CSCI, MATH 6860 FINITE ELEMENT ANALYSIS Lecture Notes: Spring 2000 Joseph E Flaherty Amos Eaton Professor Department of Computer Science Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York 12180 c 2000, Joseph E Flaherty, all rights reserved These notes are intended for classroom use by Rensselaer students taking courses CSCI, MATH 6860 Copying or downloading by others for personal use is acceptable with noti cation of the author ii CSCI, MATH 6860: Finite Element Analysis Spring 2000 Outline Introduction 1.1 Historical Perspective 1.2 Weighted Residual Methods 1.3 A Simple Finite Element Problem One-Dimensional Finite Element Methods 2.1 2.2 2.3 2.4 2.5 2.6 Introduction Galerkin's Method and Extremal Principles Essential and Natural Boundary Conditions Piecewise Lagrange Approximation Hierarchical Bases Interpolation Errors Multi-Dimensional Variational Principles 3.1 Galerkin's Method and Extremal Principles 3.2 Function Spaces and Approximation 3.3 Overview of the Finite Element Method Finite Element Approximation 4.1 4.2 4.3 4.4 4.5 4.6 Introduction Lagrange Bases on Triangles Lagrange Bases on Rectangles Hierarchical Bases Three-dimensional Bases Interpolation Errors Mesh Generation and Assembly 5.1 Introduction iii 5.2 5.3 5.4 5.5 5.6 Mesh Generation Data Structures Coordinate Transformations Generation of Element Matrices and Their Assembly Assembly of Vector Systems Numerical Integration 6.1 Introduction 6.2 One-Dimensional Gaussian Quadrature 6.3 Multi-Dimensional Gaussian Quadrature Discretization Errors 7.1 Introduction 7.2 Convergence and Optimality 7.3 Perturbations Adaptivity 8.1 Introduction 8.2 h-Re nement 8.3 p- and hp-Re nement Parabolic Problems 9.1 9.2 9.3 9.4 9.5 Introduction Semi-Discrete Galerkin Methods: The Method of Lines Finite Element Methods in Time Convergence and Stability Convection-Di usion Systems 10 Hyperbolic Problems 10.1 Introduction 10.2 Flow Problems and Upwind Weighting 10.3 Arti cial Di usion iv 10.4 Streamline Weighting 11 Linear Systems Solution 11.1 11.2 11.3 11.4 11.5 Introduction Banded Gaussian Elimination and Pro le Techniques Nested Dissection and Domain Decomposition Conjugate Gradient Methods Nonlinear Problems and Newton's Method v vi Bibliography 1] A.K Aziz, editor The Mathematical Foundations of the Finite Element Method with Applications to Partial Di erential Equations, New York, 1972 Academic Press 2] I Babuska, J Chandra, and J.E Flaherty, editors Adaptive Computational Methods for Partial Di erential Equations, Philadelphia, 1983 SIAM 3] I Babuska, O.C Zienkiewicz, J Gago, and E.R de A Oliveira, editors Accuracy Estimates and Adaptive Re nements in Finite Element Computations John Wiley and Sons, Chichester, 1986 4] K.-J Bathe Finite Element Procedures Prentice Hall, Englewood Cli s, 1995 5] E.B Becker, G.F Carey, and J.T Oden Finite Elements: An Introduction, volume I Prentice Hall, Englewood Cli s, 1981 6] M.W Bern, J.E Flaherty, and M Luskin, editors Grid Generation and Adaptive Algorithms, volume 113 of The IMA Volumes in Mathematics and its Applications, New York, 1999 Springer 7] C.A Brebia The Boundary Element Method for Engineers Pentech Press, London, second edition, 1978 8] S.C Brenner and L.R Scott The Mathematical Theory of Finite Element Methods Springer-Verlag, New York, 1994 9] G.F Carey Computational Grids: Generation, Adaptation, and Solution Strategies Series in Computational and Physical Processes in