A Computational Framework for Multiscale Structural Analysis C. Armando Duarte Dept. of Civil and Environmental Engineering Computational Science and Engineering Presented at Seoul National University Seoul, South Korea, August 2nd, 2016
Motivation: Multiscale Structural Analysis The US Air Force has expended six decades and untold resources in attempts to field a reusable hypersonic vehicle* Scientific challenge: “An inability to computationally capture the material evolution and degradation within a structural component”
X-15 (1959) *[T. G. Eason et al., 2013]
X-51a (2010)
Motivation: Multiscale Structural Analysis • The analysis of hypersonic aircrafts involves multiple spatial and temporal scales: From airframe scale to sub-assembly scale, panel scale, stiffener scale, connection scale, material scale, etc.
Lockheed Martin HTV-3X [AFRL-RB-WP-TR-2010-3069]
Motivation: Multiscale Structural Analysis • Hypersonic aircraft panels are assembled from sub-components using hundreds of fasteners or spot welds. • Multiple spatial scales: Panel, stiffeners, spot welds.
Panel 1 to Sub-Structure Attachment [AFRL-RB-WP-TR-2012-0280]
4
Motivation: Multiscale Structural Analysis • Modeling fastened connections (e.g. spot welds) with point constraints leads to mesh-dependent results [Szabó & Babuska, 1991] • Are connectors, like spot welds, critical failure sites? • Can more realistic, 3-D models be used at critical regions, while maintaining the easy-of-use of point constraints?
Panel 1 to Sub-Structure Attachment [AFRL-RB-WP-TR-2012-0280]
5
Summary—Target Problem and Overall Strategy Goal: Develop a computational framework able to capture highly localized 3-D thermo-mechanical fields at structural scale
Target Problem: • Three-dimensional modeling of fasteners (spot welds) in a representative panel • Linear and non-linear (contact and material) • Time-dependent thermo-mechanical loads Strategy: • Formulate a two-scale Generalized FEM for this class of problem • Keep global mesh coarse and resolve spot welds through enrichment functions computed in parallel. 6
Outline n
Target problem and strategy
n
Generalized finite element methods: Basic ideas
n
Bridging scales with GFEM: §
Global-local enrichments for localized nonlinearities
n
Towards modeling fasteners and contact in built-up panels
n
Summary and main conclusions
7
Early works on Generalized FEMs n n n n n n n
Babuska, Caloz and Osborn, 1994 (Special FEM). Duarte and Oden, 1995 (Hp Clouds). Babuska and Melenk, 1995 (PUFEM). Oden, Duarte and Zienkiewicz, 1996 (Hp Clouds/GFEM). Duarte, Babuska and Oden, 1998 (GFEM). Belytschko et al., 1999 (Extended FEM). Strouboulis, Babuska and Copps, 2000 (GFEM).
• Basic idea: • Use a partition of unity to build Finite Element shape functions • Review paper Belytschko T., Gracie R. and Ventura G. A review of extended/generalized finite element methods for material modeling, Mod. Simul. Matl. Sci. Eng., 2009
“The XFEM and GFEM are basically identical methods: the name generalized finite element method was adopted by the Texas school in 1995–1996 and the name extended finite element method was coined by the Northwestern school in 1999.”
Generalized Finite Element Method uGFEM (x) =
X
↵2Ih
|
ˆ ↵ '↵ (x) + u {z
}
e i=1 ↵2Ih
|
std FEM approx.
↵i (x)
m↵ XX
{z
GFEM enriched approx.
= '↵ (x)L↵i (x)
Linear FE shape function
ˆ↵, u ˜ ↵i 2 R3 u
}
• Allows construction of shape functions incorporating a-priori knowledge about solution
Enrichment function
GFEM shape function
˜ ↵i '↵ (x)L↵i (x), u
Discontinuous enrichment [Moes et al., 1999]
ωα [Oden, Duarte & Zienkiewicz, 1996]
Bridging Scales with Global-Local Enrichment Functions* Enrichment functions computed from solution of local boundary value problems: Global-Local enrichment functions n
Linear FE s hape function n
n
GFEM s hape function
Enrichment = Numerical solutions of BVP
*[Duarte and Kim, 2008]
n
n
Idea: Use available numerical solution at a simulation step to build shape functions for next step (quasi-static, transient, non-linear, etc.) Enrichment functions are produced numerically on-the-fly through a global-local analysis Use a coarse mesh enriched with Global-Local (GL) functions GFEMgl = GFEM with global-local enrichments
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Bridging Scales with Global-Local Enrichment Functions Available online at www.sciencedirect.com
Comput. Methods Appl. Mech. Engrg. 197 (2008) 487–504 www.elsevier.com/locate/cma
Analysis and applications of a generalized finite element method with global–local enrichment functions C.A. Duarte *, D.-J. Kim Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 2122 Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, USA Received 30 November 2006; received in revised form 2 June 2007; accepted 7 August 2007 Available online 14 September 2007
Abstract This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method. The procedure combines classical global–local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently. The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution. Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global–local enrichment functions, are presented. The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global–local enrichment functions. ! 2007 Elsevier B.V. All rights reserved.
