Introduction to the Finite Element Method 4E

A fully updated introduction to the principles and applications of the finite element method This authoritative and thoroughly revised and self-contained classic mechanical engineering textbook offers a broad-based overview and applications of the finite element method. This revision updates and expands the already large number of problems and worked-out examples and brings the technical coverage in line with current practices. You will get details on non-traditional applications in bioengineering, fluid and thermal sciences, and structural mechanics. Written by a world-renowned mechanical engineering researcher and author, An Introduction to the Finite Element Method, Fourth Edition, teaches, step-by-step, how to determine numerical solutions to equilibrium as well as time-dependent problems from fluid and thermal sciences and structural mechanics and a host of applied sciences.. Beginning with the governing differential equations, the book presents a systematic approach to the derivation of weak-forms (integral formulations), interpolation theory, finite element equations, solution of problems from fluid and thermal sciences and structural mechanics, computer implementation. The author provides a solutions manual as well as computer programs that are available for download. •Features updated problems and fully worked-out solutions •Contains downloadable programs that can be applied and extended to real-world situations •Written by a highly-cited mechanical engineering researcher and well-respected author Title Page Copyright Page Dedication Contents Preface to the Fourth Edition Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Symbols and Conversion Factors 1 General Introduction 1.1 Background 1.2 Mathematical Model Development 1.3 Numerical Simulations 1.4 The Finite Element Method 1.4.1 The Basic Idea 1.4.2 The Basic Features 1.4.3 Some Remarks 1.4.4 A Brief Review of the History of the Finite Element Method 1.5 The Present Study 1.6 Summary Problems References for Additional Reading 2 Mathematical Preliminaries and Classical Variational Methods 2.1 General Introduction 2.1.1 Variational Principles and Methods 2.1.2 Variational Formulations 2.1.3 Need for Weighted-Integral Statements 2.2 Some Mathematical Concepts and Formulae 2.2.1 Coordinate Systems and the Del Operator 2.2.2 Boundary Value, Initial Value, and Eigenvalue Problems 2.2.3 Integral Identities 2.2.4 Matrices and Their Operations 2.3 Energy and Virtual Work Principles 2.3.1 Introduction 2.3.2 Work and Energy 2.3.3 Strain Energy and Strain Energy Density 2.3.4 Total Potential Energy 2.3.5 Virtual Work 2.3.6 The Principle of Virtual Displacements 2.3.7 The Principle of Minimum Total Potential Energy 2.3.8 Castigliano’s Theorem I 2.4 Integral Formulations of Differential Equations 2.4.1 Introduction 2.4.2 Residual Function 2.4.3 Methods of Making the Residual Zero 2.4.4 Development of Weak Forms 2.4.5 Linear and Bilinear Forms and Quadratic Functionals 2.4.6 Examples of Weak Forms and Quadratic Functionals 2.5 Variational Methods 2.5.1 Introduction 2.5.2 The Ritz Method 2.5.3 The Method of Weighted Residuals 2.6 Equations of Continuum Mechanics 2.6.1 Preliminary Comments 2.6.2 Heat Transfer 2.6.3 Fluid Mechanics 2.6.4 Solid Mechanics 2.7 Summary Problems References for Additional Reading 3 1-D Finite Element Models of Second-Order Differential Equations 3.1 Introduction 3.1.1 Preliminary Comments 3.1.2 Desirable Features of an Effective Computational Method 3.1.3 The Basic Features of the Finite Element Method 3.2 Finite Element Analysis Steps 3.2.1 Preliminary Comments 3.2.2 Discretization of a System 3.2.3 Derivation of Element Equations: Finite Element Model 3.3 Finite Element Models of Discrete Systems 3.3.1 Linear Elastic Spring 3.3.2 Axial Deformation of Elastic Bars 3.3.3 Torsion of Circular Shafts 3.3.4 Electrical Resistor Circuits 3.3.5 Fluid Flow Through Pipes 3.3.6 One-Dimensional Heat Transfer 3.4 Finite Element Models of Continuous