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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 Nitsches Method for the Diffusion Equation             4.2 Elasticity       4.2.1 The Lagrange Multiplier Method for Elasticity           4.2.2 Nitsches 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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