{"product_id":"meshfree-and-particle-methods-9780470848005","title":"Meshfree and Particle Methods","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eMeshfree and Particle Methods is the first comprehensive book on meshfree methods, providing an overview of the major methodologies that have been developed for solid and linear mechanics. It provides a through examination of the advantages and disadvantages of various methods, with implementations and solutions techniques.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface xi\u003c\/p\u003e \u003cp\u003eGlossary of Notation xvii\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction to Meshfree and Particle Methods 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Definition of Meshfree Method 1\u003c\/p\u003e \u003cp\u003e1.2 Key Approximation Characteristics 2\u003c\/p\u003e \u003cp\u003e1.3 Meshfree Computational Model 3\u003c\/p\u003e \u003cp\u003e1.4 A Demonstration of Meshfree Analysis 4\u003c\/p\u003e \u003cp\u003e1.5 Classes of Meshfree Methods 4\u003c\/p\u003e \u003cp\u003e1.6 Applications of Meshfree Methods 8\u003c\/p\u003e \u003cp\u003eReferences 11\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Preliminaries: Strong and Weak Forms of Diffusion, Elasticity, and Solid Continua 17\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Diffusion Equation 17\u003c\/p\u003e \u003cp\u003e2.1.1 Strong Form of the Diffusion Equation 17\u003c\/p\u003e \u003cp\u003e2.1.2 The Variational Principle for the Diffusion Equation 19\u003c\/p\u003e \u003cp\u003e2.1.2.1 The Standard Variational Principle 20\u003c\/p\u003e \u003cp\u003e2.1.2.2 The Variational Equation 20\u003c\/p\u003e \u003cp\u003e2.1.2.3 Equivalence of the Variational Equation and the Strong Form 21\u003c\/p\u003e \u003cp\u003e2.1.3 Constrained Variational Principles for the Diffusion Equation 25\u003c\/p\u003e \u003cp\u003e2.1.3.1 The Penalty Method 25\u003c\/p\u003e \u003cp\u003e2.1.3.2 The Lagrange Multiplier Method 26\u003c\/p\u003e \u003cp\u003e2.1.3.3 Nitsche’s Method 28\u003c\/p\u003e \u003cp\u003e2.1.4 Weak Form of the Diffusion Equation by the Method of Weighted Residuals 29\u003c\/p\u003e \u003cp\u003e2.2 Elasticity 32\u003c\/p\u003e \u003cp\u003e2.2.1 Strong Form of Elasticity 32\u003c\/p\u003e \u003cp\u003e2.2.2 The Variational Principle for Elasticity 34\u003c\/p\u003e \u003cp\u003e2.2.3 Constrained Variational Principles for Elasticity 35\u003c\/p\u003e \u003cp\u003e2.2.3.1 The Penalty Method 35\u003c\/p\u003e \u003cp\u003e2.2.3.2 The Lagrange Multiplier Method 35\u003c\/p\u003e \u003cp\u003e2.2.3.3 Nitsche’s Method 36\u003c\/p\u003e \u003cp\u003e2.3 Nonlinear Continuum Mechanics 37\u003c\/p\u003e \u003cp\u003e2.3.1 Strong Form for General Continua 37\u003c\/p\u003e \u003cp\u003e2.3.2 Principle of Stationary Potential Energy 39\u003c\/p\u003e \u003cp\u003e2.3.3 Standard Weak Form for Nonlinear Continua 40\u003c\/p\u003e \u003cp\u003e2.A Appendix 42\u003c\/p\u003e \u003cp\u003e2.A.1 Elasticity with Discontinuities 42\u003c\/p\u003e \u003cp\u003e2.A.2 Continuum Mechanics with Discontinuities 44\u003c\/p\u003e \u003cp\u003eReferences 44\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Meshfree Approximations 45\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 MLS Approximation 45\u003c\/p\u003e \u003cp\u003e3.1.1 Weight Functions 50\u003c\/p\u003e \u003cp\u003e3.1.2 MLS Approximation of Vectors in Multiple Dimensions 53\u003c\/p\u003e \u003cp\u003e3.1.3 Reproducing Properties 56\u003c\/p\u003e \u003cp\u003e3.1.4 Continuity of Shape Functions 57\u003c\/p\u003e \u003cp\u003e3.2 Reproducing Kernel Approximation 58\u003c\/p\u003e \u003cp\u003e3.2.1 Continuous