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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 53 (1985) 67-S9NORTH-HOLLAND

ON AN It-TYPE MESH-REFINEMENT STRATEGYBASED ON MINIMIZATION OF INTERPOLATION ERRORS*

L. DEMKOWICZ, Ph. DEVLOO and J.T. ODENTexas [Ilstitllte for Computational Mechallics. Department of Aero~pace Engineering and Engineering

Mechanics. The University of Texas. Austin, TX 78712, u.s.A.

Received 15 December 1984

An adaptive finite element method is proposed which involves an automatic mesh refinement inareas of the mesh where local errors are determined to exceed a pre-assigned limit. The estimation oflocal errors is based on interpolation error bounds and extraction formulas for highly accurateestimates of second derivatives. Applications to several two-dimensional model problems are discussed.The results indicate that the method can be very effective for both regular problems and problems withstrong singularities.

l. Introduction

Adaptive finite element methods are designed to automatically improve the quality of anumerical solution by enriching the approximation in some way: moving the nodes, increasingthe local order of the approximation, refining the mesh, etc. Therefore. implicit in suchmethods is the ability to somehow assess the 'quality' of an approximate solution locally overvarious elements in a given mesh. This quality of a numerical approximation is naturallyjudged by its error in some appropriate norm-the magnitude of the difference between theexact solution u and its finite element approximation lih. Thus, an integral part of a rationaladaptive scheme is some means to' estimate a-posteriori (after one approximate solution isobtained for an initial mesh and element family) the local approximation error.

Some progress has been made in developing a-posteriori error estimates based on thecomputation of residuals, and the survey of Babuska [1] provides a detailed discussion of suchapproaches for a restricted class of problems. Alternatively, Diaz, Kikuchi and Taylor [13]have proposed an error-estimation method based on interpolation error estimates for finiteelements. This method has the advantage that it does not require the solution of local auxiliaryproblems for error indicators for each element, but, among other things, it has the disad-vantages of requiring the estimation of higher-order derivatives over each element and beingnon local.

In the present paper, an II-type adaptive scheme which employs error estimations based oninterpolation estimates is presented. In this method, a highly accurate method for calculation

• The support of this work by thc Office of Naval Research under Contract NROOOI4-84-K-0409 is gratefullyacknowledged.

0045-7825/85/$3.30 © 1985. Elsevier Science Publishers B.V. (North-Holland)

68 L. Demkowicz et aI., h-type mesh-refinement strategy

of second derivatives is employed which is based on the 'extraction formulas' of Babuska andMiller [2,3] and particularly the formulas of Demkowicz and Oden [12] for plane elasticityproblems. In addition, a simple derivation of an 'equi-distribution of error' rule is derived as abasis for error minimization in cnergy norms for a class of linear elliptic boundary-valueproblems. While we have elected to use an h-type adaptive scheme in the present exposition,the methods of error estimation and extraction are by no means limited to such strategies andcould be used in p-methods or moving mesh strategies as well.

2. Model problems

We describe our method as it applies to two model problems: a Poisson problem in theplane and a plane elasticity problem. Both are two-dimensional; in one the solution isscalar-valued and in the other it is vector-valued. While some of our theoretical developmentsare based on assumed regularity of solutions, our applications involve cases in which solutionspossess strong singularities.

2.1. A model Poisson problem

As the basic modcl problem for our considerations we have chosen the Poisson equationwith mixed boundary conditions:

- Jh = I in n.u = 0 on ru, au

an = g

(2.1)

where tu U tT = an. ru nTT = (1 and II denotes the outward normal unit to an.Introducing the notation,

B(II, v)= t Vu' Vv, L(v)= J Iv + J gv,n rr

(2,2)

we introduce the usual variational formulation of (2.1) as follows:

Find u E Hr(n) such that

B(u. v) = L(v) V v E HrCn).(2.3)

2.2. A plane elasticity problem

Given again a bounded domain n in R2 with the boundary an = tu U fT, Tu nrT = 0, weconsider the standard plane elasticity problem:

G.1u + (A + G) grad div u = -1 in n.u=O onru, 1(1I)=g onrT,

(2.4)

L. Demkowicz et al., h -type mesh -refinement strategy 69

where G, A are Lame's constants, f is a body force, t(u) denotes the stress vector correspond-ing to the displacement u, and g are prescribed tractions on rr.

Defining

Hr(fl) = {u E H I(fl) III = 0 on [~,},(2.5)

L( v) = J f' v + J g' v ,n rT

where

with D;j the Kronecker delta and the usual summation convention in force, we formulate (2.4)again in an abstract form similar to (2.3):

Find u E Hr(fl) such that

B(u. v) = L(v) V v E Hr(fl).