Systems 3.4.1 Preliminary Comments 3.4.2 Model Boundary Value Problem 3.4.3 Derivation of Element Equations: The Finite Element Model 3.4.4 Assembly of Element Equations 3.4.5 Imposition of Boundary Conditions and Condensed Equations 3.4.6 Postprocessing of the Solution 3.4.7 Remarks and Observations 3.5 Axisymmetric Problems 3.5.1 Model Equation 3.5.2 Weak Form 3.5.3 Finite Element Model 3.6 Errors in Finite Element Analysis 3.6.1 Types of Errors 3.6.2 Measures of Errors 3.6.3 Convergence and Accuracy of Solutions 3.7 Summary Problems References for Additional Reading 4 Applications to 1-D Heat Transfer and Fluid and Solid Mechanics Problems 4.1 Preliminary Comments 4.2 Heat Transfer 4.2.1 Governing Equations 4.2.2 Finite Element Equations 4.2.3 Numerical Examples 4.3 Fluid Mechanics 4.3.1 Governing Equations 4.3.2 Finite Element Model 4.4 Solid and Structural Mechanics 4.4.1 Preliminary Comments 4.4.2 Finite Element Model of Bars and Cables 4.4.3 Numerical Examples 4.5 Summary Problems References for Additional Reading 5 Finite Element Analysis of Beams and Circular Plates 5.1 Introduction 5.2 Euler–Bernoulli Beam Element 5.2.1 Governing Equation 5.2.2 Discretization of the Domain 5.2.3 Weak-Form Development 5.2.4 Approximation Functions 5.2.5 Derivation of Element Equations (Finite Element Model) 5.2.6 Assembly of Element Equations 5.2.7 Imposition of Boundary Conditions and the Condensed Equations 5.2.8 Postprocessing of the Solution 5.2.9 Numerical Examples 5.3 Timoshenko Beam Elements 5.3.1 Governing Equations 5.3.2 Weak Forms 5.3.3 General Finite Element Model 5.3.4 Shear Locking and Reduce Integration 5.3.5 Consistent Interpolation Element (CIE) 5.3.6 Reduced Integration Element (RIE) 5.3.7 Numerical Examples 5.4 Axisymmetric Bending of Circular Plates 5.4.1 Governing Equations 5.4.2 Weak Form 5.4.3 Finite Element Model 5.5 Summary Problems References for Additional Reading 6 Plane Trusses and Frames 6.1 Introduction 6.2 Analysis of Trusses 6.2.1 The Truss Element in the Local Coordinates 6.2.2 The Truss Element in the Global Coordinates 6.3 Analysis of Plane Frame Structures 6.3.1 Introductory Comments 6.3.2 General Formulation 6.3.3 Euler–Bernoulli Frame Element 6.3.4 Timoshenko Frame Element Based on CIE 6.3.5 Timoshenko Frame Element Based on RIE 6.4 Inclusion of Constraint Conditions 6.4.1 Introduction 6.4.2 Lagrange Multiplier Method 6.4.3 Penalty Function Approach 6.4.4 A Direct Approach 6.5 Summary Problems References for Additional Reading 7 Eigenvalue and Time-Dependent Problems in 1-D 7.1 Introduction 7.2 Equations of Motion 7.2.1 One-Dimensional Heat Flow 7.2.2 Axial Deformation of Bars 7.2.3 Bending of Beams: The Euler–Bernoulli Beam Theory 7.2.4 Bending of Beams: The Timoshenko Beam Theory 7.3 Eigenvalue Problems 7.3.1 General Comments 7.3.2 Physical Meaning of Eigenvalues 7.3.3 Reduction of the Equations of Motion to Eigenvalue Equations 7.3.4 Eigenvalue Problem: Buckling of Beams 7.3.5 Finite Element Models 7.3.6 Buckling of Beams 7.4 Transient Analysis 7.4.1 Introduction 7.4.2 Semidiscrete Finite Element Model of a Single Model Equation 7.4.3 The Timoshenko Beam Theory 7.4.4 Parabolic Equations 7.4.5 Hyperbolic Equations 7.4.6 Explicit and Implicit Formulations and Mass Lumping 7.4.7 Examples 7.5 Summary Problems References for Additional Reading 8 Numerical Integration and Computer Implementation 8.1 Introduction 8.2 Numerical Integration 8.2.1 Preliminary Comments 8.2.2 Natural Coordinates 8.2.3 Approximation of Geometry 8.2.4 Parametric Formulations 8.2.5 Numerical Integration 8.3 Computer Implementation 8.3.1 Introductory Comments 8.3.2 General Outline 8.3.3 Preprocessor 8.3.4 Calculation of Element Matrices (Processor) 8.3.5 Assembly of Element Equations (Processor) 8.3.6 Imposition of Boundary Conditions (Processor) 8.3.7 Solution of Equations and Postprocessing 8.4 Applications of Program FEM1D 8.4.1 General Comments 8.4.2 Illustrative Examples 8.5 Summary Problems References for Additional Reading 9 Single-Variable