2nd Edition

Meshfree Methods Moving Beyond the Finite Element Method, Second Edition

By G.R. Liu Copyright 2010
    792 Pages 503 B/W Illustrations
    by CRC Press

    792 Pages 503 B/W Illustrations
    by CRC Press

    Understand How to Use and Develop Meshfree Techniques
    An 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.

    To access MFree2D (copyright, G. R. Liu), which accompanies MESHFREE METHODS: MOVING BEYOND THE FINITE ELEMENT METHOD, Second Edition (978-1-4200-8209-8) by Dr. G. R. Liu, please go to the website: www.ase.uc.edu/~liugr

    An access code is needed to use program – to receive it please email Dr. Liu directly at: [email protected]

    Dr. Liu will reply to you directly with the code, and you can then proceed to use the software.


    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


    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


    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


    Meshfree Methods for Fluid Dynamics Problem


    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


    Boundary Meshfree Methods

    RPIM Using Polynomial Basis

    RPIM Using Radial Function Basis


    Meshfree Methods Coupled with Other Methods

    Coupled EFG/BEM

    Coupled EFG and Hybrid BEM


    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



    Techniques Used in MFree2D

    Preprocessing in MFree2D

    Postprocessing in MFree2D


    References appear at the end of each chapter.


    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)