Griebel Meshfree Methods for Partial Differential Equations V

1. Auflage 2010
ISBN: 978-3-642-16229-9
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 270 Seiten

The numerical treatment of partial differential equations with particle methods and meshfree discretization techniques is an extremely active research field, both in the mathematics and engineering communities. Meshfree methods are becoming increasingly mainstream in various applications. Due to their independence of a mesh, particle schemes and meshfree methods can deal with large geometric changes of the domain more easily than classical discretization techniques. Furthermore, meshfree methods offer a promising approach for the coupling of particle models to continuous models. This volume of LNCSE is a collection of the papers from the proceedings of the Fifth International Workshop on Meshfree Methods, held in Bonn in August 2009. The articles address the different meshfree methods and their use in applied mathematics, physics and engineering. The volume is intended to foster this highly active and exciting area of interdisciplinary research and to present recent advances and findings in this field.

Autoren/Hrsg.

Griebel, Michael

Inhaltsverzeichnis

1;Preface;5
2;Contents;6
3;Global-local Petrov-Galerkin formulations in the Meshless Finite Difference Method Slawomir Milewski, Janusz Orkisz;8
3.1;1 Introduction;8
3.2;2 Boundary value problem formulations;9
3.3;3 Meshless local Petrov-Galerkin formulations;10
3.4;4 Meshless local Petrov-Galerkin 5 (MLPG5) formulation;11
3.5;5 Basic Meshless Finite Difference Method solution approach;12
3.6;6 Combination of the MFDM and MLPG5;15
3.7;7 Higher order approximation based on correction terms;16
3.8;8 A-posteriori error analysis;17
3.9;9 Adaptive solution approach;18
3.10;10 Error indicators;18
3.11;11 HO MFDM / MLPG5 approach in 1D;19
3.12;12 HO MFDM / MLPG5 approach in 2D;20
3.13;13 Extensions of the MFDM / MLPG5 solution approach;21
3.14;14 The MFDM / MLPG7 approach;22
3.15;15 Numerical examples;24
3.16;16 Final remarks;29
3.17;References;32
4;Treatment of general domains in two space dimensions in a Partition of Unity Method Marc Alexander Schweitzer, Maharavo Randrianarivony;34
4.1;1 Introduction;34
4.2;2 Particle--Partition of Unity Method;35
4.2.1;2.1 Numerical Integration;38
4.3;3 Realization on General Domains;41
4.3.1;3.1 Domain Representation;41
4.3.2;3.2 Clipping a curved multiply connected domain;44
4.3.3;3.3 Decomposition and parametrization;46
4.3.4;3.4 Rectangle-NURBS clipping;49
4.4;4 Numerical Experiments;50
4.5;5 Concluding Remarks;54
4.6;References;54
5;Sampling Inequalities and Support Vector Machines for Galerkin Type Data Christian Rieger;57
5.1;1 Introduction;57
5.2;2 Review on sampling inequalities;58
5.2.1;2.1 Proof Sketch;61
5.3;3 Sampling Inequalities based on Weak Formulations;62
5.3.1;3.1 Sampling inequalities based on Pythagoras law;63
5.4;4 Regularization and Machine Learning;65
5.5;References;68
6;Meshfree Vectorial Interpolation Based on the Generalized Stokes Problem Csaba Gáspár;70
6.1;1 Introduction;70
6.2;2 Vectorial interpolation;71
6.2.1;2.1 Divergence-free interpolation based on the stream function;72
6.2.2;2.2 Multi-elliptic interpolation, scalar problems;73
6.3;3 Multi-elliptic divergence-free interpolation, vectorial problems;75
