Meshfree Methods: Moving Beyond the Finite Element Method

Understand How to Use and Develop Meshfree TechniquesAn Update of a Groundbreaking Work
Reflecting the significant advances made in the field since the publication of its predecessor, Meshfree Methods: Moving Beyond the Finite Element Method, Second Edition systematically covers the most widely used meshfree methods. With 70% new material, this edition addresses important new developments, especially on essential theoretical issues.
New to the Second Edition


Much more details on fundamental concepts and important theories for numerical methods
Discussions on special properties of meshfree methods, including stability, convergence, accurate, efficiency, and bound property
More detailed discussion on error estimation and adaptive analysis using meshfree methods
Developments on combined meshfree/finite element method (FEM) models
Comparison studies using meshfree and FEM


Drawing on the author’s own research, this book provides a single-source guide to meshfree techniques and theories that can effectively handle a variety of complex engineering problems. It analyzes how the methods work, explains how to use and develop the methods, and explores the problems associated with meshfree methods.

G.R. Liu is the director of the Centre for Advanced Computations in Engineering Science (ACES) and professor in the Department of Mechanical Engineering at the National University of Singapore.

Preliminaries
Physical Problems in Engineering
Solid Mechanics: A Fundamental Engineering Problem
Numerical Techniques: Practical Solution Tools
Defining Meshfree Methods
Need for Meshfree Methods
The Ideas of Meshfree Methods
Basic Techniques for Meshfree Methods
Outline of the Book
Some Notations and Default Conventions
Remarks

Meshfree Shape Function Construction
Basic Issues for Shape Function Construction
Smoothed Particle Hydrodynamics Approach
Reproducing Kernel Particle Method
Moving Least Squares Approximation
Point Interpolation Method
Radial PIM
Radial PIM with Polynomial Reproduction
Weighted Least Square (WLS) Approximation
Polynomial PIM with Rotational Coordinate Transformation
Comparison Study via Examples
Compatibility Issues: An Analysis
Other Methods

Function Spaces for Meshfree Methods
Function Spaces
Useful Spaces in Weak Formulation
G Spaces: Definition
G1h Spaces: Basic Properties
Error Estimation
Concluding Remarks

Strain Field Construction
Why Construct Strain Field?
Historical Notes
How to Construct?
Admissible Conditions for Constructed Strain Fields
Strain Construction Techniques
Concluding Remarks

Weak and Weakened Weak Formulations
Introduction to Strong and Weak Forms
Weighted Residual Method
A Weak Formulation: Galerkin
A Weakened Weak Formulation: GS-Galerkin
The Hu–Washizu Principle
The Hellinger–Reissner Principle
The Modified Hellinger–Reissner Principle
Single-Field Hellinger–Reissner Principle
The Principle of Minimum Complementary Energy
The Principle of Minimum Potential Energy
Hamilton’s Principle
Hamilton’s Principle with Constraints
Galerkin Weak Form
Galerkin Weak Form with Constraints
A Weakened Weak Formulation: SC-Galerkin
Parameterized Mixed Weak Form
Concluding Remarks

Element Free Galerkin Method
EFG Formulation with Lagrange Multipliers
EFG with Penalty Method
Summary

Meshless Local Petrov–Galerkin Method
MLPG Formulation
MLPG for Dynamic Problems
Concluding Remarks

Point Interpolation Methods
Node-Based Smoothed Point Interpolation Method (NS-PIM)
NS-PIM Using Radial Basis Functions (NS-RPIM)
Upper Bound Properties of NS-PIM and NS-RPIM
Edge-Based Smoothed Point Interpolation Methods (ES-PIMs)
A Combined ES/NS Point Interpolation Methods (ES/NS-PIM)
Strain-Constructed Point Interpolation Method (SC-PIM)
A Comparison Study
Summary

Meshfree Methods for Fluid Dynamics Problem
Introduction
Navier–Stokes Equations
Smoothed Particle Hydrodynamics Method
Gradient Smoothing Method (GSM)
Adaptive Gradient Smoothing Method (A-GSM)
A Discussion on GSM for Incompressible Flows
Other Improvements on GSM

Meshfree Methods for Beams
PIM Shape Function for Thin Beams
Strong Form Equations
Weak Formulation: Galerkin Formulation
A Weakened Weak Formulation: GS-Galerkin
Three Models
Formulation for NS-PIM for Thin Beams
Formulation for Dynamic Problems
Numerical Examples for Static Analysis
Numerical Examples: Upper Bound Solution
Numerical Examples for Free Vibration Analysis
Concluding Remarks

Meshfree Methods for Plates
Mechanics for Plates
EFG Method for Thin Plates
EFG Method for Thin Composite Laminates
EFG Method for Thick Plates
ES-PIM for Plates

Meshfree Methods for Shells
EFG Method for Spatial Thin Shells
EFG Method for Thick Shells
ES-PIM for Thick Shells
Summary

Boundary Meshfree Methods
RPIM Using Polynomial Basis
RPIM Using Radial Function Basis
Remarks

Meshfree Methods Coupled with Other Methods
Coupled EFG/BEM
Coupled EFG and Hybrid BEM
Remarks

Meshfree Methods for Adaptive Analysis
Triangular Mesh and Integration Cells
Node Numbering: A Simple Approach
Bucket Algorithm for Node Searching
Relay Model for Domains with Irregular Boundaries
Techniques for Adaptive Analysis
Concluding Remarks

MFree2D©
Overview
Techniques Used in MFree2D
Preprocessing in MFree2D
Postprocessing in MFree2D
Index
References appear at the end of each chapter.