Numerical Methods for Problems in Infinite Domains (eBook)

Front Cover 
1 
Numerical Methods for Problems in Infinite Domains 4
Copyright Page 
5 
Table of Contents 12
Dedication 6
Preface 8
PART I 18
Chapter 1. Introduction and Overview 20
1.1 Infinite (Unbounded) Domains 20
1.2 Numerical Difficulties 21
1.3 Main Numerical Methods 27
1.4 Matching Analytic and Numerical Solutions and the DtN Method 29
Chapter 2. Boundary Integral and Boundary Element Methods 36
2.1 Fundamentals 36
2.2 Standard Indirect BI/BE Methods 38
2.3 Galerkin Indirect BI/BE Methods 42
2.4 The Standard Direct BI/BE Method 43
2.5 The Galerkin Direct BI/BE Method 45
2.6 General Remarks on BI/BE Methods 46
2.7 Coupled FE-BE Methods 48
2.8 Boundary Integral Methods for Wave Problems 52
Chapter 3. Artificial Boundary Conditions and NRBCs 56
3.1 Artificial Boundary Conditions 56
3.2 The Sommerfeld Radiation Condition and Related Issues 57
3.3 Demonstration of a Fundamental Difficulty 66
3.4 General Remarks on NRBCs 68
Chapter 4. Local Non-Reflecting Boundary Conditions 72
4.1 Introduction 72
4.2 The Engquist and Majda NRBC and Related Work 73
4.3 The Bayliss and Turkel NRBC 77
4.4 The Higdon NRBC and an Equivalence Theorem 78
4.5 Additional NRBCs for the Scalar Wave Equation 80
4.6 NRBCs for Fluid-Structure Interaction Problems 82
4.7 NRBCs in Gasdynamics, Hydrodynamics and Meteorology 83
4.8 NRBCs for Elastic Waves 86
4.9 NRBCs for Electromagnetic Waves 89
Chapter 5. Nonlocal Non-Reflecting Boundary Conditions 90
5.1 Introduction 90
5.2 Various Nonlocal NRBCs 91
5.3 Dirichlet-to-Neumann NRBCs 95
5.4 Concluding Remarks on NRBCs 98
Chapter 6. Special Numerical Procedures for Unbounded and Large Domains 100
6.1 Mapping to a Finite Domain 100
6.2 The Smith Technique 102
6.3 Filtering and Damping Schemes 105
6.4 Extrapolation Schemes 108
6.5 Special Meshes 109
6.6 Sub-structuring and Domain Decomposition 110
6.7 Infinite Elements 112
6.8 Matching Numerical and Analytic Solutions 115
PART II 118
Chapter 7. The DtN Method 120
7.1 Introduction 120
7.2 An Exact Boundary Condition on an Artificial Boundary 123
7.3 The DtN Finite Element Method 125
7.4 DtN Boundary Conditions for Laplace's Equation 129
7.5 DtN Boundary Conditions for Linear Elastostatics 135
7.6 DtN Finite Element Formulation for Linear Elastostatics 139
7.7 Numerical Experiments for Laplace's Equation 141
7.8 Numerical Experiments for Elastostatics 145
Chapter 8. Computational Aspects of the DtN Method 150
8.1 Symmetry 150
8.2 Positivity 151
8.3 Nonlocality and Sparseness 153
8.4 Implementation 156
8.5 Computational Cost 158
8.6 Geometrical Symmetry 160
8.7 Convergence 160
8.8 Choosing the Computational Parameters 164
8.9 Using an Ellipse as the Artificial Boundary B 166
Chapter 9. Application of the DtN Method to Beam and Shell Problems 170
9.1 Beams and Axisymmetric Cylindrical Shells: Introduction 170
9.2 Beams and Axisymmetric Cylindrical Shells: Finite Element Formulation 172
9.3 Beams and Axisymmetric Cylindrical Shells: DtN Boundary Conditions 178
9.4 Axisymmetric Cylindrical Shells: Numerical Experiments 184
9.5 Asymmetric Shells: Introduction 187
9.6 Asymmetric Shells: Finite Element Formulation 189
9.7 Asymmetric Shells: DtN Boundary Conditions 194
9.8 Asymmetric Shells: Numerical Experiments 199
Chapter 10. The DtN Method for Time-Harmonic Waves 206
10.1 Introduction 206
10.2 The Reduced Wave Equation: DtN Boundary Conditions 206
10.3 The Reduced Wave Equation: Numerical Experiments 210
10.4 The Reduced Wave Equation: Localized DtN Boundary Conditions 215
10.5 Uniqueness and Convergence Issues 218
10.6 Elastic Waves: DtN Boundary Conditions 220
10.7 Elastic Waves: Finite Element Formulation 225
10.8 Elastic Waves: Localized DtN Boundary Conditions 225
10.9 Elastic Waves: Numerical Experiments 228
Chapter 11. The DtN Method for Time Dependent Problems 236
11.1 Introduction 236
11.2 Finite Element Formulation with a Time Dependent DtN Boundary Condition 238
11.3 Time Dependent DtN Boundary Conditions 243
11.4 A Semi-Discrete DtN Method 246
11.5 Semi-Discrete DtN Boundary Conditions 250
11.6 The Semi-Discrete DtN Method: Computational Aspects 254
11.7 The Semi-Discrete DtN Method: Numerical Examples 257
Appendix: The Finite Element Method 264
A.l A Boundary Value Problem (P) 264
A.2 The Minimization Problem (M) 264
A.3 The Problem (W) 265
A.4 Equivalence of the Problems (P), (M) and (W) 266
A.5 The Rayleigh-Ritz Method 267
A.6 The Galerkin Method 269
A.7 The Big Picture 270
A.8 The Basics of the Finite Element Method 271
A.9 The Smaller Details 274
References 276
Index 306