3, Interpolation error estimate

For both problems the abstract variational formulation is:

Find II E V such that

B(II,v)=L(v) VvEV,

where V denotes Ihe appropriate Hilbert space,A finite element approximation of (3.1) is of the form,

Find Uh E Vh such that

B(Uh, Vh) = L(Vh) V Vh E Vh,

(2.6)

(3.1)

(3.2)

where Vh denotes a finite element finite-dimensional subspace of V.Introducing the energy norm" ·111 = B(u, u). we have the following immediate estimate:

(3.3)

Particularly we can employ for Vh the finite element interpolant of the function u.Our first goal is to estimate the right-hand side of (3.3) with such a choice of Vh, i.e. the

finite element interpolant error of u. Let Th be a partition of fl into a finite element mesh suchas that in Fig. 5. Then

Ilu - vhll~= L Ilu - vhll~.Kken

70 L Demkowicz et at.. h-type mesh-refinemetll strategy

(3.4)

and the question is how to estimate the finite element interpolation error lIu - Vh IIi:, K over thesingle element K. This is to be resolved for each of the two model problems independently.

3.1. The model Poisson problem

Let x = XK + hKg, y = YK + hKTJ denote the transformation of the master element K ontoelement K (see Fig. 1). A sequence of such maps generates the tlnite element mesh. For atypical element K, we have

lilt - Vh II~.K = t [:X (It - Vh) r+ [:y (It - Vh )r dx d y

= J [~(Li - Vh )]2 + [~(u - Vh )]2 dg dTJK ag aTJ

~ Ct [U~t + U~'lld~ dTJ = Ch~ t [li~.x + ll~] dx dy

with the standard notation in use (d. Ciarlet [9]). The constant C depends only upon themaster element K. A global interpolation error bound is obtained by summing over allelemen ts K E Th :

(3.5)

Of course, a question remains how to estimate the second-order derivatives Un: and Uyy of theunknown solution u.

3.2. The plane elasticity problem

A direct calculation reveals that

"7(0,1)

,..K

(0,0)

(1,1)

-hk

D th~k

(Xk,Yk)

Ie:

(1,0) (0,0) x

Fig. l. Generation of element K by map of a master element K.

L. Demkowicz et al.• h-type mesh-refinement strategy

which can be easily bounded by

(20 + 2A )(uT.. + ub) + 20(ui, 2 + U~,.) .

Following the plan of Demkowicz and Oden [11], we obtain the estimate

7I

lIu - vhlli-. k :!S; CII ~t {(20 + 2A )(uT. 11 + u~,22) + 20(ui. 22 + 1l~.11)} dx dy, (3,6)

where C depends only upon the master element.This leads to a global estimate of the same type as (3.5).

4. Estimation of second-order derivath'es (extraction formulas)

Following the idea of Babuska and Miller [2, 3], we propose a method for estimating thesecond derivatives of the unknown solution u (u resp.) through its finite element ap-proximation Uh (Uh resp.). Although for both problems a unified approach could be presented,for the sake of clarity, we begin first with the model problem.

4.1. The model Poisson problem

Our starting point will be a derivation of a certain very special Green's function which isdefined as a solution of the following distributional different equation:

Find a function cp such that

J a2l/1 a2l/1- cp/J.l/1 = - (0, 0) - - (0,0) V l/1 E D(R 2) .

R2 ax2 ay2 (4.1)

We claim that the solution to (4.1) is cp = (1/'Trr) cos 20, where (r,O) denotes usual polarcoordinates centered at the point (0,0). The prove this, pick an arbitrary sufficiently regularfunction l/1 and let S. denote a ball with radius £ centered at (0,0). One has:

(4.2)

due to the fact that cp is harmonic in R2,,-(0, 0). Expanding l/1 and al/1/an into Taylor seriescentered at (0,0), we can prove that the right-hand side of (4.2) converges to

iPl/1 a2l/1~ (0, 0) - --"2 (0, 0)ax ay

as £ approaches O. The function cp can thus be used to derive an 'extraction formula' similar tothose in [2,3], for the difference of second-order derivatives.

So let cp = (l/'TrI.2) cos 20 and assume that the solution u to (2.1) is sufficiently regular. Again

2 L Demkowicz et al., h-rype mesh-refinement strategy

~t Se be a ball with radius e centered at an arbitrary point (io, Yo) En, and let iP be anrbitrary, sufficiently regular function. Note that (r, 0) are measured from (xo, Yo).