Mechanics and Thermal science Taylor and Francis, New York, 1997 10] G.F Carey and J.T Oden Finite Elements: A Second Course, volume II Prentice Hall, Englewood Cli s, 1983 11] G.F Carey and J.T Oden Finite Elements: Computational Aspects, volume III Prentice Hall, Englewood Cli s, 1984 vii 12] P.G Ciarlet The Finite Element Method for Elliptic Problems North-Holland, Amsterdam, 1978 13] K Clark, J.E Flaherty, and M.S Shephard, editors Applied Numerical Mathematics, volume 14, 1994 Special Issue on Adaptive Methods for Partial Di erential Equations 14] R.D Cook, D.S Malkus, and M.E Plesha Concepts and Applications of Finite Element Analysis John Wiley and Sons, New York, third edition, 1989 15] K Eriksson, D Estep, P Hansbo, and C Johnson Computational Di erential Equation Cambridge, Cambridge, 1996 16] G Fairweather Finite Element Methods for Di erential Equations Marcel Dekker, Basel, 1981 17] B Finlayson The Method of Weighted Residuals and Variational Principles Academic Press, New York, 1972 18] J.E Flaherty, P.J Paslow, M.S Shephard, and J.D Vasilakis, editors Adaptive methods for Partial Di erential Equations, Philadelphia, 1989 SIAM 19] R.H Gallagher, J.T Oden, C Taylor, and O.C Zienkiewicz, editors Finite Elements in Fluids: Mathematical Foundations, Aerodynamics and Lubrication, volume 2, London, 1975 John Wiley and Sons 20] R.H Gallagher, J.T Oden, C Taylor, and O.C Zienkiewicz, editors Finite Elements in Fluids: Viscous Flow and Hydrodynamics, volume 1, London, 1975 John Wiley and Sons 21] R.H Gallagher, O.C Zienkiewicz, J.T Oden, M Morandi Cecchi, and C Taylor, editors Finite Elements in Fluids, volume 3, London, 1978 John Wiley and Sons 22] V Girault and P.A Raviart Finite Element Approximations of the Navier-Stokes Equations Number 749 in Lecture Notes in Mathematics Springer-Verlag, Berlin, 1979 23] T.J.R Hughes, editor Finite Element Methods for Convection Dominated Flows, volume 34 of AMD, New York, 1979 ASME 24] T.J.R Hughes The Finite Element Method: Linear Static and Dynamic Finite Element Analysis Prentice Hall, Englewood Cli s, 1987 viii 25] B.M Irons and S Ahmed Techniques of Finite Elements Ellis Horwood, London, 1980 26] C Johnson Numerical Solution of Partial Di erential Equations by the Finite Element method Cambridge, Cambridge, 1987 27] N Kikuchi Finite Element Methods in Mechanics Cambridge, Cambridge, 1986 28] Y.W Kwon and H Bang The Finite Element Method Using Matlab CRC Mechanical Engineering Series CRC, Boca Raton, 1996 29] L Lapidus and G.F Pinder Numerical Solution of Partial Di erential Equations in Science and Engineering Wiley-Interscience, New York, 1982 30] D.L Logan A First Course in the Finite Element Method using ALGOR PWS, Boston, 1997 31] J.T Oden Finite Elements of Nonlinear Continua Mc Graw-Hill, New York, 1971 32] J.T Oden and G.F Carey Finite Elements: Mathematical Aspects, volume IV Prentice Hall, Englewood Cli s, 1983 33] D.R.J Owen and E Hinton Finite Elements in Plasticity-Theory and Practice Pineridge, Swansea, 1980 34] D.D Reddy and B.D Reddy Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements Number 27 in Texts in Applied Mathematics Springer-Verlag, Berlin, 1997 35] J.N Reddy The Finite Element Method in Heat Transfer and Fluid Dynamics CRC, Boca Raton, 1994 36] C Schwab P- And Hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics Numerical Mathematics and Scienti c Computation Clarendon, London, 1999 37] G Strang and