Meshfree methods for partial differential equations VI

✍ Scribed by Michael Griebel; Marc Alexander Schweitzer (eds.)

• Publisher: Springer
• Year: 2013
• Tongue: English
• Leaves: 243
• Series: Lecture notes in computational science and engineering, 89
• Category: Library

✦ Synopsis

Meshfree methods are a modern alternative to classical mesh-based discretization techniques such as finite differences or finite element methods. Especially in a time-dependent setting or in the treatment of problems with strongly singular solutions their independence of a mesh makes these methods highly attractive. This volume collects selected papers presented at the Sixth International Workshop on Meshfree Methods held in Bonn, Germany in October 2011. They address various aspects of this very active research field and cover topics from applied mathematics, physics and engineering. Read more... ESPResSo 3.1: Molecular Dynamics Software for Coarse-Grained Models / Axel Arnold, Olaf Lenz, Stefan Kesselheim -- On the Rate of Convergence of the Hamiltonian Particle-Mesh Method / Bob Peeters, Marcel Oliver, Onno Bokhove -- Peridynamics: A Nonlocal Continuum Theory / Etienne Emmrich, Richard B. Lehoucq -- Immersed Molecular Electrokinetic Finite Element Method for Nano-devices in Biotechnology and Gene Delivery / Wing Kam Liu, Adrian M. Kopacz, Tae-Rin Lee -- Corrected Stabilized Non-conforming Nodal Integration in Meshfree Methods / Marcus Rüter, Michael Hillman -- Multilevel Partition of Unity Method for Elliptic Problems with Strongly Discontinuous Coefficients / Marc Alexander Schweitzer -- HOLMES: Convergent Meshfree Approximation Schemes of Arbitrary Order and Smoothness / Agustín Bompadre, Luigi E. Perotti -- A Meshfree Splitting Method for Soliton Dynamics in Nonlinear Schrödinger Equations / Marco Caliari, Alexander Ostermann -- A Meshless Discretization Method for Markov State Models Applied to Explicit Water Peptide Folding Simulations / Konstantin Fackeldey, Alexander Bujotzek -- Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations / Gregory E. Fasshauer, Qi Ye -- A Meshfree Method for the Analysis of Planar Flows of Inviscid Fluids / Vasily N. Govorukhin -- Some Regularized Versions of the Method of Fundamental Solutions / Csaba Gáspár -- A Characteristic Particle Method for Traffic Flow Simulations on Highway Networks / Yossi Farjoun, Benjamin Seibold -- Meshfree Modeling in Laminated Composites / Daniel C. Simkins Jr., Nathan Collier

