 g/cm3 and μf = 0. The gravity was fixed to g = −1 × 103 m/s2 in the vertical direction and the geometry was discretized as a 2D problem using a mesh of 4 × 200 with three-node linear triangles and mesh size of h = 2.5 cm. Note that a Young modulus like the one used in this example corresponds to a very soft material.

( (

Figure 4.3 shows the vertical displacements in (cm) of any fluid particle as a function of time. The right image of Figure 4.3 shows the same vertical displacements of the entire domain at different time steps. The perfect vertical line corresponds to the incompressible par. This very simple result is not easy to obtain. For instance, the fractional step method (P0 = P0 |t=tn ) in equation (4.42) with exactly the same time step and the same convergence rate gives the results presented in Figure 4.4. The lack of convergence to the monolithic scheme is the reason for these oscillations.

(

(

( (

Figure 4.3. Left: vertical displacement versus time at any fluid point. Right: vertical displacement in the total domain for different time steps

Avoiding Instabilities Caused by Added Mass Effects

(

197

)

Figure 4.4. Water column over an elastic beam: vertical displacement in the total domain at different time steps for the fractional step method

The numerical solution does not converge when the interface Laplace matrix βLτf is neglected (β = 0). The best way to see the importance of this matrix is to study different situations with different densities, different Young modulus, and different time step sizes. Table 4.1 shows the performance of the algorithm for a stiff material with a Young modulus similar to steel and different density rates. We can observe that β = 0 is acceptable only for density rates larger than 6. The number of iterations to achieve the same error is equal to 20 in all cases for β = 0, but it is larger than 40 iterations for β = 0. This means that even in the case of a FSI of steel and water (ρs /ρf = 7), β = 0 must be used. Probably, only in aero-elasticity applications where the density rate is larger than 1000, the omission of the interface Laplace matrix is justified. Table 4.2 shows the same problem with different stiffness properties for the elastic domain but with the density rate

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ρs /ρf 10 7 6 5 3 1

β = 0 β=0 20 iterations 20 iterations 20 iterations More than 40 iterations 20 iterations More than 40 iterations 20 iterations Does not converge 19 iterations Does not converge 18 iterations Does not converge

Table 4.1. Iterations to achieve convergence for different density rates. Es = 2 × 101 1 dynes/cm2 , ν = 0.3, Δt = 10−5 s

fixed to one. This means equal density in the elastic solid as in the fluid. Because of the oscillatory behavior of the problem, the time step (Δt, in seconds) must be changed in order to achieve reasonable time integration with a minimum of time steps in each oscillation. We can observe that only for very high Young modulus, the case with β = 0 converge. There is an exception for E = 2 × 106 dynes/cm2 and Δt = 1 × 10−2 s. Using β = 0, the algorithm converges in 33 iterations. We must say, however, that in order to obtain an acceptable result from the time integration point of view, the correct time step for this case is less than Δt = 1 × 10−2 s. Nevertheless,

E (dynes/cm2 ) 2 × 1013 2 × 1012 2 × 1011 2 × 108 2 × 107 2 × 106 2 × 106 2 × 105

Δt (s) 0.2 × 10−5 0.5 × 10−5 1 × 10−5 1 × 10−4 1 × 10−3 1 × 10−3 1 × 10−2 1 × 10−2

β = 0 10 iterations 14 iterations 18 iterations 40 iterations 36 iterations 40 iterations 34 iterations 36 iterations

β=0 > 40 iterations Does not converge Does not converge Does not converge Does not converge Does not converge 33 iterations Does not converge

Table 4.2. Iterations to achieve convergence for different Young modulus. ρs /ρf = 1, ν = 0.3

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Δt (s) 2 × 10−5 1 × 10−5 0.75 × 10−5 0.5 × 10−5 0.25 × 10−5

199

β = 0 β=0 23 iterations 21 iterations 20 iterations > 40 iterations 18 iterations Does not converge 16 iterations Does not converge 11 iterations Does not converge

Table 4.3. Iterations to achieve convergence for different time steps. ρs /ρf = 7, Es = 2 × 101 1 dynes/cm2 , ν = 0.3

the same stiffness with Δt = 1 × 10−3 s does not converge for β = 0. This is consistent with the well-known conclusion that for equal density rates the added-mass effect is so important in all cases that without the interface Laplace matrix or any other artifice, the problem does not converge. However, the most worrisome results are those presented in Table 4.3. They correspond to a standard steel elastic modulus with a density rate equal to 7. We first use the correct time step size for this kind of problem and we see that both formulations for β = 0 and β = 0 converge reasonably well in 21 iterations. Nevertheless, halving the time step, the number of iterations with β = 0 duplicates. Decreasing the time step more and more, the method with β = 0 does not converge, while the algorithm with the interface Laplace matrix converges in a decreasing number of iterations as expected. This example shows that even when classical materials like steel and water are involved, the use of the interface Laplace matrix is recommended to avoid possible difficulties when the time step is smaller than necessary. The problems were tested with both methods: monolithic with pressure segregation and with a strongly coupled partitioned scheme with similar conclusions in both cases.

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4.7.2. Mesh sensitivity analysis The objective of this example is to show the influence of the FSI algorithm presented in this chapter for different mesh configurations. In particular, we are interested in studying the convergence rate of the coupled problem for the case of a partitioned approach and the effect of the β parameter for different mesh sizes. When solving FSI problems with large deformations using a PFEM-ALE formulation [PIN 07, IDE 06], the mesh-moving algorithm will eventually distort the mesh in such a way that the element quality becomes unacceptable. In some occasions, specially in the presence of rotations, the mesh-moving algorithm cannot deal with the extreme deformations that will finally result in elements inversion and negative Jacobians. For this reason, a robust nonlinear FSI solver should provide a way to circumvent this problem either by allowing local topological mesh changes or some kind of adaptive mesh refinement strategy. In this section, we will show how sensitive the FSI algorithm is to mesh changes and whether it affects the convergence rate of the underlying numerical technique, a basic requirement for any coupling algorithm. The test case chosen was first proposed by Neumann et al. in [NEU 06]. The sketch of the problem with dimensions and boundary conditions is depicted in Figure 4.5. The problem consists of a channel with a gradual contraction. Right

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Figure 4.5. Problem description and dimensions (cm)

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()

Figure 4.6. Velicity at inflow

before the contraction there is a flexible solid flap. The flap represents a rubber material with Young modulus E = 2.3 × 106 N/m2 , density ρs = 1500 kg/m3 and Poisson ratio ν = 0.45. The fluid is a silicon oil with density ρf = 956 kg/m3 and viscosity μ = 0.145 kg/ (ms). The fully developed laminar flow has a Reynolds number of Re = 100. The inflow velocity for the problem follows 2   πt  Vm a x 1 − cos 10 if t ≤ 10 2 v(t) = if t > 10, Vmax

(4.64)

which is depicted in Figure 4.6. Four different mesh configurations were tested. Since the test case proposed has no analytical solution or experimental data, a reference solution with a highly refined mesh was taken as the exact solution. The mesh grading around the flap varies from 0.001 for the finest reference mesh to 0.002, 0.004, and 0.008 for the coarser mesh. The meshes are shown in Figure 4.7. All fluid meshes match the solid mesh at the interface so that the solid mesh is refined and also the errors incurred in transferring loads and displacement are avoided.

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A

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Figure 4.7. Different mesh densities used in the convergence analysis. Number of elements from coarser to finer: 4380 (A), 16927 (B), 43838 (C), and 101251 (D)

Only the sensitivity with respect to the mesh size was studied. Thus, the time step size was kept constant and equal to Δt = 0.05 for all the meshes and the error computations were carried out at steady state. This means that according to equation (4.9) the value of β will only change with respect to the mesh size. It is worth mentioning for β = 0 (no interface Laplace matrix is used) that the coupled problem does not converge in the very early stages of the solution (within the first five time steps). The variable taken for the analysis was the flap tip deflection in the streamline direction corresponding to the xcoordinate at steady state. An L2 norm was used to evaluate the error: 1 L2 = |dx − rx |2 , n i=1,n

where rx is the reference displacement vector for the finer mesh. The steady state was reached at t = 25 seconds and the sum was performed for dx and rx between 25 and 35 seconds, with n = 10/δt being the total number of time steps.

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0.08 X - Displacement (m)

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Figure 4.8. Tip deflection in the x direction

Figure 4.15 depicts the tip displacement with respect to time which is in good agreement with the one shown in [NEU 06]. The error evaluated in each mesh is presented in a log– log plot in Figure 4.9 where the abscissa is the logarithmic of the error and the ordinate is the logarithmic of the mesh size. The curve slope represents the order of convergence. In this case an optimum quadratic convergence is found matching the individual order of convergence for the fluid and the solid formulation separately.

Figure 4.9. Convergence plot for the flexible flap problem

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A

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Figure 4.10. Velocity field at time steps 2.5, 5, 10, 12.5, 15, 25 s

Figure 4.10 shows the flow vectors together with the velocity modulus. The first three time steps correspond to the initial deflection. At time t = 10 s. The main vortex is detached and the maximum velocity is reached (see Figure 4.6). At this point the flap starts retracting until its steady-state position. Finally, in Figure 4.11 we show the plot of the iteration number against time for the intermediate mesh of size 0.004. The iteration counts do not vary so much for the different meshes so only this case is shown. We can see an average value between 3 and 5 iterations per time step. This compares well with the most optimum algorithm proposed in [NEU 06].

4.7.3. 3D flexible flap in a converging channel The present example is a 3D version of the test case introduced in the previous section. In Figure 4.12, the 3D geometry is depicted, and Figure 4.13 shows the surface mesh used in the solution process. The geometry is extruded in the z direction but the flap is placed at the center using a fraction of the width of the channel. In this way the flow is allowed

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to wrap around the flap. As no comparison is performed, the Young modulus E is modified to allow larger displacements: E = 1.3 × 106 dynes/cm2 . In Figure 4.14, the streamlines of velocity and the flap deflection are presented. Figure 4.15 shows the flap at its maximum deflection colored with the values of von Misses stresses. In the same figure, the tip displacement is shown for points A and B (see Figure 4.12). The large deformation of the flap triggered a re-meshing stage in the solver which was coupled to an error estimator. In this way, each new mesh was enriched with a new element size with the objective of keeping the error constant throughout the mesh. As it was shown in the previous section, the change in mesh size should not modify the convergence of the

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Figure 4.12. Problem description and dimensions (cm)

Figure 4.13. Left: the surface meshes for the full domain. Right: close up of the flexible flap

t: 1.500000 s

t: 6.500000 s

t: 8.000000 s

t: 10.000000 s

t: 17.000000 s

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Figure 4.14. Streamlines of velocity for time steps 0, 5, 8, 10, 17, and 25 s

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Fringe Levels

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5.522e+03 4.834e+03 4.146e+03 3.459e+03 2.771e+03 2.083e+03 1.395e+03 7.076e+02 1.982e+01

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Figure 4.15. Von Mises stresses at maximum deflection and flap displacement as a function of time for points A and B indicated in Figure 4.12

algorithm. Since the surface mesh size between the fluid and the solid structure does not change, the value of β is kept constant throughout the whole problem. Figure 4.16 shows different element densities across the flap. A total of 13 re-meshing stages were performed. 4.7.4. Experimental validation with aortic valve In this section, we present an experimental validation case. The geometry is shown in Figure 4.17. It consists of a channel

Figure 4.16. Tetrahedral mesh at different time steps

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Figure 4.17. Sketch and dimensions for the aortic valve validation case in meters

with a thin leaflet that pivots on one end allowing for rotation. In the middle of the channel there is an expansion into a larger cavity modeled by half a circle which forms the aortic sinus. Details for the experiment configuration and related data may be found in [STI 04]. The leaflet has a thickness of 1.5−4 m and a density of 1000 kg/m3 . For the fluid the density was assign to 1000 kg/m3 , and a constant viscosity of 0.0043 kg/(ms). The inflow is on the right-hand side of Figure 4.17 and it consists of a plug flow that follows V (t) = −27.61 + 65.25t − 48.68t2 + 11.04t3 + 0.25t4 ,

(4.65)

where t is time. These velocity values were interpolated from the actual experimental values. The lower panel of Figure 4.18 is the graphical representation for the inflow velocity. On the right-hand side of the model the pressure was imposed to p = 0 at all nodes. The rest of the boundary faces have a non-slip boundary condition. The results for the flow profile are shown in Figure 4.18. The instantaneous velocity profile is presented for five time steps and they are compared with the experimental values. The solid line represents the numerical results and the dashed line represents the experimental results. The figure on the bottom right side of the panel shows the inflow rate as

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Figure 4.19. Computed opening angle (solid line) compared to experimental results (dashed line)

a function of time. In general, we can see good agreement between the experimental and the numerical results except for some areas in the sinus. Some reasons why this may happen could be related to inexact inflow velocity and the model that we used to pivot the leaflet.

Finally, Figure 4.19 shows the opening angle as a function of time. The solid line represents the numerical results and the dashed line represents the experimental data. Again we can see good general agreement. The flat part of the curve that is observed for the numerical result between time 0.8 s and 0.9 s corresponds to the reverse flow and it indicates the contact between the leaflet and the channel wall. This contact is overpredicted by the numerical model.

