Springer Series in Computational Physics Editors: W. Beiglbock, H. Cabannes, H. B. Keller, J. Killeen, S. A. Orszag Thomasset Fran~ois Implementation of Finite Element Methods for N avier-Stokes Equations With 86 Illustrations Springer-Verlag New York Heidelberg Berlin Dr. Fran90is Thomasset Domaine de Voluceau Rocquencourt B.P.I05 F-78150 Le Chesnay France Editors: Wolf Beiglböck Henri Cabannes Institut für Angewandte Mathematik Mecanique Theoretique Universität Heidelberg Universite Pierre et Marie Curie Im Neuenheimer Feld 5 Tour 66. 4, Place Jussieu D-6900 Heidelberg 1 F-75005 Paris Federal Republic of Germany France Stephen A. Orszag H. B. Keller Department of Mathematics Applied Mathematics 101-50 Massachusetts Institute of Technology Firestone Laboratory Cambridge, Massachusetts 02139 California Institute of Technology USA Pasadena, California 91125 USA John Killeen Lawrence Livermore Laboratory P.O. Box 808 Livermore, California 94551 USA Library of Congress Cataloging in Publication Data Thomasset, Frani,(ois. Implementation of finite element for Navier Stokes equations. (Springer series in computational physics) I. Fluid dynamics. 2. Navier-Stokes equations Numerical solutions. 3. Finite element method. I. Title. II. Series. QA911.T46 515.3'53 81-9169 AACR2 © 1981 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1s t edition 1981 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer- Verlag, 175 Fifth Avenue, New York, New York 10010, U.SA 9 8 7 6 5 432 I ISBN 978-3-642-87049-1 ISBN 978-3-642-87047-7 (eBook) DOI 10.1007/978-3-642-87047-7 Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Notations ............................................ 4 Chapter 1. Elliptic Equations of Order 2: Some Standard Finite Element Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1. A I-Dimensional Model Problem: The Basic Notions. . . . . . . . . . 6 1.2. A 2-Dimensional Problem ............................. II 1.3. The Finite Element Equations. . . . . . . . . . . . . . . . . . . . . . . . . .. 14 1.4. Standard Examples of Finite Element Methods .............. 17 1.4.1. Example 1: The PI-Triangle (Courant's Triangle) ......... . 1.4.2. Example 2: The P2-Triangle . . . . . . . . . . . . . . . . . . . . . .. 20 1.4.3. Example 3: The QI-Quadrangle . . . . . . . . . . . . . . . . . . . .. 23 1.4.4. Example 4: The Q2-Quadrangle .................... 24 1.4.5. A Variational Crime: The PI Nonconforming Element ..................................... 25 1.5. Mixed Formulation and Mixed Finite Element Methods for Elliptic Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 27 1.5.1. The One Dimensional Problem ..................... 28 1.5.2. A Two Dimensional Problem ...................... 32 Chapter 2. Upwind Finite Element Schemes ................... 37 2.1. Upwind Finite Differences. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 2.2. Modified Weighted Residual (MWR) ..................... 41 2.3. Reduced Integration of the Advection Term. . . . . . . . . . . . . . . .. 43 2.4. Computation of Directional Derivatives at the Nodes .......... 48 2.5. Discontinuous Finite Elements and Mixed Interpolation .......... 50 2.6. The Method of Characteristics in Finite Elements . . . . . . . . . . . .. 55 2.7. Peturbation of the Advective Term: Bredif (1980) . . . . . . . . . . . .. 57 vi Contents 2.8. Some Numerical Tests and Further Comments. . . . . . . . . . . . . .. 58 2.8.1. One Dimensional Stationary Advection Equation (56) ........................................ 58 2.8.2. Two Dimensional Stationary Advection Equation .......... 59 2.8.3. Time Dependent Advection. . . . . . . . . . . . . . . . . . . . . . .. 68 Chapter 3. Numerical Solution of Stokes Equations . . . . . . . . . . . . .. 72 3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72 3.2. Velocity-Pressure Formulations: Discontinuous Approximations of the Pressure ......................... 75 3.2.1. uh: PI Nonconforming Triangle (§1-4-5); Ph: Piecewise Constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76 3.2.2. uh: P2 Triangle Ph: PO (=Piecewise Constant) . . . . . . . . .. 79 3.2.3. uh: "P2 + bubble" Triangle (or Modified P2); Ph: Discontinuous PI ... . . . . . . . . . . . . . . . . . . . . . . . .. 79 3.2.4. uh: Q2 Quadrangle; Ph: QI Discontinuous ............. 80 3.2.5. Numerical Solution by Penalty Methods. . . . . . . . . . . . . .. 81 3.2.6. Numerical Results and Further Comments. . . . . . . . . . . .. 85 3.3. Velocity-Pressure Formulations: Continuous Approximation of the Pressure and Velocity. . . . . . . . . . . . . . . .. 86 3.3.1. Introduction ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86 3.3.2. Examples and Error Estimates. . . . . . . . . . . . . . . . . . . . .. 86 3.3.3. Decomposition of the Stokes Problem ................ 88 3.4. Vo rtici ty-Pressure-Ve loci ty Formulations: Discontinuous Approximations of Pressure and Velocity. . . . . . . . . . . . . . . . . .. 