1st Edition

Smoothed Finite Element Methods

By G.R. Liu, Nguyen Trung Copyright 2010
    692 Pages 361 B/W Illustrations
    by CRC Press

    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.

    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.


    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