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4.7.5. Flexible valve in pulsatile flow The following example models the mechanics of a flexible valve immersed in a pulsatile flow. The valve opens and closes according to the flow rate in the channel. The objective of this test is to validate the model for a later application of the FSI algorithm proposed in real heart geometry. We realize that the heart solid mechanics is an extremely complex problem. The simplifications made in this test will allow us to evaluate the feasibility of applying the new algorithm to a more realistic approach. From the FSI point of view, the complexity of the problem lays on the fact that heart tissue and blood flow have similar densities and therefore overcoming the difficulties of the added-mass effects is important step for a successful FSI analysis of a real heart valve. For the present analysis, the fluid and solid physical properties resemble the properties of the real problem. The fluid density is ρf = 1000 kg/m3 and, although the blood exhibits non-Newtonian behavior, it is modeled here as a Newtonian flow with density μf = 0.01 kg/(ms). We used an elastic solid to model the valve with ρs = 1000 kg/m3 , E = 1.0 × 106 N/m2 and ν = 0.45. For this example, the geometry is shown in Figure 4.20. Figure 4.21 shows the surface mesh used in the computation together with a close up of the heart valve. The radius of the channel is 0.02 m which compares well with the average radius of the aorta of 0.012 m. The average thickness of the left ventricle is 0.0011 m and the tricuspid valve thickness is 0.0005 m both in the order of our approximation which is 0.001 m. The normal heart cycle at rest is 0.8 seconds which in our case was represented by a cycle of approximate 0.4 seconds. At the peak of the heartbeat the velocity in the aorta for a person at rest may reach 0.75 m/s, about seven times larger than the one proposed in our experiment but close enough for a proof of concept case study.

ALE and Fluid–Structure Interaction

free slip 0.02 0.02

inflow

valve

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Figure 4.20. Geometry description for the flexible valve problem. Dimensions in meters

ti 0.0000(H

Figure 4.21. Surface mesh and valve close up

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Figure 4.22. Velocity boundary condition

The pulsatile flow is generated with the function: v (t) = (0.5 (1 − cos (πt/0.2)))2 exp (2 (0.5 + (1 − sin (πt/0.2)))) / 64.104.

(4.66)

The problem was run for four complete cycles where the valve opens and closes once for each cycle. Owing to the large deformations of the valve, several local as well as global remeshing steps take place. The results are presented in Figures 4.23–4.25. First six different configurations of the valve are shown in Figure 4.23. The first frame represents the time of zero displacement and the last frame represents the point of maximum deflection. Figure 4.24 shows a cut plane where the velocity vectors as well as the velocity modulus were projected. In Figure 4.25 the projection of the pressure on the cut planes is shown.

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Figure 4.23. Valve deflection for time steps t=0, 0.016, 0.2, 0.22, 0.24, and 0.28

Figure 4.24. Cut plane showing velocity vectors for time steps t=0, 0.016, 0.2, 0.22, 0.24, and 0.28

Figure 4.25. Cut plane with pressure iso-contours for time steps t=0, 0.016, 0.2, 0.22, 0.24, and 0.28

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Figure 4.26. Total force transferred from the fluid to the valve as a function of time for the four complete cycles

4.8. Conclusions One of the important aims of the formulation presented is that it shows clearly how to segregate the pressure in the monolithic scheme in order to solve the FSI problems with a staggered algorithm. Based on the pressure segregation scheme, we have also proposed an interface Laplace matrix that gives excellent convergence rates for the totality of the examples performed, even in those examples where the added mass effect is important. The pressure segregation method proposed for the solution of FSI problems with special emphasis in added-mass effects has shown an excellent behavior with promising possibilities in the field of biomedical applications. The method was extended to strongly couple partitioned schemes with the same excellent results. This allows us to conclude that a correct understanding of the pressure segregation is the key issue to solve any FSI problem with a partitioned or a coupled scheme.

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The correct understanding of the pressure segregation in the monolithic scheme allows us to perfectly predict the cases where staggered methods will present convergence problems. We can conclude that the added-mass effect is not the only problem of equal densities in the solid and the fluid, the nonconvergence problem is also present where the solid stiffness is low (soft materials), which is independent of the density rate. Moreover, a serious problem of staggered schemes, that is the divergence when the time step decreases, is also solved using the new interface Laplace matrix. The examples have shown that the partitioned approach preserves the convergence rate of each separate solver (fluid and solid), namely if both have quadratic approximation in space, this rate is maintained after the coupling. 4.9. Acknowledgments This work was partially supported by projects SEDUREC and XPRES of the “Ministerio de Educación y Ciencia” of Spain. 4.10. Bibliography [BAD 08a] B ADIA S., N OBILE F., V ERGARA C., “Fluid-structure partitioned procedures based on robin transmission conditions”, Journal of Computational Physics, vol. 227, p. 7027–7051, 2008. [BAD 08b] B ADIA S., Q UAINI A., Q UARTERONI A., “Modular vs. nonmodular preconditioners for fluid-structure systems with large addedmass effect”, Computational Methods in Applied Mechanical Engineering, 2008. [CAU 05] C AUSIN P., G ERBEAU J.-F., N OBILE F., “Added-mass effect in the design of partitioned algorithms for fluid-structure problems”, Computational Methods in Applied Mechanical Engineering, vol. 194, p. 42– 44, 4506–4527, 2005. [CHO 68] C HORIN A., T EMAM R., “Numerical solution of the NavierStokes equations”, Mathematics in Computation, vol. 22, p. 745–762, 1968.

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[COD 01] C ODINA R., “Pressure stability in fractional step finite element methods for incompressible flows”, Journal of Computational Physics, vol. 170, p. 112–140, 2001. [DET 06] D ETTMER W., P ERIC D., “A computational framework for fluidstructure interaction: Finite element formulation and applications”, Computational Methods in Applied Mechanical Engineering, vol. 195, p. 5754–5779, 2006. [E.O 06] E. O NATE J. G ARCIA S. I., P IN F. D., “FIC formulations for finite element analysis of incompressible flows. Eulerian, ALE and Lagrangian approaches”, Computational Methods in Applied Mechanical Engineering, vol. 195, p. 3001–3037, 2006. [FAR 98] FARHAT C., L ESOINNE M., T ALLEC. P. L., “Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity”, Computational Methods in Applied Mechanical Engineering, vol. 157, p. 95–114, 1998. [FEL 80] F ELIPPA C., PARK K., “Staggered transient analysis procedures for coupled-field mechanical systems: formulation”, Computational Methods in Applied Mechanical Engineering, vol. 24, p. 61–111, 1980. [FEL 01] F ELIPPA C., K.C., FARHAT C., “Partitioned analysis of coupled mechanical systems”, Computational Methods in Applied Mechanical Engineering, vol. 190, p. 3247–3270, 2001. [FER 07] F ERNANDEZ M., G ERBEAU J., G RANDMONT C., “A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid”, International Journal of Numerical Methods in Engineering, vol. 69, p. 794–821, 2007. [FOR 07] F ORSTER C., WALL W., R AMM E., “Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows”, Computational Methods in Applied Mechanical Engineering, vol. 196, p. 1278–1293, 2007. [GER 03] G ERBEAU J., V IDRASCU M., “A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows”, Mathematical Model and Numerical Analysis., vol. 37, p. 631– 648, 2003. [IDE 06] I DELSOHN S. R., O NATE E., P IN F. D., C ALVO N., “Fluidstructure interaction using the particle finite element method”, Computational Methods in Applied Mechanical Engineering, vol. 195, p. 2100– 2123, 2006.

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[LET 01] L E T ALLEC P., M OURO J., “Fluid structure interaction with large structural displacements”, Computational Methods in Applied Mechanical Engineering, vol. 190, p. 3039–67, 2001. [LOH 95] L OHNER R., YANG C., C EBRAL J., B AUM J., L UO H., P ELESSONE D., C. C., “Fluid-structure interaction using a loose coupling algorithm and adaptive unstructured grids”, Vol. 95, AIAA Journal, 1995. [NEU 06] N EUMANN M., T IYYAGURA S., WALL W., R AMM E., “Robustness and efficiency aspects for computational fluid structure interaction”, Computational Science and High Performance Computing II, vol. 91, E. Krause, et al. (Eds.) Springer 2006. [ONA 04] O NATE E., “Possibilities of finite calculus in computational mechanics”, International Journal of Numerical Methods in Engineering, vol. 60, p. 255–281, 2004. [PIN 07] P IN F. D., I DELSOHN S. R., O NATE E., AUBRY R., “The ALE/lagrangian particle finite element method: A new approach to computation of free-surface flows and fluid-object interactions”, Computers and Fluids,, vol. 36, p. 27–38, 2007. [PIP 01] P IPERNO S., FARHAT C., “Partitioned procedures for the transient solution of coupled aeroelastic problems- Part II”, Computational Methods in Applied Mechanical Engineering, vol. 190, p. 3147–3170, 2001. [RUG 00] R UGONYI S., B ATHE K., “On the analysis of fully-coupled fluid flows with structural interactions—a coupling and condensation procedure”, International Journal of Computation of Civil Structural Engineering, vol. 1, p. 29–41, 2000. [RUG 01] R UGONYI S., B ATHE K., “On finite element analysis of fluid flows fully coupled with structural interactions”, Computational Methods in Applied Mechanical Engineering, vol. 2, p. 195–212, 2001. [SOU 00] S OULI M., O UAHSINE A., L EWIN L., “Arbitrary LagrangianEulerian formulation for fluid-structure interaction problems”, Computational Methods in Applied Mechanical Engineering, vol. 190, p. 659675, 2000. [STI 04] S TIJNEN J. M. A., DE H ART J., B OVENDEERD P. H. M., VAN DE V OSSE F. N., “Evaluation of a ficticious domain mehtod for predicting the dynamic response of mechanical heart valves”, Journal of Fluids and Structures, vol. 19, p. 835–50, 2004. [TEM 69] T EMAM R., “Sur l’approximation de la solution des _ équations de Navier-Stokes par la méthode des pas fractionaires (I)”, Archive of Rational Mechine Analysis, vol. 32, p. 135–153, 1969.

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[TEZ 92] T EZDUYAR T., S. M ITTAL S.E. R AY R. S., “Incompressible flow computations with stabilized bilinear and linear equal order interpolation velocity-pressure elements”, Computational Methods in Applied Mechanical Engineering, vol. 95, 1992. [TUR 98] T UREK S., Efficient Solvers for Incompressible Flow Problems: An Algorithmic Approach in View of Computational Aspects, SpringerVerlag, Berlin, 1998. [WAL 05] WALHORN E., K OLKE A., H UBNER B., D INKLER . D., “Fluidstructure coupling within a monolithic model involving free surface flows”, Computers and Structures, vol. 83, p. 2100–2111, 2005.

Chapter 5

Multidomain Finite Element Computations: Application to Multiphasic Problems

5.1. Introduction As the numerical methods and computing power have become more and more efficient, it is now possible to develop a multiphysics and multiscaling approach to study flow motion of complex fluids (that means fluid combining several phases, liquid, solid, elastic, etc.) This approach addresses industrial applications involving strong coupling between liquid–solid, gas–solid or liquid–liquid: the motion and orientation of long fiber suspensions having a polymer matrix for the production of composites; the dispersion and fragmentation of aggregates of black carbon in a polymer matrix (for the dispersion and the formation of conductive network at nanoscale); filling molds with complex fluids by taking into account the change phase of

Chapter written by Thierry C OUPEZ, Hugues D IGONNET, Elie H ACHEM, Patrice L AURE, Luisa S ILVA, Rudy VALETTE.

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components and the presence of flexible inserts. In this chapter, the proposed method aims to present a multidomain approach able to deal with the full complexity of processes and used materials. The main idea is to retain the use of monolithic formulation and coupling it some additional features that could allow a better and accurate resolution. The principle is to work on a mesh that includes all components of a multi-material flow: fluid and solid particles, fluid and elastic inserts, immiscible fluids. A multidomain formulation is introduced as follows: the interfaces between different domains are known implicitly through the values of a characteristic function defined on the whole computational domain and a specific “solver” is associated with each domain. Then, the resolution is made on an Eulerian fixed mesh. Moreover, the different domains are deformed or move under the action of velocity field or stress. These domain modifications will be made either by a Lagrangian or particle method for non-deformable bodies or by transporting the characteristic function. These numerical methods are developed in the C++ object library for finite elements (the CIMlib). In the following, we present our approach based on level set function to describe the interfaces between different phases. Most results presented here come from thesis of Bigot [BIG 01], Digonnet [DIG 01], Gruau [GRU 04], Bruchon [BRU 04], Silva [SIL 04], Basset [BAS 06], Megally [MEG 05], Leboeuf [LEB 07], Beaume [BEA 08], Hachem [HAC 09b], Ville [VIL 09], and François [FRA 09]. 5.1.1. A classification of multidomain approaches There are two main methods for solving fluid/solid or multiphase problems. Two coupled subproblems exist in this approach: the computation of fluid flow field on a moving domain (that is to say an Eulerian computation of velocity field); the displacement of solid bodies as well as their deformations

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(that is to say a Lagrangian computation of velocity field). The two distinct ways to address these problems are: 1. Only the fluid domain is. This method is based on an “accurate” description of fluid domain [HU 96]. Once the fluid velocity is computed, the stresses on the solid bodies are calculated in order to get their deformations and displacements. The fluid domain is then meshed by using, for example, Lagrangian or ALE formulation before performing computation of the next time step. Therefore, the remeshing of the interface between fluid and solid bodies is often done. This method has mainly two disadvantages, which give rise to an increase of computational cost: first, the mesh is moved at each time step and, secondly, the geometry can be complex and thus prevents the use of regular meshes. 2. The whole domain is meshed. In this approach, the weak formulation is extended from the fluid domain to the whole domain, by defining the velocity field to the solid domain. A fluid problem is then solved on all the fictitious fluid/solid domain. Glowinski et al. [GLO 01] were the first to use this technique for non-deformable solid. In this way, the rigidity condition is no more explicit but it has to be added in the weak formulation. The interest of fictitious domain methods is that the computation of Eulerian velocity field is made on the whole domain, which has a simple and fixed geometry. The mesh is fixed and is only determined once. Also note that in this latter approach the interface between solid and fluid is only defined by zero isosurface of a level set function and may intersect the element arbitrary. This interface description is rather close to that coming from the theory of mixtures and the concept of diffuse interface [AND 98, ISH 75]. Finally, the numerical methodology will be driven by how to solve the following problems: – the description of solid domain, – the computation of velocity field and stress, – the displacement and deformation of the solid domain.

