Introduction To The Finite Element Method
ME280AIntroduction to the Finite Element MethodPanayiotis PapadopoulosDepartment of Mechanical EngineeringUniversity of California, Berkeley2015 editionCopyright c 2015 by Panayiotis Papadopoulos
Contents1 Introduction to the Finite Element Method11.1Historical perspective: the origins of the finite element method . . . . . . . .11.2Introductory remarks on the concept of discretization . . . . . . . . . . . . .31.2.1Structural analogue substitution method . . . . . . . . . . . . . . . .41.2.2Finite difference method . . . . . . . . . . . . . . . . . . . . . . . . .51.2.3Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . .61.2.4Particle methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81.3Classifications of partial differential equations . . . . . . . . . . . . . . . . .91.4Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . .112 Mathematical Preliminaries132.1Sets, linear function spaces, operators and functionals . . . . . . . . . . . . .132.2Continuity and differentiability . . . . . . . . . . . . . . . . . . . . . . . . .172.3Norms, inner products, and completeness . . . . . . . . . . . . . . . . . . . .182.3.1Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .182.3.2Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .212.3.3Banach and Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . .232.3.4Linear operators and bilinear forms in Hilbert spaces . . . . . . . . .252.4Background on variational calculus . . . . . . . . . . . . . . . . . . . . . . .282.5Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .322.6Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . .343 Methods of Weighted Residuals373.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .373.2Galerkin methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .403.3Collocation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48i
3.3.1Point-collocation method . . . . . . . . . . . . . . . . . . . . . . . . .493.3.2Subdomain-collocation method . . . . . . . . . . . . . . . . . . . . .523.4Least-squares methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543.5Composite methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573.6An interpretation of finite difference methods. . . . . . . . . . . . . . . . .583.7Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .623.8Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . .694 Variational Methods714.1Introduction to variational principles . . . . . . . . . . . . . . . . . . . . . .714.2Variational forms and variational principles . . . . . . . . . . . . . . . . . . .754.3Rayleigh-Ritz method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .774.4Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .814.5Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . .845 Construction of Finite Element Subspaces875.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .875.2Finite element spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .935.3Completeness property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .975.4Basic finite element shapes in one, two and three dimensions . . . . . . . . . 1015.55.4.1One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.2Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4.3Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.4.4Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Polynomial element interpolation functions . . . . . . . . . . . . . . . . . . . 1025.5.1Interpolations in one dimension . . . . . . . . . . . . . . . . . . . . . 1025.5.2Interpolations in two dimensions . . . . . . . . . . . . . . . . . . . . . 1085.5.3Interpolations in three dimensions . . . . . . . . . . . . . . . . . . . . 1205.6The concept of isoparametric mapping . . . . . . . . . . . . . . . . . . . . . 1245.7Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 Computer Implementation of Finite Element Methods1376.1Numerical integration of element matrices . . . . . . . . . . . . . . . . . . . 1376.2Assembly of global element arrays . . . . . . . . . . . . . . . . . . . . . . . . 1426.3Algebraic equation solving in finite element methods . . . . . . . . . . . . . 147ii
6.4Finite element modeling: mesh design and generation . . . . . . . . . . . . . 1496.4.1Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506.4.2Optimal node numbering . . . . . . . . . . . . . . . . . . . . . . . . . 1516.5Computer program organization . . . . . . . . . . . . . . . . . . . . . . . . . 1526.6Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.7Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537 Elliptic Differential Equations1577.1The Laplace equation in two dimensions . . . . . . . . . . . . . . . . . . . . 1577.2Linear elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.2.1A Galerkin approximation to the weak form . . . . . . . . . . . . . . 1637.2.2On the order of numerical integration . . . . . . . . . . . . . . . . . . 