Smoothed Finite Element Methods: 1st Edition (Hardback) book cover

Smoothed Finite Element Methods

1st Edition

By G.R. Liu, Nguyen Thoi Trung

CRC Press

692 pages | 361 B/W Illus.

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Hardback: 9781439820278
pub: 2010-06-08
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pub: 2016-04-19
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Generating a quality finite element mesh is difficult and often very time-consuming. Mesh-free methods operations can also be complicated and quite costly in terms of computational effort and resources. Developed by the authors and their colleagues, the smoothed finite element method (S-FEM) only requires a triangular/tetrahedral mesh to achieve more accurate results, a generally higher convergence rate in energy without increasing computational cost, and easier auto-meshing of the problem domain. Drawing on the authors’ extensive research results, Smoothed Finite Element Methods presents the theoretical framework and development of various S-FEM models.

After introducing background material, basic equations, and an abstracted version of the FEM, the book discusses the overall modeling procedure, fundamental theories, error assessment matters, and necessary building blocks to construct useful S-FEM models. It then focuses on several specific S-FEM models, including cell-based (CS-FEM), node-based (NS-FEM), edge-based (ES-FEM), face-based (FS-FEM), and a combination of FEM and NS-FEM (αFEM). These models are then applied to a wide range of physical problems in solid mechanics, fracture mechanics, viscoelastoplasticity, plates, piezoelectric structures, heat transfer, and structural acoustics.

Requiring no previous knowledge of FEM, this book shows how computational methods and numerical techniques like the S-FEM help in the design and analysis of advanced engineering systems in rapid and cost-effective ways since the modeling and simulation can be performed automatically in a virtual environment without physically building the system. Readers can easily apply the methods presented in the text to their own engineering problems for reliable and certified solutions.


Liu and Nguyen introduce newly developed S-FEM models that combine FEM and mesh-free techniques, and explain their application to fracture mechanics, plates, piezoelectrics, heat transfer, and acoustics problems. Intended for mechanical and structural engineers, the graduate textbook describes each step in the S-FEM method and analyzes the properties of S-FEM models using smoothing domains based on cells, nodes, edges, and faces. Numerical examples are provided for an interfacial crack, elastic strain on a hollow sphere, plate buckling, an engine pedestal, and acoustic pressure distribution in a car passenger compartment.

SciTech Book News, February 2011

Table of Contents


Physical Problems in Engineering

Numerical Techniques: Practical Solution Tools

Why S-FEM?

The Idea of S-FEM

Key Techniques Used in S-FEM

S-FEM Models and Properties

Some Historical Notes

Outline of the Book

Basic Equations for Solid Mechanics

Equilibrium Equation: In Stresses

Constitutive Equation

Compatibility Equation

Equilibrium Equation: In Displacements

Equations in Matrix Form

Boundary Conditions

Some Standard Default Conventions and Notations

The Finite Element Method

General Procedure of FEM

Proper Spaces

Weak Formulation and Properties of the Solution

Domain Discretization: Creation of Finite-Dimensional Space

Creation of Shape Functions

Displacement Function Creation

Strain Evaluation

Formulation of the Discretized System of Equations

FEM Solution: Existence, Uniqueness, Error, and Convergence

Some Other Properties of the FEM Solution

Linear Triangular Element (T3)

Four-Node Quadrilateral Element (Q4)

Four-Node Tetrahedral Element (T4)

Eight-Node Hexahedral Element (H8)

Gauss Integration

Fundamental Theories for S-FEM

General Procedure for S-FEM Models

Domain Discretization with Polygonal Elements

Creating a Displacement Field: Shape Function Construction

Evaluation of the Compatible Strain Field

Modify/Construct the Strain Field

Minimum Number of Smoothing Domains: Essential to Stability

Smoothed Galerkin Weak Form

Discretized Linear Algebraic System of Equations

Solve the Algebraic System of Equations

Error Assessment in S-FEM and FEM Models

Implementation Procedure for S-FEM Models

General Properties of S-FEM Models

Cell-Based Smoothed FEM

Cell-Based Smoothing Domain

Discretized System of Equations

Shape Function Evaluation

Some Properties of CS-FEM

Stability of CS-FEM and nCS-FEM

Standard Patch Test: Accuracy

Selective CS-FEM: Volumetric Locking Free

Numerical Examples

Node-Based Smoothed FEM


Creation of Node-Based Smoothing Domains

Formulation of NS-FEM

Evaluation of Shape Function Values

Properties of NS-FEM

An Adaptive NS-FEM Using Triangular Elements

Numerical Examples

Edge-Based Smoothed FEM


Creation of Edge-Based Smoothing Domains

Formulation of the ES-FEM

Evaluation of the Shape Function Values in the ES-FEM

A Smoothing-Domain-Based Selective ES/NS-FEM

Properties of the ES-FEM

Numerical Examples

Face-Based Smoothed FEM


Face-Based Smoothing Domain Creation

Formulation of FS-FEM-T4

A Smoothing-Domain-Based Selective FS/NS-FEM-T4 Model

Stability, Accuracy, and Mesh Sensitivity

Numerical Examples

The αFEM


Idea of αFEM-T3 and αFEM-T4

αFEM-T3 and αFEM-T4 for Nonlinear Problems

Implementation and Patch Tests

Numerical Examples

S-FEM for Fracture Mechanics


Singular Stress Field Creation at the Crack-Tip

Possible sS-FEM Methods

sNS-FEM Models

sES-FEM Models

Stiffness Matrix Evaluation

J-Integral and SIF Evaluation

Interaction Integral Method for Mixed Mode

Numerical Examples Solved Using sES-FEM-T3

Numerical Examples Solved Using sNS-FEM-T3

S-FEM for Viscoelastoplasticity


Strong Formulation for Viscoelastoplasticity

FEM for Viscoelastoplasticity: A Dual Formulation

S-FEM for Viscoelastoplasticity: A Dual Formulation

A Posteriori Error Estimation

Numerical Examples

ES-FEM for Plates


Weak Form for the Reissner–Mindlin Plate

FEM Formulation for the Reissner–Mindlin Plate

ES-FEM-DSG3 for the Reissner–Mindlin Plate

Numerical Examples: Patch Test

Numerical Examples: Static Analysis

Numerical Examples: Free Vibration of Plates

Numerical Examples: Buckling of Plates

S-FEM for Piezoelectric Structures


Galerkin Weak Form for Piezoelectrics

Finite Element Formulation for the Piezoelectric Problem

S-FEM for the Piezoelectric Problem

Numerical Results

S-FEM for Heat Transfer Problems


Strong-Form Equations for Heat Transfer Problems

Boundary Conditions

Weak Forms for Heat Transfer Problems

FEM Equations

S-FEM Equations

Evaluation of the Smoothed Gradient Matrix

Numerical Example

Bioheat Transfer Problems

S-FEM for Acoustics Problems


Mathematical Model of Acoustics Problems

Weak Forms for Acoustics Problems

FEM Equations

S-FEM Equations

Error in a Numerical Model

Numerical Examples


References appear at the end of each chapter.

About the Authors

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

Nguyen Thoi Trung is a lecturer in the Department of Mechanics in the University of Science at Vietnam National University in Ho Chi Minh City. He is also the CEO of the Friends of Science and Technology (FOSAT) Group and a researcher in the Faculty of Civil Engineering at Ton Duc Thang University in Ho Chi Minh City.

Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Mechanics / General