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The chosen methodology is presented and illustrated comprehensively in the sequel. The performance of our approach is explained on elementary examples. More complex 3D examples coming from industrial applications can be found in the thesis mentioned in the introduction.

5.1.2. Notations The numerical method presented in next sections is based on the use of stable mixed formulation, which consists of continuous piecewise linear functions enriched with a bubble function for the velocity and piecewise linear functions for the pressure. In this section, a brief presentation of various notations used in our mixed finite element method is presented. Additional information can be found in the book by Brezzi and Fortin [BRE 91], which is the standard Reference for mixed finite element methods. A brief history on residual-based stabilization methods can be found in Brezzi et al. [BRE 96], the book by Donea and Huerta [DON 03], all the articles by Hughes et al. [BRO 82, HUG 95, BRE 97, HUG 06] on multiscale methods and SUPG/PSPG methods by Tezduyar [TEZ 92]. The unusual stabilized finite element method was introduced by Franca and Farhat in [FRA 95]. Codina and co-workers introduced lately recent developments of residual based stabilization methods using orthogonal subscales and time-dependent subscales [COD 00a, COD 00b, COD 01, COD 07]. 5.1.2.1. Functional and discrete spaces In summary, the weak formulation is obtained by multiplying the equation by appropriate trial functions w and integrating the equations on the computational domain Ω [ZIE 00]. In the sequel, the velocity field are defined on Sobolev space d  1 H (Ω) (d being the space dimension), the pressure in L2 (Ω) the Lebesgue space of square summable functions on Ω. The trial functions w belong to a subspace H01 (Ω) for functions

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which vanish on boundaries where there are Dirichlet boundary conditions. That reads    L2 (Ω) = q, Ω q 2 dΩ < ∞   d  H1 (Ω) = q ∈ L2 (Ω), ∇q ∈ L2 (Ω)   H01 (Ω) = q ∈ H1 (Ω), q = 0 on ∂Ω .

(5.1)

The standard scalar product in L2 (Ω) is  (f1 , f2 )Ω =

f1 f2 dΩ.

(5.2)

Ω

The Galerkin approximation consists in decomposing the domain Ω into Nel elements K such that they cover the domain and there are either disjoint or share a complete edge (or face in 3D). Using this partition Ωh , the above-defined functional spaces are approached by finite dimensional spaces spanned by continuous piecewise polynomials such that Vh =

Qh

  uh | uh ∈ C 0 (Ω)d , uh|K ∈ P 1 (K)d , ∀K ∈ Ωh

(5.3)   = ph | ph ∈ C 0 (Ω)d , ph|K ∈ P 1 (K)d , ∀K ∈ Ωh ,

where the subscript h means that they are defined on the discrete space Ωh and P1 (K) is the space of polynomials of degree 1 on element K . The above discrete spaces are restricted to linear piecewise functions, because the aim of this chapter is to point out the advantages of using linear approximations (P1 finite elements) for the accuracy and the computational cost, especially for 3D industrial applications. On this mesh, there are two types of approximation: a P1 approximation if the fields are known on the node of each element; a P0 approximation if the fields are constant on each element K .

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5.1.2.2. Bubble space It is well known that the combination of P1 −P1 approximation for the velocity and the pressure does not lead to a stable discretization since it fails to satisfy the Babuska–Brezzi condition. Many methods may be distinguished to avoid the instabilities in advection-dominated regime and the respect of the velocity–pressure compatibility condition. A very popular method was first proposed by Arnold, Brezzi and Fortin [ARN 84] for the Stokes problem. It was suggested to enrich the functional spaces with space of bubble functions known as MINI-element (see Figure 5.1). However, the bubble can take different shapes for the diffusive dominated regime and for the advection-dominated flow regime. For example, it was shown in [BRE 94, RUS 96] that upwind bubbles could be used to reproduce the SUPG stabilization. Finally, all these approaches can be summarized in the multiscale approach [HUG 98]. The stabilizing schemes from a variational multiscale point of view, consist of enriching the velocity and the pressure spaces by a space of bubbles that cures the spurious oscillations in the convection-dominated regime as well as the pressure instability. As shown in section 5.3.3, these fine scales are represented by different bubbles functions. The enrichment of discrete functional spaces is given by: Vh = Vh ⊕ V  and Qh = Qh ⊕ Q .

Velocity interpolation

Pressure interpolation

Figure 5.1. MINI-element P1+ /P1

(5.4)

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In summary, two kinds of bubble functions are used in the following. The first one bK associated with MINI-element P1+ /P1 is defined on each element K as follows: bK = 0 on ∂K ; bK (GK ) = 1, where GK is the barycenter of K and bK satisfies the orthogonality condition  ∂xk Ni ∂xl bK dΩ = 0 ∀k, l; ∀Ni ∈ P1 (K), (5.5) K

where Ni is the interpolation function associated with node i. To get the same order of integration for the bubble function as the other interpolation functions, bK is built so that it is linear on each sub-tetrahedra (or sub-triangles) of K having GK as a point and can be defined by bK = D Ni

on Ki ,

(5.6)

where D is the number of node of K , and Ki is the ith subtetrahedra. The second bubble function buK is added to weighting function space in order to stabilize convection dominant terms and it satisfies the property uh · ∇buK = −1 on K ;

buK = 0

on ∂K − ,

(5.7)

where K − is the outlet face of K defined by {x ∈ ∂K, u(x) · n > 0}. Finally, the previous formalism (5.4) for the Stokes equation and the MINI-element P1+ /P1 [PIE 95] (see Figure 5.1) gives Q = 0 and  V  = v

 3  3  ∈ C 0 (Ω) ; v  = 0 on ∂K and v  |K i ∈ P1 (Ki ) , (5.8) where the bubble function for the velocity is linear on each sub-triangle Ki of element K and vanishes at the boundary of K . 

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ALE and Fluid–Structure Interaction A V

Vy

Vx h

Figure 5.2. The mesh size in the streamline direction

5.1.2.3. Element length definition The characteristic element length h has a significant impact on the stabilizing parameters introduced in later sections. Therefore, the choice of the mesh cell is not obvious specifically in the presence of distorted mesh or highly elongated elements. It could be simply the diameter of the mesh cell or could be chosen as the mesh cell in the direction of the convection for convection–diffusion equations. The latter choice is the most recommended in the literature [FRA 01, TEZ 00], h = 2u

/D

0−1 |u · ∇Ni |

,

(5.9)

i

where D is the number of node of each element, Ni is the interpolation function associated with node i and u the local velocity in element K . Therefore, for non-uniform mesh, the mesh size h is a space function as its value depends on the element on which it is evaluated. More details about the determination of the element length taking into account anisotropy of the mesh are provided in [PRI 08]. 5.2. Characterization of different phases Let us assume for simplicity that we have two phases: fluid and solid. Consequently, the computational domain Ω is split into two subdomains, Ωf and Ωs associated respectively to fluid and solid areas. The weak formulation involves integrals over the entire computational domain, and a characteristic function is introduced in order to distinguish the two domains

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in the calculation of these integrals: Ij (x, t) = 1 if x ∈ Ωj and 0 if x ∈ / Ωj .

(5.10)

We only consider in the following the characteristic function I associated with solid domain or one of two phases (if there are two fluids) to simplify the notation. The characteristic function used in the sequel is determined from a level set or signed distance. 5.2.1. The level set or signed distance functions The characteristic function defined by relation (5.10) is approached by smooth function. The principle is to define an interface function in the whole domain from which the zero isosurface is the interface that we want to describe. This function is positive inside one subdomain and negative elsewhere. Therefore, a P1 interpolation of this function gives an accurate description of the interface. If the interface of solid domain is defined by a curve Γ, the level set function can be defined from a signed distance as  α(x) = ||x − Γ|| on Ωs (5.11) α(x) = −||x − Γ|| on Ω\Ωs . The level set function α has nice properties which allows us to improve the numerical methods moving this function: – if the interface Γ is smooth enough to ensure the definition of an unique normal vector then |∇α| = 1 ; – if n(y) is the normal vector of Γ at y then ∇α(y) = n(y). Indeed, for asmall enough s, there is α(y +sn(y)) = s and then ∇α(y), n(y) = 1. Therefore, a level set function has a welldefined gradient which gives also the normal of the interface.

A P1 approximation of a characteristic function I is easily obtained by looking at the sign of αh (xi ) at the mesh nodes xi :  1 if αh (xi ) > 0 h I (xi ) = (5.12) 0 if αh (xi ) < 0.

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However, a more regular transition is obtained by taking the following function: ⎧ = 1 if α(x) > e ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ α 1 Ie (x, t) = (5.13) if − e < α(x) < e = + ⎪ 2 2 e ⎪ ⎪ ⎪ ⎪ ⎩ = 0 if α(x) < −e, where e is the transition thickness which will depend on the mesh size. Finally, a P0 approximation of I on each element K can also be computed from the distance function α as follows: Ih |K =

+ αK , |α|K

(5.14)

+ is the sum of all positive α evaluated at nodes of where αK element K , whereas |α|K is the sum of absolute values of α.

Also note that the level set function allows to calculate the normal vector n and the curvature κ of the interface Γ:   ∇α ∇α , n = ; κ = ∇ . n = ∇ . (5.15) |∇α| |∇α| where a division by |∇α| is made in order to ensure an unitary vector n. Figure 5.3 shows the isovalues of theses functions for an interface Γ, which is a disk of radius 0.25 and center (1.0, 0.5):  α = ± |(0.25)2 − (x − 1)2 − (y − 0.5)2 |. In conclusion, we obtain a “fuzz” interface between solid and fluid domains. Even with a coarse mesh, the zero level gives a rather accurate description of interface. However, the

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Figure 5.3. The level set function α and the characteristic function Ie for a disc of radius = 0.25: isoline α = 0 (black) and. isovalues Ie for various mesh size h and thickness e: (a) h = 0.05, e = 0.01; (b) h = 0.05, e = 0.05; (c) h = 0.01, e = 0.01

method used to get the characteristic function from the level set function has an influence on the size of the transition zone between the two phases (see Figure 5.3). This characteristic function is important as it intervenes in the mixing relations which give the values of viscosity and density on the whole domain. As shown in Figure 5.3, the thickness e of transition area has to be consistent with the size h of elements around the interface. Usually this transition zone must contain at least two elements, otherwise the isolines of characteristic function will be deformed. 5.2.2. Mesh adaptation For accurate computations for a multiphasic problems, the mesh has to be refined around the interface. Indeed, adapting

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the mesh to the physical behaviors and studied phenomena is a means to improve the accuracy of numerical results. The tools developed in our team can refine the mesh around an interface Γ with anisotropic elements so as to control their numbers and their orientations [GRU 04, BAS 06, BEA 08]. To do that the level set function described in previous section is coupled to an anisotropic mesh adaptation [COU 00, GRU 05]. The mesh becomes locally refined around the zero isovalue of the level set function which enables to sharply define the interface and to save a great number of elements compared to classical isotropic refinement. This anisotropic adaptation is performed by constructing a metric map that allows the mesh size to be imposed in the direction of the distance function gradient. It is based on a topological optimization technique that, by considering the quality of the elements, improves the mesh topology. A brief review on the concept of metric map is presented. The main idea is to build a metric M (a symmetric positivedefinite matrix) that allows the creation of meshes with extremely anisotropic elements stretched along the interface. Therefore, if the metric M can be regarded as a tensor whose eigenvalues are related to the mesh sizes, and whose eigenvectors define the directions for which these sizes are applied. For example, the metric M :   100 0 M= 0 400 gives in a cartesian coordinates a mesh size h of 0.01 in the x direction and 0.05 in the y -direction. Therefore, elements align along the interface are obtained by imposing small element sizes along the distance function gradient ∇α , and keeping the same background size in the orthogonal direction ∇αT . In other words, the proposed metric takes the following form: M = m2 (∇α ⊗ ∇αT ) + 2 E,

(5.16)

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where E is the identity tensor, and m are two positive, real parameters. This

simply means that this metric leads to a mesh sizes 1/ m2 |∇α|2 + 2 and 1/ 2 in the directions ∇α and ∇αT , respectively. Since only the interface separating both subdomains is the region of interest, this pushes us to build a general metric map that is equal to an isotropic metric in the far-interface region and equal to the previously constructed metric (5.16) in the vicinity of the interface. Accordingly, the general metric map takes the following form: ⎧ e ⎪ 2 E if |α| > ⎪ ⎪ ⎪ 2 ⎨ 0 / M= (5.17)  2 ⎪ ∇α ⊗ ∇αT e N ⎪ 2 2 ⎪ − + E if |α| < , ⎪ ⎩ e |∇α|2 2 where N is the number of elements required in a specified thickness e and a parameter defining the size of the background mesh. Figure 5.4 shows an example of isotropic and anisotropic refinement around an interface. In particular, in Figure 5.4(b) it is shown that only additional nodes are locally added in refinement region, whereas the rest of domain has maintained

Figure 5.4. Examples of mesh adaptation: (a) Isotropic mesh with h = 0.01 for |α| < 0.05; (b) anisotropic mesh for e = 0.1 and N = 10

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Figure 5.5. Iterations between the metric computation and the mesh generator

at the same background size. It is an important feature that keeps low the computational time devoted to the grid generation. The proposed mesh generation algorithm works well for 2D or 3D geometries and can easily handle arbitrary geometries. A numerical example is given in Figure 5.5 to illustrate the effectiveness of the proposed algorithm. We consider two parallelepipeds in a 3D enclosure. Starting from a coarse mesh, we generate the multidomain metric and we adapt the mesh to this metric. After several iterations, we can clearly see from Figure 5.5 that the interface between these objects and the surroundings is well adapted.

