FINITE ELEMENT METHOD - Indian Institute of Space Science and Technology
FINITE ELEMENT METHOD
Abdusamad A. Salih
Department of Aerospace Engineering Indian Institute of Space Science and Technology
Thiruvananthapuram - 695547, India. salih@iist.ac.in
ii
Contents
1 Introduction
3
1.1 Finite Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Direct Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 Variational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Weighted Residual Method . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Direct Approach to Finite Element Method
7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Linear Spring System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Solution of System of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Direct Approach to Steady-Sate Heat Conduction Problem . . . . . . . . . . . . 13
3 Calculus of Variations
15
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 First Variation of Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 The Fundamental Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Maxima and Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5.1 Maxima and minima of functionals . . . . . . . . . . . . . . . . . . . . 23
3.6 The Simplest Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6.1 Essential and natural boundary conditions . . . . . . . . . . . . . . . . . 28
3.6.2 Other forms of Euler?Lagrange equation . . . . . . . . . . . . . . . . . 28
3.6.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.7 Advanced Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.7.1 Variational problems with high-order derivatives . . . . . . . . . . . . . 30
3.7.2 Variational problems with several independent variables . . . . . . . . . 31
3.8 Application of EL Equation: Minimal Path Problems . . . . . . . . . . . . . . . 31
3.8.1 Shortest distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.8.2 The brachistochrone problem . . . . . . . . . . . . . . . . . . . . . . . 32
3.8.3 Deflection of beam ? variational formulation . . . . . . . . . . . . . . . 36
iii
CONTENTS
1
3.9 Construction of Functionals from PDEs . . . . . . . . . . . . . . . . . . . . . . 38 3.10 Rayleigh?Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Weighted Residual Methods
45
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Point Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Subdomain Collocation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Least Square Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Finite Element Method
65
5.1 Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.1 Steps in FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.1.2 Selection of Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.3 One-dimensional Linear Element . . . . . . . . . . . . . . . . . . . . . . 67
5.1.4 One-dimensional Quadratic Element . . . . . . . . . . . . . . . . . . . . 70
5.2 Two-dimensional Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2.1 Linear Triangular Element . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.2 Bilinear Rectangular Element . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 Finite Element Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2
CONTENTS
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