G Fix Analysis of the Finite Element Method Prentice-Hall, Englewood Cli s, 1973 38] B Szabo and I Babuska Finite Element Analysis John Wiley and Sons, New York, 1991 39] F Thomasset Implementation of Finite Element Methods for Navier-Stokes Equations Springer Series in Computational Physics Springer-Verlag, New York, 1981 ix 40] V Thomee Galerkin Finite Element Methods for Parabolic Problems Number 1054 in Lecture Notes in Mathematics Springer-Verlag, Berlin, 1984 41] R Verfurth A Review of Posteriori Error Estimation and Adaptive MeshRe nement Techniques Teubner-Wiley, Stuttgart, 1996 42] R Vichevetsky Computer Methods for Partial Di erential Equations: Elliptic Equations and the Finite-Element Method, volume Prentice-Hall, Englewood Cli s, 1981 43] R Wait and A.R Mitchell The Finite Element Analysis and Applications John Wiley and Sons, Chichester, 1985 44] R.E White An Introduction to the Finite Element Method with Applications to Nonlinear Problems John Wiley and Sons, New York, 1985 45] J.R Whiteman, editor The Mathematics of Finite Elements and Applications V, MAFELAP 1984, London, 1985 Academic Press 46] J.R Whiteman, editor The Mathematics of Finite Elements and Applications VI, MAFELAP 1987, London, 1988 Academic Press 47] O.C Zienkiewicz The Finite Element Method Mc Graw-Hill, New York, third edition, 1977 48] O.C Zienkiewicz and R.L Taylor Finite Element Method: Solid and Fluid Mechanics Dynamics and Non-Linearity Mc Graw-Hill, New York, 1991 x 10.2 Discontinuous Galerkin Methods 29 Figure 10.2.5: Exact (line) and discontinuous Galerkin solutions of Example 10.2.2 for p = 2, and h = 1=32 Solutions with the minmod limiter (10.2.7) and an adaptive moment limiter of Biswas et al 8] are shown for p = performed with p = and J = 64 is shown in Figure 10.2.1 The entire solitary wave is shown however, the computation was performed on the center region ; =3 < x < =3 30 Hyperbolic Problems Discretization errors in the L1 norm J XZ ke( t)k = x j j =1 x ;1 j jU (x t) ; Uj (x t)jdx are presented for the solution u for various combinations of h and p in Table 10.2.2 Solutions of this nonlinear wave propagation problem appear to be converging as O(hp+1) in the L1 norm This can be proven correct for smooth solutions of discontinuous Galerkin methods 2, 11, 12] 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 -10 -8 -6 -4 -2 10 Figure 10.2.6: Solution of Example 10.2.3 at t = obtained by the discontinuous Galerkin method with p = and N = 64 J 16 32 64 128 256 p=0 2.16e-01 1.19e-01 6.39e-02 3.32e-02 1.69e-02 8.58e-03 p=1 5.12e-03 1.19e-03 2.88e-04 7.06e-05 1.74e-05 4.34e-06 p=2 1.88e-04 2.32e-05 2.90e-06 3.63e-07 4.53e-08 5.67e-09 p=3 7.12e-06 4.38e-07 2.70e-08 1.68e-09 1.04e-10 p=4 3.67e-07 1.12e-08 3.55e-10 1.10e-11 3.49e-13 Table 10.2.2: Discretization errors at t = as functions J and p for Example 10.2.3 Evaluating numerical uxes and using limiting for vector systems is more complicated than indicated by the previous scalar example Cockburn and Shu 12] reported problems when applying limiting component-wise At the price of additional computation, they applied limiting to the characteristic elds obtained by diagonalizing the Jacobian fu Biswas et al 8] proceeded in a similar manner \Flux-vector splitting" may provide a compromise between the two extremes As an example, consider the solution and ux vectors for the one-dimensional Euler equations of compressible ow (10.1.3) For this 10.2 