Meshfree Methods for Partial Differential Equations V (eBook)

Michael Griebel, Marc Alexander Schweitzer (Herausgeber)

eBook Download: PDF

2010 | 2011
VIII, 270 Seiten
Springer Berlin (Verlag)
978-3-642-16229-9 (ISBN)

Infos

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.

Inhalt

Preface 5
Contents 6
Global-local Petrov-Galerkin formulations in the Meshless Finite Difference Method Slawomir Milewski, Janusz Orkisz 8
1 Introduction 8
2 Boundary value problem formulations 9
3 Meshless local Petrov-Galerkin formulations 10
4 Meshless local Petrov-Galerkin 5 (MLPG5) formulation 11
5 Basic Meshless Finite Difference Method solution approach 12
6 Combination of the MFDM and MLPG5 15
7 Higher order approximation based on correction terms 16
8 A-posteriori error analysis 17
9 Adaptive solution approach 18
10 Error indicators 18
11 HO MFDM / MLPG5 approach in 1D 19
12 HO MFDM / MLPG5 approach in 2D 20
13 Extensions of the MFDM / MLPG5 solution approach 21
14 The MFDM / MLPG7 approach 22
15 Numerical examples 24
16 Final remarks 29
References 32
Treatment of general domains in two space dimensions in a Partition of Unity Method Marc Alexander Schweitzer, Maharavo Randrianarivony 34
1 Introduction 34
2 Particle--Partition of Unity Method 35
2.1 Numerical Integration 38
3 Realization on General Domains 41
3.1 Domain Representation 41
3.2 Clipping a curved multiply connected domain 44
3.3 Decomposition and parametrization 46
3.4 Rectangle-NURBS clipping 49
4 Numerical Experiments 50
5 Concluding Remarks 54
References 54
Sampling Inequalities and Support Vector Machines for Galerkin Type Data Christian Rieger 57
1 Introduction 57
2 Review on sampling inequalities 58
2.1 Proof Sketch 61
3 Sampling Inequalities based on Weak Formulations 62
3.1 Sampling inequalities based on Pythagoras law 63
4 Regularization and Machine Learning 65
References 68
Meshfree Vectorial Interpolation Based on the Generalized Stokes Problem Csaba Gáspár 70
1 Introduction 70
2 Vectorial interpolation 71
2.1 Divergence-free interpolation based on the stream function 72
2.2 Multi-elliptic interpolation, scalar problems 73
3 Multi-elliptic divergence-free interpolation, vectorial problems 75
3.1 The generalized Stokes problem 76
4 Solution techniques 79
4.1 Uzawa's method 79
4.2 The method of fundamental solutions 80
5 Summary and conclusions 84
References 85
Pressure XFEM for two-phase incompressible flows with application to 3D droplet problems Sven Gross 86
1 Introduction 86
2 Mathematical model 87
3 Numerical methods 88
3.1 Overview of numerical methods 88
3.2 Pressure XFEM space 88
4 Analysis of pressure XFEM space 89
4.1 Approximation order of pressure XFEM space 89
4.2 Stabilization of XFEM basis 89
5 Numerical experiment 90
References 90
Special-relativistic Smoothed Particle Hydrodynamics: a benchmark suite Stephan Rosswog 93
1 Introduction 93
2 Relativistic SPH equations from a variational principle 94
3 Artificial dissipation 96
4 Test bench 98
4.1 Test 1: Riemann problem 1 98
4.2 Test 2: Riemann problem 2 98
4.3 Test 3: Riemann problem 3 99
4.4 Test 4: Sinusoidally perturbed Riemann problem 100
4.5 Test 5: Relativistic Einfeldt rarefaction test 101
4.6 Test 6: Ultra-relativistic advection 103
5 Conclusions 105
References 106
An exact particle method for scalar conservation laws and its application to stiff reaction kinetics Yossi Farjoun, Benjamin Seibold 108
1 Introduction 108
2 Characteristic Particles and Similarity Solution Interpolant 110
3 Shock Particles 112
3.1 Evolution of Shock Particles 112
3.2 Interaction of Shock Particles 114
4 An ``Exact'' ODE Based Method 115
4.1 Approximation of the Initial Conditions 115
4.2 Integration in Time 116
5 Numerical Error Analysis of the Particle Method 116
6 Stiff Reaction Kinetics 119
7 A Particle Method for Stiff Reaction Kinetics 120
7.1 Computational Approach 122
8 Numerical Results on Reaction Kinetics 123
9 Conclusions and Outlook 125
References 126
Application of Smoothed Particle Hydrodynamics to Structure Formation in Chemical Engineering Franz Keller, Ulrich Nieken 128
1 Introduction 129
2 Smoothed Particle Hydrodynamics Method 131
2.1 Governing equations 131
2.2 Smoothed Particle Hydrodynamics 132
3 Validation of Single Processes 133
4 Simulation of the overall process 139
5 Conclusion and Outlook 141
6 Acknowledgments 142