93 3.5. Vorticity Stream-Function Formulation: Decompositions of the Biharmonic Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97 Chapter 4. Navier-Stokes Equations: Accuracy Assessments and Numerical Results .......................... 101 4.1. Remarks on the Formulation .. . . . . . . . . . . . . . . . . . . . . . . . .. 10 I 4.2. A review of the Different Methods ....................... 103 4.2.1 Velocity-Pressure Formulations: Discontinuous Approximations of the Pressure ....... . . . . . . . . . . . . .. 103 4.2.2. Velocity-Pressure Formulations: Continuous Approximations of the Pressure .. . . . . . . . . . . . . . . . . . .. 104 4.2.3. Vo rticity-Pressure-Velocity Formulations: Discontinuous Approximations of Pressure and Velocity ...................................... 104 4.2.4. Vorticity Stream-Function Formulation ............... 106 Contents vii 4.3. Some Numerical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 107 4.3.1. The Square Wall Driven Cavity Flow . . . . . . . . . . . . . . . .. 107 4.3.2. An Engineering Problem: Unsteady 2-D Flow Around and In an Air-Intake. . . . . . . . . . . . . . . . . . . . . .. 116 Chapter 5. Computational Problems and Bookkeeping . . . . . . . . . . .. 120 5.1. Mesh Generation ................................... 120 5.2. Solution of the Nonlinear Problems ....................... 123 5.2.1. Successive Approximations (or Linearization) with Under Relaxation .............................. 123 5.2.2. Newton-Raphson Algorithm . . . . . . . . . . . . . . . . . . . . . .. 124 5.2.3. Conjugate Gradient Method (with Scaling) for . Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 125 5.2.4. A Splitting Technique for the Transient Problem . . . . . . . .. 127 5.3. Iterative and Direct Solvers of Linear Equations. . . . . . . . . . . . .. 127 5.3.1. Successive Over Relaxation . . . . . . . . . . . . . . . . . . . . . . .. 127 5.3.2. Cholesky Factorizations . . . . . . . . . . . . . . . . . . . . . . . . .. 128 5.3.3. Out of Core Factorizations . . . . . . . . . . . . . . . . . . . . . . .. 131 5.3.4. Preconditioned Conjugate Gradient . . . . . . . . . . . . . . . . .. 133 Appendix 1. The Patch Test of the PI Nonconforming Triangle: Sketchy Proof of Convergence . . . . . . . . . . . .. 136 Appendix 2. Numerical llIustration . . . . . . . . . . . . . . . . . . . . . . . . .. 138 Appendix 3. The Zero Divergence Basis for 2-D PI Nonconforming Elements ................................... 142 Three Dimensional Case . . . . . . . . . . . . . . . . . . . . . . .. 144 References 148 Index ............................................... 161 Introduction In structure mechanics analysis, finite element methods are now well estab lished and well documented techniques; their advantage lies in a higher flexibility, in particular for: (i) The representation of arbitrary complicated boundaries; (ii) Systematic rules for the developments of stable numerical schemes ap proximating mathematically wellposed problems, with various types of boundary conditions. On the other hand, compared to finite difference methods, this flexibility is paid by: an increased programming complexity; additional storage require ment. The application of finite element methods to fluid mechanics has been lagging behind and is relatively recent for several types of reasons: (i) Historical reasons: the early methods were invented by engineers for the analysis of torsion, flexion deformation of bearns, plates, shells, etc ... (see the historics in Strang and Fix (1972) or Zienckiewicz (1977». (ii) Technical reasons: fluid flow problems present specific difficulties: strong gradients,l of the velocity or temperature for instance, may occur which a finite mesh is unable to properly represent; a remedy lies in the various upwind finite element schemes which recently turned up, and which are reviewed in chapter 2 (yet their effect is just as controversial as in finite differences). Next, waves can propagate (e.g. in ocean dynamics with shallowwaters equations) which will be falsely distorted by a finite non regular mesh, as Kreiss (1979) pointed out. We are concerned in this course with the approximation of incompressible, viscous, Newtonian fluids, i.e. governed by N avier Stokes equations. We leave aside many problems of practical interest which have been solved by finite element techniques: For transonic flows with shocks, see Bristeau (1977, 1978), Glowinski and Pironneau (1976) and the references of these papers. I Shocks will not be considered in this course. 