Problems in Two Dimensions 9.1 Introduction 9.2 Boundary Value Problems 9.2.1 The Model Equation 9.2.2 Finite Element Discretization 9.2.3 Weak Form 9.2.4 Vector Form of the Variational Problem 9.2.5 Finite Element Model 9.2.6 Derivation of Interpolation Functions 9.2.7 Evaluation of Element Matrices and Vectors 9.2.8 Assembly of Element Equations 9.2.9 Post-computations 9.2.10 Axisymmetric Problems 9.3 Modeling Considerations 9.3.1 Exploitation of Solution Symmetries 9.3.2 Choice of a Mesh and Mesh Refinement 9.3.3 Imposition of Boundary Conditions 9.4 Numerical Examples 9.4.1 General Field Problems 9.4.2 Conduction and Convection Heat Transfer 9.4.3 Axisymmetric Systems 9.4.4 Fluid Mechanics 9.4.5 Solid Mechanics 9.5 Eigenvalue and Time-Dependent Problems 9.5.1 Finite Element Formulation 9.5.2 Parabolic Equations 9.5.3 Hyperbolic Equations 9.6 Summary Problems References for Additional Reading 10 2-D Interpolation Functions, Numerical Integration, and Computer Implementation 10.1 Introduction 10.1.1 Interpolation Functions 10.1.2 Numerical Integration 10.1.3 Program FEM2D 10.2 2-D Element Library 10.2.1 Pascal’s Triangle for Triangular Elements 10.2.2 Interpolation Functions for Triangular Elements Using Area Coordinates 10.2.3 Interpolation Functions Using Natural Coordinates 10.2.4 The Serendipity Elements 10.3 Numerical Integration 10.3.1 Preliminary Comments 10.3.2 Coordinate Transformations 10.3.3 Numerical Integration over Master Rectangular Element 10.3.4 Integration over a Master Triangular Element 10.4 Modeling Considerations 10.4.1 Preliminary Comments 10.4.2 Element Geometries 10.4.3 Mesh Refinements 10.4.4 Load Representation 10.5 Computer Implementation and FEM2D 10.5.1 Overview of Program FEM2D 10.5.2 Preprocessor 10.5.3 Element Computations (Processor) 10.5.4 Applications of FEM2D 10.5.5 Illustrative Examples 10.6 Summary Problems References for Additional Reading 11 Flows of Viscous Incompressible Fluids 11.1 Introduction 11.2 Governing Equations 11.3 Velocity–Pressure Formulation 11.3.1 Weak Formulation 11.3.2 Finite Element Model 11.4 Penalty Function Formulation 11.4.1 Preliminary Comments 11.4.2 Formulation of the Flow Problem as a Constrained Problem 11.4.3 Lagrange Multiplier Model 11.4.4 Penalty Model 11.4.5 Time Approximation 11.5 Computational Aspects 11.5.1 Properties of the Matrix Equations 11.5.2 Choice of Elements 11.5.3 Evaluation of Element Matrices in the Penalty Model 11.5.4 Post-computation of Stresses 11.6 Numerical Examples 11.7 Summary Problems References for Additional Reading 12 Plane Elasticity 12.1 Introduction 12.2 Governing Equations 12.2.1 Plane Strain 12.2.2 Plane Stress 12.2.3 Summary of Equations 12.3 Virtual Work and Weak Formulations 12.3.1 Preliminary Comments 12.3.2 Principle of Virtual Displacements in Vector Form 12.3.3 Weak-Form Formulation 12.4 Finite Element Model 12.4.1 General Comments 12.4.2 FE Model Using the Vector Form 12.4.3 FE Model Using Weak Form 12.4.4 Eigenvalue and Transient Problems 12.4.5 Evaluation of Integrals 12.4.6 Assembly of Finite Element Equations 12.4.7 Post-computation of Strains and Stresses 12.5 Elimination of Shear Locking in Linear Elements 12.5.1 Background 12.5.2 Modification of the Stiffness Matrix of Linear Finite Elements 12.6 Numerical Examples 12.7 Summary Problems References for Additional Reading 13 3-D Finite Element Analysis 13.1 Introduction 13.2 Heat Transfer 13.2.1 Preliminary Comments 13.2.2 Governing Equations 13.2.3 Weak Form 13.2.4 Finite Element Model 13.3 Flows of Viscous Incompressible Fluids 13.3.1 Governing Equations 13.3.2 Weak Forms 13.3.3 Finite Element Model 13.4 Elasticity 13.4.1 Governing Equations 13.4.2 Principle of Virtual Displacements 13.4.3 Finite Element Model 13.5 Element Interpolation Functions and Numerical Integration 13.5.1 Fully Discretized Models and Computer Implementation 13.5.2 Three-Dimensional Finite Elements 13.5.3 Numerical Integration 13.6 Numerical Examples 13.7 Summary Problems References for Additional Reading Index