Application of Green's identity leads to the following equation:

J {a(cp+iP) au}+ u-(cp+iP)- .ilS, an an

\ssuming, e.g. that iP is CI around (xo, Yo), letting e go to 0, and exploiting the limitingIroperties of rjJ, we obtain

Next, we replace u in (4.3) by its finite element approximation Uh and thereby obtain an:xtraction formula that provides the combination of second derivatives at (xo, Yo). The choiceIf iP is somewhat arbitrary. One may choose iP such that cp+ iP = 0 on ru and a(cp + iP)/an = 0III rr, eliminating in this way the unknown portion of the boundary term in (4.3). This isailed a 'blending' technique. Alternatively, we can use a 'cut-off' method if we define iP sohat cp + iP is only locally different from 0 around the point (xo, Yo). In this case,alculation of (4.3) can be implemented locally, a feature which is convenient from a:omputational point of view. Numerical experiments are discussed in Section 6.

Finally, we reiterate that (4.3) combined with (2.1) allows us to calculate second derivativesIf u at any point of n. We also note that that choice of rjJ = (lhrr) sin 20 leads to an:xtraction formula identical with (4.3) but with the left-hand side replaced by~a2u(xo, Yo)/ax ay,

1.2. The plane elasticity problem

Our starting point is once again the determination of special Green's functions which areolutions to special auxiliary problems.

We summarize key results in the following:

~ROPOSITION 4,1 (Demkowicz and aden [12]). Let v = (u, v) be an arbitrary C2 function.men the following identities hold:

(i)

. (COS 20 sin 20)if ip = ~; ----;.2 ;

(4.4)

L DemkolVicz et aI., "-type mesh-refinC'ment strategy 73

(ii) (4.5)

f = (_ sin 20. cos 20) .I II' 2' 2 'r r

(iii) (4.6)

3G+"\ .where a = 2( G + ,,\) ,

(iv) (4.7)

with the same constant a as before.

Proof of the proposition, although lengthy, is straightforward. It involves only the obser-vation that all II' above satisfy homogeneous equilibrium equations and requires an in-vestigation of the limit of the respective boundary terms as in the case of the model Poissonproblem.

Based on these four particular choices of 11', we can derive four analogous extractionformulas for the respective combinations of second derivatives of the solution of the planeelasticity problem. More precisely, for this model problem we have

{u,v}(Xo,Yo)= f (tp+iP)'f+ f O'jj,j(iP)'u- J {t(tp+iP)'u-(tp+iP)·t(u)}.n n an (4.8)

Here II' is one of four extraction functions, and {u, v} denote the corresponding combination ofsecond-order derivatives. As before, the four extraction formulas (4.8) with u replaced by itsfinite element approximation Uk, when combined with the partial differential equation (2.4),provide conditions sufficient to provide good estimates of all second derivatives of the solutionu.

5. Mesh refinement strategy

For both model problems, the tinal interpolation error estimate is of the form.

(5.1)

74 L Demkowicz et al.. h -type mesh -refinement strategy

where IV is the propcr combination of squarcs of sccond-order derivatives of Ii. If the mesh issufficiently fine the estimate (5.1) may be considered as an approximation to the followingfunctional]

J(h)=cj wh2dxdy./l

(5.2)

Here h = hex, y) denotes a continuous 'mesh density function', the ptecewlse constant ap-proximation of which corresponds to (5.1) when we define

h(x,y)=h(( if(x,y)EK (finiteelementK). (5.3)

A direct minimization of (5.2) leads to the trivial result h = 0, or in other words, the finer themesh, the smaller the error. To arrive at a useful result, it is clear that a constraint must beimposed on the density function h, A natural constraint is

L 1:2 dx dy = N = constant. (5.4)

The interpretation is evident. When h is replaced by its piecewise constant approximation(5.3), the left-hand side of (5.4) is exactly equal to the number of finite elements. Thus thefixed number N in (5.4) is intcrpreted as the number of finitc elements in a given mesh.

We thus arrive at the following constrained minimization problem: Minimize] (h) of (5.2)subject to (5.4). An application of Lagrange multipliers leads to the optimality condition

where

L (wh - :3)oh = 0 , (5.5)

(5.6)

where a is a Lagrange multiplier (up to a multiplicative factor) associated with the constraint(5.4). Taking II = (a/w)1/4 and substituting into (5.4) determines a as a function of N.

5.1, Mesh -refinement technique

When applied to the piecewise constant h of (5.3), condition (5.6) reduces to the fundamen-tal relation

h'i L w dx dy = a 'V K E Th . (5.7)

This condition suggests the following recursive mesh-refinement technique:

- Given an initial mesh, instead of fixing the number of elements N, choose a value for a and

L Demkowicz et at.. h -type mesh -refil/emelll strategy 75

calculate

dr JaK =ht K wdxdy

for each element K for some approximatioll of w.- Reduce the diameter 11K of all elements K for which aK > a according to formula (5.7).- Resolve the boundary-value problem on the new mesh and again estimate the function w.-Continue this procedure until the quantity aK is everywhere less or equal to a.

