FEM Strang

7/27/2019 FEM Strang

1/8

Proceedings of the International Congress of MathematiciansVancouver, 1974

The Finite Element MethodLinear andNonlinear Applications

Gilbert StrangL Numerical analysis is a crazy mixture of pure and applied mathematics. Itasks us lo do two things at once, and on the surface they do appear complementary:(i) to propose a good algorithm, and (ii) to analyze it. In principle, the analysis

should reveal what makes the algorithm good, and suggest how to make it better.For some problemscomputing the eigenvalues of a large matrix, for example,which used to be a hopeless messthis combination of invention and analysis hasactually succeeded. But for partial differential equations, which come to us in suchterrible variety, there seems to be a long way to go.We want to speak about an algorithm which, at least in its rapidly developingextensions to nonlinear problems, is still new and flexible enough to be improvedby analysis. It is known as thefiniteelement method, and was created to solve theequations of elasticity and plasticity. In this instance, the "numerical analysts"were all engineers. They needed a better technique than finite differences, especiallyfor complicated systems on irregular domains, and they found one. Their methodfalls into the framework of the Ritz-Galerkin technique, which operates with problems in "variational form"starting either from an extremum principle, or fromthe weak form of the differential equation, which is the engineer's equation ofvirtual work. The key idea which has made this classical approach a success is to usepiecewise polynomials as trialfunctions in the variational problem.1We plan to begin by describing the method as it applies to linear problems. Because the basic idea is mathematically sound, convergence can be proved and theerror can be estimated. This theory has been developed by a great many numerical^he most important applications are still to structural problems, but no longer to the design ofairplanes; that has been superseded by the safety of nuclear reactors.

1975, Canadian Mathemat ical Congress429


2/8

4 3 0 GILBERT STRANG

analysts, and we can summarize only a few of the most essential pointsthe conditions which guarantee convergence, and which govern its speed. This linear analysis has left everyone happier, and some divergent elements have been thrownout, b ut the me thod itself has not been enormously changed. For n onlinear problemsthe situation is entirely different. It seems to me that numerical analysts, especiallythose in optimization and nonlinear systems, can still make a major contribution.The time is actually a little short, because the large-scale programs for plasticity,buckling , an d no nlinea r elasticity are already being written. But everyone is agreedthat they are tremendously expensive, and that new ideas are needed.

No nlinea r pro blem s present a new challenge also to the analyst wh o is concernedwith error estimates. The main aim of this paper is to describe some very fragmentary results (111) and several open questions (IV). We are primarily interested in those nonlinearities which arise, in an otherwise linear problem, when thesolution is required to satisfy an inequality constraint. This is typical of the problem sin plasticity. The solution is still determined by a variational principle, but theclass of admissible functions becomes a convex set instead of a subspace. In otherwo rds, the equ ation of virtual work becomes a variational inequality.

A t the end w e look in still a different direction, at linear programm ing constrainedby differential equations. He re we need not only good algorithms and a propernumerical analysis, but also answers to the more fundamental questions of existence, uniqueness, and regularity.H. Linear equations. The finite element method applies above all to ellipticboundary value problems, which we write in the following form : Find u in thespace of admissible functions V such that

(1) a{u9v) = l{v) for all v in V.STANDARD EX AMPLE. JJ {uxvx + uyvy) dx dy = JJ fv dx dy for all v in 3tf\{Q).This is the weak form of Poisson's equation Au = f. Because the expressiona{u 9 v) is in this case sym metric an d p ositive definite, the p rob lem is equivalentt o : Minimize J{v) = a{v, v) 2l{v) over the admissible space V. The "strainenergy" a{v9 v) is the natural norm in which to estimate the error.The error comes from changing to a finite-dimensional problem : Find uh in Shsuch that

(2) ah{uh9 vk) = lh{vh) for all vh in Sh.It is this problem which the computer actually solves, once it is given a basis$N f r the space Sh. Very briefly, it has to form the stiffness matrix K{j =ah {


3/8

THE FINITE ELEMENT METHOD 431

Our plan in this section is to summarize four of the main points in the theory ofconvergence. Each of them is concerned with the change in solution when there is achange in the problemwhen the admissible space V is replaced by Sh9 or thegiven a and / are approximated by ah and lh. To give some kind of order to thediscussion, we formulate all four as applications of the "fundamental theorem ofnumerical analysis" :Consistency + Stability o Convergence.

