Meshfree Methods Moving Beyond the Finite Element Method, Second Edition

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

G¹_h 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.

Biography

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.

Praise for the First Edition:

"This book addresses some of the current important issues, both positive and negative, related to mesh free methods, which should prove beneficial to researchers, engineers, and students who are interested in venturing into this area of research. … This is the first book published that comprehensively covers mesh free methods."
—Zentralblatt MATH, Vol. 1031 (2004/06)