FINITE ELEMENTSIN

COMPUTATIONAL MECHANICSFEICOM-85

Proceedings of the International Conference held at theIndian Institute of Technology, Bombay, India,

2-6 December 1985

Edited by

TARUNKANTDepartmellt oj Civil Engineering

ludiall lnstiwte oj Techllology, BombayPOlVai, Bombay - 400 076, India

PERGAMON PRESSOXFORD· NEW YORK· TORONTO, SYDNEY· FRANKFURT

Copyright © 1985 Pergamon Press Ltd.Proc. Int. Conf. Finite Elements in ComputationalMechanics, Bombay, India, 2-6 December 1985

ADAPTIVE FINITE ELEMENTMETHODS FOR COMPLEX

PROBLEMS IN SOLID AND FLUIDMECHANICS

J. T. Oden and L. DemkowiczTexas Institl4te for Computational Mechanics, The University of Texas

at Austill, USA

ABSTRACT

This paper addresses the general topic of adaptive methods for automaticallyenhancing the quality 6f numerical solutions to linear and nonlinear boundary-value problems in solid and fluid mechanics, and reviews some of the recentwork of the author and his collaborators on this subject.

KEYWORDS

Adaptive Finite Element Methods, A Posteriori Error Estimates.

~.INTRODUCTION

The basic objective of an adaptive finite element method is to improve thequality of an initial finite element approximation by automatically changingthe ·model: refining the mesh, moving mesh nodal points, enriching the localorder of approximation, etc. Thus, all adaptive methods must attempt to re-solve two basic issues: 1) how is the quality of the approximate solution tobe measured? and 2) how does one adapt the model to improve the quality ofthe approximation?

The first question is generally resolved by attempting to measure the localapproximation error in some appropriate norm. The error, of course, is thedifference between the exact solution u and a finite element approximationu of u ~on a given ·mesh. Since u is not known, the problem of assessingtRe quality of an approximation reduces to one.of a-posteriori error estima-tion: the determination of estimates of the error using computed finiteelement solutions. A number of important papers on various schemes for a-posteriori error estimation has been contributed by Babuska and his collabora-tors (1978a, 1978b, l~e~:.

Once an estimate of the distribution of the error is available, the difficultquestion of how to best modify the model to improve accuracy arises. Thereare three general approaches:

3

4

h-Methods. Here the mesh is refined; the mesh size h is reduced andthe number of elements in the mesh is increased in regions of large error.

p-Methods. Here the mesh is fixed but the degree p of the polynomialshape functions is increased over elements in which a high error is indicated.

Moving Mesh Methods. In these methods, the number of nodes and the typeof finite element remains constant during the adaptive process and the nodalpoints are moved to regions of high error.

Of course, one can also employ combinations of these strategies. But thecorrect strategy for use of combined methods is apparently a delicate issueand one in which much additional study needs to be done.

We shall describe here two methods for error estimation and show how thesecan be implemented in each of the three adaptive schemes listed above.

A-POSTERIORI ERROR ESTIMATES

We describe two classes of a-posteriori error estimation, one based on thecomputation of element residuals and the other based on interpolation errorestimates. The former class of methods was introduced by Demkowicz, Oden,and Strouboulis (1984) and includes several of the results of Babuska andcolleagues (1978a, 1978b, 1983), and the latter was first advocated by Diaz,Kikuchi, and Taylor (1983) and investigated by Demkowicz and Oden (1985a,1985b, 1965c). ..

Residual Methods

Consider the abstract boundary-value problem, Find u in V such that

= for all v in V

where

(l)

A

vf

a (possibly nonlinear) operator from a reflexive Banach spaceof admissible functions V into its dual v*an arbitrary test function in Vgiven data in V* *duality pairing on V x V

*This problem is equivalent to the abstract problem: Au = f in VA Galerkin approximation of (1) consists of seeking a function u

hin a

finite dimensional subspace Vh

of V such that

h for all in (2)

The residuaL rh is the degree with which the approximation uh fails tosatisfy the original conditions on the solution:

(3)

*Since the residual belongs to the dual space V and not necessa*ily V, itsmagnitude must be measured with respect to the norm 11 ·11 * on V :

sUP'

v 6 vl\vlI

= sup Ilv II ~ lh'

(4)

5

where II ·11 is the norm in VIn some of the error estimators that we have developed, we use the followingprocedure to approximate the supremum in (4):

• The original finite element approximation uh is computed in a spaceVh

= V6 of spanned by low-order (say, linear) piecewise polynomialsHape functions, resulting in the residual r).n

• The full space Vspace Vh' p > 1 ,

is approximated by a higher-order finite elementspanned by piecewise polynomials of degree p .