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5.2.3. Displacement of the level set function The level set function is defined as the signed distance to the interface and is used to initialize the material interface in the problem, since its position is given by the zero isosurface of α. Therefore, the advection of this isosurface will describe the time evolution of the interface. Finally, all the isolines can be convected with the velocity field, although the position of the interface is always given by the isovalue zero. Also note that the convected function α does no more satisfy the regularity properties of the initial distance function. Assuming that the velocity field u is known and defined in the whole domain Ω, motion of the interface (or any level of α) is given by ⎧ ∂α ⎪ ⎪ + u.∇α = 0 ⎪ ⎪ ⎨ ∂t (5.18) α(x, 0) = α0 in Ω ⎪ ⎪ ⎪ ⎪ ⎩ α(x, t) = g on Γ− , where Γ− is the inlet boundary. The evolution equation (5.18) takes the signed distance as the initial value. However the solution of (5.18) is no longer a signed distance even if the zero isosurface is correctly convected and remains able to split the space into two subdomains. Since, the gradient of advected α may become very steep (especially in problems with free surfaces) and numerical instabilities may occur. Classically, it is necessary to reinitialize the distance function [BAS 06]. 5.2.3.1. The reinitialization As the displacement by equation (5.18) does not ensure the conservation of level set gradient, the most common method used to overcome this problem is to solve a Hamilton–Jacobi equation [PEN 99] in order to get a more regular level set

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function with the same zero isosurface of α(x; t). Let us introduce a virtual time τ , we search β(x; τ ) that has the same zero values: ⎧ ⎨ ∂β + s(|∇β| − 1) = 0 ∂τ (5.19) ⎩ β(x, τ = 0) = α(x, t), where the function s is the sign of β (which initially equals α) so that a new distance function is built from the invariant region α = 0 for which s = 0. In practice, the Hamilton–Jacobi equation (5.19) is solved as soon as the level set becomes too sharp ( |∇α| is far from 1). Therefore, this constraint adds an extra step in the numerical scheme for numerical computation involving free surfaces or interfaces. This additional step can be avoided by remarking that equation (5.19) is also an advection equation. Indeed, by changing the gradient term, the equation reads ⎧ → ⎨ ∂β + − U .∇β = s ∂τ ⎩ β(x, τ = 0) = α(x, t),

(5.20)

→ − where a convective velocity , U = s∇β/|∇β|, is introduced.

5.2.3.2. Convective reinitialization The main idea in this section is to explain how the reinitialization step can be avoided [VIL 09]. For this purpose, the parameter λ is introduced as follows: λ =

∂τ . ∂t

(5.21)

Let us consider the time marching schemes of physical time step Δt associated with the pseudo time step Δτ evaluated as: |U |Δτ ∼ h.

Multidomain Finite Element Computations

Since |U | ∼ 1, then λ will be chosen to be equal to sequel, and thus ∂β ∂β =λ ∂t ∂τ

h Δt

237

in the

(5.22)

and by rewriting equation (5.20), ∂β + λs(|∇β| − 1) = 0. ∂t

(5.23)

In an Eulerian framework for a moving domain, the partial time derivation, ∂t β is simply changed in a total derivative, ∂t β +u.∇β . In this way, the reinitialization equation is written in a moving framework driven by the fluid velocity. Finally, using again the level set function α as the main variable, equation (5.18) is replaced by a “convected reinitialization” equation ⎧ ⎨ ∂α + u.∇α + λs(|∇α| − 1) = s ∂t ⎩ α(x, 0) = α0 ,

(5.24)

where s = s(α) and is equal to 1, −1, and 0, respectively, according to sign of α, in order to recover equation (5.18) in the neighborhood of 0. 5.2.3.3. The local distance function The role of the convective velocity can be limited to a neighbourhood of the interface. In this way, numerical instabilities are reduced as only a few isosurface around the interface are moved. Then in the following, the classical level set function is replaced a function which is both self-determining and self-truncating. So, a sinusoidal filter is used so that the new function coincides with the level set function only near the

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y

2 E /π x E

Le

ve

ls

et

Sinusoidal level set

Figure 5.6. Graphical comparison between the signed distance function and sinusoidal distance function. The thickness E allows us to connect the sinusoidal function having an unitary tangent in 0 and a zero tangent in E

interface (see Figure 5.6). It reads π

2E sin sx−Γ α(x) = if  x − Γ ≤ E π 2E (5.25) 2E if  x − Γ > E, α(x) = s π where s = 1, −1, 0 in Ωs , Ωf and Γ, respectively, and the scalar E depends on the mesh size. With this definition, the level set gradient ∇ · α varies continuously from 0 to 1:  π 2 α . |∇α| = 1 − (5.26) 2E Finally, the evolution of the zero isoline is given by the following transport equation which has the property of keeping a unitary gradient near the interface: / 0 ⎧ 

⎪ ⎨ ∂α + u · ∇α + λs |∇α| − 1 − π α 2 = 0 (5.27) ∂t 2E ⎪ ⎩ α(t = 0, x) = α0 (x), where λ depends on the mesh size and time step.

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5.2.3.4. Parameters of reinitialized and truncated convective equations Equation (5.27) is strongly nonlinear and the metric con→ − vective velocity U = s∇α/|∇α| can be introduced in order to be able to linearize and solve it. Moreover, indications on the values of the various parameters s, E, λ which depend on the mesh size (assuming an uniform and isotropic mesh) can be given. The linearized equation is written ⎧  π 2 ⎪ → → − ⎨ ∂α + (u + λ− U ).∇α = λs | U | 1 − α ∂t 2E (5.28) ⎪ ⎩ α(x, t = 0) = α0 (x), where a new parameter intervenes. In fact, the sign s plays an important role in this equation. As the interface is not known explicitly. The function s has to vanish on some elements around zero isosurface. In the following, we use the following relations: s (α) =

α si |α| > and s (α) = 0 si |α| ≤ , |α|

(5.29)

and ∼ n1 h with n1 = 1 or 2. In the same way, the metric → − convective velocity U is only defined for |∇α| = 0. Therefore, its expression is regularized so that it tends to zero as the level gradient vanishes: − → U = s

∇α , + (1 − )|∇α|

(5.30)

→ − and | U | is added on the rhs of (5.28) in order to take into account of this regularization.

In equation (5.20), the fictitious time τ is related to the met→ − ric convective velocity U which has an unitary norm. From → − a pratical point of view, we take | U |Δτ = Δτ = h, the fictitious time remaining of order h. Then h (or a fraction) is chosen as fictitious time and it becomes λ = dτ /dt ∼ h/Δt.

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For the last parameter E , a multiple of h is chosen : E ∼ n2 h with n2 = 10 or 20. In summary, equation (5.28) behaves as follows: – if |α(x)| ≤ , α is only convected; – if ≤ |α(x)| ≤ E , the function is simultaneously reinitialized and convected; – if E ≤ |α(x)| α tends in time to a constant value corresponding to the minimal or maximal values of the sinusoidal level set : ±2E/π . The finite element resolution of this equation is discussed in the next section: the time discretization uses an implicit scheme whereas the spatial discretization requires a stabilization method. In summary, this method (called Streamline Upwind Petrov Galerkin (SUPG) or Residual-Free Bubbles (RFB)) is built by adding a diffusion term in the direction of the flow motion. 5.3. Stabilized finite element formulations It is known that the Galerkin approximation of the Navier– Stokes equations may fail because of two reasons. First, for convection dominated flows, for which it appears layers where the solution and its gradient exhibit rapid variation, the classical Galerkin approach leads to oscillations of the solution in theses layer regions which can spread quickly and pollute the entire solution domain. Secondly, the use of inappropriate combinations of interpolation functions to represent the velocity and pressure fields [FRA 98, NES 99] yields unstable schemes. These associated instabilities are usually circumvented by addition of stabilization terms to the Galerkin formulation. These stabilization terms can be obtained by using multiscale formalism [HUG 98, HAC 09b]. In this way, one gets the stabilized expression for the Navier–Stokes equations (5.55) in which stabilizing parameters are introduced. The determination of suitable stabilization parameters has attracted

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a significant amount of attention and research. The values of theses parameters are related to the shape of local bubbles and are related to various methods. Therefore, the selection of the optimal bubble function reproduces the appropriate choice of stability parameters [BRE 94, CAN 96, CAN 96]. As we deal with anisotropic mesh, we propose to compute the stabilizing parameter by a static condensation of bubble function on each element. In this way, we overcome the choice of a suitable definition of the characteristic length h for nonisotropic element [TEZ 00]. Finally, we recall the SUPG weak formulation for convection equation (5.18) and its link with multiscale approach, which is explained for the Navier–Stokes equations. 5.3.1. Multidomain Navier–Stokes equations For two-phase flow with different densities and viscosities, the Navier–Stokes equations of unsteady incompressible Newtonian flow are written as follows: ⎧   ∂u ⎪ ⎪ + u . ∇u = ρ g + ∇ . σ ρ ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ∇ . u = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σ = −pI + 2η ( ˙ u) ⎪ ⎪ ⎪ [[u]]Γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [[σ.n]]Γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u

= 0 = 0

on external boundary Γext , (5.31) where u is the velocity, p the pressure, η the dynamic viscosity, and ρ the density. In our monolithic approach, physical data as density and viscosity are here non-constant and depend on a phase function α (for example, the level set function defined in = uΓ

ext

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the previous section) For a two-phase flow, the simpler mixing law consists to mean all the densities or viscosities in presence ρ = Ie (α)ρ1 + (E − Ie (α)) ρ2 ; η = Ie (α)η1 + (E − Ie (α)) η2 , (5.32) where the subscript 1 or 2 is related to number associated to each phase. For three-phase flow, two level set functions α1 and α2 are introduced and now the mixing law reads     ρ = Ie (α1 )ρ1 + E − Ie (α1 ) ρ2 Ie (α2 ) + ρ3 E − Ie (α2 )     η = Ie (α1 )η1 + E − Ie (α1 ) η2 Ie (α2 ) + η3 E − Ie (α2 ) . (5.33)

Let us note that the variational formulation is particularly suitable because the jump conditions at the interface Γ is automatically satisfied. The accuracy of this approach will essentially depend on the size e of the transition zone and the choice of the characteristic function, Ie , which is used for the mixing law (5.32).

5.3.2. Weak formulation of incompressible Navier– Stokes equations If there are Dirichlet boundary conditions on external boundaries, the weak form of the system (5.31) consists in d   finding u, p ∈ H1 (Ω) × L2 (Ω) such that   ⎧  Ω + ρ (u · ∇u, w)  Ω + σ(p, u) : ( ˙ w)  Ω ρ (∂t u, w) ⎪ ⎪ ⎪ ⎨

= f, w  ∀w  ∈ H01 (Ω) ⎪ Ω ⎪ ⎪ ⎩ (∇ · u, q)Ω = 0 ∀q ∈ L2 (Ω),

(5.34)

where η and ρ are defined on the whole domain, thanks to the mixing law (5.32).

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The Galerkin discrete problem therefore consists in solving the following mixed problem on the discrete spaces defined in (5.3):  h , qh ) ∈ Vh,0 × Qh Find a pair (uh , ph ) ∈ Vh ×Qh , such that ∀ (w   ⎧  h )Ω + ρ (uh · ∇uh , w  h )Ω + 2η ( ˙ uh ) : ( ˙w h) Ω ρ (∂t uh , w ⎪ ⎪ ⎪ ⎨

, w − (p , ∇ · w  ) = f  h h h Ω ⎪ Ω ⎪ ⎪ ⎩ (∇ · uh , qh )Ω = 0. (5.35)

5.3.3. Stable multiscale Navier–Stokes equations

variational

approach

for

In this section the general equations of time-dependent Navier–Stokes equation are solved. The stabilizing schemes from a variational multiscale point of view are described and presented. The velocity and the pressure spaces are enriched by a space of bubbles that cures the spurious oscillations in the convection-dominated regime as well as the pressure instability. Following [HUG 98, HAC 09b] and decomposition (5.4), we consider an overlapping sum decomposition of the velocity and the pressure fields into resolvable coarse scale and unresolved fine scale u = uh + u and p = ph + p . Likewise, we regard the  =w h + w  same decomposition for the weighting functions w  and q = qh +q . The unresolved fine scales are usually modeled using residual-based terms that are derived consistently. The static condensation involves in substituting the fine-scale solution into the large-scale problem providing additional terms, tuned by a local time-dependent stabilizing parameter, that enhance the stability and accuracy of the standard Galerkin

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formulation for the transient nonlinear Navier–Stokes equations. Thus, the mixed-finite element approximation of problem (5.35) can read  q) ∈ V0 × Q0 Find a pair (u, p) ∈ V × Q, such that: ∀ (w, ⎧   h + w  ) Ω ρ ∂t (uh + u ), (w ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ +ρ (uh + u ) · ∇(uh + u ), (w h + w  ) Ω ⎪ ⎪ ⎪ ⎪ ⎨   + 2η ( ˙ uh + u  ) : ( ˙w h + w  ) Ω ⎪ ⎪

⎪   ⎪    ⎪ , (w ⎪ − (p + p ), ∇ · ( w  + w  ) = f  + w  ) h h h ⎪ Ω ⎪ Ω ⎪ ⎪   ⎪ ⎩ ∇ · (u + u  ), (q + q  ) = 0. h

h

Ω

(5.36) These equations are split into two subproblems by separating the two scales. Integrating by parts within each element, we obtain the so-called coarse-scale problem    ⎧   h Ω + ρ (uh + u  ) · ∇(uh + u  ), w h Ω ρ ∂t (uh + u  ), w ⎪ ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ + 2η ( ˙ uh ) : ( ˙w  h ) Ω − (ph + p ), ∇ · w h Ω ⎨

⎪  ⎪ = f, w h ∀w  h ∈ Vh,0 ⎪ ⎪ ⎪ Ω ⎪ ⎪   ⎩ ∇ · (uh + u  ), qh Ω = 0 ∀qh ∈ Qh,0 (5.37) and the fine-scale problem    ⎧    K + ρ (uh + u  ) · ∇(uh + u  ), w  K ρ ∂t (uh + u  ), w ⎪ ⎪ ⎪ ⎪ ⎪     ⎪ ⎪ + 2η ( ˙ u  ) : ( ˙w   ) K − (ph + p ), ∇ · w  Ω ⎨

⎪  , w ⎪ = ∀w   ∈ V0 f  ⎪ ⎪ ⎪ Ω ⎪ ⎪  ⎩  ∇ · (uh + u  ), q  Ω = 0 ∀q  ∈ Q0 . (5.38) To derive our stabilized formulation, we first solve the finescale problem, defined on the sum of element interiors and written in terms of the time-dependant large-scale variables.