References 142
Numerical validation of a constraints-based multiscale simulation method for solids Konstantin Fackeldey, Dorian Krause, Rolf Krause 144
1 Introduction 144
2 Coupling with projection-based constraints 145
2.1 Molecular Dynamics 145
2.2 Multiscale Coupling 145
2.3 A method for weak coupling conditions 146
2.4 Damping in zero-temperature simulations 149
3 Numerical Validation 149
3.1 Comparison with pointwise constraints 150
3.2 Energy and reflection measurements 151
3.3 Mode-I fracture simulation 154
4 Conclusion 156
References 156
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
1 Introduction 158
2 Governing equations 160
3 Numerical methods 162
3.1 Particle Method for the Boltzmann equation 162
3.2 Meshfree particle method for the Navier-Stokes equations 162
4 Hybrid method 163
4.1 Adaptive grid refinement 164
4.2 Selection of time steps 165
4.3 Coupling condition 166
4.4 Coupling Algorithm 168
5 Numerical examples 168
5.1 CPU time 170
6 Conclusion 171
References 173
Accuracy and Robustness of Kinetic Meshfree Method Konark Arora, Suresh M. Deshpande 175
1 Introduction 175
2 Least Squares Meshfree Method 176
3 Method of calculation of Weights in 2-D 178
4 Higher Order Accuracy in meshfree methods 179
5 Kinetic Meshfree Method for Euler Equations 181
6 Higher order accuracy by combining Defect Correction with Entropy Variables (q-LSKUM) 181
7 Results and Discussion 182
8 Conclusion 186
References 187
Kinetic meshless methods for unsteady moving boundaries V. Ramesh, S. Vivek, S. M. Deshpande 191
1 Introduction 191
2 Least Squares Kinetic Upwind Method on Moving Nodes 192
3 Formulation of LSKUM_MN 193
4 Advantages of LSKUM_MN 196
5 Results and Discussion 196
5.1 Turbomachinery cascades 196
5.2 Store separation 203
6 Conclusions 206
7 Acknowledgements 207
References 207
Efficient cloud refinement for kinetic meshless methods M. Somasekhar, S. Vivek, K. S. Malagi, V. Ramesh, S. M. Deshpande 209
1 Introduction 209
2 LSKUM: a meshfree solver 210
3 Adaptive Cloud Refinement (ACR) 211
4 Automatic Connectivity Update(ACU) 212
5 Results and Discussions 213
5.1 Transonic test case NACA0012 215
5.2 Supersonic test case NACA0012 216
5.3 Subsonic test case NACA0012 217
6 Conclusions 218
7 Acknowledgements 218
References 218
Fast exact evaluation of particle interaction vectors in the finite volume particle method Nathan J. Quinlan, Ruairi M. Nestor 220
1 Introduction 220
2 The Finite Volume Particle Method 221
2.1 Derivation and properties 221
2.2 The particle interaction vectors 224
2.3 Boundary conditions 225
3 A new choice for the particle weight function 225
4 Implementation 227
5 Validation 228
6 Application to free surface flow 229
7 Conclusions 233
References 233
Parallel summation of symmetric inter-particle forces in smoothed particle hydrodynamics Johannes Willkomm, H. Martin Bücker 236
1 Introduction 236
2 Implementation of smoothed particle hydrodynamics 237
3 Symmetric inter-particle forces on a grid of cells 239
4 Parallel symmetric summation algorithm 240
5 Experimental results 245
6 Conclusion 248
References 248
Meshfree Wavelet-Galerkin Method for Steady-State Analysis of Nonlinear Microwave Circuits Alla Brunner 250
1 Introduction 250
2 Wavelet-Galerkin method 251
2.1 Bubnov-Galerkin projection method 251
2.2 Haar-wavelets basis 252
3 Meshfree Wavelet-Galerkin Method 253
3.1 The network equations formulation 254
3.2 Solution of the linear subnetwork 256
3.3 Solution of the nonlinear subnetwork 257
4 Solution of the network equations 258
5 Illustrative examples 260
5.1 Simulation results of the Broadband amplifier 260
5.2 Simulation results of the Schmitt-Trigger circuit 261
6 Conclusions 262
References 262

Produkt-Details

• Erscheint lt. Verlag: 4.11.2010
• Reihe/Serie: Lecture Notes in Computational Science and Engineering
• Reihe/Serie: Lecture Notes in Computational Science and Engineering
• Zusatzinfo: VIII, 270 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik ► Bauwesen
• Schlagworte: element-free Galerkin method • engineering applications • meshfree discretizations • Partial differential equations • partition of unity method • reproducing kernel particle methods • smoothed particle hydrodynamics • stochastic particle methods
• ISBN-10: 3-642-16229-0 / 3642162290
• ISBN-13: 978-3-642-16229-9 / 9783642162299