Reproducing Kernel Approximation 58\u003c\/p\u003e \u003cp\u003e3.2.2 Discrete RK Approximation 62\u003c\/p\u003e \u003cp\u003e3.3 Differentiation of Meshfree Shape Functions and Derivative Completeness Conditions 67\u003c\/p\u003e \u003cp\u003e3.4 Properties of the MLS and Reproducing Kernel Approximations 68\u003c\/p\u003e \u003cp\u003e3.5 Derivative Approximations in Meshfree Methods 73\u003c\/p\u003e \u003cp\u003e3.5.1 Direct Derivatives 73\u003c\/p\u003e \u003cp\u003e3.5.2 Diffuse Derivatives 74\u003c\/p\u003e \u003cp\u003e3.5.3 Implicit Gradients and Synchronized Derivatives 74\u003c\/p\u003e \u003cp\u003e3.5.4 Generalized Finite Difference Methods 79\u003c\/p\u003e \u003cp\u003e3.5.5 Non-ordinary State-based Peridynamics under the Correspondence Principle, and RK Peridynamics 80\u003c\/p\u003e \u003cp\u003eReferences 83\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Solving PDEs with Galerkin Meshfree Methods 87\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Linear Diffusion Equation 87\u003c\/p\u003e \u003cp\u003e4.1.1 Penalty Method for the Diffusion Equation 90\u003c\/p\u003e \u003cp\u003e4.1.2 The Lagrange Multiplier Method for the Diffusion Equation 92\u003c\/p\u003e \u003cp\u003e4.1.3 Nitsche’s Method for the Diffusion Equation 95\u003c\/p\u003e \u003cp\u003e4.2 Elasticity 98\u003c\/p\u003e \u003cp\u003e4.2.1 The Lagrange Multiplier Method for Elasticity 101\u003c\/p\u003e \u003cp\u003e4.2.2 Nitsche’s Method for Elasticity 102\u003c\/p\u003e \u003cp\u003e4.3 Numerical Integration 105\u003c\/p\u003e \u003cp\u003e4.4 Further Discussions on Essential Boundary Conditions 107\u003c\/p\u003e \u003cp\u003eReferences 108\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Construction of Kinematically Admissible Shape Functions 111\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Strong Enforcement of Essential Boundary Conditions 111\u003c\/p\u003e \u003cp\u003e5.2 Basic Ideas, Notation, and Formal Requirements 112\u003c\/p\u003e \u003cp\u003e5.2.1 Basic Ideas 112\u003c\/p\u003e \u003cp\u003e5.2.2 Formal Requirements 112\u003c\/p\u003e \u003cp\u003e5.2.3 Comment on Procedures 114\u003c\/p\u003e \u003cp\u003e5.3 Transformation Methods 114\u003c\/p\u003e \u003cp\u003e5.3.1 Full Transformation Method: Matrix Implementation 114\u003c\/p\u003e \u003cp\u003e5.3.2 Full Transformation Method: Row-Swap Implementation 117\u003c\/p\u003e \u003cp\u003e5.3.3 Mixed Transformation Method 120\u003c\/p\u003e \u003cp\u003e5.3.4 The Sparsity of Transformation Methods 121\u003c\/p\u003e \u003cp\u003e5.3.5 Preconditioners in Transformation Methods 121\u003c\/p\u003e \u003cp\u003e5.4 Boundary Singular Kernel Method 123\u003c\/p\u003e \u003cp\u003e5.5 RK with Nodal Interpolation 125\u003c\/p\u003e \u003cp\u003e5.6 Coupling with Finite Elements on the Boundary 126\u003c\/p\u003e \u003cp\u003e5.7 Comparison of Strong Methods 127\u003c\/p\u003e \u003cp\u003e5.8 Higher-Order Accuracy and Convergence in Strong Methods 130\u003c\/p\u003e \u003cp\u003e5.8.1 Standard Weak Form 130\u003c\/p\u003e \u003cp\u003e5.8.2 Consistent Weak Formulation One (CWF I) 131\u003c\/p\u003e \u003cp\u003e5.8.3 Consistent Weak Formulation Two (CWF II) 134\u003c\/p\u003e \u003cp\u003e5.9 Comparison Between Weak Methods and Strong Methods 135\u003c\/p\u003e \u003cp\u003eReferences 136\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Quadrature in Meshfree Methods 137\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Nomenclature and Acronyms 137\u003c\/p\u003e \u003cp\u003e6.2 Gauss Integration: An Introduction to Quadrature in Meshfree Methods 138\u003c\/p\u003e \u003cp\u003e6.3 Issues with Quadrature in Meshfree Methods 140\u003c\/p\u003e \u003cp\u003e6.4 Introduction to Nodal integration 142\u003c\/p\u003e \u003cp\u003e6.5 Integration Constraints and the Linear Patch Test 144\u003c\/p\u003e \u003cp\u003e6.6 Stabilized Conforming Nodal Integration 148\u003c\/p\u003e \u003cp\u003e6.7 Variationally Consistent Integration 154\u003c\/p\u003e \u003cp\u003e6.7.1 Variational Consistency Conditions 154\u003c\/p\u003e \u003cp\u003e6.7.2 Petrov–Galerkin Correction: VCI 157\u003c\/p\u003e \u003cp\u003e6.8 Quasi-Conforming SNNI for Extreme Deformations: Adaptive Cells 159\u003c\/p\u003e \u003cp\u003e6.9 Instability in Nodal Integration 160\u003c\/p\u003e \u003cp\u003e6.10 Stabilization of Nodal Integration 161\u003c\/p\u003e \u003cp\u003e6.10.1 Notation for Stabilized Nodal Integration 163\u003c\/p\u003e \u003cp\u003e6.10.2 Modified Strain Smoothing 164\u003c\/p\u003e \u003cp\u003e6.10.3 Naturally Stabilized Nodal Integration 166\u003c\/p\u003e \u003cp\u003e6.10.4 Naturally Stabilized Conforming Nodal Integration 168\u003c\/p\u003e \u003cp\u003eNotes 168\u003c\/p\u003e \u003cp\u003eReferences 169\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Nonlinear Meshfree Methods 173\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Lagrangian Description of the Strong Form 174\u003c\/p\u003e \u003cp\u003e7.2 Lagrangian Reproducing Kernel Approximation and Discretization 177\u003c\/p\u003e \u003cp\u003e7.3 Semi-Lagrangian Reproducing Kernel Approximation and Discretization 180\u003c\/p\u003e \u003cp\u003e7.4 Stability of Lagrangian and Semi-Lagrangian Discretizations 185\u003c\/p\u003e \u003cp\u003e7.4.1 Stability Analysis for the Lagrangian RK Equation of Motion 185\u003c\/p\u003e \u003cp\u003e7.4.2 Stability Analysis for the Semi-Lagrangian RK Equation of Motion 187\u003c\/p\u003e \u003cp\u003e7.4.3 Critical Time Step Estimation for the Lagrangian Formulation 189\u003c\/p\u003e \u003cp\u003e7.4.4 Critical Time Step Estimation for the Semi-Lagrangian Formulation 191\u003c\/p\u003e \u003cp\u003e7.4.5 Numerical Tests of Critical Time Step Estimation 192\u003c\/p\u003e \u003cp\u003e7.5 Neighbor Search Algorithms 196\u003c\/p\u003e \u003cp\u003e7.6 Smooth Contact Algorithm 198\u003c\/p\u003e \u003cp\u003e7.6.1 Continuum-Based Contact Formulation 198\u003c\/p\u003e \u003cp\u003e7.6.2 Meshfree Smooth Curve Representation 201\u003c\/p\u003e \u003cp\u003e7.6.3 Three-Dimensional Meshfree Smooth Contact Surface Representation and Contact Detection by a Nonparametric Approach 204\u003c\/p\u003e \u003cp\u003e7.7 Natural Kernel Contact Algorithm 207\u003c\/p\u003e \u003cp\u003e7.7.1 A Friction-like Plasticity Model 209\u003c\/p\u003e \u003cp\u003e7.7.2 Semi-Lagrangian RK Discretization and Natural Kernel Contact Algorithms 210\u003c\/p\u003e \u003cp\u003eNotes 212\u003c\/p\u003e \u003cp\u003eReferences 215\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Other Galerkin Meshfree Methods 219\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Smoothed Particle Hydrodynamics 219\u003c\/p\u003e \u003cp\u003e8.1.1 Kernel Estimate 220\u003c\/p\u003e \u003cp\u003e8.1.2 SPH Conservation Equations 224\u003c\/p\u003e \u003cp\u003e8.1.2.1 Mass Conservation (Continuity Equation) 224\u003c\/p\u003e \u003cp\u003e8.1.2.2 Equation of Motion 225\u003c\/p\u003e \u003cp\u003e8.1.2.3 Energy Conservation Equation 227\u003c\/p\u003e \u003cp\u003e8.1.3 Stability of SPH 228\u003c\/p\u003e \u003cp\u003e8.2 Partition of Unity Finite Element Method and h-p Clouds 232\u003c\/p\u003e \u003cp\u003e8.3 Natural Element Method 234\u003c\/p\u003e \u003cp\u003e8.3.1 First-Order Voronoi Diagram and Delaunay Triangulation 234\u003c\/p\u003e \u003cp\u003e8.3.2 Second-Order Voronoi Cell and Sibson