Discontinuous Galerkin Methods 31 and related di erential systems, the ux vector is a homogeneous function that may be expressed as f (u) = Au = fu (u)u: (10.2.9a) Since the system is hyperbolic, the Jacobian A may be diagonalized as described in Section 10.1 to yield f (u) = P;1 Pu where the diagonal matrix =6 contains the eigenvalues of A 3 u;c 7=4 5: u u+c (10.2.9b) (10.2.9c) m p The variable c = @p=@ is the speed of sound in the uid The matrix decomposed into components = + + ; can be (10.2.10a) where + and ; are, respectively, composed of the non-negative and non-positive components of i j ij i = : : : m: Writing the ux vector in similar fashion using (10.2.9) i = f (u) = P;1( + + ;)Pu = f (u)+ + f (u); : (10.2.10b) (10.2.10c) Split uxes for the Euler equations were presented by Steger and Warming 26] Van Leer 27] found an improvement that provided better performance near sonic and stagnation points of the ow The split uxes are evaluated by upwind techniques Thus, at an interface x = xj , f + is evaluated using Uj (xj t) and f ; is evaluated using Uj+1(xj t) Calculating uxes based on the solution of Riemann problems is another popular way of specifying numerical uxes for vector systems To this end, let w(x=t uL uR ) be the solution of a Riemann problem for (10.1.1a) with the peicewise-constant initial data (10.1.25) The solution of a Riemann problem \breaking" at (xj tn) would be w((x ; xj )=(t ; tn) Uj (xj tn) Uj (xj+1 tn)) Using this, we would calculate the numerical ux at (xj t), t > tn , as F(Uj (xj tn) Uj+1(xj tn)) = f (w(0 Uj (xj tn) Uj+1(xj tn)): (10.2.11) 32 Hyperbolic Problems Example 10.2.4 Let us calculate the numerical ux based on the solution of a Riemann problem for Burgers' equation (10.1.16) Using the results of Example 10.1.8) we know that the solution of the appropriate Riemann problem is if Uj Uj+1 > > Uj > >U > j +1 if Uj Uj +1 < < if Uj < Uj+1 > w(0 Uj Uj+1) = > : > Uj if Uj > Uj+1 < (Uj + Uj+1)=2 > > > :U j +1 if Uj > Uj +1 < (Uj + Uj +1 )=2 < (The arguments of Uj and Uj+1 are all (xj tn) These have been omitted for clarity.) With f (u) = u2=2 for Burgers' equation, we nd the numerical ux > Uj =2 if Uj Uj +1 > > > U =2 if U U < > j +1 < j j +1 if Uj < Uj+1 > F (Uj Uj+1) = > : > Uj2 =2 if Uj > Uj +1 < (Uj + Uj +1 )=2 > > > : U =2 if U > U < (U + U )=2 < j j +1 j j +1 j +1 Letting u+ = max(u 0) u; = min(u 0) we can write the numerical ux more concisely as F (Uj Uj+1) = max (Uj+)2 =2 (Uj; )2=2]: +1 When used with a piecewise-constant basis and forward Euler time integration, the resulting discontinuous Galerkin scheme is identical to Godunov's nite di erence scheme 18] This was the rst di erence scheme to be based on the solution of a Riemann problem This early work and a subsequent work of Glimm 17] and Chorin 9] stimulated a great deal of interest in using Riemann problems to construct numerical ux functions A summary of a large number of choices appears in Cockburn and Shu 12] 10.3 Multidimensional Discontinuous Galerkin Methods Let us extend the discontinuous Galerkin method to multidimensional conservation laws of the form ut + r f (u)x = b(x y z t u) (x y z) t>0 (10.3.1a) where f (u) = f (u) g(u) h(u)] (10.3.1b) 10.3 Multidimensional Discontinuous Galerkin Methods and r f (u) = f (u)x + g(u)y + h(u)z : 33 (10.3.1c) The solution u(x y z t) componenets of the ux vector f (u), g(u), and h(u) and the loading b(x y z t u) are m-vectors and is a bounde region of