Typically we may start with a coarse mesh and choose for the following iteration stepa = minK aK and thereby fix the maximal element size for the entire procedure.

Once the criterion for refinement is met in one element, the actual refinement process isdone by splitting quadrilateral elements into sub-quadrilaterals in a systematic way. Nodalvalues of nodes at the intersection of boundaries of a sub-element with a larger element areobtained by averaging, while interior sub-element nodes are assigned new values of thesolution that must be computed anew. Two steps in a typical refinement sequence areillustrated in Fig. 2. As the elemcnt E is connected to a larger elcment, its first node isconstrained to be the average of the values of the second and third node of the element A.

Denote by {1£, 2£, 3£, 4£} the 4 nodal values of the solution over element E and{lA, 2A, 3A• 4A} the nodal values over A Then, the variational principle for element E can bewritten as

Because IE = H2A + 3A) and 3A = 4£ (Fig, 2) and therefore IE = H2A + 4£) we can write

C I 0

+

divide F...-----.......4_ 3~4. 3.

E2, GA 1,

1. 2 F H

C 0

divide B

BA

Fig. 2. Example of a two-step refinement.

76

.lr

fhus,

L. Demkowicz et al., h-type mesh-refinement strategy

[SUEl'[~El'[ ~][~E HllE 1= [SllE II[~E]'[ FE I.We have,

[SUE l'[~H liE] = [SIIE J'[ FE J .

fhe 4 variables associated with I~E are unconstrained, and [i J is still symmetric; [~] and [FE]:an still be assembled in the usual fashion. The advantage of this procedure is that the:onstrained variable is eliminated at element level, thus saving execution time and storagc·equirements.

For the second refinement illustrated in Fig. 2 the error estimate for element F exceeded.he threshold value. But before this element could be refined, elements D and A had to be refinedn order to not violate the assumption that not more than 2 elements can be connected to a side of)nc element.

5.2. Order of convergence

In the case of non-uniform meshes, it is natural to consider the rate of convergence of thefinite element method as a function of the total number of degrees of freedom. For uniformmeshes, Nh2,.. - meas(n) and, therefore, (5.1) may be rewritten in the form

1111- vhll~ ~ C' meas n J IV dx dy . ~./1 N

(5.8)

We shall now investigate the order of convergence in the case of optimal meshes con-.;tructed by the technique described above.

Considering again our 'continuous model', we assert that for the optimal mesh distribution,5pecified by h = hOPI' one has

J Wh2dXdy=J IVh4'~dXdY=O"J ~dxdY=O"N.n n h nh

Simultaneously

N = J ~dx dy = 0'-1/2 J Wl/2 dx dy.

nh n

Thus, combining (5.9) with (5.10), we conclude that for the optimal mesh distribution,

J2 I

1111- Vhll~ ~ C meas n( . wl12 dx dY) . -.lJ N

(5.9)

(5.10)

(5.11 )

rhus, for regular solutions (i.e .. those for which the estimate (5.8) is valid) the rate of:onvergence for optimal meshes is precisely the same as for the uniform mesh. The critical

L. DemkolVicz et aI., h-type mesh-refinement strategy 77

(6.1)

difference is that the constant f n W dx dy in (5,8) has been replaced in (5.9) by a generallysmaller factor {fn wl12dx dy)2,

This particular result indicates that for regular solutions the definition of given-optimalmeshes introduced by Babuska and Vogelius (4] and based on estimates of the convergencerate is of limited value since every uniform mesh is 'given-optimal' as it provides the same rateof convergence as the optimal mesh. This is not the case, of course, when the estimate (5.8) isnot applicable, i.e. in the case of irregular u.

In the next section, we present some simple numerical experiments confirming the expected,asymptotic rate of convergence.

6. Numerical examples

6.1. Some computational aspects

The algorithm described in the preceding section has been implemented in a test code usedto analyze several example problems. Of special interest is the quality of the local errorestimate that can be provided by the interpolation error strategy described earlier. We shalldiscuss this aspect of the results subsequently.

To implement the mesh refinement scheme, we begin by specifying an initial mesh and anerror threshold number a which represents an upper bound of the local error that is to betolerated. An initial solution is obtained and estimates of the energy norm of the error overeach element are computed using, e,g. (3.4), i.e ..

Ilu- uhll~,K = aK = hi t W(lIh) dx dy, W(lIh) = uL ..+ u~'YY'

Only those elements K for which aK ;;?: a are refined.Our test code employs four-node bilinear (QI) isoparametric elements. When aK ;;?: a, the

mesh is refined in such a way that no more than two elements can be connected to the side ofan element.

6.1.1. Local error estimatorsImplementation of (6.1) requires that we construct an estimate of the second-derivative

function w. We describe three techniques for accomplishing this for bilinear element ap-proximations of the two-dimensional Poisson problem.