An approximation of the residual 1 . d d'r, lS constructe accor lng to.1

+ supII v~ \I

< 1rh,< 1(5)

where C is a constant, va is an element of Vand v~ is an arbitrary element of V~ . Ifhfor a partition Th of elements K

realizing the sup in (4)is the mesh size (i.e.

h = = diameter (K)

we generally have

so that. it makes sense asymr-tot:ically (as h -..0) to approximateby sup

1sup

As an example of how this procedure can be implemented, consider the modelproblem,

Find u in V = (v 6 H1(1l) ; v = a on f1 } such that

In

\7U• \7vdx = In fv dx + I gv ds for all v in Vf2

( 6)

This is the variational form of the model Poisson problem,

-~u = f in n c m2

u = o on r lC anau

g on r 2C;)n=an

(7)

6

2 1with 6 = 6 , H (n) the usual Sobolev space of functions with derivatives in2 - -

L (n) , and an = r1 U r2 .

We define

Vh = {vh G V : vhlK G Ql(K) , IT ~ UK}

V~(K) = {v~ G Pp(K)}

V~O(K) = {v~ G V~(K) ; P > 1 , v~ interpolant v~ = O}

(8)

where Ql(K) is the usual set of bilinear functions defined on a quadri-lateral element K and P (K) is the space of polynomials of degree pdefined on K P

The residual satisfies

(9)

E {f (-6u~ - f)v~ dxK K 1 1

faUh aU*h

+ . ~(- - - ) vP dsdK\an an dn h

1

JdUh(- - g)

aKA r dn2

where u~ is the (coarse-grid-initiall finite element approximation of udetermined using the srace V

hand ~h is the function~l o~ vP defined

.as indicated, with aU*h/an an approx!mation of dul n o~talned~rom anadjacent element to K.

It is not enoughelement K. Inwhich will boundzero at the same

Ih .'to simply calculate RK as an indicator of the error ingeneral, we wish to have an indicator ~ of the errorthe local error above and below and wbic~ will converge torate as the actual error; e.g.

clI

< {C EK

7

(l2)

where C is a (hopefully) known constant. Though this estimate is global,

we use \I K111 K as an estimate of the local error over each element K.,. I

In general, reducing \I K11l,Kimplies a reduction in II rhll* which (parti-cularly for linear self-adjoint problems on Hilbert spaces) implies a reduc-

tion in II u-~~ \I .

Interpolation Err0r Estimates

It is well known (see, e.g. Oden and Carey, 1983l that for linear ellipticproblems the approximation error II ehll~= II r-uhllv can be bounded aboveby the so-called interpolation error,

II eh IIV ~ C II u-v~ Ilv for all1

in VIvh h

For the model problem (6), for example,

~n V (u-u~)I ~• V(u-u ) dxh

lu-u~ Il, < C inf lu-v~ Il,n= vh

G Vh

(13)

(l4)

If u is smooth enough, a local interpolation e~~o~ estimate can be derivedof the type (for Ql-elements) •.

where

(15)

W dx dx dy (l6)

The basic problem we face when attempting to make use of any of these estimatesis that we must calculate the higher order derivatives of the unknown solutionusing only available information, i.e., through use of the currently availablefinite element solution u~. There are numerous a priori techniques forestimating the second derivatives u ,u or u ,but many are somewhat

xx xy yyintuitive and not all are based on rigorous estimates. Exceptions are thetechniques based on so-called "extraction formulas" introduced by Babuskaand Miller (1984a, 1984b). Following their idea one can prove that, if uis regular enough, then the second derivatives at an arbitrary point (x

O'

8

yo) satisfy

a2u a2u~ (xO' yO) - --Z (xO' YO)ax oy In 6~ u dxdy

J (~+~) f dxdy -J u ;nn an (¢' + ~) ds + J (~+~) aaudsan n (17)

are the polar coordinates centered at the

of the same order of1

by uh

. For example,

for the first order approximation we can "extract" the difference of secondorder derivatives with O(h2) order of convergence! Formula (17), when com-bined with equation (7), allows us to calculate each of the derivatives

. 1 sin28separately. Also, by choos1ng ~ = n --z-- in the same formula, we can

a2 r"extract" the mixed derivative aX~y (xO' YO) .

