Meshfree and Particle Methods

Provides thorough coverage of essential concepts and state-of-the-art developments in the field

Meshfree and Particle Methods is the first book of its kind to combine comprehensive, up-to-date information on the fundamental theories and applications of meshfree methods with systematic guidance on practical coding implementation. Broad in scope and content, this unique volume provides readers with the knowledge necessary to perform research and solve challenging problems in nearly all fields of science and engineering using meshfree computational techniques.

The authors provide detailed descriptions of essential issues in meshfree methods, as well as specific techniques to address them, while discussing a wide range of subjects and use cases. Topics include approximations in meshfree methods, nonlinear meshfree methods, essential boundary condition enforcement, quadrature in meshfree methods, strong form collocation methods, and more. Throughout the book, topics are integrated with descriptions of computer implementation and an open-source code (with a dedicated chapter for users) to illustrate the connection between the formulations discussed in the text and their real-world implementation and application. This authoritative resource:

• Explains the fundamentals of meshfree methods, their constructions, and their unique capabilities as compared to traditional methods
• Features an overview of the open-source meshfree code RKPM2D, including code and numerical examples
• Describes all the variational concepts required to solve scientific and engineering problems using meshfree methods such as Nitsche’s method and the Lagrange multiplier method
• Includes comprehensive reviews of essential boundary condition enforcement, quadrature in meshfree methods, and nonlinear aspects of meshfree analysis
• Discusses other Galerkin meshfree methods, strong form meshfree methods, and their comparisons

Meshfree and Particle Methods: Fundamentals and Applications is the perfect introduction to meshfree methods for upper-level students in advanced numerical analysis courses, and is an invaluable reference for professionals in mechanical, aerospace, civil, and structural engineering, and related fields, who want to understand and apply these concepts directly, or effectively use commercial and other production meshfree and particle codes in their work.

Ted Belytschko, the former Walter P Murphy and McCormick Institute Professor of Northwestern University, is one of the world’s most renowned researchers in computational mechanics and meshfree methods. He is the originator of the Element-Free Galerkin (EFG) Methods, and his paper Element-Free Galerkin Methods published in 1994 remains the most widely cited paper on the subject.

J. S. Chen is Distinguished Professor and William Prager Chair Professor in the Department of Structural Engineering & Department of Mechanical and Aerospace Engineering at The University of California, San Diego. His research interests are in computational solid mechanics and multiscale materials modeling, with focus on meshfree methods and advanced finite element methods.

Michael Hillman a Principal Scientist at Karagozian and Case Inc., and the former L. Robert and Mary L. Kimball Professor and Associate Professor of Civil Engineering at The Pennsylvania State University. His research interests are in computational solid mechanics, fundamental advancement of meshfree methods, and enhanced and novel meshfree methods.

Preface

Glossary of Notation

Chapter 1: Introduction to Meshfree and Particle Methods           

1.1 Definition of Meshfree Method         

1.2 Key Approximation Characteristics    

1.3 Meshfree Computational Model        

1.4 A Demonstration of Meshfree Analysis           

1.5 Classes of Meshfree Methods             

1.6 Applications of Meshfree Methods   

References

Chapter 2: Preliminaries: Strong and Weak Forms of Diffusion, Elasticity and Solid Continua

