Smoothed Finite Element Methods

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 (aFEM). 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.

Introduction 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 SolutionDomain 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 FieldMinimum 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 ModelsImplementation 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 Introduction Creation of Node-Based Smoothing Domains Formulation of NS-FEMEvaluation of Shape Function Values Properties of NS-FEMAn Adaptive NS-FEM Using Triangular Elements Numerical Examples Edge-Based Smoothed FEM Introduction Creation of Edge-Based Smoothing Domains Formulation of the ES-FEMEvaluation of the Shape Function Values in the ES-FEM A Smoothing-Domain-Based Selective ES/NS-FEM Properties of the ES-FEMNumerical Examples Face-Based Smoothed FEM Introduction 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 aFEM Introduction Idea of aFEM-T3 and aFEM-T4 aFEM-T3 and aFEM-T4 for Nonlinear Problems Implementation and Patch TestsNumerical Examples S-FEM for Fracture Mechanics Introduction Singular Stress Field Creation at the Crack-TipPossible sS-FEM Methods sNS-FEM ModelssES-FEM ModelsStiffness Matrix Evaluation J-Integral and SIF EvaluationInteraction Integral Method for Mixed ModeNumerical Examples Solved Using sES-FEM-T3 Numerical Examples Solved Using sNS-FEM-T3 S-FEM for Viscoelastoplasticity Introduction Strong Formulation for ViscoelastoplasticityFEM for Viscoelastoplasticity: A Dual FormulationS-FEM for Viscoelastoplasticity: A Dual FormulationA Posteriori Error Estimation Numerical Examples ES-FEM for Plates Introduction Weak Form for the Reissner–Mindlin Plate FEM Formulation for the Reissner–Mindlin Plate ES-FEM-DSG3 for the Reissner–Mindlin PlateNumerical Examples: Patch Test Numerical Examples: Static Analysis Numerical Examples: Free Vibration of Plates Numerical Examples: Buckling of Plates S-FEM for Piezoelectric Structures Introduction Galerkin Weak Form for PiezoelectricsFinite Element Formulation for the Piezoelectric Problem S-FEM for the Piezoelectric ProblemNumerical Results S-FEM for Heat Transfer Problems Introduction Strong-Form Equations for Heat Transfer Problems Boundary Conditions Weak Forms for Heat Transfer ProblemsFEM Equations S-FEM Equations Evaluation of the Smoothed Gradient Matrix Numerical ExampleBioheat Transfer Problems S-FEM for Acoustics Problems Introduction Mathematical Model of Acoustics Problems Weak Forms for Acoustics ProblemsFEM Equations S-FEM Equations Error in a Numerical Model Numerical Examples Index 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.