1667.2.3The patch test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.3Best approximation property of the finite element method . . . . . . . . . . 1747.4Error sources and estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777.5Application to incompressible elastostatics and Stokes’ flow . . . . . . . . . . 1807.6Suggestions for further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.7Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1878 Parabolic Differential Equations1918.1Standard semi-discretization methods . . . . . . . . . . . . . . . . . . . . . . 1928.2Stability of classical time integrators . . . . . . . . . . . . . . . . . . . . . . 1998.3Weighted-residual interpretation of classical time integrators . . . . . . . . . 2038.4Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2059 Hyperbolic Differential Equations2079.1The one-dimensional convection-diffusion equation . . . . . . . . . . . . . . . 2079.2Linear elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2139.3Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219iii
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List of Figures1.1B.G. Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2R. Courant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21.3R.W. Clough (left) and J. Argyris (right) . . . . . . . . . . . . . . . . . . . .21.4An infinite degree-of-freedom system . . . . . . . . . . . . . . . . . . . . . . .31.5A simple example of the structural analogue method . . . . . . . . . . . . . .41.6The finite difference method in one dimension . . . . . . . . . . . . . . . . .51.7A one-dimensional finite element approximation . . . . . . . . . . . . . . . .61.8A one-dimensional kernel function Wl associated with a particle method . . .82.1Schematic depiction of a set V . . . . . . . . . . . . . . . . . . . . . . . . . .14Example of a set that does not form a linear space . . . . . . . . . . . . . . .15Mapping between two sets . . . . . . . . . . . . . . . . . . . . . . . . . . . .162.22.302.4A function of class C (0, 2) . . . . . . . . . . . . . . . . . . . . . . . . . . .182.5Distance between two points in the classical Euclidean sense . . . . . . . . .192.6The neighborhood Nr (u) of a point u in V . . . . . . . . . . . . . . . . . . .20A continuous piecewise linear function and its derivatives . . . . . . . . . . .25A linear operator mapping U to V . . . . . . . . . . . . . . . . . . . . . . . .262.72.82.9A bilinear form on U V. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 A functional exhibiting a minimum, maximum or saddle point at u u . . .3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.437The domain Ω of the Laplace-Poisson equation with Dirichlet boundary Γuand Neumann boundary Γq . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.328An open and connected domain Ω with smooth boundary written as the unionof boundary regions Ωi3.22740Linear and quadratic approximations of the solution to the boundary-valueproblem in Example 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .47The point-collocation method . . . . . . . . . . . . . . . . . . . . . . . . . . .49v
3.5The point collocation method in a square domain . . . . . . . . . . . . . . . .503.6The subdomain-collocation method . . . . . . . . . . . . . . . . . . . . . . . .533.7Polynomial interpolation functions used for region (xl x, xl2 x]2in theweighted-residual interpretation of the finite difference method . . . . . . . .3.8Polynomial interpolation functions used for region [0, x1 x] in the weighted2residual interpretation of the finite difference method . . . . . . . . . . . . .3.95959Interpolation functions for a finite element approximation of a one-dimensionaltwo-cell domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .614.1Piecewise linear interpolations functions in one dimension . . . . . . . . . .794.2Comparison of exact and approximate solutions . . . . . . . . . . . . . . . .805.1Geometric interpretation of Fourier coefficients . . . . . . . . . . . . . . . .925.2A finite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .945.3A finite element-based interpolation function . . . . . . . . . . . . . . . . . .955.4Finite element vs. exact domain . . . . . . . . . . . . . . . . . . . . . . . . .965.5Error in the enforcement of Dirichlet boundary conditions due to the differencebetween the exact and the finite element domain . . . . . . . . . . . . . . . .965.6A potential violation of the integrability (compatibility) requirement . . . . .975.7A function u and its approximation uh in the domain (x̄, x̄ h) . . . . . . .985.8Pascal triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.9Finite element domains in one dimension . . . . . . . . . . . . . . . . . . . . 1015.10 Finite element domains in two dimensions . . . . . . . . . . . . . . . . . . . 1015.11 Finite element domains in three dimensions . . . . . . . . . . . . . . . . . . 