2.1 Diffusion Equation

2.1.1 Strong Form of the Diffusion Equation

2.1.2 The Variational Principle for the Diffusion Equation

2.1.3 Constrained Principles for the Diffusion Equation

2.1.4 Weak Form of the Diffusion Equation by the Method of Weighted Residuals

2.2 Elasticity

2.2.1 Strong Form of Elasticity

2.2.2 The Variational Principle for Elasticity

2.2.3 Constrained Variational Principles for Elasticity

2.3 Nonlinear Continuum Mechanics

2.3.1 Strong Form for General Continua

2.3.2 Principle of Stationary Potential Energy

2.3.3 Standard Weak Form for Nonlinear Continua

Appendix

References

Chapter 3: Meshfree Approximations      

3.1 Moving Least Squares (MLS) Approximation  

3.1.1 Weight Functions  

3.1.2 MLS Approximation of Vectors in Multiple dimensions         

3.1.3 Reproducing Properties     

3.1.4 Continuity of Shape Functions        

3.2 Reproducing Kernel Approximation  

3.2.1 Continuous Reproducing Kernel Approximation       

3.2.2 Discrete Reproducing Kernel Approximation             

3.3 Differentiation of Meshfree Shape Functions and Derivative Completeness Conditions              

3.4 Properties of the Moving Least Squares and Reproducing Kernel Approximations        

3.5 Derivative Approximations in Meshfree Methods       

3.5.1 Direct Derivatives

3.5.2 Diffuse Derivatives

3.5.3 Implicit Gradients and Synchronized Derivatives

3.5.4 Generalized Finite Difference Methods

3.5.5 Non-ordinary State-based Peridynamics under the Correspondence Principle

References

Chapter 4: Solving PDEs with the Galerkin Meshfree Methods     

4.1 Linear Diffusion Equation      

4.1.1 Penalty Method for Diffusion Equation       

4.1.2 The Lagrange Multiplier Method for Diffusion Equation       

4.1.3 Nitsche’s Method for the Diffusion Equation            

4.2 Elasticity      

4.2.1 The Lagrange Multiplier Method for Elasticity          

4.2.2 Nitsche’s Method for Elasticity       

4.3 Numerical Integration            

4.4 Further Discussions on Essential Boundary Conditions

References

Chapter 5: Construction of Kinematically Admissible Shape Functions      

5.1 Strong Enforcement of Essential Boundary Conditions             

5.2 Basic Ideas, Notation, and Formal Requirements         

5.2.1 Basic Ideas             

5.2.2 Formal Requirements         

5.2.3 Comment on Procedures   

5.3 Transformation Methods      

5.3.1 Full Transformation Method: Matrix Implementation           

5.3.2 Full Transformation Method: Row-Swap Implementation    

5.3.3 Mixed Transformation Method       

5.3.4 The Sparsity of Transformation Methods    

5.3.5 Preconditioners in Transformation Methods            

5.4 Boundary Singular Kernel      

5.5 Reproducing Kernel with Nodal Interpolation              

5.6 Coupling with Finite Elements on The Boundary         

5.7 Comparison of Strong Methods         

5.8 Higher-Order Convergence in Strong Methods            

5.8.1 Standard Weak Form          

5.8.2 Consistent Weak Formulation One (CWF I)

5.8.3 Consistent Weak Formulation Two (CWF II)              

5.9 Comparison Between Weak Methods and Strong Methods    

References

Chapter 6: Quadrature in Meshfree Methods

6.1 Nomenclature and Acronyms             

6.2 Gauss integration: an Introduction to Quadrature in Meshfree Methods

6.3 Issues with Quadrature in Meshfree Methods             

6.4 Introduction to Nodal Integration     

6.5 Integration Constraints and the Linear Patch Test      

6.6 Stabilized Conforming Nodal Integration        

6.7 Variationally Consistent Integration 

6.7.1 Variational Consistency Conditions

6.7.2 Petrov-Galerkin Correction: A Variationally Consistent Integration  

6.8 Quasi-Conforming SNNI for Extreme Deformations: Adaptive Cells     

6.9 Instability in Nodal Integration           

6.10 Stabilization of Nodal Integration    

6.10.1 Notation for Stabilized Nodal Integration                

6.10.2 Modified Strain Smoothing            

6.10.3 Naturally Stabilized Nodal Integration       

6.10.4 Naturally Stabilized Conforming Nodal Integration              

References

Chapter 7: Nonlinear Meshfree Methods

7.1 Lagrangian Description of the Strong Form    

7.2 Lagrangian Reproducing Kernel Approximation and Discretization      

7.3 Semi-Lagrangian Reproducing Kernel Approximation and Discretization           

7.4 Stability of Lagrangian and Semi-Lagrangian Discretizations

7.4.1 Stability Analysis for the Lagrangian RK Equation of Motion

7.4.2 Stability Analysis for the Semi-Lagrangian RK Equation of Motion

7.4.3 Critical Time Step Estimation for the Lagrangian Formulation

7.4.4 Critical Time Step Estimation for the Semi-Lagrangian Formulation 

7.4.5 Numerical Tests of Critical Time Step Estimation     

7.5 Neighbor Search Algorithms

7.6 Smooth Contact Algorithm   

7.6.1 Continuum-Based Contact Formulation       

7.6.2 Meshfree Smooth Curve Representation    

7.6.3 Three-Dimensional Meshfree Smooth Contact Surface Representation and Contact Detection by a Non-parametric Approach             

7.7 Natural Kernel Contact algorithm      

References

Chapter 8: Other Galerkin Meshfree Methods     

8.1 Smoothed Particle Hydrodynamics    

8.1.1 Kernel Estimate     

8.1.2 SPH Conservation Equations            

8.1.3 Stability of SPH      

8.2 Partition of Unity Finite Element Method and h-p Clouds       

8.3 Natural Element Method      

8.3.1 First Order Voronoi Diagram and Delaunay Triangulation    

8.3.2 Second Order Voronoi Cell and Sibson Interpolation             

8.3.3 Laplace Interpolant (Non-Sibson Interpolation)        

References

Chapter 9: Strong Form Collocation Meshfree Methods  

9.1 The Meshfree Collocation Method    

9.2 Approximation Functions and Convergence for Strong Form Collocation

9.2.1 Radial Basis Functions

9.2.2 Moving Least-square and Reproducing Kernel

9.2.3 Reproducing Kernel Enhanced Local Radial Basis

9.3 Weighted Collocation Methods and Optimal Weights for Strong Form Collocation      

9.4 Gradient Reproducing Kernel Collocation Method      

9.5 Subdomain Collocation for Heterogeneity and Discontinuities              

9.6 Comparison of Nodally Integrated Galerkin Meshfree Methods and Nodally Collocated Strong Form Meshfree Methods

9.6.1 Performance of Galerkin and Collocation Methods

9.6.2 Stability of Node-Based Galerkin and Collocation Methods 

References

Chapter 10: RKPM2D: A Two-dimensional Implementation of RKPM

10.1       Reproducing Kernel Particle Method: Approximation and Weak Form      

10.1.1    Reproducing Kernel Approximation         

10.1.2    Galerkin Formulation     

10.2       Domain Integration        

10.2.1    Gauss Integration           

10.2.2    Variationally Consistent Nodal Integration           

10.2.3    Stabilized Nodal Integration Schemes     

10.3       Computer Implementation                         

10.3.1    Domain Discretization    

10.3.2    Quadrature Point Generation     

10.3.3    RK Shape Function Generation   

10.3.4    Stabilization Methods    

10.3.5    Matrix Evaluation and Assembly

10.3.6    Description of Subroutines in RKPM2D   

10.4       Getting Started 

10.4.1    Input File Generation

10.4.2    Executing RKPM2D         

10.4.3    Post-processing

10.5       Numerical Examples       

10.5.1    Plotting the RK Shape Functions

10.5.2    Patch Test          

10.5.3    Cantilever Beam Problem            

10.5.4    Plate with a Hole Problem           

Appendix

References

Index

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