(1) Estimates based on gradient jumps. As the mesh consists of rectangular bilinearelements, a direct calculation of Un: and Uyy reveals that they are identically zero for eachelement, independent of the solution. This is remedied by extending the element slightlybeyond its boundary and integrating the square of the gradient jump along the boundary ofthe element. In this way we obtain the first estimator,

lu~liXl' = ({K I [aa:~]12 dS) 112

An advantage of this method is that it is suited for regular as well as irregular meshes.

78 L. Demkowicz et al.. IHype mesh -refinement strategy

Note a factor h difference between this formula and a similar formula obtained byZienkiewicz et al. [17].

(2) Estimate based on a finite difference stensile. Every node in a square mesh is surroundedby eight degrees of freedom. The values at a given node j and surrounding nodes can be usedto estimate the second derivative at j [10]. When the second derivatives are computed at allnodes, those values can be used to estimate the Hrseminorm over each element. This leads tothe estimator,

This estimate, although theoretically justifiable, has the disadvanlage that it is difficult toimplement on irregular meshes.

(3) Estimate based on the extraction formula. The extraction formulas described earlier canbe used to compute second derivatives of the function at the centroid of each element. Ablending function ip is chosen so that I{> + ip = 0 on an (see Cavendish [8]).

In estimating the second derivatives at the centroid of element K, the singular integral IK I{>fis computed analytically for every quadratic f. In the finite element code, f K I{>f is ap-proximated by f K I{>ft where f~ is a quadratic approximation of f over the element K. Wethen apply the formulas,

Thus, for example.

AK = area of element K.

In order to implement the above extraction formulas, it is necessary to construct anappropriate cut-off or blending function ip. In our calculations, we have applied the bivariateblending functions of Gordon [14] and Gordon and Hall [15]. This is accomplished as follows:Consider first an element K surrounded by a patch nK of eight elements as shown in Fig. 3.The patch nK is mapped to a unit square 0 by a mapping T-1

, i.e. T(O) = nk (anyone-to-one, onto map is suitable). This mapping is then used to define the restriction of afunction I{>* to ao:

l{> *Ivo : i/Q -+ IR: I{>*(r, s) = (I{> 0 T )(r, s) .

Identify by I{>t I{>}, I{>~, I{>~ the values of l{> * on the bottom, left side, right side and upper side,

L Demkowicz et al., II-type mesll-refillelllelll strateg}'

K

Fig. 3. A patch of nine elements surrounding element K.

79

respectively (see Fig. 4). Those functions on the boundary depend on one paramelcr only, e.g.

<p~:[O,1]~R.

Now <p * : Q~ R can be constructed with bivariate blending, i.e.

<p*(r, s) = (I - r)<pHs) + r<p~(s) + (1- s)<pi:(r) + s<p~(r)

- (1- r)(I- s)<pi:(O)- r(l- s)<p~(O)- rs<p~(l)- (I - r)s<p~(l).

0· (t-,4) = 0· (.)(0,1)1 (I ,(1,1)

(0,0) ( 1,0)

Fig. 4. Ip. is the bivariate blending of Ipb, .;i, Ip~ • .;;.

80 L. Demkowicz et al.. h-type mesh-refinement strategy

The function iP : ilk - R follows now naturally as

We notc that the mapping T can also be constructed by using the concept of blendingfunctions. In some of the actual calculations given latcr ilK has been set equal to il forsimplicity.

In the cases where the solution is known, the computed values of the H 2-seminorm arecompared with the true seminorm and with the energy norm of thc interpolation error. Inevery case, only the integrated values over the entire mesh are compared.

6,1.2. Solution of the system of eqliationsThere are two reasons for using iterative methods in the case of adaptive methods:(a) A very good approximation to the solution is known from the previous coarse-grid

analysis. Thus, an accurate initial iterate can increase the speed of convergence of the iterativeprocedure,

(b) Due to the highly nonregular node-numbering which arises in the refinement process,the sparse assembled matrix has a large bandwidth. Gauss elimination becomes thusinefficient.

