Griebel / Schweitzer Meshfree Methods for Partial Differential Equations III

1. Auflage 2007
ISBN: 978-3-540-46222-4
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 57, 304 Seiten

Reihe: Lecture Notes in Computational Science and Engineering

Meshfree methods for the numerical solution of partial differential equations are becoming more and more mainstream in many areas of applications. This volume represents the state-of-the-art in meshfree methods. It consists of articles which address the different meshfree techniques, their mathematical properties and their application in applied mathematics, physics and engineering.

Autoren/Hrsg.

Griebel, Michael

Schweitzer, Marc Alexander

Inhaltsverzeichnis

1;Preface;5
2;Contents;6
3;Local Maximum-Entropy Approximation Schemes;8
3.1;1 Introduction;8
3.2;2 Formulation;9
3.3;3 Properties;17
3.4;4 Other Interpretations: Relative Entropy;19
3.5;5 Applications in Galerkin Methods;21
3.6;6 Conclusions;22
3.7;References;22
4;Genetic Algorithms for Meshfree Numerical Integration;24
4.1;1 Introduction;24
4.2;2 Problem Formulation and Discretization;26
4.3;3 Numerical Integration;37
4.4;4 Numerical Examples;41
4.5;5 Concluding Remarks;46
4.6;References;46
5;An RBF Meshless Method for Injection Molding Modelling;48
5.1;1 Model Equations;48
5.2;2 Pressure Field;50
5.3;3 Front Displacement;56
5.4;4 Conclusions;62
5.5;References;62
6;Strain Smoothing for Stabilization and Regularization of Galerkin Meshfree Methods;64
6.1;1 Introduction;64
6.2;2 Conforming Strain Smoothing for Rank Instability;65
6.3;3 Additional Stabilization for Spurious Nonzero Energy Modes;69
6.4;4 Gradient Strain Smoothing Regularization for Material Instability in Strain Localization;72
6.5;5 Application to Fragment Penetration Problems;77
6.6;6 Conclusions;80
6.7;References;81
7;Fuzzy Grid Method for Lagrangian Gas Dynamics Equations;83
7.1;1 Smooth Particle Hydrodynamics;83
7.2;2 Fuzzy Grid;85
7.3;3 The Unsymmetrical Approach to Modify Weights;86
7.4;4 The Symmetrical Approach to Modify Weights;87
7.5;5 Numerical Examples;88
7.6;6 Conclusion;92
7.7;References;92
8;New Shape Functions for Arbitrary Discontinuities without Additional Unknowns;93
8.1;1 Introduction;93
8.2;2 Level-Set Method;95
8.3;3 Moving Least-Squares Method;95
8.4;4 Design of Special Shape Functions;97
8.5;5 Coupling;100
8.6;6 Numerical Results;102
8.7;7 Conclusion;107
8.8;References;108
9;A Meshless BEM for 2-D Stress Analysis in Linear Elastic FGMs;110
9.1;1 Introduction;110
9.2;2 Formulation of Boundary-Domain Integral Equations;112
9.3;3 Transformation of Domain-Integrals into Boundary Integrals;114
9.4;4 System of Linear Algebraic Equations;115
9.5;5 Computation of Stresses;116
9.6;6 Numerical Example;118
9.7;7 Conclusions;122
9.8;References;123
10;A Particle-Partition of Unity Method Part VII: Adaptivity;125
10.1;1 Introduction;125
10.2;2 Partition of Unity Method;126
10.3;3 Adaptive Multilevel Solution;133
10.4;4 Numerical Results;142
10.5;5 Concluding Remarks;150
10.6;References;151
11;Enriched Reproducing Kernel Particle Approximation for Simulating Problems Involving Moving Interfaces;152
11.1;1 Introduction;152
11.2;2 Enriched Functional Approximations;153
11.3;3 Definition of the Continuous Shape Functions with Discontinuous Derivatives;159
11.4;4 Enrichment Function Expressed by a Polynomial;161
11.5;5 Numerical Results;163
11.6;6 Conclusion;166
11.7;References;166
12;Deterministic Particle Methods for High Dimensional Fokker- Planck Equations;168
12.1;1 A Polymer Model;168
12.2;2 Meshfree Solution Methods;173
12.3;3 QMC Product Measures;175
12.4;4 QMC Simulations;181
12.5;References;185
13;Bridging Scale Method and Its Applications;187
13.1;1 Bridging Scale Fundamentals;187
13.2;2 Reduction of the MD Domain;192
13.3;3 Cauchy-Born Rule;199
13.4;4 Atomistic Model;200
13.5;5 Dynamic Crack Propagation in Three Dimensions;200
13.6;6 Discussion on the Bridging Scale Method;205
13.7;References;206
14;A New Stabilized Nodal Integration Approach;208
14.1;1 Introduction;208
14.2;2 Formulation;209
14.3;3 Results;214
14.4;4 Discussion;217
14.5;References;218
15;Multigrid and M-Matrices in the Finite Pointset Method for Incompressible Flows;219
15.1;1 Introduction;219
15.2;2 FPM for the Incompressible Navier-Stokes Equation;220
15.3;3 Finite Difference Approximation;221
15.4;4 Conditions for the Existence of Positive Stencils;224
15.5;5 Optimization for Interior Points;228
15.6;6 Setting Up the System Matrix;229
15.7;7 Algebraic Multigrid;230
15.8;8 Numerical Results;231
15.9;9 Conclusions and Outlook;233
15.10;References;234
16;Assessment of Generalized Finite Elements in Nonlinear Analysis;235
16.1;1 Introduction;235
16.2;2 Formulation;236
16.3;3 Numerical Examples;241
16.4;4 Conclusion;246
16.5;References;246
17;A Meshfree Method for Simulations of Interactions between Fluids and Flexible Structures;248
17.1;1 Introduction;248
17.2;2 Governing Equations;249
17.3;3 Numerical Schemes;251
17.4;4 Numerical Examples;257
17.5;5 Conclusion;261
17.6;References;263
18;Goal Oriented Error Estimation for the Element Free Galerkin Method;264
18.1;1 Introduction;264
18.2;2 Basics on Output Oriented Error Estimation in FEM;266
18.3;3 Extension to Element Free Galerkin;269
18.4;4 Numerical Results;274
18.5;5 Concluding Remarks;279
18.6;References;280
19;Bubble and Hermite Natural Element Approximations;282
19.1;1 Introduction;283
19.2;2 Review of the Natural Element Method;284
19.3;3 Hierarchical Bubble Functions in the Natural Element Method;286
19.4;4 Hermite-Natural Element Formulation;292
19.5;5 Conclusion;295
19.6;References;296
20;Color Plates;298