Meshfree and Particle Methods

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

Autorentext
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.

Klappentext
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.

Zusammenfassung
Meshfree 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.

Inhalt
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 Refere…

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