Boundary Integral Equation Methods for Solids and Fluids

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ISBN: 9780471971849 | 0471971847
Cover: Hardcover
Copyright: 7/9/1999

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The boundary element method is more appropriate than the finite element method to tackle linear, wave propagation, infinite domain, mobile boundaries and unknown boundaries problems. In some engineering applications, both methods are combined. This book presents the mathematical basis of this method and its computer implementation. Numerous applications to fluid mechanics, mechanics of solids, acoustics and electromagnetism are developed.

Preface

(2)

Acknowledgements

xiii

(2)

Introduction

References

xix

1 Basic principles and fields of application

(14)

1.1 Basic principles

(2)

1.2 Overview of fields of application

(2)

1.3 Heat conduction

(1)

1.4 Fluid flow

(2)

1.5 Electromagnetism

(1)

1.6 Linear elastostatics

(6)

Part I STATICS

(136)

2 Integral equations for the Poisson equation

(24)

2.1 Integral representation

(7)

2.2 Boundary integral equation

(3)

2.3 Exterior problems and conditions at infinity

(3)

2.4 Indirect integral formulations

(2)

2.5 Two-dimensional problems

(3)

Appendix 2.A Some useful formulas

(1)

Appendix 2.B Derivation of the fundamental solution of the Laplacian

(2)

Appendix 2.C Integrability of r,(n)/r^(2)

(1)

References

(1)

3 Boundary element method for potential problems

(28)

3.1 Discretization of the boundary

(6)

3.2 Discretization of the unknowns

(1)

3.3 Assembly and solution of the discrete problem

(4)

3.4 Evaluation of nonsingular integrals

(6)

3.5 Evaluation of singular integrals

(5)

3.6 Evaluation of domain integrals

(2)

3.7 Using subregions

(3)

References

(1)

4 Integral equations and representations for elastostatics

(42)

4.1 Maxwell-Betti reciprocity identity

(1)

4.2 Fundamental solutions of linear elastostatics

(5)

4.3 Integral representation formulas

(2)

4.4 Displacement boundary integral equation

(5)

4.5 The boundary element method in elastostatics

(2)

4.6 Integral equations for exterior problems

(3)

4.7 The boundary element method

(7)

4.8 The direct approach for singular element integration

(3)

4.9 Elasticity in plane strain

(3)

Appendix 4.A The three-dimensional Kelvin solution

(4)

Appendix 4.B Integral of the Kelvin traction vector

102

(2)

Appendix 4.C Half-space fundamental solution

104

(2)

Appendix 4.D Fundamental solution for two bonded half-spaces

106

(1)

References

107

(2)

5 Representation of gradients and stresses on the boundary

109

(34)

5.1 Motivation

109

(3)

5.2 Definition using a limiting process

112

(1)

5.3 Second-order regularization

113

(2)

5.4 Integration by parts and first-order regularization

115

(3)

5.5 Extension to elastostatics

118

(4)

5.6 Indirect boundary integral equations

122

(4)

5.7 Comments

126

(2)

5.8 Boundary element discretization

128

(3)

5.9 The direct approach

131

(4)

Appendix 5.A Integral identities for the Kelvin solution

135

(3)

Appendix 5.B Proof of expression (5.16) of the free term

138

(1)

Appendix 5.C Proof of expression (5.28) for the elastic free term

139

(2)

References

141

(2)

6 Some classical general results

143

(8)

6.1 Fundamental problems of potential theory

143

(1)

6.2 Singular boundary integral equations of elasticity theory

144

(1)

6.3 Green tensors for bounded domains

145

(1)

6.4 Fundamental problems of elasticity theory

146

(1)

References

147

(4)

Part II WAVE AND EVOLUTION PROBLEMS

151

(68)

7 Scalar and elastic wave problems in the time domain

151

(32)

7.1 Wave equation and fundamental solutions

151

(3)

7.2 Integral representation formulas

154

(2)

7.3 Boundary integral equations

156

(3)

7.4 Exterior problems and conditions at infinity

159

(3)

7.5 The boundary element method in dynamics

162

(6)

7.6 Governing equations of linear elastodynamics

168

(2)

7.7 Elastodynamic fundamental solutions

170

(4)

7.8 Integral representation of an elastodynamic state

174

(2)

7.9 Boundary integral equations in elastodynamics

176

(2)

7.10 Exterior problems and conditions at infinity

178

(3)

References

181

(2)

8 Scalar and elastic wave problems in the frequency domain

183

(26)

8.1 Helmholtz equation and fundamental solutions

184

(2)

8.2 Integral equations and representations

186

(2)