1 cos2Here ~ = n-:r- where (r, 6)

rpoint (xO' YO) under consideration and ~ is an arbitrary, regular func-

tion. By the proper choice of $, one can eliminate the boundary terms in(l7). Of course, u on the right-hand side of (17) remains still unknown,but when replaced by its element approximations u~ results in a formula

for approximation of second-derivatives at (xO

' YO)

accuracy as the L2-error in the approximation of u

One method we have used successfully in applying the estimate (l5) is toconstruct the function $ using a bivariate blending function of Gordonand Hall (1971, 19?3) type.

Note that we still have a global estimate although we "apply it" to K

ll2lu - uh 1,n < C 2 I 2L hK \u 2,KK

MESH REFINEMENT STRATEGIES BASED ON THE A POSTERIORI ERROR ESTIMATES

While many issues ~emain open in the area of reliable a-posteriori errorestimation, still further complications exist in designing efficient adap-tivc algorithms bascd on these estimates. Thc baslc prohlcm can assumc theform of an optimal control problem in which one has to attain a discreteapproximation which is optimal in some sense determined by the error measurasand the strategy used to reduce error. The entire problem is further compli-cated by the fact that our a-posteriori estimates are global in nature (par-ticularly the residual-type estimates discussed earlier) even though theyare used locally as a basis for local enrichments of the solution.

In this section we describe three methods developed by Oden and colleagues,(1985), Demkowicz and Oden (1985), Demkowicz, Oden, and Strouboulis (l98~),and Demkowicz, Oden, and Devloo (l985).

An h-Method

Consider a quadratic mesh and the associated error estimate (l8). If we

I)

define a function hex, y) specifying a "density" of the mesh by:

hex, y) ~ if (x, y) G K (l9)

the error estimate (18) may be written in the form of a functional

J(h) 2h W dxdy , (20)

where W is given by (l6) .

This leads to a natural minimization problem:

Find h = hopt minimizing the functional (20) and subjected to the constraint:

N (21)

where N is simply the number of elements. The solution of this problem is

h2 f W dxdyK K

const. for every element K (22)

which results in a very simple iterative - mesh refinement scheme, providedwe can estimate the function W. For additional details we refer toDemkowicz, Oden, and Devloo (1985).

A p-Method (see Oden and others, 1985i Demkowicz, Oden, Strouboulis, 1985)

It is difficult to predict the rate of convergence of local interpolationerror in the case of the p-version since it depends only on the unkno~regularity of the solution (Babuska and all, 1981), and it is almost im-possible to say anything about a local order of convergence of the error orthe.residual-type estimate. One way that we have used estimates of the type(l2) effectively is to employ the following steps:

1. Solve local problem (ll) and determine local contribution I~Kll Kto the global error estimate. '

2. Normalize the local indicators I~Kll K by subdividing by thelargest one. '

3. If

o < I~Kll,K < 0l

IS < I~Kll,K < 152

°2 < I4>K1l,K < 1

the first order approximation is retained.

a second-order approximation is used.

a third-order approximation is used.

Here the numbers 01 and 02 are chosen rather arbitrarily andponds to first-order approximation on a uniform mesh.

1~h corres-

10

Moving Mesh Strategy (see Demkowicz and Oden, 1985a)

A popular moving finite element method, due to Miller, is based on an 12_residual estimate in its basic form which finds justification in only theone-dimensional case. In Demkowicz and Oden (l985) we present a moving meshstrategy based on the interpolation error estimate that can be considered asa generalization of the Brackbill and Saltzman strategy (1982).

We consider a fixed mesh of elements K that is mapped onto a distortedmesh in such a way as to minimize the interpolation error. We may penalizethe interpolation error functional further by appending to it additionalterms to smooth the mesh and prevent the Jacobian of the map from vanishing.