6.3.1;3.1 The generalized Stokes problem;76
6.4;4 Solution techniques;79
6.4.1;4.1 Uzawa's method;79
6.4.2;4.2 The method of fundamental solutions;80
6.5;5 Summary and conclusions;84
6.6;References;85
7;Pressure XFEM for two-phase incompressible flows with application to 3D droplet problems Sven Gross;86
7.1;1 Introduction;86
7.2;2 Mathematical model;87
7.3;3 Numerical methods;88
7.3.1;3.1 Overview of numerical methods;88
7.3.2;3.2 Pressure XFEM space;88
7.4;4 Analysis of pressure XFEM space;89
7.4.1;4.1 Approximation order of pressure XFEM space;89
7.4.2;4.2 Stabilization of XFEM basis;89
7.5;5 Numerical experiment;90
7.6;References;90
8;Special-relativistic Smoothed Particle Hydrodynamics: a benchmark suite Stephan Rosswog;93
8.1;1 Introduction;93
8.2;2 Relativistic SPH equations from a variational principle;94
8.3;3 Artificial dissipation;96
8.4;4 Test bench;98
8.4.1;4.1 Test 1: Riemann problem 1;98
8.4.2;4.2 Test 2: Riemann problem 2;98
8.4.3;4.3 Test 3: Riemann problem 3;99
8.4.4;4.4 Test 4: Sinusoidally perturbed Riemann problem;100
8.4.5;4.5 Test 5: Relativistic Einfeldt rarefaction test;101
8.4.6;4.6 Test 6: Ultra-relativistic advection;103
8.5;5 Conclusions;105
8.6;References;106
9;An exact particle method for scalar conservation laws and its application to stiff reaction kinetics Yossi Farjoun, Benjamin Seibold;108
9.1;1 Introduction;108
9.2;2 Characteristic Particles and Similarity Solution Interpolant;110
9.3;3 Shock Particles;112
9.3.1;3.1 Evolution of Shock Particles;112
9.3.2;3.2 Interaction of Shock Particles;114
9.4;4 An ``Exact'' ODE Based Method;115
9.4.1;4.1 Approximation of the Initial Conditions;115
9.4.2;4.2 Integration in Time;116
9.5;5 Numerical Error Analysis of the Particle Method;116
9.6;6 Stiff Reaction Kinetics;119
9.7;7 A Particle Method for Stiff Reaction Kinetics;120
9.7.1;7.1 Computational Approach;122
9.8;8 Numerical Results on Reaction Kinetics;123
9.9;9 Conclusions and Outlook;125
9.10;References;126
10;Application of Smoothed Particle Hydrodynamics to Structure Formation in Chemical Engineering Franz Keller, Ulrich Nieken;128
10.1;1 Introduction;129
10.2;2 Smoothed Particle Hydrodynamics Method;131
10.2.1;2.1 Governing equations;131
10.2.2;2.2 Smoothed Particle Hydrodynamics;132
10.3;3 Validation of Single Processes;133
10.4;4 Simulation of the overall process;139
10.5;5 Conclusion and Outlook;141
10.6;6 Acknowledgments;142
10.7;References;142
11;Numerical validation of a constraints-based multiscale simulation method for solids Konstantin Fackeldey, Dorian Krause, Rolf Krause;144
11.1;1 Introduction;144
11.2;2 Coupling with projection-based constraints;145
11.2.1;2.1 Molecular Dynamics;145
11.2.2;2.2 Multiscale Coupling;145
11.2.3;2.3 A method for weak coupling conditions;146
11.2.4;2.4 Damping in zero-temperature simulations;149
11.3;3 Numerical Validation;149
11.3.1;3.1 Comparison with pointwise constraints;150
11.3.2;3.2 Energy and reflection measurements;151
11.3.3;3.3 Mode-I fracture simulation;154
11.4;4 Conclusion;156
11.5;References;156
12;Coupling of the Navier-Stokes and the Boltzmann equations with a meshfree particle and kinetic particle methods for a micro cavity Sudarshan Tiwari, Axel Klar;158
12.1;1 Introduction;158
12.2;2 Governing equations ;160
12.3;3 Numerical methods;162
12.3.1;3.1 Particle Method for the Boltzmann equation;162