Multidomain Finite Element Computations

245

Then we substitute the fine-scale solution back into the coarse problem (5.37), thereby eliminating the explicit appearance of the fine-scale while still modeling their effects. At this stage, three important remarks have to be made: i) when using linear interpolation functions, the second derivatives vanish as well as all terms involving integrals over the element interior boundaries; ii) as the fine-scale space is assumed to be orthogonal to the finite element space, the crossed viscous terms vanish in (5.37) and (5.38) [COU 96]. 5.3.3.1. The fine scale subproblem Rearranging the terms of equation (5.38) leads to ⎧       Ω + ρ (uh + u  ) · ∇u  , w  Ω ρ ∂t u  , w ⎪ ⎪ ⎪ ⎨       + 2η ( ˙ u  ) : ( ˙w   ) Ω + ∇p , w   Ω = RM , w   Ω ∀w   ∈ V0 ⎪ ⎪ ⎪     ⎩ ∇ · u  , q  Ω = RC , q  Ω ∀q  ∈ Q0 , (5.39) with RM and RC the momentum and continuity residuals, respectively, RM = f − ρ∂t uh − ρ(uh + u ) · ∇uh − ∇ph RC = −∇ · uh .

(5.40)

Here, some assumptions have to be made in order to deal with the time-dependency and the nonlinearity of the momentum equation of the subscale system (5.39): i) the subscales are not tracked in time, therefore, quasistatic subscales are considered here (see [DUB 99] for a justification of this choice); however, the subscale equation remains quasi time-dependent since it is driven by the largescale time-dependent residual; ii) the convective velocity of the nonlinear term may be approximated using only large-scale part so that (uh +u  )·∇(uh + u  ) ≈ uh · ∇(uh + u  ).

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ALE and Fluid–Structure Interaction

Consequently, the fine-scale problem reduces to the following: ⎧         Ω + 2η ( ˙ u  ) : ( ˙w   ) Ω + ∇p , w  Ω ρ uh · ∇u  , w ⎪ ⎪ ⎪ ⎨   = RM , w  Ω ∀w   ∈ V0 ⎪ ⎪ ⎪    ⎩  ∇ · u  , q  Ω = RC , q  Ω ∀q  ∈ Q0 . (5.41) With regard to classical condensation for the Stokes equations [PIE 95, NOR 98, PER 00], two important extensions can be identified. The first one consists of considering the advection terms in equation (5.41) and the second one is that the smallscale pressure is included. These two extensions are essential for simulating high convection-dominated flows. Indeed, it is known from the works of Wall et al. [WAL 00], Tezduyar and Osawa [TEZ 00] that considering the small-scale pressure as an additional variable enables to complete the continuity condition on the small-scale level. It provides additional stability especially when increasing Reynolds number. However, solving the small-scale equation for both the velocity and the pressure is somewhat complicated. Franca and co-workers [FRA 98] proposed a separation technique of the small-scale unknowns. They replaced the small-scale continuity equation by the small-scale pressure Poisson equation (PPE). Since only the effect of the small-scale pressure Poisson equation on the large-scale equation must be retained, Franca and Oliveira [FRA 03] showed that rather than solving this equation, it could be approximated by way of an additional term in the fashion of a stabilizing term as follows: p ≈ τC RC .

(5.42)

In this presentation, we adopt the definition proposed by Codina in [COD 00a] for the stabilizing coefficient, /   01/2  μ 2 c2 uK 2 τC = + , ρ c1 h

(5.43)

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where c1 and c2 are two constants independent of h (h being the characteristic length of the element). Once this stabilizing coefficient τC has been defined, expression (5.42) can be inserted into the large-scale equation (5.37). Then, it remains to deal with the small-scale momentum equation. Codina has shown in [COD 00a] that the small-scale velocity is exclusively driven by the residual of the large-scale momentum equation and not by the residual of the continuity equation. Consequently, in order to eliminate the effects of the smallscale pressure in the small-scale momentum equation, we impose p = 0. Finally, the method can be regarded as a combination of a stable formulation (MINI-element) and a stabilizing strategy. Indeed, the stable formulation, introduced usually to solve the Stokes problem, is applied to the velocity field while the fine-scale pressure is modeled using a stabilizing method.

Now, it remains to solve the small-scale momentum equation. Following Masud and Khurram [MAS 04] and without loss of generality, the fine-scale fields can be expanded using bubble functions on individual elements, u  =



u K bK

and

w =

K ∈Ω h



w 

 K bK ,

(5.44)

K ∈Ω h

where bK represents the bubble functions, u K denotes the  K vector of coefficients for the fine-scale velocity field and w represents the coefficients for the fine scale weighting function. Inserting expressions (5.44) into the fine-scale momentum equation (5.41) yields

 ρ uh · ∇bK u K , bK w 

K ∈Th

=



 K K

 + 2η (b ˙ K u K ) : (b ˙ Kw 

 K ∈Ω h

RM , bK w 



 K K



 K) K

.

(5.45)

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ALE and Fluid–Structure Interaction

Since the bubble functions vanish on element boundaries, the previous expression simplifies into ∀K ∈ Ωh     ρ uh · ∇bK u K , bK w K + 2η (b ˙ K u K ) : (b ˙ Kw  K   = RM , bK w  K K .



 K) K

(5.46)  K out of Taking the constant vector of coefficients u K and w  K , we obtain the integral and exploiting arbitrariness of w u K =

bK   · (RM , bK )K ρ (uh · ∇bK , bK )K + 2η (b ˙ K ) : (b ˙ K) K

∀K ∈ Ωh .

(5.47)

Assuming that the large-scale momentum residual RM is constant, the fine-scale velocity on each element K can read u  |K = τK RM

∀K ∈ Ωh ,

(5.48)

where τK is the stabilization parameter, which has been naturally obtained after the resolution of the fine-scale subproblem,  bK K bK dΩ   τK = ∀K ∈ Ωh . ρ (uh · ∇bK , bK )K + 2η (b ˙ K ) : (b ˙ K) K (5.49) The effect of the bubble is now condensed in this elemental parameter. Obviously, the choice of the bubble functions affects the value of the stability parameter. In expression (5.49), both convection and viscous regime are represented. However, using the same bubble function for the trial function and the weighting function leads to the cancellation of the convection term. Indeed, under the assumption that uh is piecewise constant, the choice of the MINI-element yields: (uh · ∇bK , bK )K = 0

∀K ∈ Ωh .

(5.50)

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249

As pointed out in [BRE 94], a way to recover the convection term is to resort to upwind bubbles. Such a choice enables to reproduce naturally the coefficient of the SUPG stabilization method. This issue has also been highlighted by Masud et al. in [MAS 04], they propose to use different order of interpolation functions for the trial and the weighting functions in the skew part of (5.49). In order to extract the structure of the stability parameter τK , we employ a combination of standard bubble function bK and upwind functions buK in the fine-scale  , field w w   |K = w 

 ∗ K bK

=w 

 K

(bK + buK ) .

(5.51)

  into (5.47) leads to the modified Introducing the modified w form of the stabilization parameter τK , τK

 bK K b∗k dΩ    =  ρ uh · ∇bK , buK K + 2η (b ˙ K ) : (b ˙ K) K

∀K ∈ Ωh .

(5.52) As we use linear interpolations, the upwind part drops out directly in the viscous term. 5.3.3.2. The coarse scale subproblem Let us consider the coarse-scale problem of the expression (5.37) including the assumptions made for the fine-scale fields,   ⎧  h )Ω + (ρuh · ∇uh , w  h )Ω + ρuh · ∇u  , w h Ω ρ (∂t uh , w ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ + 2η ( ˙ uh ) : ( ˙w  h ) Ω − (ph , ∇ · w  h )Ω ⎨

  ⎪ ⎪ − p , ∇ · w  h Ω = f, w ∀w  h ∈ Vh,0 h ⎪ ⎪ ⎪ Ω ⎪ ⎪   ⎩ (∇ · uh , qh )Ω + ∇ · u  , qh Ω = 0 ∀qh ∈ Qh,0 . (5.53) Applying integration by parts to the third terms in the first equation of (5.53) and to the second term in the second equation, then substituting the expressions of both the fine-scale

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ALE and Fluid–Structure Interaction

pressure (5.42) and the fine-scale velocity (5.48), we get ⎧  h )Ω + (ρuh · ∇uh , w  h )Ω ρ (∂t uh , w ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ (τK RM , ρuh ∇w  h )K + 2η ( ˙ uh ) : ( ˙w h) Ω ⎪ − ⎪ ⎪ ⎪ K ∈Ω h ⎪ ⎪ ⎨ − (p , ∇ · w  h )Ω + (τC RC , ∇ · w  h )K h K ∈Ω ⎪ h

⎪ ⎪ ⎪ , w ⎪ = f  ∀w  h ∈ Vh,0 h ⎪ ⎪ Ω ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (∇ ·  u , q ) − (τK RM , ∇qh )K = 0 ∀qh ∈ Qh,0 . ⎪ h h Ω ⎩ K ∈Ω h

(5.54) Instead of evaluating τK using equation (5.52) by integration on the element K , one can choose a constant representative value called τ˜K on each element. This value can be the average of τK on the element, or by a suitable choice of bubble function τK is directly constant. Finally, substituting the residual of the momentum equation and expanding all the additional terms, we obtain a modified coarse-scale equations expressed solely in terms of coarse-scale functions. The new modified problem can now be decomposed into four main terms: the first one is the Galerkin contribution, the second and the third terms take into account the influence of the fine-scale velocity on the finite element components and the last term models the influence of the fine-scale pressure onto the large-scale problem,   ρ (∂t uh + uh .∇uh , w  h )Ω + 2η ( ˙ uh ) : ( ˙w h) Ω

− (ph , ∇.w  h )Ω + (∇.uh , qh )Ω − f, w h Ω % &' ( Galerkin terms (5.55)

+ τ˜K ρ(∂t uh + uh .∇uh ) + ∇ph − f, ρuh ∇w h K ∈Ω h

%

K

&' Upwind stabilization terms

(

Multidomain Finite Element Computations

+





τ˜K ρ(∂t uh + uh .∇uh ) + ∇ph − f, ∇qh

K ∈Ω h

%

&'

251

K

(

Pressure stabilization terms

+

K ∈Ω h

%

τC (∇ · uh , ∇ · w  h )K = 0 &'

∀w  h ∈ Vh,0 ,

∀qh ∈ Qh,0

(

grad-div stabilization term

When compared with the Galerkin method (5.35), the proposed stable formulation involves additional integrals that are evaluated elementwise. These additional terms, obtained by replacing the approximated u  and p into the large-scale equation, represent the effects of the sub-grid scales and they are introduced in a consistent way to the Galerkin formulation. All of these terms enable to overcome the instability of the classical formulation arising in convection dominated flows and to satisfy the inf-sup condition for the velocity and pressure interpolations. Moreover, the last term in equation (5.55) provides additional stability at high Reynolds number [BEH 93]. For the sake of simplicity in the notation and for a better representation of all the additional terms in equation (5.55), the condensation procedure of the small scale into the large scale is masked under these stabilizing parameters. However, from the implementation point of view, the structure of the stabilizing parameters is computed naturally via the elementlevel matrices. There is a large amount of literature concerning the choice and design of the stability parameters. The most common used definition for the transient Navier–Stokes problems and linear elements comes from [TEZ 00, COD 02b, COD 02a], τ˜K ∼ τK = 

2 Δt

2

 +

1 4η h2

2

 +

4|uk | h

2

(5.56)

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ALE and Fluid–Structure Interaction

where the second and third terms are respectively the classical MINI-element and upwind stabilization terms. 5.3.3.3. Time advancing The time derivative is discretized using a simple first-order Euler formulae while an implicit scheme is used for the other terms. To illustrate this point, let us focus on the coarse-scale momentum equation including the small-scale pressure simplification. In this case, the weak form of the implicit scheme reads 0 /  n+1 uhn+1 − uhn ρ ,w h + (ρuh · ∇uh , w  h )n+1 + ρuh · ∇u  , w h Ω Ω n Δt Ω     + 2η ( ˙ uhn+1 ) : ( ˙w  h ) Ω − pn+1 , ∇ · w  h h Ω

  + τC ∇ · uhn+1 , ∇ · w  h Ω = f n , w h , Ω

(5.57) where the superscript n denotes the current time iteration while the exponent (n + 1) represents the next time level we want to compute. The resulting implicit scheme (5.57) is obviously nonlinear because of the nonlinear nature of the convective terms. In order to circumvent this issue, we resort to a classical Newton–Raphson linearization procedure. The implicit Newton–Raphson iterative scheme reads 0 /  n,i+1 uhn,i+1 −uhn ρ ,w  h + (ρuh · ∇uh , w  h )n,i+1 + ρuh · ∇u , w h Ω Ω n Δt Ω



n,i+1 + 2η ( ˙ uh ) : ( ˙w  h ) − pn,i+1 , ∇ · w  h h Ω

Ω

+ τC ∇ · uhn,i+1 , ∇ · w h = f n , w h , Ω

Ω

(5.58) where the exponent (n, i + 1) denotes the (i + 1) th iteration of the Newton–Raphson procedure which uses as initial guess the solution at time level n. The linearization of convective

