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s. Amini. P. J. Harris, D. T. Wilton
Coupled Boundary and Finite Element Methods for the Solution of the Dynamic Fluid-Structure Interaction Problem
i
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J. Argyris . K.-J. Bathe' A. S. Cakmak . J. Connor' R. McCrory C. S. Desai· K.-P. Holz . F. A. Leckie' G. Pinder' A. R. S. Pont J. H. Seinfeld . P. Silvester· P. Spanos' W. Wunderlich . S. Yip Authors
Dr. Siamak Amini Dept. of Mathematics and Computer Science University of Salford Salford, Lancashire, M 5 4 WT
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Dr. Paul John Harris School of Mathematics and Statistics University of Birmingham Edgbaston, Birmingham, B15 2TT
UK
Dr. David T. Wilton Dept. of Mathematics and Statistics Polytechnic South West Plymouth, Devon, PL4 8AA
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Preface This text considers the problem of the dynamic fluid-structure interaction between a finite elastic structure and the acoustic field in an unbounded fluid-filled exterior domain. The exterior acoustic field is modelled through a boundary integral equation over the structure surface. However, the classical boundary integral equation formulations of this problem either have no solutions or do not have unique solutions at certain characteristic frequencies (which depend on the surface geometry) and it is necessary to employ modified boundary integral equation formulations which are valid for all frequencies. The particular approach adopted here involves an arbitrary coupling parameter and the effect that this parameter has on the stability and accuracy of the numerical method used to solve the integral equation is examined. The boundary integral analysis of the exterior acoustic problem is coupled with a finite element analysis of the elastic structure in order to investigate the interaction between the dynamic behaviour of the structure and the associated acoustic field. Recently there has been some controversy over whether or not the coupled problem also suffers from the non-uniqueness problems associated with the classical integral equation formulations of the exterior acoustic problem. This question is resolved by demonstrating that .the solution to the coupled problem is not unique at the characteristic frequencies and that it is necessary to employ an integral equation formulation valid for all frequencies. Numerical results are presented and discussed for both the uncoupled acoustic problem and the coupled fluid-structure interaction problem for a number of axisymmetric and fully three-dimensional problems. In particular, the method is applied to the coupled problem of a piezoelectric ring sonar transducer transmitting an acoustic signal in water for which reasonable agreement between the theoretical predictions and some experimental results is observed.
Contents 1 INTRODUCTION
1
2 INTEGRAL EQUATION FORMULATIONS OF THE EXTERIOR HELMHOLTZ PROBLEM
3
2.1 2.2
2.3 2.4
Introduction.................................. 3 Basic Integral Equation Formulations . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Indirect Integral Equation Formulations of the Exterior Helmholtz Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , 6 2.2.2 Direct Integral Equation Formulations of the Exterior Helmholtz Problem. . . . . . . . . . . . . . . 7 9 Basic Integral Equation Theory . . . . . . Improved Integral Equation Formulations. 15 2.4.1 Indirect Formulations . . . . . . . . 16 2.4.2 The Method of Schenck and Related Direct Formulations . 17 18 2.4.3 The Burton and Miller Formulation . . . 2.4.4 Modified Green's Function Formulations 22 2.4.5 Comparison of Approaches . . . . . . . . 23
3 NUMERICAL SOLUTION OF THE EXTERIOR HELMHOLTZ PROBLEM 25 3.1
3.2 3.3
3.4 3.5
Numerical Methods for Solving Integral Equations. 3.1.1 Nystrom Method . . . . . . 3.1.2 Degenerate Kernel Method. . . . . . . . . 3.1.3 Projection Methods. . . . . . . . . . . . . 3.1.4 An Application of the Collocation Method Surface Representation . . . . . Numerical Quadrature . . . . . . . . 3.3.1 Axisymmetric Elements . . . 3.3.2 Three-Dimensional Elements. The Choice of the Coupling Parameter Numerical Results. . . . . . . . . . . .
25 25 26 27 29 32 35 36 38 40 50
4 THE DYNAMIC FLUID-STRUCTURE INTERACTION PROBLEM 57 4.1 4.2
4.3 4.4 4.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite Element Analysis of the Structure . . . . . . . . . . . . . . 4.2.1 Finite Element Analysis of an Axisymmetric Structure .. 4.2.2 Finite Element Analysis of a Three-Dimensional Structure The Coupled Equations of Motion . . . . . . The Conditioning of the Coupled Equations Numerical Results . . . . . . . . . . . . . . .
57 58 59 62 65 68 71
VI
5 THE DETERMINATION OF THE RESPONSE FROM SONAR TRANSDUCERS 84 5.1 An Introduction to Piezoelectric Sonar Transducers . 5.2 Loading on the Structure. . . . . . . . . . . . 5.3 Experimental Determination of the Response. 5.4 Results and Conclusions . . . . . . . . . . . .
REFERENCES APPENDIX A. An Analytical Solution for a Hollow Elastic Sphere in an Acoustic Medium
84 85 86 87 95
106