2 Introduction Viscous non Newtonian fluids are studied by Nickell, Tanner and Caswell (1974), Engelman (1979) (with application to blood flow past prosthetic valves), Zienckiewicz and Godbole (1975). Chavent (1979), Jaffre (1979b) considered flows in porous media with applications to oil production. Moving meshes (to deal with free surfaces or multi fluid problems) have been seldom used: see Hughes, Liu and Zimmermann (1978, generalized ALE technique). An alternative technique using a fixed mesh for the simulation of two fluid flows is proposed by Dervieux and Thomasset (1979). Further applications can be found e.g. in Zienckiewicz (1977). The plan of the course is as follows: in the first chapter some useful finite element methods are presented. Next in chapter 2 the upwinding techniques in finite elements are reviewed and in chapter 3 the various formulations available to deal with the incompressibility conditions are presented. In chapter 4, the techniques of previous chapters are combined for the ap proximation of N avier Stokes equations; some numerical results are pre sented, although it is felt that further tests should be necessary, in order to allow definite assessments about the performance of the various methods. Finally chapter 5 surveys the different techniques related to automatic mesh generation, and solution of systems of linear or non linear equations. Unless the contrary is specified2 the methods presented in this paper extend to three dimensional problems without other difficulties, than pro gramming, solving very large systems of equations, and interpreting the results (which by the way are not small difficulties). The actual challenge is the application of finite element methods to large 3D problems will require the development of parallel processors; Pironneau (1979) suggested some methods for the solution of Navier Stokes equations using parallel processors with SIMD architecture. For the theory of finite element methods, the reader is referred, e.g. to Ciarlet (1978), Babuska and Aziz (1972), Strang and Fix (1972), Oden and Reddy (1976b), Girault and Raviart (1979), de Boor (1974), Miller (1978), Raviart (1979); for mathematical problems concerning partial differential equations, see for instance Agmon (1965), Lions (1969, 1977), Necas (1967), Temam (1977). Finite element methods offer a great flexibility to solve general partial differential equations; on the other hand, the preliminary investments in programmation are important, and cannot be efficiently developed by iso lated programmers. Thus in order to remedy the development of redundant programs by various institutions and industries. I.R.I.A. created the club MODULEF3 about five years ago. The goal of this club is to offer a structure where the members, from industry and university, can meet and share their 2e.g. the methods with the stream function are 2-dimensional. 3 Begis and Perronnet (1979). Introduction 3 experience and packages; libraries of routines are already developed concern ing: mesh generation; solvers of linear systems; analysis of structural mecha nics problems; a few modules (at this date) concern fluid mechanics (Navier Stokes equations). Further effort should be done for easier learning facilities and readability of Fortran programs. For the realization of this course I am gratefully indebted to a number of persons: first of all to my friends at LN.R.LA. for helpful discussions and supporting fellowship, specially to A. Dervieux, J. F. Bourgat and M. O. Bristeau who supplied some figures; to J. Periaux from Avions-Marcel Dassault/Breguet-Aviation who supplied a numerical example in an in dustrial configuration; to my friends of Departement d'Analyse Numerique of Universites Paris 6 and Paris 11; to B. L. Hua from L.O.P./Museum, for helpful comments; to Mme Kurinckx-Longree for the efficient and beautiful typing of the manuscript; to my wife Annic for her patient support during these last few months. Thanks are also due to the Von Karman Institute for Fluid Dynamics (Belgium) where this course was displayed (V.K.L Lecture series, Computa tional Fluid Dynamics, March 26-29, 1980. Finally am gratefully indebted to Pro R. Temam, to whom lowe my initiation to Numerical Analysis and who introduced me to V.K.L, and to Pro R. Glowinski and Pr. P. A. Raviart for their encouraging and stimulating discussions. Notations Q = physical (bounded) domain of calculation f=aQ=boundary of Q fo, fl' f2. .. =parts of f (in the case of the flow around an obstacle, fo is the exterior boundary) t= time variable x= {XI' x2} or {XI' x2, x3} = generic point in Q dx=dxI dX2 or dXI dX2 dX3 = infinitesimal area ds = infinitesimal length on f L2(Q)=set of square integrable functions ={v: Jolv(x)12dx<+oo} H\Q)= {v EL2(Q)= Jolgrad Vl2 dx< + oo} Hci(Q)=set of functions in HI(Q) vanishing on the boundary ={vEHI(Q): vlr=O} H(div, Q)= {v= {VI' v2}: Vi EL2(Q), divvEL2(Q)} Ho(div, Q)= {vEH(div, Q): divv=O in Q, v·n=O on f} n=n(x)=unit vector normal to the boundary at a point x, pointing outwards of the domain. u,O=solutions of model problems: Poisson's equation and advection- diffusion equation u=velocity field (u in I dimension) p=pressure aij = stress tensor f = {.t;} = body force = diffusivity coefficient K Kij =diffusivity tensor = kinematic viscosity p J/I = stream function w=vorticity (or a relaxation parameter, according to the context). A. = value of p (or w) on the boundary. v, v, q = trial functions