In the method discussed in Demkowicz and Oden (l98Sa) the optimal mesh resultsfrom minimizing the functional

J I + aI + 13Jo (23)where

J 2 2 2 210 = ~ A (x + y + x~ + Ys)d~dn (24)

K K n n

J 1 2 2 2 222I = k A J [(Xr;+ Ys)u + (x + Y )ur;r;]d!=;dn (25)K K nn n n

f 2 1J = E (xsY + x y) J d~dn (26)A Ann nK K

'Here a and 8 are constant parameters, 1 is the Dirichlet integralwhich treats as a ~onstraint the invertibili2y of the transformation definingthe mesh. In the absence of other terms, its minimization yields a conformalmesh and then the method is merely a generalization of the well-known con-formal-map mesh generators. The functional K forces the mesh lines to beorthogonal and this .,orthogonality allows us to simplify the form of the·error functional I .

NUMERICAL EXPERIMENTS

In this section we present some representative examples.

A Heat Conduction Problem With a Moving Domain

C. Where:u

We first consider a linear transient heat conduction problem defined on amoving domain n = (0, 4 + O.lt) x (0, 3) with purely homogenequs Dirichletboundary and inifial data. We choose the data on the right-hand side of theequation to correspond with a prescribed exact solution: .

2lO e-S(x - 1 - 0.2t) x (4 + O.lt - x) Y (3 - y)

c for 0 < t < 0.5for t > 0.5

This problem has been solved for a mesh with 24 elements (see Fig. 1). Thetime step has been chosen ~t = 0.1 and the solution was computed for 20

11

time steps. The constants 0l and 02 described in the previous chapterhave been chosen as follows:

= 1/2 .

Figure 1 shows the mesh enrichment for terms t = 0.5, 1.0, 1.5 and 2.0while Figs. 2 and 3 present the computed first-order ..nd enriched solutionson the section AA (see Fig. 1) compared to the exact solution. For additionaldetails we refer to Demkowicz, Oden, and Strouboulis (1984).

A Stokes Problem

The second problem we consider deals with the application of the moving meshstrategy described earlier rather than the penalty-formulation. In thisexample Q is the rectangle (0, 2) x (0, 1) and we define onthis domain aninitial uniform 16 x 8 mesh of rectangular elements. Dirichlet boundaryconditions are used with the velocity u prescribed as Uo = (l, 0) alongthe top edge and u = (0, 0) along the-remaining sides (oriven cavity prob-lem).

The optimal mesh results from a minimization of the functional

NJ = 10 + L a.li + 8J + yKi=l 1.

Where ai' i = 1, 2, 8 and Y stand for positive (given)is defined by (25) with u replaced by the i-th componentfield, J

Ois a similar term for the hydrostatic pressure

Figure 4 presents a computed mesh for ~1 = u2

= 4, B = 4,

A Model Elliptic Problem

real numbers, Iiof the velocitygiven previously.y = 0.1

~.

As an example of the application of the h-mathod combined with the interpola-tion error estimate, we have solved the model elliptic problem (7) withhomogeneous Dirichlet boundary conditions. The right-hand side of the eq~a-tions correspond to the following (exact) solution defined in n = (0, 1) :

u(x, y) = ~(x)~(y) , where

~(x)-£e + Ax + B

with A and B chosen in that way that $(0) = $(1) = O. With xo = 0.55,YO = 0.5, £ = 0.02, and £ = 0.05; after 19 mesh refinements we have ob-x . ytained a mesh presented in Fig. 5. The error together with its exact and com-putational error estimates are shown in Fig. 6.

ACKNOWLEDGEMENT

This work was supported by the U.S. Office of Naval Research under ContractNOOOl4-84-K-0409. Helpful discussions of the work reported here with Messrs.

12

T. Strouboulis and Ph. Devloo are gratefully acknowledged.

REFERENCES

Babuska, I. and W.C. Rheinboldt (1978). Error estimates for adaptive finiteelement computations. SIAM J. Numer. Anal., 15(4).

Babuska, I. and W.C. Rheinboldt (1978). A-posteriori error estimates for thefinite element method. International Journal for Numerical Methods inEngineering, ~, 1597-1615.

Babuska, I., A. Miller, and M. Vogelius (1983). Adaptive methods and errorestimation for elliptic problems of structural mechanics. Inst. for Phys.Science and Technology, University of Maryland, Lab. for Numerical Analy-sis, Technical Note BN-I009.

