Civil Engineering Association

Meshfree Methods for Partial Differential Equations V (Lecture Notes in Computational Science and Engineering)
By Michael Griebel, Marc Alexander Schweitzer


Publisher: Springer
Number Of Pages: 250
Publication Date: 2010-11-25
ISBN-10 / ASIN: 3642162282
ISBN-13 / EAN: 9783642162282


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.



Table of Contents

Preface
Contents
Global-local Petrov-Galerkin formulations in the Meshless Finite Difference Method
1 Introduction
2 Boundary value problem formulations
3 Meshless local Petrov-Galerkin formulations
4 Meshless local Petrov-Galerkin 5 (MLPG5) formulation
5 Basic Meshless Finite Difference Method solution approach
6 Combination of the MFDM and MLPG5
7 Higher order approximation based on correction terms
8 A-posteriori error analysis
9 Adaptive solution approach
10 Error indicators
11 HO MFDM / MLPG5 approach in 1D
12 HO MFDM / MLPG5 approach in 2D
13 Extensions of the MFDM / MLPG5 solution approach
14 The MFDM / MLPG7 approach
15 Numerical examples
16 Final remarks
References
Treatment of general domains in two space dimensions in a Partition of Unity Method
1 Introduction
2 Particle--Partition of Unity Method
2.1 Numerical Integration
3 Realization on General Domains
3.1 Domain Representation
3.2 Clipping a curved multiply connected domain
3.3 Decomposition and parametrization
3.4 Rectangle-NURBS clipping
4 Numerical Experiments
5 Concluding Remarks
References
Sampling Inequalities and Support Vector Machines for Galerkin Type Data
1 Introduction
2 Review on sampling inequalities
2.1 Proof Sketch
3 Sampling Inequalities based on Weak Formulations
3.1 Sampling inequalities based on Pythagoras law
4 Regularization and Machine Learning
References
Meshfree Vectorial Interpolation Based on the Generalized Stokes Problem
1 Introduction
2 Vectorial interpolation
2.1 Divergence-free interpolation based on the stream function
2.2 Multi-elliptic interpolation, scalar problems
3 Multi-elliptic divergence-free interpolation, vectorial problems
3.1 The generalized Stokes problem
4 Solution techniques
4.1 Uzawa's method
4.2 The method of fundamental solutions
5 Summary and conclusions
References
Pressure XFEM for two-phase incompressible flows with application to 3D droplet problems
1 Introduction
2 Mathematical model
3 Numerical methods
3.1 Overview of numerical methods
3.2 Pressure XFEM space
4 Analysis of pressure XFEM space
4.1 Approximation order of pressure XFEM space
4.2 Stabilization of XFEM basis
5 Numerical experiment
References
Special-relativistic Smoothed Particle Hydrodynamics: a benchmark suite
1 Introduction
2 Relativistic SPH equations from a variational principle
3 Artificial dissipation
4 Test bench
4.1 Test 1: Riemann problem 1
4.2 Test 2: Riemann problem 2
4.3 Test 3: Riemann problem 3
4.4 Test 4: Sinusoidally perturbed Riemann problem
4.5 Test 5: Relativistic Einfeldt rarefaction test
4.6 Test 6: Ultra-relativistic advection
5 Conclusions
References
An exact particle method for scalar conservation laws and its application to stiff reaction kinetics
1 Introduction
2 Characteristic Particles and Similarity Solution Interpolant
3 Shock Particles
3.1 Evolution of Shock Particles
3.2 Interaction of Shock Particles
4 An ``Exact'' ODE Based Method
4.1 Approximation of the Initial Conditions
4.2 Integration in Time
5 Numerical Error Analysis of the Particle Method
6 Stiff Reaction Kinetics
7 A Particle Method for Stiff Reaction Kinetics
7.1 Computational Approach
8 Numerical Results on Reaction Kinetics
9 Conclusions and Outlook
References
Application of Smoothed Particle Hydrodynamics to Structure Formation in Chemical Engineering
1 Introduction
2 Smoothed Particle Hydrodynamics Method
2.1 Governing equations
2.2 Smoothed Particle Hydrodynamics
3 Validation of Single Processes
4 Simulation of the overall process
5 Conclusion and Outlook
6 Acknowledgments
References
Numerical validation of a constraints-based multiscale simulation method for solids
1 Introduction
2 Coupling with projection-based constraints
2.1 Molecular Dynamics
2.2 Multiscale Coupling
2.3 A method for weak coupling conditions
2.4 Damping in zero-temperature simulations
3 Numerical Validation
3.1 Comparison with pointwise constraints
3.2 Energy and reflection measurements
3.3 Mode-I fracture simulation
4 Conclusion
References
Coupling of the Navier-Stokes and the Boltzmann equations with a meshfree particle and kinetic particle methods for a micro cavity
1 Introduction
2 Governing equations
3 Numerical methods
3.1 Particle Method for the Boltzmann equation
3.2 Meshfree particle method for the Navier-Stokes equations
4 Hybrid method
4.1 Adaptive grid refinement
4.2 Selection of time steps
4.3 Coupling condition
4.4 Coupling Algorithm
5 Numerical examples
5.1 CPU time
6 Conclusion
References
Accuracy and Robustness of Kinetic Meshfree Method
1 Introduction
2 Least Squares Meshfree Method
3 Method of calculation of Weights in 2-D
4 Higher Order Accuracy in meshfree methods
5 Kinetic Meshfree Method for Euler Equations
6 Higher order accuracy by combining Defect Correction with Entropy Variables (q-LSKUM)
7 Results and Discussion
8 Conclusion
References
Kinetic meshless methods for unsteady moving boundaries
1 Introduction
2 Least Squares Kinetic Upwind Method on Moving Nodes
3 Formulation of LSKUM_MN
4 Advantages of LSKUM_MN
5 Results and Discussion
5.1 Turbomachinery cascades
5.2 Store separation
6 Conclusions
7 Acknowledgements
References
Efficient cloud refinement for kinetic meshless methods
1 Introduction
2 LSKUM: a meshfree solver
3 Adaptive Cloud Refinement (ACR)
4 Automatic Connectivity Update(ACU)
5 Results and Discussions
5.1 Transonic test case NACA0012
5.2 Supersonic test case NACA0012
5.3 Subsonic test case NACA0012
6 Conclusions
7 Acknowledgements
References
Fast exact evaluation of particle interaction vectors in the finite volume particle method
1 Introduction
2 The Finite Volume Particle Method
2.1 Derivation and properties
2.2 The particle interaction vectors
2.3 Boundary conditions
3 A new choice for the particle weight function
4 Implementation
5 Validation
6 Application to free surface flow
7 Conclusions
References
Parallel summation of symmetric inter-particle forces in smoothed particle hydrodynamics
1 Introduction
2 Implementation of smoothed particle hydrodynamics
3 Symmetric inter-particle forces on a grid of cells
4 Parallel symmetric summation algorithm
5 Experimental results
6 Conclusion
References
Meshfree Wavelet-Galerkin Method for Steady-State Analysis of Nonlinear Microwave Circuits
1 Introduction
2 Wavelet-Galerkin method
2.1 Bubnov-Galerkin projection method
2.2 Haar-wavelets basis
3 Meshfree Wavelet-Galerkin Method
3.1 The network equations formulation
3.2 Solution of the linear subnetwork
3.3 Solution of the nonlinear subnetwork
4 Solution of the network equations
5 Illustrative examples
5.1 Simulation results of the Broadband amplifier
5.2 Simulation results of the Schmitt-Trigger circuit
6 Conclusions
References