Multidomain Finite Element Computations

253

terms consists of keeping only first-order terms at the (i + 1)th Newton iteration as follows: (uh · ∇uh ) n,i+1



= uhn,i + (uhn,i+1 − uhn,i ) · ∇ uhn,i + (uhn,i+1 − uhn,i ) = uhn,i · ∇uhn,i+1 + uhn,i+1 · ∇uhn,i − uhn,i · ∇uhn,i



+ uhn,i+1 − uhn,i · ∇ uhn,i+1 − uhn,i ≈ uhn,i · ∇uhn,i+1 + uhn,i+1 · ∇uhn,i − uhn,i · ∇uhn,i ,

(5.59) with the value of the velocity at the previous Newton iteration. Since we use quasi-static bubble functions, the third term of equation (5.58) reduces to   n,i+1 ≈ uhn,i · ∇u n,i+1 . (5.60) uh · ∇u uhn,i

The complete linearized Newton–Raphson scheme finally reads 0 / uhn,i+1 n,i n,i+1 n,i+1 n,i ρ + uh · ∇uh + uh · ∇uh , w h Δtn Ω



n,i n,i+1  n,i+1 + ρuh · ∇u ,w  h + 2η ( ˙ uh ) : ( ˙w h) Ω



− pn,i+1 ,∇ · w h h



Ω

+ τC ∇ · uhn,i+1 , ∇ · w h

  uhn n,i n,i n  = f + ρ n + uh · ∇uh , w h . Δt Ω

Ω

Ω

(5.61)

5.3.3.4. Matrix formulation of the problem When applied to both the coarse-scale system and the finescale system, the previous scheme gives rise to a linear system that remains to be solved. This system can be put naturally

254

ALE and Fluid–Structure Interaction

under the following matrix form: ⎡ ⎤ ⎡ ⎤ ⎤⎡ Aww tAwb tAwq Bw uh ⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎣ Awb Abb tAbq ⎦ ⎣ u  ⎦ = ⎣ Bb ⎦ , Awq Abq 0 ph Bq where

(5.62)

0 uhn,i+1 Aww (uh ) = ρ + uhn,i · ∇uhn,i+1 + uhn,i+1 · ∇uhn,i , w h Δtn Ω



+ 2η ( ˙ uhn,i+1 ) : ( ˙w  h ) + τC ∇ · uhn,i+1 , ∇ · w h /

Ω



  Abb (u  ) = ρ uhn,i · ∇u n,i+1 , w   + 2η ( ˙ u n,i+1 ) : ( ˙w  ) Ω Ω

/

un,i+1 h Awb (uh ) = ρ + uhn,i · ∇uhn,i+1 + uhn,i+1 · ∇uhn,i , w  Δtn

Awq (uh ) = − ∇ · uhn,i+1 , qh   Abq (u ) = − ∇ · u n,i+1 , qh Ω

Ω

0  Ω

Ω

  uhn n,i n,i n  Bw = f + ρ n + uh · ∇uh , w h Δt Ω   n  u Bb = f n + ρ hn + uhn,i · ∇uhn,i , w  Δt Ω Bq = 0.

(5.63) We can notice that in the present case of the quasi-static bubble assumption, the following simplification holds:

n,i+1 tAwb (u  ) = ρ un,i · ∇ u , w  . (5.64) h h Ω

The static condensation process, previously detailed, which consists in solving the second line involving u  and inserting the solution into the first and third lines of system (5.62)

Multidomain Finite Element Computations

255

results into the condensed matrix scheme for large-scale unknowns uh and ph that reads      ˜w A˜ww tA˜wq uh B (5.65) = ˜q ph A˜wq A˜qq B with A˜ww = Aww − tAwb A−1 bb Awb

tA˜wq = tAwq − tAwb A−1 bb tAbq

A˜wq = Awq − Abq A−1 bb Awb

A˜qq = −Abq A−1 bb tAbq

˜w = Bw − tAwb A−1 Bb B bb

˜q = −Abq A−1 Bb . B bb

(5.66) Taking into account locally the influence of unresolved fine scales upon the resolved large scales has introduced new stabilizing terms and modified the components of all the matrices while the effect of the fine-scale pressure has been added directly to the first matrix by a stabilizing term. This matrix formulation can be put in relation with the classic static condensation of stokes MINI-element as follows: i) the modified terms A˜ww and A˜wq incorporate the upwind −1 stabilization terms provided by tAwb A−1 bb Awb and Abq Abb Awb ; ii) similarly, the modified operators tA˜wq and A˜qq contain the −1 pressure stabilization terms tAwb A−1 bb tAbq and Abq Abb tAbq ; iii) eventually, the right-hand side components have been modified to ensure consistency by means of tAwb A−1 bb Bb and Abq A−1 B . b bb 5.3.4. Stabilized equation

weak

formulation

for

convection

With the same formalism, we recover a discrete stabilized weak formulation for convection equation, which is analogus to those obtained by SUPG or RFB methods. As there is no incompressibility equation and the unknown α is a scalar, the

256

ALE and Fluid–Structure Interaction

formal computations become simpler. Let us start from the discrete weak formulation of equation (5.18) with an implicit time scheme,   αh − αh− , wh + (uh .∇αh , wh ) = 0 ∀ wh ∈ Q0h . (5.67) Δt The solution α and the weighting function w of the above equation can be split into two terms on each element K :   α = αh + αK bK and w = wh + wK buK (5.68) with the same notations as those already written in relations (5.44) and (5.51). Therefore, the stabilization parameter τK satisfies the relation  bK K buK dΩ  τK =  ∀K ∈ Ωh . (5.69) uh · ∇bK , buK K Now, the stabilized weak formulation reads    αh −αh−  αh −αh− +uh .∇αh , w + +uh .∇αh , uh .∇wh =0. τK Δt Δt k∈Ω h

(5.70) By taking an upwind bubble function which satisfies uh .∇buK = −1 on the outlet faces of K defined by (uh .n > 0), we obtain 1 huK , τK = (5.71) 3 ||u|K || where u|K in the mean velocity in K and huK is the size of the biggest edge of K parallel to u|K . In [HAC 09a], the same strategy is used to stabilize the discrete weak formulation of the heat diffusion equation. In this way, numerical instabilities due to thermal shock are prevented. 5.4. Multiphasic problems with fluid–air and fluid– fluid interface The decoupled approach consists in solving at each time step the following steps:

Multidomain Finite Element Computations

257

1. computations of viscosity and density in the whole domain by using mixing relations (5.32); 2. computation of velocity field by solving multidomain Navier–Stokes equations (5.31) with Newton iterations; 3. displacement of the interface by solving a transport equation (5.28); 4. mesh refinement around the interface. The last step is not mandatory, but it allows to deal with a mesh with a moderate number of elements (especially in 3D) without losing a good description of the interface. This step adds a loss of accuracy when field solutions are interpolated on the new mesh. A possible alternative is to perform periodically this local refinement when the interface leaves the area having the smallest mesh size. 5.4.1. Two-dimensional Zalesak’s problem First, the performance of the our convected level set method is checked in order to underline the influence of mesh size and time step [VIL 09]. This is done by looking at the rotation of a slotted disk (Zalesak’s problem) in a rotation velocity field  vx = −2 π ( y − 0.5) (5.72) vy = 2 π ( x − 0.5). The computations are made in square domain of size 1: the slotted circle of radius 0.2 is initially located at (0.25, 0.5); the slot length is 0.35 (see Figure 5.7). This test is severe with regard to the diffusion error as the shape of the corner has to be preserved during the transport. Three simulations were performed with different meshes: 50 × 50; 100 × 100 and 200 × 200 with respectively time steps 0.0025; 0.001 and 0.0005. The results are plotted in Figures 5.8. In the first two tests, with a 50 × 50 and a 100 × 100 meshes, the sharp corners are lost because of the coarse mesh precision. On the third case, with a 200 × 200 mesh, a good accuracy of the method is observed on the corners’s advection.

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ALE and Fluid–Structure Interaction

Initial time

t = 0.25 s

t = 0.5 s

t = 0.75 s

t=1s

Figure 5.7. 2D Zalesak’s problem: slotted disk rotation

These examples show that the accuracy increases as the mesh size h decreases.

5.4.2. Gravitational flow An illustration of an fluid/air interface can be made by looking at the classical problem of the collapse of a water column [ANA 99, GAS 97, MAR 00, MAR 52]: the fluid is initially maintained between two walls; at initial time a wall is removed and the fluid falls down under the effect of gravity force. Our approach is purely Eulerian and requires to make computation on a large domain (we have to evaluate the final area occupied by the fluid at the end of the calculation) and

T=0s

a)

T=1s

b)

c)

Figure 5.8. Disk deformation after one rotation; influence of mesh size h: (a) 50 × 50 mesh (b) 100 × 100 mesh (c) 200 × 200 mesh

Multidomain Finite Element Computations

259

determine the flow motion in the air. The rheological properties of two products (water, air) are ρe = 1370 kg/m3 ; ηe = 1 × 10−3 Pa.s; ρa = 1.194 kg/m3 ; ηa = 1.85 × 10−5 Pa.s .

Let us remark that the viscosity ratio is ηf /ηa ∼ 102 and density ratio ρf /ρa ∼ 103 . As the viscosity of water is rather low, the Reynolds number related to the air motion can be important if the initial height ho is too high. If we want to replicate the experiments of Martin et al. [MAR 52] about the laminar spreading of a water column, the computation are made for a rather small initial height (that is equivalent after scaling to increase the viscosity of each phase). The initial position and the boundary conditions are given in Figure 5.9, for y = 0, 2h0 , ∂y ux = uy = 0 for x = 0, L, ux = ∂x uy = 0.

(5.73)

A mixed condition on the walls is used in order to allow slip. Indeed, we have to deal with the triple point which belongs to both liquid and air and has also a zero velocity field because it belongs to the wall (non-slip condition). uy = 0 ux= 0 ho

g

2h o

uy = 0 L

Figure 5.9. Geometry and boundary conditions

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ALE and Fluid–Structure Interaction

If computations are made for more viscous fluids (as the concrete [DUF 05, LAU 07b]), the falling is slower and different boundary conditions can be used for the air and the fluid respectively, for y = 0 u = 0 in Γext ∩ Ωf and σ · n = 0 in Γext ∩ Ωa . (5.74) This allows to take into account of a fountain effect assuming that it is not necessary to compute accurately the air motion. In Figure 5.10 computation are made for Reynolds number around 100 and the level set is plotted for various time. Let us remark that this method captures perfectly the drop of air when the wave comes back into the cavity after touching the right wall. Similar 3D computations for the filling of a beam or the falling of Abraham cone for Bingham fluids can be found in [LAU 07b]. Finally, the ability of the method to compute surface evolution of high Reynolds flows (around 105 ) is pointed out on a 3D example of damp breaking [FRA 09, CRU 07]. Other examples of damp breaking problem for high Reynolds numbers can be found in [LUB 03, MAR 06b]. We consider the breaking of a cubic water column in a 3D domain as depicted in Figure 5.9 with length L = 0.42 m and height ho = 0.228 m and width W = 0.228 m (the dimensions come from an experimental apparatus [CRU 07]). The strong deformation of the interface and the unsteady character of the flow presented in this test case are essential for validating the proposed stabilized finite element method. Figure 5.11 shows snapshots of the water surface position at different time step. The water cubic column collapses and accelerates toward the air due to the pressure difference between the adjacent water and air. We see that from the time the water front reaches the maximum height on a side wall, it falls back and creates some propagates waves toward the opposite wall.

Multidomain Finite Element Computations

Figure 5.10. Evolution of interface in 2D: L = 4 ho , the mesh size is h = 0.02, = 0.02 and E = 2. Isovalues of level set function (the zero isoline is in black) for t = 20, 40, 60, and 80

261

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ALE and Fluid–Structure Interaction

Figure 5.11. Interface positions at different time step: computation is made on 100,000 tetrahedral element, Δ = 0.001; (a) t = 0; (b) t = 0.2 s; (c) t = 0.624 s; (d) t = 0.752 s

5.4.3. The surface tension A natural question arising with this approach is to see how a constraint can be applied on an interface defined by a mixing area. Example of a such situation can be encountered by taking into account of the surface tension which corresponds to a stress jump between the two phases: [[σ · n]]Γ = γκn,

(5.75)

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263

where γ is the surface tension (N/m), κ the curvature (m−1 ) and n normal vector to Γ. With the notation used for the system (5.34), the variational formulation which deals with this jump reads     Du ,w  ˙ w)  Ω − (p, ∇ · w)  Ω + ρ 2 η ( ˙ u) : ( Dt Ω = (ρg , w)  Ω + (γκn, w)  Γ.

(5.76)

Finally, this new term is added to other volume forces. The major difficulty comes from the replacement of the surface integral by an integral over a volume, because the interface is not explicitly known. As proposed in [BÉL 98, BRA 92], the surface tension can be taken into account by introducing a volume force acting in a thin strip around the interface (this force vanishes outside this thin strip). This force can be expressed as follows: f = γ κ n δΓ  γ κ ∇Ie (α),

(5.77)

where δΓ is the Dirac function associated with the interface Γ and Ie is a characteristic function defined from the level set function such as n δΓ = ∇Ie . According to the usual approaches [BEN 06, BRU 07, FOR 06], we have ⎧ = 0 if α < −e ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = 1 if α > e Ie (α) = (5.78) ⎪ ⎪   ⎪ ⎪ α 1 α 1 ⎪ ⎪ 1 + + sin(π ) if − e < α < e. ⎩ = 2 e π e Moreover, thanks to relation (5.15), the curvature κ can be computed from the level set function α. However, if the level set function (α) has a linear approximation P1 , the normal vector is constant by element (P0 approximation) and therefore the curvature computation requires a special treatment. To avoid this problem, it is usual [REN 02] to express this

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force f as the divergence of a tensor T defined by T = γ (E − n ⊗ n) δΓ .