8.3 Indirect integral formulations and the fictitious eigenvalue problem

188

(3)

8.4 The boundary element method for frequency-domain problems

191

(2)

8.5 Frequency-domain elastodynamics

193

(4)

8.6 Integral equations and representations for frequency-domain elastodynamics

197

(2)

8.7 Numerical examples in time-harmonic elastodynamics

199

(9)

References

208

(1)

9 Diffusion, fluid flow

209

(10)

9.1 Diffusion equation

209

(3)

9.2 Fluid flow

212

(3)

References

215

(4)

Part III ADVANCED TOPICS

219

(80)

10 Symmetric Galerkin boundary integral formulations

219

(30)

10.1 Basic idea

219

(1)

10.2 Direct Galerkin formulation of Dirichlet and Neumann problems

220

(4)

10.3 Indirect Galerkin formulation of Dirichlet and Neumann problems

224

(1)

10.4 Dirichlet and Neumann scalar wave problems

225

(1)

10.5 Some convergence results

226

(2)

10.6 Numerical implementation

228

(3)

10.7 Galerkin integral formulation for mixed boundary-value problems

231

(3)

10.8 Fluid-structure coupling in acoustics

234

(2)

10.9 Linear elasticity

236

(4)

10.10 Numerical examples

240

(4)

Appendix 10.A Regularized expressions of I(1) and I(2)

244

(1)

References

245

(4)

11 Exploitation of geometrical symmetries

249

(18)

11.1 Introduction

249

(1)

11.2 Geometrical symmetry in boundary-value problems

250

(7)

11.3 Boundary integral equations and geometrical symmetry

257

(2)

11.4 Discretization of the integral equation

259

(1)

11.5 Integral representations at interior points

260

(2)

11.6 Exploitation of symmetry: recapitulation

262

(2)

11.7 Numerical example in elastodynamics

264

(1)

References

265

(2)

12 Domain differentiation of boundary integral equations

267

(32)

12.1 Motivation

267

(2)

12.2 Differentiation in a domain perturbation

269

(3)

12.3 Differentiation of an objective function

272

(1)

12.4 Governing boundary integral equation for the first-order domain derivative of the state

273

(5)

12.5 Governing boundary integral equation for the second-order domain derivative of the state

278

(3)

12.6 Comments about the derivative integral equations

281

(3)

12.7 Boundary element discretisation

284

(2)

12.8 Derivative boundary integral equations in elastostatics

286

(2)

12.9 The adjoint state method

288

(2)

12.10 Example: energy release rate in fracture mechanics

290

(1)

Appendix 12.A Derivative of the differential area element

290

(1)

Appendix 12.B Proof of identity (12.36)

291

(1)

Appendix 12.C Expression of (D(ij)f)^(*), (D(i)f)^(*), (D(i)u(i))^(*)

292

(2)

Appendix 12.D Symmetric expression for K(1)(x, u; (Theta), (Mu))

294

(1)

References

295

(4)

Part IV SOLID MECHANICS

299

(46)

13 Linear fracture mechanics

299

(24)

13.1 Inapplicability of the displacement boundary integral equation

299

(2)

13.2 The multiregion approach

301

(1)

13.3 Displacement discontinuity approach: cracks of arbitrary shape in infinite media

302

(3)

13.4 Displacement discontinuity approach: planar cracks in infinite media

305

(3)

13.5 Displacement discontinuity approach for a bounded body

308

(2)

13.6 Evaluation of stress intensity factors

310

(4)

13.7 Numerical examples

314

(2)

13.8 Boundary element formulation for the energy approach in fracture mechanics

316

(4)

References

320

(3)

14 Initial strains and stresses -- elastoplasticity

323

(22)

14.1 Elastostatic problems with initial strain or stress distributions

323

(4)

14.2 Elastic fields generated by inclusions

327

(2)

14.3 Small-strain elastoplasticity

329

(5)

14.4 Examples

334

(7)

References

341

(4)

APPENDICES

345

(42)

Appendix A Differential operators and integration by parts on surfaces

345

(4)

Appendix B Interpolation functions and numerical integration

349

(10)

B.1 One-dimensional interpolation

350

(1)

B.2 Two-dimensional interpolation

351

(5)

B.3 Gaussian numerical integration rules

356

(1)

References

357

(2)

Appendix C Bibliography

359

(28)

C.1 Books on BIE/BEM

359

(2)

C.2 Articles

361

(24)

C.3 General purpose books

385

(2)

Author index

387

(2)

Subject index

389

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Boundary Integral Equation Methods for Solids and Fluids

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Table of Contents

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