Introduction to the Finite Element Method - University of California ...

[Pages:233]ME280A

Introduction to the Finite Element Method

Panayiotis Papadopoulos

Department of Mechanical Engineering University of California, Berkeley

2015 edition

Copyright c 2015 by Panayiotis Papadopoulos

Contents

1 Introduction to the Finite Element Method

1

1.1 Historical perspective: the origins of the finite element method . . . . . . . . 1

1.2 Introductory remarks on the concept of discretization . . . . . . . . . . . . . 3

1.2.1 Structural analogue substitution method . . . . . . . . . . . . . . . . 4

1.2.2 Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.4 Particle methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Classifications of partial differential equations . . . . . . . . . . . . . . . . . 9

1.4 Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mathematical Preliminaries

13

2.1 Sets, linear function spaces, operators and functionals . . . . . . . . . . . . . 13

2.2 Continuity and differentiability . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Norms, inner products, and completeness . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Banach and Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.4 Linear operators and bilinear forms in Hilbert spaces . . . . . . . . . 25

2.4 Background on variational calculus . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Methods of Weighted Residuals

37

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Galerkin methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Collocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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3.3.1 Point-collocation method . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3.2 Subdomain-collocation method . . . . . . . . . . . . . . . . . . . . . 52 3.4 Least-squares methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 Composite methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.6 An interpretation of finite difference methods . . . . . . . . . . . . . . . . . 58 3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.8 Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Variational Methods

71

4.1 Introduction to variational principles . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Variational forms and variational principles . . . . . . . . . . . . . . . . . . . 75

4.3 Rayleigh-Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5 Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5 Construction of Finite Element Subspaces

87

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Finite element spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3 Completeness property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.4 Basic finite element shapes in one, two and three dimensions . . . . . . . . . 101

5.4.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4.3 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.4 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.5 Polynomial element interpolation functions . . . . . . . . . . . . . . . . . . . 102

5.5.1 Interpolations in one dimension . . . . . . . . . . . . . . . . . . . . . 102

5.5.2 Interpolations in two dimensions . . . . . . . . . . . . . . . . . . . . . 108

5.5.3 Interpolations in three dimensions . . . . . . . . . . . . . . . . . . . . 120

5.6 The concept of isoparametric mapping . . . . . . . . . . . . . . . . . . . . . 124

5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6 Computer Implementation of Finite Element Methods

137

6.1 Numerical integration of element matrices . . . . . . . . . . . . . . . . . . . 137

6.2 Assembly of global element arrays . . . . . . . . . . . . . . . . . . . . . . . . 142

6.3 Algebraic equation solving in finite element methods . . . . . . . . . . . . . 147

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6.4 Finite element modeling: mesh design and generation . . . . . . . . . . . . . 149 6.4.1 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4.2 Optimal node numbering . . . . . . . . . . . . . . . . . . . . . . . . . 151

6.5 Computer program organization . . . . . . . . . . . . . . . . . . . . . . . . . 152 6.6 Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

7 Elliptic Differential Equations

157

7.1 The Laplace equation in two dimensions . . . . . . . . . . . . . . . . . . . . 157

7.2 Linear elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

7.2.1 A Galerkin approximation to the weak form . . . . . . . . . . . . . . 163

7.2.2 On the order of numerical integration . . . . . . . . . . . . . . . . . . 166

7.2.3 The patch test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

7.3 Best approximation property of the finite element method . . . . . . . . . . 174

7.4 Error sources and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.5 Application to incompressible elastostatics and Stokes' flow . . . . . . . . . . 180

7.6 Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 186

7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

8 Parabolic Differential Equations

191

8.1 Standard semi-discretization methods . . . . . . . . . . . . . . . . . . . . . . 192

8.2 Stability of classical time integrators . . . . . . . . . . . . . . . . . . . . . . 199

8.3 Weighted-residual interpretation of classical time integrators . . . . . . . . . 203

8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9 Hyperbolic Differential Equations

207

9.1 The one-dimensional convection-diffusion equation . . . . . . . . . . . . . . . 207

9.2 Linear elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

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iv

List of Figures

1.1 B.G. Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 R. Courant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 R.W. Clough (left) and J. Argyris (right) . . . . . . . . . . . . . . . . . . . . 2 1.4 An infinite degree-of-freedom system . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 A simple example of the structural analogue method . . . . . . . . . . . . . . 4 1.6 The finite difference method in one dimension . . . . . . . . . . . . . . . . . 5 1.7 A one-dimensional finite element approximation . . . . . . . . . . . . . . . . 6 1.8 A one-dimensional kernel function Wl associated with a particle method . . . 8

2.1 Schematic depiction of a set V . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Example of a set that does not form a linear space . . . . . . . . . . . . . . . 15 2.3 Mapping between two sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 A function of class C0(0, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.5 Distance between two points in the classical Euclidean sense . . . . . . . . . 19 2.6 The neighborhood Nr(u) of a point u in V . . . . . . . . . . . . . . . . . . . 20 2.7 A continuous piecewise linear function and its derivatives . . . . . . . . . . . 25 2.8 A linear operator mapping U to V . . . . . . . . . . . . . . . . . . . . . . . . 26 2.9 A bilinear form on U ? V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.10 A functional exhibiting a minimum, maximum or saddle point at u = u . . . 28

