Givoli Numerical Methods for Problems in Infinite Domains

1. Auflage 2013
ISBN: 978-1-4832-9108-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 33, 316 Seiten, Web PDF

Reihe: Studies in Applied Mechanics

This volume reviews and discusses the main numerical methods used today for solving problems in infinite domains. It also presents in detail one very effective method in this class, namely the Dirichlet-to-Neumann (DtN) finite element method. The book is intended to provide the researcher or engineer with the state-of-the-art in numerical solution methods for infinite domain problems, such as the problems encountered in acoustics and structural acoustics, fluid dynamics, meteorology, and many other fields of application. The emphasis is on the fundamentals of the various methods, and on reporting recent progress and forecasting future directions. An appendix at the end of the book provides an introduction to the essentials of the finite element method, and suggests a short list of texts on the subject which are categorized by their level of mathematics.

Autoren/Hrsg.

Givoli, D.

Inhaltsverzeichnis

1;Front Cover
;1
2;Numerical Methods for Problems in Infinite Domains;4
3;Copyright Page
;5
4;Table of Contents;12
5;Dedication;6
6;Preface;8
7;PART I;18
7.1;Chapter 1. Introduction and Overview;20
7.1.1;1.1 Infinite (Unbounded) Domains;20
7.1.2;1.2 Numerical Difficulties;21
7.1.3;1.3 Main Numerical Methods;27
7.1.4;1.4 Matching Analytic and Numerical Solutions and the DtN Method;29
7.2;Chapter 2. Boundary Integral and Boundary Element Methods;36
7.2.1;2.1 Fundamentals;36
7.2.2;2.2 Standard Indirect BI/BE Methods;38
7.2.3;2.3 Galerkin Indirect BI/BE Methods;42
7.2.4;2.4 The Standard Direct BI/BE Method;43
7.2.5;2.5 The Galerkin Direct BI/BE Method;45
7.2.6;2.6 General Remarks on BI/BE Methods;46
7.2.7;2.7 Coupled FE-BE Methods;48
7.2.8;2.8 Boundary Integral Methods for Wave Problems;52
7.3;Chapter 3. Artificial Boundary Conditions and NRBCs;56
7.3.1;3.1 Artificial Boundary Conditions;56
7.3.2;3.2 The Sommerfeld Radiation Condition and Related Issues;57
7.3.3;3.3 Demonstration of a Fundamental Difficulty;66
7.3.4;3.4 General Remarks on NRBCs;68
7.4;Chapter 4. Local Non-Reflecting Boundary Conditions;72
7.4.1;4.1 Introduction;72
7.4.2;4.2 The Engquist and Majda NRBC and Related Work;73
7.4.3;4.3 The Bayliss and Turkel NRBC;77
7.4.4;4.4 The Higdon NRBC and an Equivalence Theorem;78
7.4.5;4.5 Additional NRBCs for the Scalar Wave Equation;80
7.4.6;4.6 NRBCs for Fluid-Structure Interaction Problems;82
7.4.7;4.7 NRBCs in Gasdynamics, Hydrodynamics and Meteorology;83
7.4.8;4.8 NRBCs for Elastic Waves;86
7.4.9;4.9 NRBCs for Electromagnetic Waves;89
7.5;Chapter 5. Nonlocal Non-Reflecting Boundary Conditions;90
7.5.1;5.1 Introduction;90
7.5.2;5.2 Various Nonlocal NRBCs;91
7.5.3;5.3 Dirichlet-to-Neumann NRBCs;95
7.5.4;5.4 Concluding Remarks on NRBCs;98
7.6;Chapter 6. Special Numerical Procedures for Unbounded and Large Domains;100
7.6.1;6.1 Mapping to a Finite Domain;100
7.6.2;6.2 The Smith Technique;102
7.6.3;6.3 Filtering and Damping Schemes;105