12.3.2;3.2 Meshfree particle method for the Navier-Stokes equations;162
12.4;4 Hybrid method;163
12.4.1;4.1 Adaptive grid refinement;164
12.4.2;4.2 Selection of time steps;165
12.4.3;4.3 Coupling condition;166
12.4.4;4.4 Coupling Algorithm;168
12.5;5 Numerical examples;168
12.5.1;5.1 CPU time;170
12.6;6 Conclusion;171
12.7;References;173
13;Accuracy and Robustness of Kinetic Meshfree Method Konark Arora, Suresh M. Deshpande;175
13.1;1 Introduction;175
13.2;2 Least Squares Meshfree Method;176
13.3;3 Method of calculation of Weights in 2-D;178
13.4;4 Higher Order Accuracy in meshfree methods;179
13.5;5 Kinetic Meshfree Method for Euler Equations;181
13.6;6 Higher order accuracy by combining Defect Correction with Entropy Variables (q-LSKUM);181
13.7;7 Results and Discussion;182
13.8;8 Conclusion;186
13.9;References;187
14;Kinetic meshless methods for unsteady moving boundaries V. Ramesh, S. Vivek, S. M. Deshpande;191
14.1;1 Introduction;191
14.2;2 Least Squares Kinetic Upwind Method on Moving Nodes;192
14.3;3 Formulation of LSKUM_MN;193
14.4;4 Advantages of LSKUM_MN;196
14.5;5 Results and Discussion;196
14.5.1;5.1 Turbomachinery cascades;196
14.5.2;5.2 Store separation;203
14.6;6 Conclusions;206
14.7;7 Acknowledgements;207
14.8;References;207
15;Efficient cloud refinement for kinetic meshless methods M. Somasekhar, S. Vivek, K. S. Malagi, V. Ramesh, S. M. Deshpande;209
15.1;1 Introduction;209
15.2;2 LSKUM: a meshfree solver;210
15.3;3 Adaptive Cloud Refinement (ACR);211
15.4;4 Automatic Connectivity Update(ACU);212
15.5;5 Results and Discussions;213
15.5.1;5.1 Transonic test case NACA0012;215
15.5.2;5.2 Supersonic test case NACA0012;216
15.5.3;5.3 Subsonic test case NACA0012;217
15.6;6 Conclusions;218
15.7;7 Acknowledgements;218
15.8;References;218
16;Fast exact evaluation of particle interaction vectors in the finite volume particle method Nathan J. Quinlan, Ruairi M. Nestor;220
16.1;1 Introduction;220
16.2;2 The Finite Volume Particle Method;221
16.2.1;2.1 Derivation and properties;221
16.2.2;2.2 The particle interaction vectors;224
16.2.3;2.3 Boundary conditions;225
16.3;3 A new choice for the particle weight function;225
16.4;4 Implementation;227
16.5;5 Validation;228
16.6;6 Application to free surface flow;229
16.7;7 Conclusions;233
16.8;References;233
17;Parallel summation of symmetric inter-particle forces in smoothed particle hydrodynamics Johannes Willkomm, H. Martin Bücker;236
17.1;1 Introduction;236
17.2;2 Implementation of smoothed particle hydrodynamics;237
17.3;3 Symmetric inter-particle forces on a grid of cells;239
17.4;4 Parallel symmetric summation algorithm;240
17.5;5 Experimental results;245
17.6;6 Conclusion;248
17.7;References;248
18;Meshfree Wavelet-Galerkin Method for Steady-State Analysis of Nonlinear Microwave Circuits Alla Brunner;250
18.1;1 Introduction;250
18.2;2 Wavelet-Galerkin method;251
18.2.1;2.1 Bubnov-Galerkin projection method;251
18.2.2;2.2 Haar-wavelets basis;252
18.3;3 Meshfree Wavelet-Galerkin Method ;253
18.3.1;3.1 The network equations formulation;254
18.3.2;3.2 Solution of the linear subnetwork;256
18.3.3;3.3 Solution of the nonlinear subnetwork;257
18.4;4 Solution of the network equations;258
18.5;5 Illustrative examples;260
18.5.1;5.1 Simulation results of the Broadband amplifier;260
18.5.2;5.2 Simulation results of the Schmitt-Trigger circuit;261
18.6;6 Conclusions;262
18.7;References;262