3.1 An open and connected domain with smooth boundary written as the union of boundary regions i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 The domain of the Laplace-Poisson equation with Dirichlet boundary u and Neumann boundary q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Linear and quadratic approximations of the solution to the boundary-value problem in Example 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 The point-collocation method . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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3.5 The point collocation method in a square domain . . . . . . . . . . . . . . . . 50

3.6 The subdomain-collocation method . . . . . . . . . . . . . . . . . . . . . . . . 53

3.7

Polynomial

interpolation

functions

used

for

region

(xl

-

x 2

,

xl

+

x 2

]

in

the

weighted-residual interpretation of the finite difference method . . . . . . . . 59

3.8

Polynomial

interpolation

functions

used

for

region

[0,

x1

+

x 2

]

in

the

weighted-

residual interpretation of the finite difference method . . . . . . . . . . . . . 59

3.9 Interpolation functions for a finite element approximation of a one-dimensional

two-cell domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Piecewise linear interpolations functions in one dimension . . . . . . . . . . 79 4.2 Comparison of exact and approximate solutions . . . . . . . . . . . . . . . . 80

5.1 Geometric interpretation of Fourier coefficients . . . . . . . . . . . . . . . . 92 5.2 A finite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.3 A finite element-based interpolation function . . . . . . . . . . . . . . . . . . 95 5.4 Finite element vs. exact domain . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 Error in the enforcement of Dirichlet boundary conditions due to the difference

between the exact and the finite element domain . . . . . . . . . . . . . . . . 96 5.6 A potential violation of the integrability (compatibility) requirement . . . . . 97 5.7 A function u and its approximation uh in the domain (x?, x? + h) . . . . . . . 98 5.8 Pascal triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.9 Finite element domains in one dimension . . . . . . . . . . . . . . . . . . . . 101 5.10 Finite element domains in two dimensions . . . . . . . . . . . . . . . . . . . 101 5.11 Finite element domains in three dimensions . . . . . . . . . . . . . . . . . . 102 5.12 Linear element interpolation in one dimension . . . . . . . . . . . . . . . . . 103 5.13 One-dimensional finite element mesh with piecewise linear interpolation . . . 103 5.14 Standard quadratic element interpolations in one dimension . . . . . . . . . . 104 5.15 Hierarchical quadratic element interpolations in one dimension . . . . . . . . 105 5.16 Hermitian interpolation functions in one dimension . . . . . . . . . . . . . . 107 5.17 A 3-node triangular element . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.18 Higher-order triangular elements (left: 6-node element, right: 10-node element)110 5.19 A transitional 4-node triangular element . . . . . . . . . . . . . . . . . . . . 111 5.20 Area coordinates in a triangular domain . . . . . . . . . . . . . . . . . . . . 112 5.21 Four-node rectangular element . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.22 Interpolation function N1e for a = b = 1 (a hyperbolic paraboloid) . . . . . . . 114

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5.23 Three members of the serendipity family of rectangular elements . . . . . . . 114

5.24 Pascal's triangle for two-dimensional serendipity elements (before accounting for any interior nodes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.25 Three members of the Lagrangian family of rectangular elements . . . . . . . 115

5.26 Pascal's triangle for two-dimensional Lagrangian elements . . . . . . . . . . 116

5.27 A general quadrilateral finite element domain . . . . . . . . . . . . . . . . . . 117

5.28 Rectangular finite elements made of two or four joined triangular elements . 117

5.29

A

simple

potential

3-

or

4-node

triangular

element

for

the

case

p

=

2

(u,

u s

,

u n

dofs at nodes 1, 2, 3 and, possibly, u dof at node 4) . . . . . . . . . . . . . . . 118

5.30 Illustration of violation of the integrability requirement for the 9- or 10-dof

triangle for the case p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.31

A

12-dof

triangular

element

for

the

case

p=

2

(u,

u s

,

u n

dofs

at

nodes

1, 2, 3

and

u n

at

nodes

4, 5, 6)

.............................

119

5.32

Clough-Tocher

triangular

element

for

the

case

p

=

2

(u,

u s

,

u n

dofs

at

nodes

1, 2, 3 and

u n

at nodes 4, 5, 6)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

120

5.33 The 4-node tetrahedral element . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.34 The 10-node tetrahedral element . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.35 The 6- and 15-node pentahedral elements . . . . . . . . . . . . . . . . . . . . 123

5.36 The 8-node hexahedral element . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.37 The 20- and 27-node hexahedral elements . . . . . . . . . . . . . . . . . . . . 124

5.38 Schematic of a parametric mapping from e to e . . . . . . . . . . . . . . . 125

5.39 The 4-node isoparametric quadrilateral . . . . . . . . . . . . . . . . . . . . . 127

5.40 Geometric interpretation of one-to-one isoparametric mapping in the 4-node

quadrilateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.41 Convex and non-convex 4-node quadrilateral element domains . . . . . . . . 130

5.42 Relation between area elements in the natural and physical domain . . . . . . 130

5.43 Isoparametric 6-node triangle and 8-node quadrilateral . . . . . . . . . . . . . 131

5.44 Isoparametric 8-node hexahedral element . . . . . . . . . . . . . . . . . . . . 132

6.1 Two-dimensional Gauss quadrature rules for q1, q2 1 (left), q1, q2 3 (center), and q1, q2 5 (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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