• 117 citations • Adopted by researchers at o National Laboratories (AFRL, Sandia) o Universities in the USA, Germany, Spain, Brazil, India
Keywords: Generalized finite element method; Extended finite element method; Meshfree; Global–local; Fracture mechanics; Multiple cracks
1. Introduction The effectiveness of partition of unity methods like hp clouds [25,26], generalized [7,19,20,32,40,49,50], and extended [10,34] finite element methods relies, to a great extent, on the proper selection of enrichment functions. Customized enrichment functions can be used to model local features in a domain like cracks [15,21,34,35, 39,41,55], edge singularities [20], boundary layers [17], inclusions [54], voids [50,54], microstructures [33,47], etc., instead of strongly refined meshes with elements faces/ *
Corresponding author. Tel.: +1 217 244 2830; fax: +1 217 333 3821. E-mail address:
[email protected] (C.A. Duarte).
edges fitting the local features, as required in the finite element method. This has lead to a growing interest on this class of methods by the engineering community. The generalized/extended FEM has the ability to analyze crack surfaces arbitrarily located within the mesh (across finite elements). Fig. 1b shows the representation of a three-dimensional crack surface using the generalized finite element method presented in [23,45]. The generalized FEM enjoys, for this class of problems, the same level of flexibility and user friendliness as meshfree methods while being more computationally efficient. In simple two-dimensional cases, a crack can be accurately modeled in partition of unity methods using enrichment functions from the asymptotic expansion of the
0045-7825/$ - see front matter ! 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.08.017
11
GFEMgl for Problems with Localized Non- Linearities • Model Problem: Simulation of propagating cracks using cohesive fracture models tcoh ft b
Γ Gt
Non-linear traction- separation law
C0
ΩG
tG = t
Cohesive(crack( Γ
coh
CDk
Ccoh
Γ Gu
u
uG = u
Find u 2 H 1 (⌦G ), such that 8 u 2 H 1 (⌦G ) Z
⌦G
=
Z
rs ( u) : ⌦G
(u) dV +
u · b dV +
Z
t G
Z
coh
[[u]] · tcoh ([[u]]) dS + ⌘
u · t¯ dS + ⌘
Z
u G
Z
u G
u · u dS
¯ dS u·u 12
Global-Local Enrichments for Problems with Localized Non-Linearities • Three-Point Bending Beam P, u 80
Expected crack path
150 50
stress-free crack
50
50
700 All lengths in [mm]
•
Bi-linear intrinsic CZM [ Park et al. 2 008]
Typical FEM discretization [Park et al. 2008]
Goals: • Solve problem on a coarse global mesh. • Few non-linear iterations at global scale. 13
Global-Local Enrichments for Problems with Localized Non-Linearities* Let unG 2 SnG (⌦) be a GFEM approximation of global problem at load step n. Global-local enrichments are used in the definition of SnG (⌦).
Compute a rough and cheap estimate of global solution at next load step. Define un+1 G,0 =
n+1 n n uG
un+1 G,0 is used to prescribe boundary conditions for a non-linear local problem as defined next. *[Jongheon Kim and C.A. Duarte, ijnme, 2015]
14
Global-Local Enrichments for Problems with Localized Non-Linearities § Solve following non-linear local problem at load step n+1 using, e.g., hp-GFEM
n+1 n+1 n+1 Find un+1 2 S (⌦ ) such that, 8 u 2 S L L L L L (⌦L )
Z
⌦L
+ +⌘
un+1 : L
rs
Z
Z
L\
(
un+1 dV + L
u t L \( G [ G )
u L\ G
un+1 L
)
Z
coh
n+1 coh [[un+1 ]] · t ([[u L L L ]]) dS + ⌘
un+1 · un+1 dS = L L
¯ dS + ·u
Z
L\
(
Z
⌦L
u t L \( G [ G )
)
un+1 · b dV + L un+1 L
·
[tn+1 G
⇣
Z
t L\ G
un+1 G,0
⌘
Z
u L\ G
n+1 un+1 · u dS L L
un+1 · t¯ dS L
+ un+1 G,0 ] dS 15
Global-Local Enrichments for Problems with Localized Non-Linearities • Global space enriched with non-linear local solution
n+1 ↵ (x)
= '↵ (x)un+1 L (x)
n+1 gl,n+1 gl FEM un+1 (x) 2 S ' u , ↵ 2 I (⌦ ) = S + ↵ G ↵ G G G 9 8 n+1, (x) > u ↵ uL > > > = < n+1, where ugl,n+1 (x) = , u↵ , v ↵ , w ↵ 2 R v ↵ uL (x) ↵ > > > > ; : n+1, w ↵ uL (x)
• Discretization spaces updated on-the-fly with global-local enrichment functions
16
Global-Local Enrichments for Problems with Localized Non-Linearities • Linear global problem at load step n+1 n+1 n+1 n+1 Find un+1 2 S (⌦ ) such that, 8 u 2 S G G G G G (⌦G )
Z
⌦G
=
Z
un+1 : G
rs ⌦G
un+1 dV + G
un+1 · b dV + G
Z
t G
Z
coh
n+1 n+1 n+1 [[un+1 G ]] · CD (uL )[[uG ]] dS + ⌘
un+1 · t¯ dS + ⌘ G
Z
u G
Z
u G
un+1 · un+1 dS G G
¯ dS un+1 ·u G
The cohesive secant matrix is given by Cn+1 D
2
n+1, CD =4 0 0
0 n+1, CD 0
3
0 5 0 n+1, CD
where n+1, CD
n+1 t> t