✦ Table of Contents

Cover......Page 1
Meshfree Methods for Partial Differential Equations VI......Page 4
Preface......Page 6
Contents......Page 8
Meshfree Modeling in Laminated Composites......Page 10
1 Introduction......Page 11
2 Characteristics......Page 12
3 Methods and Algorithms......Page 14
4 Advanced Electrostatics......Page 15
4.1 Dielectric Contrasts......Page 16
4.2 MEMD......Page 17
4.3 The ICC Algorithm for Dielectric Interfaces......Page 19
5 Rigid Bodies......Page 20
6 Dynamic Bonding......Page 22
7 Lattice Boltzmann......Page 24
8 Correlator......Page 26
9 Conclusions and Outlook......Page 29
References......Page 30
1 Introduction......Page 33
2 The HPM Method for Shallow Water......Page 35
3 Theoretical Estimates and Consequences......Page 37
4.1 Burgers' Solution for d=1......Page 39
4.2 Cosine Vortex over Topography for d=2......Page 40
5.1 Optimal Global Smoothing......Page 42
5.2 Optimal Number of Particles per Cell......Page 44
5.3 The Role of the Global Smoothing Order q......Page 46
5.4 The Role of the Strang–Fix Order p......Page 49
6 Conclusion......Page 50
References......Page 51
1 Introduction......Page 52
2.1 Bond-Based Model......Page 54
2.2 Linearization......Page 56
2.3 State-Based Model......Page 57
2.4 Other Nonlocal Models in Elasticity Theory......Page 59
3.1 Linear Bond-Based Model in L2......Page 60
3.3 Nonlinear Bond-Based Model......Page 62
3.4 State-Based Model and Nonlocal Vector Calculus......Page 63
4.1 Limit of Vanishing Nonlocality......Page 64
4.2 Composite Material and Two-Scale Convergence......Page 65
5 Numerical Approximation......Page 66
6.1 Simulation of Nanofibres......Page 68
6.2 Simulation of Cracks......Page 69
References......Page 70
2 Vascular Nanoparticle Transport......Page 74
3 Nanodiamond Platform for Gene Delivery......Page 76
4 Concluding Remarks......Page 79
References......Page 80
1 Introduction......Page 81
2.1 Strong Form of the Model Problem......Page 83
2.3 Galerkin Discretization......Page 84
3.1 Numerical Integration in Galerkin Methods......Page 85
3.2 Correction of the Integration Error......Page 86
4.1 The Reproducing Kernel Particle Method (RKPM)......Page 87
4.2 Stabilized Conforming and Non-conforming Nodal Integration......Page 88
4.3 The Assumed Strain Tensor......Page 90
5.1 Beam Problem......Page 92
5.2 Tube Problem......Page 95
References......Page 97
1 Introduction......Page 99
2 Model Problem and Robust Solvers......Page 101
3 Multilevel Partition of Unity Method......Page 102
3.1 Enrichment Functions......Page 104
3.2 Essential Boundary Conditions......Page 106
4 Numerical Experiments......Page 107
5 Concluding Remarks......Page 113
References......Page 114
1 Introduction......Page 117
2 Prolegomena......Page 118
3.1 Local Maximum-Entropy Approximation Schemes......Page 119
3.2 Definition of High-Order Local Maximum Entropy Approximation Schemes......Page 120
3.3 Dual Formulation of HOLMES......Page 122
5 Examples......Page 123
5.1 Elastic Rod Under Sine Load......Page 124
5.2 Elastic Beam Under Sine Load......Page 126
5.3 Membrane Under Sine Load......Page 127
5.4 Kirchhoff Plate......Page 128
6 Summary and Conclusions......Page 130
References......Page 131
1 Introduction......Page 132
2 Soliton Dynamics......Page 134
3 Meshfree Approximation......Page 135
4.1 Approximation of the Ground State Solution......Page 136
4.2 Choice of the Interpolation Points and the Shape Parameter......Page 137
5.1 Splitting Methods......Page 138
5.2 Computation of the Semiflows......Page 139
5.3 Time Evolution of the Solution......Page 140
6 Numerical Experiments......Page 141
References......Page 144
1 Introduction......Page 145
2 Basics of Conformation Dynamics......Page 146
2.1 Discretization of the State Space......Page 148
2.2 Decoupled Markov States......Page 149
2.3 Nearly Decoupled Markov States and Water......Page 150
3 Simulation of Trialanine in an Explicit Solvent......Page 152
References......Page 156
1 Introduction......Page 159
1.1 Problem Setting......Page 160
2 Kernel-Based Collocation Method......Page 162
2.1 Approximation of SPDEs......Page 163
2.2 Convergence Analysis......Page 164
3 Galerkin Finite Element Method......Page 165
4.2 Relationship Between the Two Methods......Page 167
4.3 Competitive Advantages......Page 169
5 Numerical Examples......Page 170
Appendix A. Reproducing Kernels and Gaussian Fields......Page 172
References......Page 174
1 Introduction......Page 175
2 Governing Equations......Page 176
3 Numerical Method......Page 177
3.1 Approximation of the Velocity Field......Page 178
3.3 Parallel Algorithm......Page 180
4 Numerical Experiments......Page 181
References......Page 183
1 Introduction......Page 185
2 The Method of Fundamental Solutions......Page 186
3 The Regularized Method of Fundamental Solutions......Page 188
4.1 Preliminary Lemmas......Page 190
4.2 Estimations for the Solutions of Multi-elliptic Equations......Page 193
4.3 Estimations for Boundary Functions......Page 196
4.4 Error Estimations for the RMFS......Page 197
5 A Numerical Example......Page 200
6 Summary and Conclusions......Page 201
References......Page 202
1 Introduction......Page 203
2.1 Macroscopic Traffic Models......Page 205
2.2 Traffic Networks......Page 206
3.1 Characteristic Particles and Similarity Interpolant......Page 208
3.2 Representation of Shocks......Page 209
4 Generalizing Particleclaw to Network Flows......Page 210
4.1 Virtual Domain, Excess Area, Virtual Area, and Area Credit......Page 211
4.2 Synchronization Step......Page 212
4.3 Representation of Virtual Area by Particles......Page 215
5.1 Bottleneck Test Case......Page 217
5.2 Simulation of a Diamond Network......Page 219
6 Conclusions and Outlook......Page 220
References......Page 222
1 Introduction......Page 224
2.1 Crack Morphology......Page 225
2.2 Visibility Condition......Page 227
3.1 Hierarchical Multiscale Modeling......Page 231
3.2 External Knowledge Enhancement......Page 232
4 Results......Page 234
5 Conclusions......Page 235
References......Page 236
Editorial Policy......Page 237
Lecture Notesin Computational Science and Engineering
......Page 239
Texts in Computational Science and Engineering
......Page 243