1. The classical Ritz-Galerkin case. The energy a{v,v) is symmetric positivedefinite ;Sh is a subspace of V\ ah = a and lh = I.Since every vh is an admissible v, we may com pare (1) and (2) : a{u 9 vh) = a{uh9 vh).This means that in the "energy inner product," uh is the projection of u onto thesubspace Sh. In other words, the positive definiteness of a{v9 v) implies two properties at once [1, p. 40] : The projection uh is no larger than u itself,(3) a{u h9 uh) S a{u 9 u)9and at the same time uh is as close as possible to w :(4) a{u uh, u - uh) S a{u - vh, u - vh) for all vh in Sh .Property (3) represents stability ; the approximations are uniformly bounded. Giventhat u can be approximated by the subspace S in this Ritz-Galerkin context,consistency is the same as approximabilityconvergence follows immediatelyfrom (4).2. The indefinite case, u is only a stationary point of the functional J{v). Thiscorresponds to the use of Lagrange mutipliers in optimization; the form a{v9v)can take either sign, and v may include two different types of unknownsbothdisplacements and stresses, in the "mixed method" and "hybrid method."Consistency reduces as before to approximation by polynomials. But stability isno longer automatic; even the simplest indefinite form J{v) = i>iv2which has aunique stationary point at the origin, if Kis the plane R2 will collapse on the one-dimensional subspace given by v2 = 0. Therefore, for each finite element space Shand each functional J{v), it has to be proved tha t a degeneracy of this kind does notoccur.The proper stability condition is due to Babuska and Brezzi :(5) sup|a(v, w)\ ^ c II w II.

I l e l l = l " "Brezzi has succeeded in verifying this condition for several important hybrid elements. For other applications the verification is still incomplete, and the convergence of stationary pointswhich is critical to the whole theory of optimizationremains much harder to prove than the convergence of minima.3. The modified Galerkin method, a and /are changed to ah and lh (numericalintegration of the stiffness matrix and load vector), and vh may lie outside V (non-wise polynom ials and for the proof of their approximation properties. Perhaps the favorites, wh enderivatives of order m appear in the energy a(v, v), are the polyn om ials of degree w + 1 .


4/8


conforming elements).The effect on u can be estimated by combining (1) and (2):(6) ah{u - uh9u- uh) = {ak - a) (, u - uh) - {lh - l){u- uh).S tability, in this situation, means a lower bound for the left side:(7) ah{u - uh9u- uh) ^ ca{u - uh9u - uh).Consistency is translated into an upper bound for the right side, and it is checkedby applying the patch test: Whenever the solution is in a "state of constant strain"the highest derivatives in a{u 9 u) are all constantthen uh must coincide with M.3The patch test applies especially to nonconforming elements, for which a{vh9 vh)= oo ; the derivatives of vh introduce delta-functions, which are simply ignored inthe approximate energy ah. This is extremely illegal, but still the test is sometimespassed and the approximation is consistent. Convergence was established by theauthor for one such element, and Raviart, Ciarlet, Crouzeix, and Lesaint haverecently made the list much more complete.4. Superconvergence. E xtra accuracy of the finite element approximation atcertain points of the domain. It was recognized very early that in some specialcasesu" =/with linear elements, or u"" = / with cubicsthe computed uh isexactly correct at the nodes. (The Green's function lies in Sh.) And even earlier therearose the difficulty of interpreting the finite element output in a more generalproblem; uh and its derivatives can be evaluated at any point in the domain, butwhich points do we choose? This question is as important as ever to the engineers.In many problems the error u uh oscillates within each element, and there mustbe points of exceptional accuracy. Thome discovered superconvergence at thenodes of a regular mesh, for ut = uXX9 and his analysis has been extended by D ouglas, Dupont, Bramble, and Wendroff. It is not usually carried out in our context ofconsistency and stability, but perhaps it could be: Consistency is checked by apatch test at the superconvergence points, to see which polynomial solutions andwhich derivatives are correctly reproduced, and stability needs to be established inthe pointwise sense.4HI. Variational inequalities. What happens when a constraint such as v g cj) isenforced on the admissible functions v, so that the functional J{v) is minimized onlyover a convex subset K of the original space VI This occurs naturally in plasticitytheory, when v represents the stress; wherever the yield limit is reached, thedifferential equation (Hooke's law) is replaced by plastic flow. Fo r the minimizingu, the "free boundary" which marks out this plastic region u = is not known inadvance.5 Since such a solution u lies on the edge of the convex set K9 J{u) ^J{u + e{vu)) is guaranteed only for e ^ 0. This translates into the variational

