Ted Belytschko, J. S. Chen

Meshfree and Particle Methods

Fundamentals and Applications

• Gebundenes Buch

Produktbeschreibung

Meshfree methods is a 'hot' topic. Over the last five years, there has been considerable research activity in the field especially in the US and more recently in Europe. This title provides vital information about this developing field including:
An integrated treatment of the fundamental methods such as least square approximations, partition of unity methods and kernel methods
An overview of the major methodologies that have been developed for solid and linear mechanics
Implementations and solution techniques
A thorough examination of the advantages and disadvantages of various methods

Produktdetails

• Verlag: Wiley & Sons
• Artikelnr. des Verlages: 14584800000
• 1. Auflage
• Seitenzahl: 352
• Erscheinungstermin: 2. Februar 2024
• Englisch
• Abmessung: 259mm x 179mm x 24mm
• Gewicht: 822g
• ISBN-13: 9780470848005
• ISBN-10: 0470848006
• Artikelnr.: 13785548

Autorenporträt

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.

Inhaltsangabe

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