Smoothed Finite Element Methods  book cover
1st Edition

Smoothed Finite Element Methods

ISBN 9781439820278
Published June 8, 2010 by CRC Press
692 Pages 361 B/W Illustrations

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Book Description

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.

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.

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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.


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