Meshfree Methods : Moving Beyond the Finite Element Method, Second Edition book cover
2nd Edition

Meshfree Methods
Moving Beyond the Finite Element Method, Second Edition

ISBN 9781138372702
Published September 10, 2018 by CRC Press
792 Pages 503 B/W Illustrations

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Book Description

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:

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.

Table of Contents


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.

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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)