Babuska, I., B.A. Szabo, and I.N. Katz (1981). The p-version of the finiteelement method. SIAM J. Numer. Anal., 18(3), 515-545.

Babuska, I. and A. ~(fller (1984). The poSt-processing approach in the finiteelement method - part l: calculation of displacements, stresses and otherhigher derivatives of the displacements. Int. J. Numer. Methods Eng., 20,1085-1109.

Babuska, I. and A. Miller (1984). The post-processing approach in the finiteelement method -part 2: the calculation of stress intensity factors.Int. J. Numer. Methods Eng., 20, 1111-1129.

Brackbill, J.V. and J. Saltzman (1982). Adaptive zoning for singula~ problemsin two dimensions. J. Computational Physics, 46, 342-368.·

Carey, G.F. and J.T. Oden (1983). Finite Elements: A Second Course, PrenticeHall, Englewood Cliffs, N.J.

Demkowicz, L., J.T. Oden, and T. Strouboulis (1984). Adaptive methods forflow problems with moving boundaries. I. Variational principles and a-posteriori estimates. Computer Methods in Applied Mechanics and Engineer~ing., 46_, 217-251.

Demkowicz, L. and J.T. Oden (to appear). On a mesh optimization method basedon a minimization of interpolation error. Int. J. Eng. Sci.

Demkowicz, L., J.T. Oden and Ph. Devloo (to appear). On an h-type mesh re-finement strategy based on minimization of interpolation errors. CpmputerMethods in Applied Mechanics and Engineering.

Demkowicz, L. and J.T. Oden (to appear). Extraction methods for second-derivative i1 finite element approximations of linear elasticity problems.Communications in Applied Numerical Methods.

Diaz, A.R., N. Kikuchi, and J.E. Taylor (1983). A method of grid optimizationfor finite element methods. C~mputer Methods in Applied Mechanics andEngineering, ~, 2Y-45.

Gordon, W.J. (1971). Blending function methods of bivariate and multivariateinterpolation and approximation. SIAM J. Num. Anal., 8, 158-177.

Gordon, W.J. and C.A. Hall (1973). Transfinite element methods: blendingfunction interpolation over arbitrary curved element domains. Numer. Math.,£!.' 109-129.

Oden, J.T. and G.F. Carey (1981). Finite Elements: Mathematical Aspects,Prentice-Hall, Englewood Cliffs, N.J.

Oden, J.T., L. Demkowicz, 7. Strouboulis, and Ph. Devloo (to appear). Adap-tive methods for problems in solid and fluid mechanis. In T. Babuska andO.C. Zienkiewicz (Eds.), Adaptive Methods and Error Refinement in FiniteElement Computation, John Wiley and Sons, Ltd., London.

13

A. A A.

t=0.5 t= 1.0

A. A

t = 1.5 t=2.0

~1oord6r ~ 30 order

Figure 1. Mesh enrichment for the heat conduction problem

~i f\ 0 ~.0. TIME=2.00TIME=0.5 0M--- 1st Order -1st Order

~1 II \\Enriched ~ Enriched

0- Exact 0 -Exact.

0 ..N \

\\\;1 j/ I \\ ~j / j \ \0.,..

\1/ II ~

~ -0- -.,-0.00 1.00 2.00 3.00 4.00 O.O~ 1.00 2.00 3.00 4.00

Figure 2. Heat conduction problem.Computed solutions on section AA forp = I and adaptive correction fort = 0.5.

Figure 3. Heat conduction problem.Computed solutions on section AA forp = 1 and adaptive correction fort = 2,0.

14

Figure 4. Driven cavity problem. Optimal mesh after 8 FErecalculations; 0. = 4., B = 4., y = Q. I.

Ln II error II

\\

fJ" computational error~~\ estlmato basod on

....,\ tho extraction formula

.,'..'".\'-.\\,

.",\"':.'',.\

'-:""':\~.

Ln(Nl/2)2.0 .'\". ,

".

error estimate ~\

1.0o

1.0

-1.0

-2.0

Figure 5. Refined mesh after19 FE recalculations.

Figure 6. Convergence of the interpo-lation-error estimate based on the ex-traction formula.

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