Interpolation 235\u003c\/p\u003e \u003cp\u003e8.3.3 Laplace Interpolant (Non-Sibson Interpolation) 236\u003c\/p\u003e \u003cp\u003eReferences 237\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Strong Form Collocation Meshfree Methods 241\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 The Meshfree Collocation Method 242\u003c\/p\u003e \u003cp\u003e9.2 Approximations and Convergence for Strong Form Collocation 245\u003c\/p\u003e \u003cp\u003e9.2.1 Radial Basis Functions 245\u003c\/p\u003e \u003cp\u003e9.2.2 Moving Least Squares and Reproducing Kernel Approximations 246\u003c\/p\u003e \u003cp\u003e9.2.3 Reproducing Kernel Enhanced Local Radial Basis 247\u003c\/p\u003e \u003cp\u003e9.3 Weighted Collocation Methods and Optimal Weights 248\u003c\/p\u003e \u003cp\u003e9.4 Gradient Reproducing Kernel Collocation Method 251\u003c\/p\u003e \u003cp\u003e9.5 Subdomain Collocation for Heterogeneity and Discontinuities 253\u003c\/p\u003e \u003cp\u003e9.6 Comparison of Nodally-Integrated Galerkin Meshfree Methods and Nodally Collocated Strong Form Meshfree Methods 255\u003c\/p\u003e \u003cp\u003e9.6.1 Performance of Galerkin and Collocation Methods 255\u003c\/p\u003e \u003cp\u003e9.6.2 Stability of Node-Based Galerkin and Collocation Methods 256\u003c\/p\u003e \u003cp\u003eReferences 258\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 RKPM2D: A Two-Dimensional Implementation of RKPM 261\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Reproducing Kernel Particle Method: Approximation and Weak Form 261\u003c\/p\u003e \u003cp\u003e10.1.1 Reproducing Kernel Approximation 261\u003c\/p\u003e \u003cp\u003e10.1.2 Galerkin Formulation 262\u003c\/p\u003e \u003cp\u003e10.2 Domain Integration 264\u003c\/p\u003e \u003cp\u003e10.2.1 Gauss Integration 264\u003c\/p\u003e \u003cp\u003e10.2.2 Variationally Consistent Nodal Integration 265\u003c\/p\u003e \u003cp\u003e10.2.3 Stabilized Nodal Integration Schemes 266\u003c\/p\u003e \u003cp\u003e10.2.3.1 Modified Stabilized Nodal Integration 267\u003c\/p\u003e \u003cp\u003e10.2.3.2 Naturally Stabilized Nodal Integration 268\u003c\/p\u003e \u003cp\u003e10.3 Computer Implementation 269\u003c\/p\u003e \u003cp\u003e10.3.1 Domain Discretization 269\u003c\/p\u003e \u003cp\u003e10.3.2 Quadrature Point Generation 272\u003c\/p\u003e \u003cp\u003e10.3.3 RK Shape Function Generation 273\u003c\/p\u003e \u003cp\u003e10.3.4 Stabilization Methods 278\u003c\/p\u003e \u003cp\u003e10.3.5 Matrix Evaluation and Assembly 281\u003c\/p\u003e \u003cp\u003e10.3.6 Description of subroutines in RKPM2D 285\u003c\/p\u003e \u003cp\u003e10.4 Getting Started 287\u003c\/p\u003e \u003cp\u003e10.4.1 Input File Generation 288\u003c\/p\u003e \u003cp\u003e10.4.1.1 Model 290\u003c\/p\u003e \u003cp\u003e10.4.1.2 RK 294\u003c\/p\u003e \u003cp\u003e10.4.1.3 Quadrature 295\u003c\/p\u003e \u003cp\u003e10.4.2 Executing RKPM2D 295\u003c\/p\u003e \u003cp\u003e10.4.3 Post-Processing 295\u003c\/p\u003e \u003cp\u003e10.5 Numerical Examples 297\u003c\/p\u003e \u003cp\u003e10.5.1 Plotting the RK Shape Functions 297\u003c\/p\u003e \u003cp\u003e10.5.2 Patch Test 298\u003c\/p\u003e \u003cp\u003e10.5.3 Cantilever Beam Problem 300\u003c\/p\u003e \u003cp\u003e10.5.4 Plate With a Hole Problem 303\u003c\/p\u003e \u003cp\u003e10.A Appendix 310\u003c\/p\u003e \u003cp\u003eReferences 313\u003c\/p\u003e \u003cp\u003eIndex 315\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49402438648151,"sku":"9780470848005","price":99.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9780470848005.jpg?v=1730480399","url":"https:\/\/bookcurl.com\/products\/meshfree-and-particle-methods-9780470848005","provider":"Book Curl","version":"1.0","type":"link"}