The Jacobi conjugate-gradient method is an iterative procedure with excellent convergencerates and can be readily implemented in our refinement process. Indeed, any conjugate-gradient method retrieves the exact solution in at most N iterations (N = number of equa-tions), Moreover, the conjugate-gradient method is known to be fairly insensitive to theavailable machine precision. The speed of convergence of the conjugate-gradient methoddepends strongly upon the conditioning of the matrix, and, therefore, it is natural to applypreconditioning to the original set of equations, i.e. instead of solving

Au = b,

we solve instead

where the condition number of 0-1 A is better than the condition number of A. Note that ifQ-I = A-I, then the conditioning of 0-1.4 is unity. A necessary condition for applying the CGacceleration algorithm to 0-1 A is the existence of a symmetric positive definite matrix Zsuch that ZQ-1A is symmetric positive definite. In the case of the Jacobi conjugate-gradientalgorithm, Q-I = D-1

, where D = diagonal(A). A natural choice for Z is D, but anotherpossibility is Z = A (Young et al. 116]). The resulting conjugate-gradient algorithm is then:

Let u(O) be any estimate of the solution; set

L. Demkowicz et aI., II-tYfJl' mesh -refinement strategy

Step 1. Set

81

(S(Il), Zo(n»)a =

n (S(Il-I), zs(n-I»'

A == (s(n), zs(n»)n (p(n), ZQ-1Ap(n»)'

u(n+l) == u(n) + AnP(n) .

if II == 0if II ~ 1 .

II ~ I.

II ~ 0,

Step 2. Compute a reasonable error measure,

if Ilerroril <Tol stop

Otherwise, set 11 == n + 1 and go to Step 1.

In our implementation, the iterations at each refinement step were restarted with index 0,but with u(O) equal to the solution of the previous grid. Our experimcnts indicated that thenumber of ilcrations required for convergencc were morc or Icss constant for each step.independent of the number of equations.

6.2. The model Poisson problem

Two elliptic problems with Known solutions were implemented in the program:(1) Model Problem I (MPI), with exact solution

u(x, y) == rp(x, Xo, Ex)rp(y, Yo, Ey),

rp(x, Xo, E) == 1 _I exp( I 2 ) + Ax + B,exp(E) (x - Xo) + E

where A, B are such that rp(O) == 0 == cp(I). In the examples: Xo == 0.55, Ex == 0.02, Yo == 0.5,Ey == 0.05.

(2) Model Problem 2 (MP2), with exact solution

II (x, y) == t//(X, Xn, Cx )l/J( y, Yn. Cy), Xo .=::; O. C < 0 .

l/J(x, xO, C) == (x + xol' + Ax + B,

A B are such that 1{1(0, XO. C) == 1{1(1. Xn, C) == n. In the examples: Xo == -0.03. C == -0.25.Yo == -0.03, Cy == -0.25.

82 L. Demkowicz et at.. h-type mesh-refinement strategy

Figs. 5 and 6 show meshes which were obtained using our refinement strategy. Shadingindicates elements with largest error estimate at the 18th stcp using different error indicators.In these calculations, a maximum number of elemcnls that can be rcfined pcr step is alsoimposed. If this maximum is chosen smaller. i.e. if less elements are refined per step. the meshwill be closer to the 'optimal'. but the computation time will, of course, bc much larger.Figurc 5 contains the adaptivc grid corresponding to thc second model problem with boundarylayer singularity. A maximum of 4 elements could be refincd per step. Fig. 6 contains thcadaptive grid corresponding to the first model problem. A maximum of 16 clements wererefincd per step.

6.2.1. Verification of the extraction form u/asThe extraction formulas give very sharp approximations (3 digits of accuracy) of thc second

derivatives at interior points in the domain. At points close to the boundary, the second-orderderivative estimates were polluted due to the fact that the Green's formula does not holdexactly for if; and Uh'

From thc formula for the second derivatives,

a2

11 a2

11 Jf Jf----:2 - ~ = (q; + if;)f dx dy + i.1if;1I dx dy ,ax ay n ()

I I I I I I_.ml m-Jr I I II

Fig. 5. Adaptive grid for Model Problem 2.

we conclude that:

L. Dl'mkowicz et al .. IHype mesh -rl'fittl'melJl strategy

, ,I

I I

I l-

I JI T

T

III iI r I I I

Fig. 6. Adaptive grid for Model Problem 1.

83

Whcn the point (xu, Yo) at which we calculate the second derivatives approaches the boundary,the values of 1I.1r,iiIIL2 increase, resulting in a poorer approximation.

Fig. 7 shows a very close agreement between the exact valucs of the H2-seminorm andthose approximated by our extraction formula for MPl. This figure shows the evolution of thelogarithm of two norms with respect to the logarithm of the square root of the number ofnodes. Indeed In(YNN) (NN is the number of nodes) is roughly proportional to (n(1/h)(It = element diameter). The full line in the figure corresponds to

1/2

(~ltillil~'K )

and the dashed line corresponds to

where

84 L Demkowicz et at.. II -type mt'sh -refilll'melll strategy

/ .. I error I 5.

0.5

0.1

25 100 500

1/2 Ln(NN) NN: number of nodes

Fig. 7. Close agrcemcnt between lul2 and IUhl~"J(MPI).

