Introduction to the Finite Element Method 4E

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