The Finite Element Method

This much-anticipated second edition introduces the fundamentals of the finite element method featuring clear-cut examples and an applications-oriented approach. Using the transport equation for heat transfer as the foundation for the governing equations, this new edition demonstrates the versatility of the method for a wide range of applications, including structural analysis and fluid flow. Much attention is given to the development of the discrete set of algebraic equations, beginning with simple one-dimensional problems that can be solved by inspection, continuing to two- and three-dimensional elements, and ending with three chapters describing applications. The increased number of example problems per chapter helps build an understanding of the method to define and organize required initial and boundary condition data for specific problems. In addition to exercises that can be worked out manually, this new edition refers to user-friendly computer codes for solving one-, two-, and three-dimensional problems. Among the first FEM textbooks to include finite element software, the book contains a website with access to an even more comprehensive list of finite element software written in FEMLAB, MAPLE, MathCad, MATLAB, FORTRAN, C++, and JAVA - the most popular programming languages. This textbook is valuable for senior level undergraduates in mechanical, aeronautical, electrical, chemical, and civil engineering. Useful for short courses and home-study learning, the book can also serve as an introduction for first-year graduate students new to finite element coursework and as a refresher for industry professionals. The book is a perfect lead-in to Intermediate Finite Element Method: Fluid Flow and Heat and Transfer Applications (Taylor & Francis, 1999, Hb 1560323094).

Preface

1. INTRODUCTION

1.1 Background
1.2 History
1.3 Orientation
1.4 Closure
References

2. THE METHOD OF WEIGHTED RESIDUALS AND
GALERKIN APPROXIMATIONS

2.1 Background
2.2 Classical Solutions
2.3 The Weak Statement
2.4 Closure
Exercises
References

3. THE FINITE ELEMENT METHOD IN ONE DIMENSION

3.1 Overview
3.2 Shape Functions
3.2.1 Linear Elements
3.2.2 Quadratic Elements
3.2.3 Cubic Elements
3.3 Steady Conduction Equation
3.3.1 Galerkin Formulation
3.3.2 Variable Diffusion and Boundary Convection
3.4 Axisymmetric Heat Conduction
3.5 Natural Coordinate System
3.6 Time Dependence
3.6.1 Spatial Discretization
3.6.2 Time Discretization
3.7 Matrix Formulation
3.8 Solution Methods
3.9 Closure
Exercises
References

4. THE TWO-DIMENSIONAL TRIANGULAR ELEMENT

4.1 Overview
4.2 The Mesh
4.3 Shape Functions (Linear, Quadratic)
4.3.1 Linear Shape Functions
4.3.2 Quadratic Shape Functions
4.4 Area Coordinates
4.5 Numerical Integration
4.6 Diffusion in a Triangular Element
4.7 Steady-State Diffusion with Boundary Convection
4.8 The Axisymmetric Conduction Equation
4.9 The Quadratic Triangular Element
4.10 Time-Dependent Diffusion Equation
4.11 Bandwidth
4.12 Mass Lumping
4.13 Closure
Exercises
References

5. THE TWO-DIMENSIONAL QUADRILATERAL ELEMENT

5.1 Background
5.2 Element Mesh
5.3 Shape Functions
5.3.1 Bilinear Rectangular Element
5.3.2 Quadratic Rectangular Elements
5.4 Natural Coordinate System
5.5 Numerical Integration using Gaussian Quadratures
5.6 Steady-State Conduction with Boundary Convection
5.7 The Quadratic Quadrilateral Element
5.8 Time-Dependent Diffusion
5.9 Computer Program Exercises
5.10 Closure
Exercises
References

6. ISOPARAMETRIC TWO-DIMENSIONAL ELEMENTS

6.1 Background
6.2 Natural Coordinate System
6.3 Shape Functions
6.3.1 Bilinear Quadrilateral
6.3.2 Eight-Noded Quadratic Quadrilateral
6.3.3 Linear Triangle
6.3.4 Quadratic Triangle
6.3.5 Directional Cosines
6.4 The Element Matrices
6.5 Inviscid Flow Example
6.6 Closure
Exercises
References

7. THE THREE-DIMENSIONAL ELEMENT

7.1 Background
7.2 Element Mesh
7.3 Shape Functions
7.3.1 Tetrahedron
7.3.2 Hexahedron
7.4 Numerical Integration
7.5 One Element Heat Conduction Problem
7.5.1 Tetrahedron
7.5.2 Hexahedron
7.6 Time-Dependent Heat Conduction with Radiation
and Convection
7.6.1 Radiation
7.6.2 Shape Factors
7.7 Closure
Exercises
References

8. FINITE ELEMENTS IN SOLID MECHANICS

8.1 Background
8.2 Two-Dimensional Elasticity - Stress-Strain
8.3 Galerkin Approximation
8.4 Potential Energy
8.5 Thermal Stresses
8.6 Three-Dimensional Solid Elements
8.7 Closure
Exercises
References

9. APPLICATIONS TO CONVECTIVE TRANSPORT

9.1 Background
9.2 Potential Flow
9.3 Convective Transport
9.4 Nonlinear Convective Transport
9.5 Groundwater Flow
9.6 Lubrication
9.7 Closure
Exercises
References

10. INTRODUCTION TO FLUID FLOW

10.1 Background
10.2 Viscous Incompressible Flow with Heat Transfer
10.3 The Penalty Function Algorithm
10.4 Application to Natural Convection
10.5 Summary
Exercises
References

APPENDICES

A. Matrix Algebra
B. Units
C. Thermophysical Properties of Some Common Materials
D. Notation
E. Computer Programs
E.1 MESH-1D, FEM-1D
E.2 MESH-2D, FEM-2D
E.3 FEM-3D
E.4 FEMLAB
E.5 MATLAB, MATHCAD, MAPLE

INDEX