Topics in Boundary Element Research - GBV

Topics in Boundary Element Research Edited by C. A. Brebbia

Volume 3: Computational Aspects

With 126 Figures and 28 Tables

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Contents

1 NUMERICAL CONVERGENCE OF BOUNDARY SOLUTIONS IN TRANSIENT HEAT CONDUCTION PROBLEMS 1 Abstract 1

1.1 Introduction 1 1.2 Boundary Element Approximations 2 1.2.1 Motivation 2 1.2.2 Boundary Integral Equations 2 1.2.3 Boundary Element Discretization 4 1.2.4 Interpolation of Functions 5 1.2.5 Discretization of the Equations 5 1.2.̂ Problem with Boundary Condition of the Linear Radiation Type . . . 6 1.3 Convergence and Stability 8 1.3.1 Preliminary Estimates 8 1.3.2 Theorem on Convergence and Stability 11 1.3.3 Numerical Demonstrations 12 1.4 Singularities and Unbounded Domains 16 1.4.1 Re-entrant Corners 17 1.4.2 Slit Boundary 19 1.4.3 Interzonal Singularity 20 1.4.4 Infinite Domain 21 1.5 Conclusions 23

Acknowledgements 23 References 23

2 NEW INTEGRAL EQUATION APPROACH TO VISCOELASTIC PROBLEMS 25

2.1 Introduction 25 2.2 Preliminaries 26 2.3 Boundary Integral Equation in Space and Time 27 2.4 Alternative Approach — Incremental Formulation 30 2.5 Numerical Results and Discussion 31 2.6 Concluding Remarks 34

Acknowledgements 34 References 35

XII CONTENTS

3 NUMERICAL INTEGRATION 36 3.1 Introduction 36 3.2 Integration over One-Dimensional Domains 37 3.2.1 Basic Rules 38 3.2.2 Gauss Formulas 40 3.2.3 Singular Integrals 41 3.3 Integration for Two-Dimensional Problems 44 3.3.1 Regulär Boundary Integrals 44 3.3.2 Singular Boundary Integrals 45 3.3.3 Domain Integrals 46 3.4 Integration for Three-Dimensional Problems 48 3.4.1 Regulär Boundary Integrals 48 3.4.2 Singular Boundary Integrals 49 3.4.3 Domain Integrals 49

Acknowledgements 49 References 49

4 COMPUTATIONAL ASPECTS OF THE BOUNDARY ELEMENT METHOD 51

4.1 Introduction 51 4.2 Formulation and Numerical Treatment 53 4.3 A Boundary Element Program Organization 57 4.4 Storage and Management of Data 63 4.4.1 Variables in a Boundary Element Program 63 4.4.2 Data Structure 64 4.5 Data Input 73 4.6 Computation of the Matrices 84 4.6.1 Computation of the Integration Coefficients 84 4.6.1.1 Analytical and Semianalytical Integration over an Element Containing

the Collocation Point 86 4.6.1.2 Special Numerical Integration Techniques

over an Element Containing the Collocation Point 89 4.6.1.3 Adaptive Numerical Integration Schemes 93 4.6.1.4 Computation of the Diagonal Terms of the Matrix B 96 4.6.2 Boundary Conditions and Assemblage of the Matrices 106 4.6.3 Symmetries 112 4.6.4 Piecewise Heterogeneous Bodies . 114 4.6.4.1 Condensation Process 115 4.6.4.2 Ordering of the System of Equations 118 4.7 Solution of the System of Equations 118

References 127 Other Bibliography 131

5 THE EDGE FUNCTION METHOD (E.F.M.) FOR CRACKS, CAVITIES AND CURVED BOUNDARIES IN ELASTOSTATICS 132

5.1 Introduction 132 5.2 The Edge Function Method (E.F.M.) (a Qualitative Description) . . . . 133

CONTENTS XIII

5.3 Complex Displacement Method for Elastostatics 134 5.4 Displacements and Stresses for Arbitrary Cartesian Axes 135 5.5 Basic Problems 137

References 158 Appendix A — Harmonie Fitting 158 Appendix B - Data Inputs for 2-D Elastostatic Program "EQUINP".. 163 Appendix C - Program EQUINP 166

6 THEORETICAL AND PRACTICAL ASPECTS OF MULTIGRID METHODS IN BOUNDARY ELEMENT CALCULATIONS 168 Summary 168

6.1 Introduction 168 6.2 Boundary Integral Equations 170 6.3 Approximation of Boundary Integral Equations 174 6.4 Practical Aspects of Multigrid Methods 176 6.5 Smoothing Property of the Relaxation Process 182 6.6 Theoretical Aspects of Multigrid Methods 184 6.7 Numerical Results 186 6.8 Conclusions and Recommendations 187

References 188

7 COMPLEX VARIABLE BOUNDARY ELEMENTS IN COMPUTATIONAL MECHANICS 191

7.1 Introduction 191 7.2 A Complex Variable Boundary Element Approximation Model 191 7.3 The Analytical Function Defined by the Approximator d> (z) 200 7.4 A Constant Boundary Element Method 202 7.5 The Complex Variable Boundary Element Method 203 7.6 Approximation Error from the CVBEM 206 7.7 A CVBEM Modeling Strategy to Reduce Approximation Error 209 7.8 Expansion of the Hk Approximation Function 213 7.9 Upper Half Plane Boundary Value Problems 217 7.10 The Approximate Boundary for Error Analysis 219 7.11 Locating Additional Nodal Points on r 223 7.12 Sources and Sinks 229 7.13 Regional Inhomogeneity 230 7.14 The Poisson Equation 231 7.15 Computer-Aided-Analysis and the CVBEM 232

References 234

8 POTENTIAL PROBLEMS 235 8.1 Introduction 235 8.2 Review of the Theory 236 8.3 Numerical and Computational Aspects 238 8.3.1 Integration 238 8.3.2 Infinite Regions 239 8.3.3 Robin Boundary Conditions 240

XIV CONTENTS

8.3.4 Symmetry 241 8.4 Description of the Computer Program 241 8.4.1 Main Program and Data Structure 241 8.4.2 Subroutine INPUT 244 8.4.3 Subroutine GENER 246 8.4.4 Subroutine SYMMET 247 8.4.5 Subroutine GAUSS 247 8.4.6 Subroutine FMAT 248 8.4.7 Subroutine INTE 250 8.4.8 Subroutine SLNPD 251 8.4.9 Subroutine REORD 253 8.4.10 Subroutine INTER 253 8.4.11 Subroutine OUTPT 254 8.5 Applications 254 8.5.1 Pipe of Elliptical Cross-Section 255 8.5.2 Circular Cavity in Infinite Medium 256 8.5.3 Rectangular Concrete Column 260

References 264

9 ELASTOSTATIC PROBLEMS 265 9.1 Introduction 265 9.2 Outline of Theory 265 9.3 Numerical and Computational Procedures 270 9.3.1 Integration 270 9.3.2 Extemal Problems in Infinite Regions 271 9.3.3 Stresses on the Boundary 273 9.3.4 Surface Traction Discontinuities 273 9.3.5 Symmetry 274 9.4 Computer Program 275 9.4.1 Main Program and Data Structure 276 9.4.2 Subroutine INPUT 278 9.4.3 Subroutine MATRX 281 9.4.4 Subroutine FUNC 283 9.4.5 Subroutine SLNPD 285 9.4.6 Subroutine OUTPT 286 9.4.7 Subroutine FENC 288 9.4.8 Examples 288

References 294

SUBJECT INDEX 295