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Meshfree Methods for Partial Differential Equations IV (Lecture Notes in Computational Science and Engineering)
By Michael Griebel, Marc Alexander Schweitzer

Publisher: Springer
Number Of Pages: 412
Publication Date: 2008-11-01
ISBN-10 / ASIN: 3540799931
ISBN-13 / EAN: 9783540799931
Binding: Paperback

The numerical treatment of partial differential equations with particle methods and meshfree discretization techniques is a very active research field both in the mathematics and engineering community. 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 proceedings papers of the Fourth International Workshop on Meshfree Methods held in September 2007 in Bonn. The articles address the different meshfree methods (SPH, PUM, GFEM, EFGM, RKPM, etc.) and their application in applied mathematics, physics and engineering. The volume is intended to foster this very active and exciting area of interdisciplinary research and to present recent advances and results in this field.

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Meshfree Methods for Partial Differential Equations III (Lecture Notes in Computational Science and Engineering)
By Michael Griebel, Marc A. Schweitzer (Editors)

Publisher: Springer
Number Of Pages: 312
Publication Date: 2006-11-16
ISBN-10 / ASIN: 3540462147
ISBN-13 / EAN: 9783540462149
Binding: Paperback

Meshfree methods for the numerical solution of partial differential equations are becoming more and more mainstream in many areas of applications. Their flexiblity and wide applicability are attracting engineers, scientists, and mathematicians to this very dynamic research area. 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.

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Meshfree Methods for Partial Differential Equations II (Lecture Notes in Computational Science and Engineering)
By Michael Griebel, Marc A. Schweitzer

Publisher: Springer
Number Of Pages: 303
Publication Date: 2005-01-12
ISBN-10 / ASIN: 3540230262
ISBN-13 / EAN: 9783540230267
Binding: Paperback

The numerical treatment of partial differential equations with particle methods and meshfree discretization techniques is a very active research field both in the mathematics and engineering community. 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 Second International Workshop on Meshfree Methods held in September 2003 in Bonn. The articles address the different meshfree methods (SPH, PUM, GFEM, EFGM, RKPM, etc.) and their application in applied mathematics, physics and engineering. The volume is intended to foster this new and exciting area of interdisciplinary research and to present recent advances and results in this field.

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