Table of Contents

Preface xi

Glossary of Notation xvii

1 Introduction to Meshfree and Particle Methods 1

1.1 Definition of Meshfree Method 1

1.2 Key Approximation Characteristics 2

1.3 Meshfree Computational Model 3

1.4 A Demonstration of Meshfree Analysis 4

1.5 Classes of Meshfree Methods 4

1.6 Applications of Meshfree Methods 8

References 11

2 Preliminaries: Strong and Weak Forms of Diffusion, Elasticity, and Solid Continua 17

2.1 Diffusion Equation 17

2.1.1 Strong Form of the Diffusion Equation 17

2.1.2 The Variational Principle for the Diffusion Equation 19

2.1.2.1 The Standard Variational Principle 20

2.1.2.2 The Variational Equation 20

2.1.2.3 Equivalence of the Variational Equation and the Strong Form 21

2.1.3 Constrained Variational Principles for the Diffusion Equation 25

2.1.3.1 The Penalty Method 25

2.1.3.2 The Lagrange Multiplier Method 26

2.1.3.3 Nitsche’s Method 28

2.1.4 Weak Form of the Diffusion Equation by the Method of Weighted Residuals 29

2.2 Elasticity 32

2.2.1 Strong Form of Elasticity 32

2.2.2 The Variational Principle for Elasticity 34

2.2.3 Constrained Variational Principles for Elasticity 35

2.2.3.1 The Penalty Method 35

2.2.3.2 The Lagrange Multiplier Method 35

2.2.3.3 Nitsche’s Method 36

2.3 Nonlinear Continuum Mechanics 37

2.3.1 Strong Form for General Continua 37

2.3.2 Principle of Stationary Potential Energy 39

2.3.3 Standard Weak Form for Nonlinear Continua 40

2.A Appendix 42

2.A.1 Elasticity with Discontinuities 42

2.A.2 Continuum Mechanics with Discontinuities 44

References 44

3 Meshfree Approximations 45

3.1 MLS Approximation 45

3.1.1 Weight Functions 50

3.1.2 MLS Approximation of Vectors in Multiple Dimensions 53

3.1.3 Reproducing Properties 56

3.1.4 Continuity of Shape Functions 57

3.2 Reproducing Kernel Approximation 58

3.2.1 Continuous Reproducing Kernel Approximation 58

3.2.2 Discrete RK Approximation 62

3.3 Differentiation of Meshfree Shape Functions and Derivative Completeness Conditions 67

3.4 Properties of the MLS and Reproducing Kernel Approximations 68

3.5 Derivative Approximations in Meshfree Methods 73

3.5.1 Direct Derivatives 73

3.5.2 Diffuse Derivatives 74

3.5.3 Implicit Gradients and Synchronized Derivatives 74

3.5.4 Generalized Finite Difference Methods 79

3.5.5 Non-ordinary State-based Peridynamics under the Correspondence Principle, and RK Peridynamics 80

References 83

4 Solving PDEs with Galerkin Meshfree Methods 87

4.1 Linear Diffusion Equation 87

4.1.1 Penalty Method for the Diffusion Equation 90

4.1.2 The Lagrange Multiplier Method for the Diffusion Equation 92

4.1.3 Nitsche’s Method for the Diffusion Equation 95

4.2 Elasticity 98

4.2.1 The Lagrange Multiplier Method for Elasticity 101

4.2.2 Nitsche’s Method for Elasticity 102

4.3 Numerical Integration 105

4.4 Further Discussions on Essential Boundary Conditions 107

References 108

5 Construction of Kinematically Admissible Shape Functions 111

5.1 Strong Enforcement of Essential Boundary Conditions 111

5.2 Basic Ideas, Notation, and Formal Requirements 112

5.2.1 Basic Ideas 112

5.2.2 Formal Requirements 112

5.2.3 Comment on Procedures 114

5.3 Transformation Methods 114

5.3.1 Full Transformation Method: Matrix Implementation 114

5.3.2 Full Transformation Method: Row-Swap Implementation 117

5.3.3 Mixed Transformation Method 120

5.3.4 The Sparsity of Transformation Methods 121

5.3.5 Preconditioners in Transformation Methods 121

5.4 Boundary Singular Kernel Method 123

5.5 RK with Nodal Interpolation 125

5.6 Coupling with Finite Elements on the Boundary 126

5.7 Comparison of Strong Methods 127

5.8 Higher-Order Accuracy and Convergence in Strong Methods 130

5.8.1 Standard Weak Form 130

5.8.2 Consistent Weak Formulation One (CWF I) 131

5.8.3 Consistent Weak Formulation Two (CWF II) 134

5.9 Comparison Between Weak Methods and Strong Methods 135

References 136

6 Quadrature in Meshfree Methods 137

6.1 Nomenclature and Acronyms 137

6.2 Gauss Integration: An Introduction to Quadrature in Meshfree Methods 138

6.3 Issues with Quadrature in Meshfree Methods 140

6.4 Introduction to Nodal integration 142

6.5 Integration Constraints and the Linear Patch Test 144

6.6 Stabilized Conforming Nodal Integration 148

6.7 Variationally Consistent Integration 154

6.7.1 Variational Consistency Conditions 154

6.7.2 Petrov–Galerkin Correction: VCI 157