1025.12 Linear element interpolation in one dimension . . . . . . . . . . . . . . . . . 1035.13 One-dimensional finite element mesh with piecewise linear interpolation . . . 1035.14 Standard quadratic element interpolations in one dimension . . . . . . . . . . 1045.15 Hierarchical quadratic element interpolations in one dimension . . . . . . . . 1055.16 Hermitian interpolation functions in one dimension . . . . . . . . . . . . . . 1075.17 A 3-node triangular element . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.18 Higher-order triangular elements (left: 6-node element, right: 10-node element)1105.19 A transitional 4-node triangular element . . . . . . . . . . . . . . . . . . . . 1115.20 Area coordinates in a triangular domain . . . . . . . . . . . . . . . . . . . . 1125.21 Four-node rectangular element . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.22 Interpolation function N1e for a b 1 (a hyperbolic paraboloid) . . . . . . . 114vi
5.23 Three members of the serendipity family of rectangular elements . . . . . . . 1145.24 Pascal’s triangle for two-dimensional serendipity elements (before accountingfor any interior nodes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.25 Three members of the Lagrangian family of rectangular elements . . . . . . . 1155.26 Pascal’s triangle for two-dimensional Lagrangian elements . . . . . . . . . . 1165.27 A general quadrilateral finite element domain . . . . . . . . . . . . . . . . . . 1175.28 Rectangular finite elements made of two or four joined triangular elements . 117 u u5.29 A simple potential 3- or 4-node triangular element for the case p 2 (u, , s ndofs at nodes 1, 2, 3 and, possibly, u dof at node 4) . . . . . . . . . . . . . . . 1185.30 Illustration of violation of the integrability requirement for the 9- or 10-doftriangle for the case p 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . u u5.31 A 12-dof triangular element for the case p 2 (u, ,dofs at nodes 1, 2, 3 s n uat nodes 4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .and n u udofs at nodes5.32 Clough-Tocher triangular element for the case p 2 (u, , s n uat nodes 4, 5, 6) . . . . . . . . . . . . . . . . . . . . . . . . . .1, 2, 3 and n5.33 The 4-node tetrahedral element . . . . . . . . . . . . . . . . . . . . . . . . . .1181191201215.34 The 10-node tetrahedral element . . . . . . . . . . . . . . . . . . . . . . . . . 1225.35 The 6- and 15-node pentahedral elements . . . . . . . . . . . . . . . . . . . . 1235.36 The 8-node hexahedral element . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.37 The 20- and 27-node hexahedral elements . . . . . . . . . . . . . . . . . . . . 1245.38 Schematic of a parametric mapping from Ωe to Ωe . . . . . . . . . . . . . . . 1255.39 The 4-node isoparametric quadrilateral . . . . . . . . . . . . . . . . . . . . . 1275.40 Geometric interpretation of one-to-one isoparametric mapping in the 4-nodequadrilateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.41 Convex and non-convex 4-node quadrilateral element domains . . . . . . . . 1305.42 Relation between area elements in the natural and physical domain . . . . . . 1305.43 Isoparametric 6-node triangle and 8-node quadrilateral . . . . . . . . . . . . . 1315.44 Isoparametric 8-node hexahedral element . . . . . . . . . . . . . . . . . . . . 1326.1Two-dimensional Gauss quadrature rules for q1 , q2 1 (left), q1 , q2 3 (cen-ter), and q1 , q2 5 (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141vii
6.2Integration rules in triangular domains for q 1 (left), q 2 (center), andq 3 (right). At left, the integration point is located at the barycenter ofthe triangle and the weight is w1 1; at center, the integration points arelocated at the mid-edges and the weights are w1 w2 w3 1/3; at right,one integration point is located at the barycenter and has weight w1 27/48,while the other three are at points with coordinates (0.6, 0.2, 0.2), (0.2, 0.6, 0.2),and (0.2, 0.2, 0.6), with associated weights w2 w3 w4 25/48. . . . . . . 1426.3Finite element mesh depicting global node and element numbering, as well asglobal degree of freedom assignments (both degrees of freedom are fixed at node1 and the second degree of freedom is fixed at node 7) . . . . . . . . . . . . . 1456.4Profile of a typical finite element stiffness matrix ( denotes a non-zero entryor a zero entry having at least one non-zero entry below and above it in thecolumn to which it belongs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486.5Representative examples of symmetries in the domains of differential equations(corresponding symmetries in the boundary conditions, loading, and equationsthemselves are assumed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.6Two possible ways of node numbering in a finite element mesh . . . . . . . . 1527.1The domain Ω of the linear elastostatics problem . . . . . . . . . . . . . . . . 