6.2.2. Verification of the expected rate of convergenceSince our model problems are regular, the asymptotic rate of convergence of the adaptive

mesh-refinemenl program should be the same as the rate of convergence of a uniform meshrefinement. Figs. 8 and 9 show the computed rates of convergence from the model problem inthe cases of adaptive mesh refinement and uniform mesh refinement. Both meshes wererefined up to 1000 elements.

Fig. 8 shows the comparison of different errors and error estimates applied to theapproximate solution of the first model problem. The different lines correspond to:

1f2

(1): (~h1<IU'~'K ) uniform refinement (exact),

1/2

(2): (~h1<luliK) adaptive refinement (exact),

1/2

(3): (~h1«luli.lkf) adaptive refinement,

1/2

(4): (L h1<C1ul~~'kf) adaptive refinement,K

L Demkow;cz et al .. IJ-type mesh-refinement strategy 85

5.0

N nodes100025 100

17

10

III I error I

100

\\\\\\...J

'\

'\ \\

\ ',\

6' J" \@-/ , \

\ \, \" \\ \"", \'\,,\

'\\...

50025

0.1

0.5

In I error I

Fig. 8. Diffcrcnl error cstimates for MPI. Fig. 9. Different error estimates for MP2.

1/2

(5): (~hkOllli~lk)2) adaptive refinement.

1/2

(6): (~(IIi-uhltK) adaptive refinement.

Observe that the asymptotic rates of convergence of all estimates are equal. except for (5).However, the refined mesh based on estimate (5) was equivalent to the mesh based on the trueinterpolation error.

Fig. 9 shows a similar comparison for the second elliptic model problem. The sameconclusions IlOid. Estimate 3 was not calculated in this example.

6.3. The plane elasticity problem

Although this problem does not fit wilhin our theoretical framework, the adaptive mesh-refinement program is still expected to work. We considered a symmetrically cracked-plateproblem (Fig. 10). As the problem is symmetric, we modelled only one quarter of the domainthat was discretized. Fig. II shows a refined grid obtained for the cracked-plate problem. Amaximum of 4 elements were refined per step.

L Demkowicz et aI., h-type mesh-refinement strategy

, rn

lO.5m O.5m

Fig. 10. Symmctrically cracked-plate problcm.

The adaptive scheme refined the mesh heavily in the crack region, which actually increased:he computed error estimate. Therefore, during the first refinement steps, the adaptive:Jrogram does an apparently poorer job than simple uniform refinement. Also, as the true;olution is not in H 2(n), the asymptotic rate of convergence is not necessarily the same as forJniform refinement. The curves in Fig. 12 correspond to:

IIII

--

+ •+Fig. 11. Refined grid for the cracked-plate problem.

L. Demkowicz et aI., h -type mesh -refinement strategy 87

I" I error I

0.1

0.5 ,25 100 1000 N nodes

Fig. 12. Error estimalcs for thc crackcd-platc problem.

1/2

(I): (L: hk(11I12~llY) uniform refinement.K

112

(2): (L: hk(11I12~IAY) uniform refinement.K

1/2

(3): (~ h kClIl12~1d) adaptive refinement,

112

(4): (~hi(11I12~'Aj) adaptive refinement.

Note the difference in the asymptotic rates of convergence between the adaptive procedureand uniform refinement. The 'kinks' in line (3) correspond to a refinement at the crack tip.Due to the nature of the implementation of the Dirichlet boundary condition, the singularity isaccentuated after each refinement at the crack tip. We also conclude that Estimator 2 is moresensitive to singular behavior than Estimator 1.

Fig. 13 is a plot of the II and v displacements along the x-axis through Ihe crack tip. Notethe high resolution of the singular behavior around the crack tip.

L Demkowicz et al., h-type mesh-refinement strategy

1.830

:>'

~/, v-dlaplacement

x=O,O

. Concluding remarks

Fig. 13. u and v displacement at Ihe crack tip.

x=1,O

In this paper we have proposed an adaptive JHype finite element method based ono1terpolation error estimates. A crucial factor for using such estimates in an adaptive scheme ishe capability of computing precise approximations of second partial derivatives. We ac-omplish this by developing and implementing so-called extraction formulas and we demon-trate these ideas by treating some simple model problems. Also, we estimate the rate-of-onvergence of such schemes and, on the basis of numerical experiments, we comparedaptivity results with those obtained by 'ad-hoc' methods. Some of these crude methods yieldurprisingly close results to our more precise methods for selected example problems.

A number of issues deserve more study:(1) Numerical efficiency. The implementation of the extraction ideas in a numerically

,fficient way is an important issue. Our inclination at this point is to derive usable 'cut-off'unctions rather than 'blending' functions for such implementation. In extracting estimates ofecond derivatives at points close to the boundary, boundary terms should also be integrated1 order to avoid irregular behavior of the function cp.(2) The conseclIlive finite element resolutions required in our schemes suggest the use of

lUltigrid methods (e.g. Bank and Sherman (7]) which are usually based on isometric schemes.\n extensive study of such methods would seem to be worthwhile.