6.8 Quasi-Conforming SNNI for Extreme Deformations: Adaptive Cells 159

6.9 Instability in Nodal Integration 160

6.10 Stabilization of Nodal Integration 161

6.10.1 Notation for Stabilized Nodal Integration 163

6.10.2 Modified Strain Smoothing 164

6.10.3 Naturally Stabilized Nodal Integration 166

6.10.4 Naturally Stabilized Conforming Nodal Integration 168

Notes 168

References 169

7 Nonlinear Meshfree Methods 173

7.1 Lagrangian Description of the Strong Form 174

7.2 Lagrangian Reproducing Kernel Approximation and Discretization 177

7.3 Semi-Lagrangian Reproducing Kernel Approximation and Discretization 180

7.4 Stability of Lagrangian and Semi-Lagrangian Discretizations 185

7.4.1 Stability Analysis for the Lagrangian RK Equation of Motion 185

7.4.2 Stability Analysis for the Semi-Lagrangian RK Equation of Motion 187

7.4.3 Critical Time Step Estimation for the Lagrangian Formulation 189

7.4.4 Critical Time Step Estimation for the Semi-Lagrangian Formulation 191

7.4.5 Numerical Tests of Critical Time Step Estimation 192

7.5 Neighbor Search Algorithms 196

7.6 Smooth Contact Algorithm 198

7.6.1 Continuum-Based Contact Formulation 198

7.6.2 Meshfree Smooth Curve Representation 201

7.6.3 Three-Dimensional Meshfree Smooth Contact Surface Representation and Contact Detection by a Nonparametric Approach 204

7.7 Natural Kernel Contact Algorithm 207

7.7.1 A Friction-like Plasticity Model 209

7.7.2 Semi-Lagrangian RK Discretization and Natural Kernel Contact Algorithms 210

Notes 212

References 215

8 Other Galerkin Meshfree Methods 219

8.1 Smoothed Particle Hydrodynamics 219

8.1.1 Kernel Estimate 220

8.1.2 SPH Conservation Equations 224

8.1.2.1 Mass Conservation (Continuity Equation) 224

8.1.2.2 Equation of Motion 225

8.1.2.3 Energy Conservation Equation 227

8.1.3 Stability of SPH 228

8.2 Partition of Unity Finite Element Method and h-p Clouds 232

8.3 Natural Element Method 234

8.3.1 First-Order Voronoi Diagram and Delaunay Triangulation 234

8.3.2 Second-Order Voronoi Cell and Sibson Interpolation 235

8.3.3 Laplace Interpolant (Non-Sibson Interpolation) 236

References 237

9 Strong Form Collocation Meshfree Methods 241

9.1 The Meshfree Collocation Method 242

9.2 Approximations and Convergence for Strong Form Collocation 245

9.2.1 Radial Basis Functions 245

9.2.2 Moving Least Squares and Reproducing Kernel Approximations 246

9.2.3 Reproducing Kernel Enhanced Local Radial Basis 247

9.3 Weighted Collocation Methods and Optimal Weights 248

9.4 Gradient Reproducing Kernel Collocation Method 251

9.5 Subdomain Collocation for Heterogeneity and Discontinuities 253

9.6 Comparison of Nodally-Integrated Galerkin Meshfree Methods and Nodally Collocated Strong Form Meshfree Methods 255

9.6.1 Performance of Galerkin and Collocation Methods 255

9.6.2 Stability of Node-Based Galerkin and Collocation Methods 256

References 258

10 RKPM2D: A Two-Dimensional Implementation of RKPM 261

10.1 Reproducing Kernel Particle Method: Approximation and Weak Form 261

10.1.1 Reproducing Kernel Approximation 261

10.1.2 Galerkin Formulation 262

10.2 Domain Integration 264

10.2.1 Gauss Integration 264

10.2.2 Variationally Consistent Nodal Integration 265

10.2.3 Stabilized Nodal Integration Schemes 266

10.2.3.1 Modified Stabilized Nodal Integration 267

10.2.3.2 Naturally Stabilized Nodal Integration 268

10.3 Computer Implementation 269

10.3.1 Domain Discretization 269

10.3.2 Quadrature Point Generation 272

10.3.3 RK Shape Function Generation 273

10.3.4 Stabilization Methods 278

10.3.5 Matrix Evaluation and Assembly 281

10.3.6 Description of subroutines in RKPM2D 285

10.4 Getting Started 287

10.4.1 Input File Generation 288

10.4.1.1 Model 290

10.4.1.2 RK 294

10.4.1.3 Quadrature 295

10.4.2 Executing RKPM2D 295

10.4.3 Post-Processing 295

10.5 Numerical Examples 297

10.5.1 Plotting the RK Shape Functions 297

10.5.2 Patch Test 298

10.5.3 Cantilever Beam Problem 300

10.5.4 Plate With a Hole Problem 303

10.A Appendix 310

References 313

Index 315

About the Author

Ted Belytschko, the former Walter P. Murphy and McCormick Institute Professor of Northwestern University, was one of the world’s most renowned researchers in computational mechanics and meshfree methods. He was 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 is 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.