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ME280A Introduction to the Finite Element Method

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ME280A Introduction to the Finite Element Method 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 Suggestionsforfurtherreading. ... 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 Innerproducts .............................. 21 2.3.3 BanachandHilbertspaces . 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 Suggestionsforfurtherreading. ... 34 3 Methods of Weighted Residuals 37 3.1 Introduction.................................... 37 3.2 Galerkinmethods................................. 40 3.3 Collocationmethods ............................... 48 i 3.3.1 Point-collocation method . 49 3.3.2 Subdomain-collocation method . 52 3.4 Least-squaresmethods . .. .. 54 3.5 Compositemethods................................ 57 3.6 An interpretation of finite difference methods . ... 58 3.7 Exercises...................................... 62 3.8 Suggestionsforfurtherreading. ... 69 4 Variational Methods 71 4.1 Introduction to variational principles . .. 71 4.2 Variational forms and variational principles . ... 75 4.3 Rayleigh-Ritzmethod .............................. 77 4.4 Exercises...................................... 81 4.5 Suggestionsforfurtherreading. ... 84 5 Construction of Finite Element Subspaces 87 5.1 Introduction.................................... 87 5.2 Finiteelementspaces............................... 93 5.3 Completenessproperty .............................. 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 Theconceptofisoparametricmapping . 124 5.7 Exercises...................................... 133 6 Computer Implementation of Finite Element Methods 137 6.1 Numerical integration of element matrices . .. 137 6.2 Assemblyofglobalelementarrays. 142 6.3 Algebraic equation solving in finite element methods . 147 ii 6.4 Finite element modeling: mesh design and generation . .. 149 6.4.1 Symmetry................................. 150 6.4.2 Optimal node numbering . 151 6.5 Computerprogramorganization . 152 6.6 Suggestionsforfurtherreading. ... 153 6.7 Exercises...................................... 153 7 Elliptic Differential Equations 157 7.1 The Laplace equation in two dimensions . 157 7.2 Linearelastostatics ................................ 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 Thepatchtest .............................. 171 7.3 Best approximation property of the finite element method . ..... 174 7.4 Errorsourcesandestimates . 177 7.5 Application to incompressible elastostatics and Stokes’ flow . ...... 180 7.6 Suggestionsforfurtherreading. ... 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 Linearelastodynamics .............................. 213 9.3 Exercises...................................... 219 iii 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 .......................... 14 V 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 (u) of a point u in ................... 20 Nr V 2.7 A continuous piecewise linear function and its derivatives . .... 25 2.8 A linear operator mapping to ........................ 26 U V 2.9 A bilinear form on ............................ 27 U×V 2.10 A functional exhibiting a minimum, maximum or saddle point at u = u∗ ... 28 3.1 An open and connected domain Ω with smooth boundary written as the union of boundary regions ∂Ωi ............................. 37 3.2 The domain Ω of the Laplace-Poisson equation with Dirichlet boundary Γu and Neumann boundary Γq ............................ 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 v 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 (x ∆x , x + ∆x ] in the l − 2 l 2 weighted-residual interpretation of the finite difference method ........ 59 ∆x 3.8 Polynomial interpolation functions used for region [0, x1 + 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 e 5.22 Interpolation function N1 for a = b =1 (a hyperbolic paraboloid) ....... 114 vi 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 ∂u ∂u 5.29 A simple potential 3- or 4-node triangular element for the case p =2 (u, , ∂s ∂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 ∂u ∂u 5.31 A 12-dof triangular element for the case p =2 (u, , dofs at nodes 1, 2, 3 ∂s ∂n ∂u and at nodes 4, 5, 6) ............................. 119 ∂n ∂u ∂u 5.32 Clough-Tocher triangular element for the case p =2 (u, , dofs at nodes ∂s ∂n ∂u 1, 2, 3 and at nodes 4, 5, 6) .......................... 120 ∂n 5.33 The 4-node tetrahedral element .........................
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