(5.79)

Using that δΓ = |∇Ie |, we find [BÉL 98] that T =γ

|∇Ie |2 E − ∇Ie ⊗ ∇Ie |∇Ie |

and finally the right term in equation (5.79) reads   (γ κ n, w)  Γ = (γ κ ∇Ie , w)  Ω = γ T : (w)  Ω.

(5.80)

(5.81)

We check the validity of this approach on the Young–Laplace law (a classical test for interfacial problems with surface tension). A drop with an initial rectangular shape is placed in a viscous liquid. The drop will get a spherical shape because of surface tension. At equilibrium, the velocity vanishes and the jump condition becomes γ Δp = 2 , R

(5.82)

where R is the radius of the final circle. Figure 5.12 shows the evolution of the interface and velocity field for γ = 0.1 and η1 = η2 = 1. In step 2 of our algorithm, the Stokes equation is solved because gravity and inertia are neglected. We can add an additional step in this algorithm in order to preserve the volume conservation of droplet initial [FOR 06]. It is sufficient to displace uniformly the zero isoline. Other examples in 2D and 3D can be found in [BEN 06, FOR 06, MAR 06a]. 5.5. Immersion of solid bodies in fluid We consider the specific case for which the second phase is a solid body and two situations are studied: the solid body has a known explicit motion that can induce flow motion; the motion of solid bodies are induced by the flow motion.

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Figure 5.12. Shapes of the drop and velocity fields for different time (the length vector is normalized and the color gives the norm value): (a) t = 0; (b) t = 5; (c) t = 5; (d) t = 10. The parameter values are η1 = η2 = 1, γ = 0.1 and Δt = 0.05

For both cases, an Eulerian approach is proposed and computations are made on a domain which contains both phases. This approach is usually called “fictitious domain” method [GLO 94, KHA 00]. 5.5.1. Immersed body having an imposed velocity We start by considering the case where the immersed objects have imposed velocities. For example, this situation can correspond to a rotating inner cylinder in the Couette– Taylor geometry or the addition of solid and rigid insert which changes the flow motion. In the case of a solid object with an imposed movement, this approach is particularly interesting because it is not necessary to mesh the computational domain at each time step.

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Figure 5.13. Contraction geometry: the height is 1, the length 5, and the contraction is located at x = 2.5 and y = 0.8. In the second picture, the solid domain is bounded by a zero isoline

As seen in Figure 5.13, the contraction can be described either by meshing directly the real geometry (Figure 5.13a), or from a rectangular domain with two subdomains (a fluid domain and solid domain with a zero velocity, Figure 5.13b). The boundary is defined by the zero isovalue of a function (which is rather easy to compute for a rectangular domain). Let us note that in the case of multidomain approach, the mesh is not substantially larger because the computational domain is meshed coarsely in the solid domain and it is refined only in a thin strip around the zero isoline defining the boundary of immersed object. To impose Dirichlet boundary conditions (u = 0) on solid domain, the characteristic function (5.13) is changed in order to have a mixing law only in the solid domain: ⎧ = 1 if α(x) > e ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ α if 0 < α(x) < e = Ie (x, t) = (5.83) e ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = 0 if α < 0 with a solid viscosity ηs ∼ 100 ηf .

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267

Finally, the solid domain is taken into account as follows: – if α > e, the Dirichlet condition u = 0 is imposed; – for 0 < α < e, one uses a mixing law (5.32) for viscosity. In this way, elements that cross zero-isovalue of α becomes “rigid” as their viscosity is larger than in the fluid domain. Computations are made in a geometry of height 1, length 5, and the contraction takes place at coordinate x = 2.5 and y = 0.8. The boundary conditions on vertical walls reads ux (x = 0, y) = (y 2 − 1)/2, uy (x = 0, y) = 0 ; σ(x = 5, y).n = 0. (5.84) The velocity and pressure fields computed with the second approach are shown in Figure 5.14. The comparison of velocity fields calculated at the exit of the contraction is plotted in Figure 5.15.

The same method can also be used if the solid body has a non-zero velocity. This can be illustrated on the Couette– Taylor problem (the fluid motion between two concentric cylinders in uniform rotations), because the computed solution can be compared to an analytic solution for a laminar flow: ur = 0

and

uθ = Ar +

B r

(ωi − ωe ) Re2 Ri2 , Re2 − Ri2 (5.85) where ωi,e and Ri,e are the rotations and radii of the inner and outer cylinders, respectively. The computation is made on multidomain mesh depicted in Figure 5.16(a) and the comparison plotted in Figure 5.16(b) between analytic and computed radial velocity shows an excellent agreement.

with A =

ωe Re2 − ωi Ri2 ; Re2 − Ri2

B=

For the Couette–Taylor problem, the inner cylinder occupies always the same location, but it may happen that the solid body having an imposed speed moves in computing domain. In this case, the level set function associated with the

268

ALE and Fluid–Structure Interaction Ux 0.00

0.50

1.0

1.5

2.0

2.5

Uy 0.00

0.50

1.0

P 0

75.0

5.00

Figure 5.14. Velocity and pressure field computed with multidomain method: (a) ux , (b) uy , (c) pressure

solid body will be computed at each time step. As explained in the next section, it is more convenient to move the solid body by using a Lagrangian displacement. Examples of such 3D computations are shown in [VAL 07]. 5.5.2. Velocity-pressure formulation for solid bodies in a fluid We consider the case where the velocity of solids is not imposed because it is derived from the hydrodynamic forces induced by fluid flow. To resolve this problem, Patankar et al. [PAT 00] have proposed to extend the Navier–Stokes to the

Multidomain Finite Element Computations 3.0

269

contraction multidomain

2.5

Ux

2.0

1.5

1.0

0.5

0.0 0.80

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

y Figure 5.15. Comparison of ux computed with the two approaches at the contraction exit (x = 4)

whole domain by using Lagrange multiplier for the rigid motion constraint ( ˙ v ) = 0 on Ωs . Then the solid domain is treated as a fluid subjected to an additional rigidity constraint and one ends with the system of equations: ⎧   ∂u ⎪ ⎪ + u . ∇u = ρs g + ∇ . σ ρs ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ∇ . u = 0 ⎪ ⎪ ⎨ ( ˙ u) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [[u]]Γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [[σ.n]] Γ ρs being the solid body density.

= 0 = 0 = 0,

(5.86)

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0.5

Utheta

0.0

−0.5

analytic multidomain −1.0 0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

r Figure 5.16. Computation of the Couette solution for values: ωi = −1; ωe = 1 ; Ri = 0.5; Re = 1. (a) multidomain mesh; (b) comparison between the analytic solution and the computed solution

1.00

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271

Therefore, the stress tensor σ in the solid has the following form: σ = ηs ( ˙ u) − pI + ( ˙ λ). (5.87) Thanks to rigid motion constraint, the two first terms vanish, ηs acts as a penalization factor of the constraint ( ˙ u) = 0, and the symmetrical tensor ( ˙ λ) is the Lagrange multiplier associated with this constraint. 5.5.3. Weak formulation of fluid/solid system With Dirichlet boundary conditions on external walls, the weak formulation  Ω becomes   for the whole domain  ∈ H01 (Ω) × L2 (Ω) × find u, p, λ such that ∀ v , q, μ H1 (Ωs (t)) ⎧      Du  ⎪ ⎪ ⎪ −  g ,  v u ) : ( ˙ v ) − (p, ∇ ·  v ) + ρ 0 = 2 η ( ˙ ⎪ Ω Ω ⎪ Dt ⎪ Ω ⎪

⎪ ⎪ ⎨  + ( ˙ λ) : ( ˙ v) Ωs ⎪ ⎪ ⎪ 0 = (∇ · u, q)Ω ⎪ ⎪ ⎪ ⎪ ⎪   ⎪ ⎩ 0 = ( ˙ μ) : ( ˙ v) Ω , s (5.88) where ρ and η are defined on Ωs thanks to mixing relations (5.32). With respect to formulation (5.34), there is an integral over Ωs which allows to take into account the rigidity condition. 5.5.4. Discrete formulation and Uzawa algorithm The augmented Lagrangian methods and an Uzawa-type algorithm [FOR 85] are used to solve equations (5.88) without increasing the size of linear system. Indeed, we simply add a second term in the rhs of the Navier–Stokes equations presented in the previous section. In this way, we can solve in the same iteration the problem of nonlinearity, time integration and computation of the Lagrange multiplier: an implicit time

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scheme is used which the Uzawa and Newton methods solve in a single loop of the nonlinearity and rigidity constraint. At each time step tn the procedure is the following: 1. initialization with values obtained at the previous time step: uh0 = uh (tn−1 ), p0h = ph (tn−1 ), λ0h = 0, u∗h = uh (tn−1 )

2. at step k find ukh and pkh with system:  ⎧ 

  ukh ⎪ k−1 k . ⎪ , w  + ρ ∇ u u , w  + 2 η ( ˙ u ) : ( ˙ w  ) ρ ⎪ h h h h h h ⎪ Ω Δt ⎪ Ω ⎪ Ω ⎪   ⎪ ⎪  ⎪ u 0 ⎨ − p k , ∇ · w h  h Ω = (ρ g , w  h )Ω + ρ h , w h Δt Ω ⎪



⎪ ⎪ k−1 k−1 k−1 ⎪  + ρ ∇ u .  u , w  − I ( ˙ λ ) : ( ˙ w  ) ⎪ s h h h h h ⎪ ⎪ Ω Ω ⎪ ⎪ ⎪   ⎩ ∇ · uhk , qh = 0 (5.89) k−1 k k    3. update λ : λh = λh + ηs uh

4. check ||uhk−1 − uhk || < 1 and ||Is ( ˙ uhk )|| < 2 to stop the loop on k . We can remark that in system (5.88), the field λ is only defined on the solid domain Ωs . For practical reasons, this field is extended to the whole domain Ω because only the part inside Ωs is taken into account, thanks to the characteristic function. 5.5.5. Lagrangian particle displacement The various fictitious domain approaches give the velocity field at any point of the whole domain (solid and fluid). But we have not yet disscussed the evolution of the solid domain (that is to say the of particle displacements). Indeed, once the problem is discretized in time and space, there are two tasks

Multidomain Finite Element Computations

273

to do at each time step: the computation of velocity field and the determination of new particle positions. We have shown in previous sections how to move the interface by convecting the level set function. However, for a rigid particle, it is sufficient to move a few points (one for a sphere, two for an axisymmetric solid and three for any solid) to get the new position. Knowing this new position, it is now possible to compute again the new level set function without deformation or volume dissipation. This is the approach proposed in [MEG 05, BEA 08] to move the particles. For example, the displacement of gravity center can be done by using the following equation:  + Δt) = X(t)   X(t + Δt u(X(t), t).

(5.90)

Moreover, powerful algorithm is required in order to interpolate the velocity at selected coordinates related to moving particles (that is to say “find in which element belongs these ponts”) because an important computational time is necessary for large meshes. This explicit time integration scheme can be improved by taking into account the velocity at previous time steps. For example, a second-order Adams–Bashfort scheme [HWA 04, BEA 08] can be chosen:

Δt  + Δt) = X(t)    − Δt), t − Δt)) . X(t + 3 u(X(t), t) − u(X(t 2 (5.91)

5.5.6. A sphere in a shear flow Our approach is checked on the simple case of a sphere in a shear flow for which there is an analytic solution [BAT 72]. The analytic formulas for a sphere of radius a in a shear flow (γ˙ y, 0, 0) are in cartesian coordinates (the origin is also the

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center of the sphere): x2 y γ a 5 xy 2 γ a 5 y x A − + γy; u = γ ˙ A − y r2 2 r r2 2 r

3 xyz xy a = γ˙ 2 A; p = p0 − 5 η γ 2 r r r (5.92)

ux = γ˙ uz

with A=−

5 a 3 5 a 5 + . 2 r 2 r

The numerical computation are performed in a rectangular (or parallelepipedic in 3D) geometry of height h and length L (see Figure 5.17). A simple shear flow is imposed by setting the horizontal velocity of the upper and lower walls: ux (x, 0) = −u; ux (x, 1) = u and uy (x, 0) = uy (x, 1) = 0. (5.93) Moreover, a condition on vertical walls is added in order to satisfy a Couette flow conditions: uy (0, y) = uy (2, y) = 0 and ∂x ux (0, y) = ∂x ux (2, y) = 0. (5.94)

The comparison between 2D and 3D numerical results and analytic formulas are plotted in Figure 5.18. In agreement with a previous study [LAU 07a], the 3D computations (for a sphere) confirms that accurate results are obtained with five Uzawa iterations and a penalization ηs = 102 ηf . There is also a difference between 2D and 3D computations, because 2D calculations correspond rather to a flow around a cylinder. u h L

−u

Figure 5.17. Geometry and notations: a spherical particle in a shear flow γ˙ = 2u/h . The horizontal velocities ±u are imposed on the upper and lower walls. The sphere radius is labeled by a, the rectangle height is h, and the length L

Multidomain Finite Element Computations

0.15

3D 2D th Couette

ux

0.10

0.05

0.00 0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

r 0.

uy

−.005

−.01

−.015 3D 2D th −.02 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

r Figure 5.18. Comparison between 2D and 3D computed solutions and analytic formulas for u = 0.5, h = 1, L = 2h, W = h, η = 1 and a = 0.05. The velocity is plotted along the line z = 0.5, x = 1 + r cos π/4 and y = 0.5 + r cos π/4

275

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ALE and Fluid–Structure Interaction