3The patch test is also an ideal way to check that a finite element program is actually working.Convergence in Loo has b een proved by Scott since the Con gress; it was one of the outstanding

problems in the linear theory.5This boundary cannot be found by solving the original linear problem and then replacing u

bymin(tf,


5/8

THE FINITE ELEMENT METHOD 4 3 3

inequality which determines u:(8) a{u 9 v - u) ^ l{v - u) for all v in K.

In the finite element method, we minimize an approximate functional Jh{v) =t*k(v9 v) ~ 2/(v) over afinite-dimensionalconvex set Kh. For example, the piece-wise polynomials may be constrained by vh ^


6/8


from equations to inequalities produces a new term :(10) \\u - Uh\\* ^ \\u - vh\\2 + 2 J ( / + Au) {u-vh + uk- V).We may choose any vh in Kh and any v in Tand for simplicity we have specializedto / = lh = J/v and a = ah = j " | Vv|2. The new term is automatically zero in theelastic part, where Au=f, but elsewhere/ + Au > 0.To estimate (10), we take v = cj) and vh = u2which lies in Kh because it cannotexceed (j) at the nodes, where it agrees with u. (In the case of quadratic polynomials,some members of Kh will go above the yield limit ^ within the triangles, but we haveto be generous enough to permit that; it does not hurt the error estimate, andanyway it is only constraints on vh at nodal "checkpoints" which can be enforcedin practice.) With this choice of v and vh9 the terms in (10) are

(i) With Courant's linearfiniteelements :fu - uj ||2 ~ h\ \{f + Au) {u - ui) ~ h\ \{f + Au) {u h -c/>)S0.(ii) With quadraticfiniteelements :

\\u - uj\\2 ~ h\ \{ f + Au) {u - uj) ~ h\ \{ f + Au) {u h -


7/8

THE FINITE ELEMENT METHOD 435(iii) To find a quick way of solving, with adequate accuracy, the obstacle problem which arises at each time step.We believe that these are among the most important questions in nonlinearfinite element analysis and that answers can be found.A second class of problems, of an entirely different type, arises if we are interested only in the multiple X of the load/w hich will induce plastic collapse. This isknown as limit analysis, and no longer requires us to follow the loading history. Inplace of minimizing a quadratic functional, the problem falls into the framework ofinfinite-dimensional linear programming. Here is a typical example, with unknownstresses afj{x9 y) and m ultiplier X: Maximize /I, subject to :equilibrium : {da^/dx^) = Xfj in Q, 2 07/ n% = tei o n 3ft andpiecewise linear yield conditions : 2 &?/ On ^ ca inQ9 ^ a M.Suppose we make this problem finite-dimensionalby assuming that the stresses(and also the displacements, which are the unknowns in the dual program) belongto piecewise polynomial spaces Sh. The continuous linear programming problemis then approximated, in a completely natural way, by a discrete one [3]. But weknow nothing about the rate of convergence.

References1. G. Strang and G. Fix, An analysis of the finite element method, Prentice-Hall, EnglewoodCliffs, N. J.5 1973.2. R. Falk, Error estimates for the approximation of a class of variational inequalities, Math.Comp, (to appear).3. E.Anderheggen and H. Knpfel, Finite element limit analysis using linear programming, Int.J. S olids and S tructures 8 (1972), 1413-1431.MASSACHUSETTS INSTITUTE OF TECHNOLOGY

CAMBRIDGE, MASSACHUSETTS 02139, U.S.A.


8/8

Documents

FEM Strang