1587.2Zero-energy modes for the 4-node quadrilateral with 1 1 Gaussian quadrature 1697.37.4Zero-energy modes for the 8-node quadrilateral with 2 2 Gaussian quadrature 170Schematic of the patch test (Form A) . . . . . . . . . . . . . . . . . . . . . . 1727.5Schematic of the patch test (Form B) . . . . . . . . . . . . . . . . . . . . . . 1737.6Schematic of the patch test (Form C) . . . . . . . . . . . . . . . . . . . . . . 1747.7Geometric interpretation of the best approximation property as a closest-point7.8projection from u to Uh in the sense of the energy norm . . . . . . . . . . . . 177Illustration of volumetric locking in plane strain when using 3-node triangularelements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.9The simplest convergent planar element for incompressible elastostatics/Stokes’flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.1Schematic depiction of semi-discretization (left) and space-time discretization(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1928.2Integration of (8.36) in the domain (tn , t] . . . . . . . . . . . . . . . . . . . . 199viii
8.3Amplification factor r as a function of λ t for forward Euler, backward Eulerand the exact solution of the homogeneous counterpart of (8.34) . . . . . . . 2029.1Plots of the solution (9.3) of the steady-state convection-diffusion equation forL 1, ū 1 and Péclet numbers P e 0.1 and P e 10. . . . . . . . . . . . 2089.2Finite element discretization for the one-dimensional convection-diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2099.3Finite element solution for the one-dimensional convection-diffusion equationfor c 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2109.4Finite element solution for the one-dimensional convection-diffusion equationfor c 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2109.5A schematic depiction of the upwind Petrov-Galerkin method for the convectiondiffusion equation (continuous line: Bubnov-Galerkin, broken line: PetrovGalerkin) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212ix
IntroductionThis is a set of notes written as part of teaching ME280A, a first-year graduate course onthe finite element method, in the Department of Mechanical Engineering at the Universityof California, Berkeley.Berkeley, CaliforniaP.P.August 2015x
Chapter 1Introduction to the Finite ElementMethod1.1Historical perspective: the origins of the finite element methodThe finite element method constitutes a general tool for the numerical solution of partialdifferential equations in engineering and applied science. Historically, all major practicaladvances of the method have taken place since the early 1950sin conjunction with the development of digital computers.However, interest in approximate solutions of field equationsdates as far back in time as the development of the classicalfield theories (e.g. elasticity, electro-magnetism) themselves.The work of Lord Rayleigh1 (1870) and W. Ritz2 (1909) on variational methods and the weighted-residual approach taken byB.G. Galerkin3 (1915) and others form the theoretical framework to the finite element method. With a bit of a stretch, onemay even claim that K. Schellbach’s approximate solution toPlateau’s problem (find a surface of minimum area enclosed byFigure 1.1. B.G. Galerkina given closed curve in three dimensions) by triangulation, which dates back to 1851, is a1John William Strutt, 3rd Baron Rayleigh (1842–1919) was a British physicist.Walther Ritz (1878–1909) was a Swiss theoretical physicist.3Boris Grigoryevich Galerkin (1871-1945) was a Russian mathematician and mechanician.21
2Introductionrudimentary application of the finite element method.Most researchers agree that the era of the finite elementmethod begins with a lecture presented in 1941 by R. Courant4to the American Association for the Advancement of Science.In his work, Courant used the Ritz method and introducedthe pivotal concept of spatial discretizatio
6.3 Finite element mesh depicting global node and element numbering, as well as global degree of freedom assignments (both degrees of freedom are fixed at node 1 and the second degree of freedom is fixed at node 7) . . . . . . . . . . . . . 145
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Finite element analysis DNV GL AS 1.7 Finite element types All calculation methods described in this class guideline are based on linear finite element analysis of three dimensional structural models. The general types of finite elements to be used in the finite element analysis are given in Table 2. Table 2 Types of finite element Type of .
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2.7 The solution of the finite element equation 35 2.8 Time for solution 37 2.9 The finite element software systems 37 2.9.1 Selection of the finite element softwaresystem 38 2.9.2 Training 38 2.9.3 LUSAS finite element system 39 CHAPTER 3: THEORETICAL PREDICTION OF THE DESIGN ANALYSIS OF THE HYDRAULIC PRESS MACHINE 3.1 Introduction 52