5.5.7. Two rigid spheres in stokes flow We study the movement of solid bodies and the action of the hydrodynamic interaction forces in the simple case of two spheres in a shear 2D flow when gravity and inertia are neglected. This configuration has already been studied in [HWA 04] with other boundary conditions. Let us consider the same geometry as in the previous section and two identical spheres are placed in the superior and inferior mid-plane and move toward the right and the left side respectively. We compare two procedures for moving rigid bodies the first using the transport equation for the level set function (5.18) and the second with the Lagrangian displacement (5.90). For the first method, it is important to compute accurately the velocity field at the fluid/solid interface. The previous section has shown the validity of our approach for a single particle, but the influence of mesh size on short range hydrodynamic interaction forces is studied. Indeed, this forces prevent the between solid particles when they become closer. 5.5.7.1. Influence of mesh First, a configuration with two symmetrically located particles is selected as a test problem. The two particles are close enough to check our ability to compute accurately hydrodynamic interaction and analyze the influence of mesh size. The centers of two particles with the same radius a = 0.15 are located at point (0.873, 0.627) and (1.127, 0.373) and velocity field are computed for various meshes. The main mesh characteristics are reported in Table 5.1. The first three meshes have a homogenous mesh size while the last mesh Mh is built from M1 by increasing the number of elements in the vicinity of spheres (see Figure 5.19). The velocity components are plotted along on the diagonal (see Figure 5.19) that goes through the particle centers. The

Multidomain Finite Element Computations

Mesh M1 M2 M3 Mh

Element number 12 161 50 506 201 922 14 116

Node number 6 218 25 543 101 097 7 196

277

Mesh size 0.02 0.01 0.005

Table 5.1. Mesh descriptions

computations are made for a penalty factor ηs = 102 ηf and six Uzawa iterations. The computations show uniform convergence with mesh refinement and the three meshes M2 , M3 , and Mh give rather comparable results. More precisely, Figure 5.20 points out that the computed velocity is less dependent on mesh size for particle centers than near its boundaries. 5.5.7.2. Particle displacements In this section, we compare the Lagrangian displacement with the method using a transport equation. The two particles with radius a = 0.12 are initially placed at points (0.5, 0.6) and (1.5, 0.4). The two particles move toward each other, then

Figure 5.19. Computational geometry: two particles in a shear flow and the boundary-fitted mesh Mh . The velocity field is plotted along the red line

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ALE and Fluid–Structure Interaction

0.5 0.4 0.3 0.2 0.1

ux

0.0

−0.1 −0.2 −0.3 −0.4 −0.5 0.0

M1 M2 M3 Mh

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.6

0.7

0.8

0.9

1.0

x

0.15 0.10 0.05

uy

0.00

−0.05 −0.10 −0.15 0.0

M1 M2 M3 Mh

0.1

0.2

0.3

0.4

0.5 x

Figure 5.20. Comparison of the velocity field along the diagonal that crosses the two spheres. Vertical lines mark the sphere boundaries

Multidomain Finite Element Computations

279

0.70 0.65 0.60 0.55

Y

0.50 0.45 0.40 0.35 0.30 0.25 0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

X Figure 5.21. Orbits of particle centers: the particles are initially in (0.5,0.6) and (1.5, 0.4)

they are pushed back to avoid a collision because of hydrodynamic repelling forces. Finally, they come back to their initial vertical location. Orbits of the particle centers are plotted in Figure 5.21. An isolated sphere in a shear flow creates a small vortex because it spins (see Figure 5.22a). However, if they are close together, they move in a large movement of rotation to prevent a collision (see Figure 5.22b). One can compare the sphere displacements for the two methods by plotting the zero isoline of level set function at various times. The time step used for both methods is Δt = 0.05 and computations are made with mesh M3 . We compare

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Figure 5.22. Focus on velocity field: t = 2.5; t = 6.5

the particle positions if the level set is convected or the particle centers is moved by a Lagrangian method. Finally, we point out that (see Figure 5.23 ): – without interaction between particles, there is no difference between the two methods (t = 2.5); – as the two particles become closer, the two methods do no longer coincide. In fact, the RFB (or SUPG) method needs an accurate computation of the velocity near the particle boundaries whereas

Figure 5.23. In red zero isovalue of level set function moves with a RFB method and in black zero isovalue of level set function computed after a Lagrangian update of particle center: t = 2.5; t = 5; t = 10; t = 15

Multidomain Finite Element Computations

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with the Lagrangian method it is sufficient to compute accurately the velocity at particle center. Therefore, without a better description of interface, the second method gives better results. Finally, we have shown that the most efficient method involving couple finite element computation of velocity to a Lagrangian method for particle displacement. For more concentrated suspensions, we must take into account collisions and slipping between particles. This can be done either in the variational formulation or when updating the particle position [LAU 05, BEA 08]. There are actually current developments on this subject: for example in [WAC 09] fictitious domain method is coupled with discrete element method (granular solver) in order to handle with particles having complex shapes and short range repulsive forces.

5.6. Conclusion We have presented here the foundations of a purely monolithic approach to solve the Eulerian multiphase problems. It provides conclusive results on problems with two fluids or a fluid and rigid solid. An important work has been done on calculating the displacement of the interface defined by a level set function. Its extension to more specific fluid–structure problems is under development. The first step is to allow the deformation of solid domain. For example, the linearized constitutive equations of an elastic solid under the assumption of small deformations can be written as a mixed displacement/pressure formulation by introducing a compressibility modulus [KLA 99], and thus we can use the same methodology as for the Navier–Stokes equations. The approach involves transform fluid structure interaction problem to a biphasic problem starts to be checked. In this approach, which relies on the theory of mixtures [BOW 76, KRI 97], it is necessary to model the interaction between phases and this can

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be done by adding a friction force. The advantage of this biphasic approach is that it can be extended to problems of phase change (crystalization, cooling, etc.).

5.7. Acknowledgements The authors thank the partners of consortium REM3D that have sponsered most of thesis mentioned in this chapter. The Companies are in alphabetical order: Arkema, DOW Chemicals, Essilor International, FCI, Plastic Omnium Auto Extérieur, Schneider Electric, Snecma Propulsion Solide, and Transvalor.

5.8. Bibliography [ANA 99] A NAGNOSTOPOULOS J., B ERGELES G., “Three-dimensional modeling of the flow and the interface surface in a continuous casting mold model”, Metallurgical and Materials Transaction, vol. 30B, p. 1095–1105, 1999. [AND 98] A NDERSON D., M C FADDEN G., W HEELER A., “Diffuse interface methods in fuid mechanics”, Annual Review of Fluid Mechanics, vol. 30, p. 139–165, 1998. [ARN 84] A RNOLD D., B REZZI F., F ORTIN M., “A stable finite element for the Stokes equations”, Calcolo, vol. 23, num. 4, p. 337–344, 1984. [BÉL 98] B ÉLIVEAU A., F ORTIN A., D EMAY Y., “A numerical method for the deformation of two-dimensional drops with surface tension”, International Journal for Computational Fluid Dynamics, vol. 10, p. 225–240, 1998. [BAS 06] B ASSET O., Simulation numérique d’écoulements multi-fluides sur grille de calcul, PhD thesis, Ecole Nationale Supérieure des Mines de Paris, 2006. [BAT 72] B ATCHELOR G., G REEN J., “The hydrodynamic interaction of two small freely-moving spheres in a linear flow field”, Journal of Fluid Mechanics, vol. 56, p. 375–400, 1972. [BEA 08] B EAUME G., Modélisation et simulation numérique directe de l’écoulement d’un fluide complexe, PhD thesis, Ecole nationale supérieure des mines de Paris, 2008.

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[BEH 93] B EHR M. A., F RANCA L., T EZDUYAR T., “Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows”, Computational Methods in Applied Mechanical Engineering, vol. 104, num. 1, p. 31–48, 1993. [BEN 06] B ENMOUSSA K., F ORTIN A., “An adaptive remeshing strategy for free-surface fluid flow oroblems. Part II: The three-dimensional case”, Journal Polymer Engineering, vol. 26, p. 59–85, 2006. [BIG 01] B IGOT E., Simulation tridimensionnelle du remplissage de corps minces par injection, PhD thesis, Ecole nationale supérieure des mines de Paris, 2001. [BOW 76] B OWEN R., Theory of Mixtures, vol. 3 of Continuum Physics, A. C. Eringen, Ed., Academic Press, New York, NY, 1976. [BRA 92] B RACKBILL J., K OTHE D., Z EMACH C., “A continum method for modeling surface tension”, Journal of Computational Physics, vol. 100, p. 335–383, 1992. [BRE 91] B REZZI F., F ORTIN M., Mixed and Hybrid Finite Element Methods, Num. 15 Springer Series in Computational Mathematics, Springer-Verlag, New York, 1991. [BRE 94] B REZZI F., R USSO A., “Choosing bubbles for advection-diffusion problems”, Mathematical Models and Methods in Applied Sciences, vol. 4, p. 571–587, 1994. [BRE 96] B REZZI F., F RANCA L. P., H UGHES T. J. R., R USSO A., “Stabilization techniques and subgrid scales capturing”, Conference of the State of the Art in Numerical Analysis, York, England, April 1996.  [BRE 97] B REZZI F., F RANCA L., H UGHES T. J. R., R USSO A., “b = g”, Computational Methods in Applied Mechanical Engineering, vol. 145, p. 329–339, 1997. [BRO 82] B ROOKS A., H UGHES T., “Streamline upwind/Petrov-Galerkin formulations for convectivedominated flows with particular emphasis on incompressible Navier-Stokes equations”, Computational Methods in Applied Mechanical Engineering, vol. 32, p. 199–259, 1982. [BRU 04] B RUCHON J., Etude de la formation d’une structure mousse par simulation directe de l’expansion de bulles dans une matrice liquide polymère, PhD thesis, Ecole nationale supérieure des mines de Paris, 2004. [BRU 07] B RUCHON J., F ORTIN A., B OUSMINA M., B ENMOUSSA K., “Direct 2D simulation of small gas bubble clusters: From the expansion step to the equilibrium state”, International Journal for Namerical Methods in Fluids, vol. 54, p. 73–101, 2007.

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List of Authors

David J. BENSON University of California at San Diego (USCD) San Diego USA Thierry COUPEZ CEMEF Sophia Antipolis France Facundo DEL PIN Livermore Software Technology, LSTC Livermore USA Hugues DIGONNET CEMEF Sophia Antipolis France Elie HACHEM CEMEF Sophia Antipolis France

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Sergio IDELSOHN Universitat Politècnica de Catalunya, CIMNE Barcelona Spain Patrice LAURE CEMEF Sophia Antipolis France Riccardo ROSSI Universitat Politècnica de Catalunya, CIMNE Barcelona Spain Michael SHÄFER Damstadt Technical University Germany Luisa SILVA CEMEF Sophia Antipolis France Mhamed SOULI Lille University, LML Villeneuve d’Ascq France Rudy VALETTE CEMEF Sophia Antipolis France

Index

contraints 101 convection 6, 59, 226, 240, 255 convergence 70, 121, 145, 168, 277 Crank-Nicolson 117, 135

A added mass 166, 175, 186, 211 ALE 2, 39, 52, 83, 157 algorithm advection 25, 57, 64, 226 contact 23, 41, 54, 92, 210 coupling 2, 32, 52, 58, 74, 83 Donor Cell 29, 34, 59 Gauss Seidel 142 monolithic 146, 166, 174, 241 Newton 145, 149, 252, 272 partitioned 147, 166, 176, 216 transport 4, 29, 34, 115, 238 assembling 135

D, E, F damping 10, 21, 70, 81 dimension 5, 19, 62, 200 explicit 7, 18, 40, 146 coupling 67, 148, 193 finite element 77, 222 G Galerkin 110, 172, 225, 240 gas compressible 69 ideal 20, 83, 112 Gauss quadrature 14, 134 Gauss theorem 16 gravity 95, 152, 206 Green-Lagrange 139

B, C coefficient damping 10, 16, 70, 81, 135, 155 elasticity 122, 167 matrix 118 Poisson 120, 140, 156, 196, 246 stiffness 16, 21, 62, 135, 216 compressibility 117, 174, 281 artificial 117 condensation 169, 176, 182, 241 conservation 3, 24, 55, 111 mass 8, 57, 112, 169

H, I Hermite interpolation 145

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hydrodynamic equations 52 forces 52, 268, 276, 279 hydrostatic 91, 101 hyperelasticity 122 incompressibility 74, 112, 118, 255 instability 67, 226, 243 interface forces 56, 58, 63, 65, 75 reconstruction 39, 42, 44 spring 62 interpolation 19, 41, 116, 173, 228 isotropic 33, 111, 159, 232, 241 J, K, L Jacobian 3, 9, 145, 200 Krylov 136, 147 law Fourier 97, 111 Hooke 123 material 121, 122, 127 level set 39, 41, 138, 222, 235 M matrix mass 13, 172 stiffness 131, 135, 172, 179 method elliptic 143, 145, 153 impplicit 22, 117, 131, 146, 240 penalty 60, 66 modes hourglass 14, 16

modulus bulk 42, 62, 67, 122 Young 122, 159, 196, 205 monolithic 146, 166, 174, 241 N, O Navier-Stokes 55, 83, 240 Neumann 19, 120, 191 Newmark 70, 136, 194 Newton 25, 145, 252 Newton-Raphson 149, 262 number Mach 112 Reynolds 114, 156, 201 P, Q Piola-Kirchhoff 128 piston 52, 73 PISO 121 projection 7, 28, 168 R, S, T residual 171, 224, 247 sloshing 22, 84, 91 spring 62, 77 stability 10, 21, 136, 241 thermodynamic 83, 112 triangle 90, 96, 227 U, V velocity potential 26, 76 viscosity 11, 56, 151, 170 VOF 39, 42 vortex 159, 204, 279