7.6.4;6.4 Extrapolation Schemes;108
7.6.5;6.5 Special Meshes;109
7.6.6;6.6 Sub-structuring and Domain Decomposition;110
7.6.7;6.7 Infinite Elements;112
7.6.8;6.8 Matching Numerical and Analytic Solutions;115
8;PART II;118
8.1;Chapter 7. The DtN Method;120
8.1.1;7.1 Introduction;120
8.1.2;7.2 An Exact Boundary Condition on an Artificial Boundary;123
8.1.3;7.3 The DtN Finite Element Method;125
8.1.4;7.4 DtN Boundary Conditions for Laplace's Equation;129
8.1.5;7.5 DtN Boundary Conditions for Linear Elastostatics;135
8.1.6;7.6 DtN Finite Element Formulation for Linear Elastostatics;139
8.1.7;7.7 Numerical Experiments for Laplace's Equation;141
8.1.8;7.8 Numerical Experiments for Elastostatics;145
8.2;Chapter 8. Computational Aspects of the DtN Method;150
8.2.1;8.1 Symmetry;150
8.2.2;8.2 Positivity;151
8.2.3;8.3 Nonlocality and Sparseness;153
8.2.4;8.4 Implementation;156
8.2.5;8.5 Computational Cost;158
8.2.6;8.6 Geometrical Symmetry;160
8.2.7;8.7 Convergence;160
8.2.8;8.8 Choosing the Computational Parameters;164
8.2.9;8.9 Using an Ellipse as the Artificial Boundary B;166
8.3;Chapter 9. Application of the DtN Method to Beam and Shell Problems;170
8.3.1;9.1 Beams and Axisymmetric Cylindrical Shells: Introduction;170
8.3.2;9.2 Beams and Axisymmetric Cylindrical Shells: Finite Element Formulation;172
8.3.3;9.3 Beams and Axisymmetric Cylindrical Shells: DtN Boundary Conditions;178
8.3.4;9.4 Axisymmetric Cylindrical Shells: Numerical Experiments;184
8.3.5;9.5 Asymmetric Shells: Introduction;187
8.3.6;9.6 Asymmetric Shells: Finite Element Formulation;189
8.3.7;9.7 Asymmetric Shells: DtN Boundary Conditions;194
8.3.8;9.8 Asymmetric Shells: Numerical Experiments;199
8.4;Chapter 10. The DtN Method for Time-Harmonic Waves;206
8.4.1;10.1 Introduction;206
8.4.2;10.2 The Reduced Wave Equation: DtN Boundary Conditions;206
8.4.3;10.3 The Reduced Wave Equation: Numerical Experiments;210
8.4.4;10.4 The Reduced Wave Equation: Localized DtN Boundary Conditions;215
8.4.5;10.5 Uniqueness and Convergence Issues;218
8.4.6;10.6 Elastic Waves: DtN Boundary Conditions;220
8.4.7;10.7 Elastic Waves: Finite Element Formulation;225
8.4.8;10.8 Elastic Waves: Localized DtN Boundary Conditions;225
8.4.9;10.9 Elastic Waves: Numerical Experiments;228
8.5;Chapter 11. The DtN Method for Time Dependent Problems;236
8.5.1;11.1 Introduction;236
8.5.2;11.2 Finite Element Formulation with a Time Dependent DtN Boundary Condition;238
8.5.3;11.3 Time Dependent DtN Boundary Conditions;243
8.5.4;11.4 A Semi-Discrete DtN Method;246
8.5.5;11.5 Semi-Discrete DtN Boundary Conditions;250
8.5.6;11.6 The Semi-Discrete DtN Method: Computational Aspects;254
8.5.7;11.7 The Semi-Discrete DtN Method: Numerical Examples;257
9;Appendix: The Finite Element Method;264
9.1;A.l A Boundary Value Problem (P);264
9.2;A.2 The Minimization Problem (M);264
9.3;A.3 The Problem (W);265
9.4;A.4 Equivalence of the Problems (P), (M) and (W);266
9.5;A.5 The Rayleigh-Ritz Method;267
9.6;A.6 The Galerkin Method;269
9.7;A.7 The Big Picture;270
9.8;A.8 The Basics of the Finite Element Method;271
9.9;A.9 The Smaller Details;274
10;References;276
11;Index;306