(3) Singular problems. Our scheme applied to the simple example of a cracked-panel,roblem in plane elasticity, although actually falling outside Ihc general framework of our

L Demkowicz et (//., II-type mesh -refillemelll strategy 89

theory, yielded surprisingly good results (compared with Babuska [6] or Rivara [14]).However, from the theoretical point of view. a different estimate should be derived as a basisfor possible mesh refinements. This remains to be done.

Finally, the extraction formulas are important themselves as a basis for certain post-processing techniques and should be generalized for other problems, such as general elliptic(not necessarily self-adjoint) boundary value problems with possible variable coefficients, andcertain nonlinear cases.

References

(1J I.Babuska, Feedback. adaptivity and a-posteriori estimates in tJnite elements: aims, theory and cxpericnce, Tech.Note BN-1022, Institute for Physical Science and Technology, University of Maryland, College Park, MD,June 1984.

[2J I. Babuska, and A Miller. The post-processing approach in the finite element method-Part 1: Calculation ofdisplacements, stresses and other higher derivatives of the displacemcnts, Internal. J. Numer. Meths. Engrg. 2020 (1984) 1085-1109.

[3J I. Babuska and A Miller, The post-processing approach in the finite element method-Part 2: The calculationof stress intensity factors, Internal. J. Numer. Meths. Engrg. 20 (1984) 1111-1129.

[4J I. Babuska and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundaryvalue problems. Tech. Note BN-1006, Institute for Physical Science and Technology, University of Maryland.College Park. MD. Octobcr 1983.

[5J I. Babuska. W. Gui and B. Szabo. Performance of the II, p and h-p vcrsions of the tJnitc element method,University of Maryland. College Park, MD, 1984.

(6) I. l3abuska, A Miller and M. Vogelius, Adaptive methods and error cstimation for elliptic problems ofstructural mechanics, Tech. Notc BN· 1009, Institute for Physical Science and Technology. University ofMaryland, Collcge Park, MD, June 1983.

[71 R.E. Bank and AH. Sherman, An adaptive, multi-Icvcl method for elliptic boundary value problems,Computing 26 (1981) 9 I-illS.

18) J.e. Cavendish and e.A. Hall. A new class of transitional blended finite elements for the analysis of solidstructurcs, Internal. J. Numer. Meths. Engrg. 20 (1984) 241-253.

(9) Ph.G. Ciarlet, Thc Finite Element Method for Elliptic Problems (North-Holland, New York, 1978).IIOJ L. Demkowicz. A. K,lrafiat and T, Liszka, On somc convergence results for FDM with irregular mesh,

Comput. Meths. Appl. Mech. Engrg. 42 (1984) 34>-355.[11] L. Demkowicz and J.T. Oden, On a moving mesh strategy based on an interpolation error estimate technique.

Internat. J. Engrg. Sci. (1985) (to appear).[12) L. Demkowicz and J.T. Oden, Extraction methods for second-derivatives in finite elcmcnt approximations of

linear elasticity problems, Comm. AppI. Mech. Engrg. 1 (1) (1985) (to appear).(13) AR. Diaz, N. Kikuchi and J.E. Taylor, A method of grid optimization for finite element methods, Comput.

Meths. App!. Mech. Engrg. 41 (1983) 29-45.[14) W.J. Gordon. Blending function methods for bivariate and multivariate interpolation and approximation.

SIAM J. Numer. Anal. 8 (1977) 158-177.[15J W.J. Gordon and e.A Hall, Construction of curviliner coordinate systems and applications to mesh

generation, Internat. J. Numer. Meths. Engrg. 7 (1973) 461-477.[16] M.e. Rivara. Adaptive finite element refinement and fully irregular and conforming triangulations, ARFEC,

Lisbon, June 1984.[17] E. Stein, D. Bischoff, G. Brand and L. Plank, Methods for convergence acceleration by uniform and adaptive

refining and coarsening of finite-element meshes, ARFEC, Lisbon, June 1984.118] D.M. Young, L.J. Hayes and K.e. Jea, Generalized conjugate gradient accelcration of iterative methods. Part

I: The symmetrizable case, Tech. Rep!. CNA-162, Ccnter for Numerical Analysis. TIle University of Texas atAustin, August 1980.

[19J O.e. Zienkiewicz, D.W. Kelly, J. Gago and \. Babuska. Hierarchical finite element approaches, errorestimates and adaptive refinement, Internat. J. Numer. Meths. Engrg. 19 (1983) 1593-1619.