Pardllel Compuutional Fluid Dynamics:Implementations and Results Using Parallel ComputersA. Beer. J. Periaux. N. Satofuka and S. Taylor (Editors)Ci 1995 Elsevier Science B. V. All rights reserved.

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Parallel Adaptive hp Finite ElementApproximations For Stokesian Flows: Adaptive

Strategies, Load Balancing and DOlllainDecoillposition Solvers

Abani Patra'and .J. T. Odent

Texas Institute For Computational and Applied MathematicsUniversity of Texas at Austin

A ustin.TX-78712

Abstract

This paper summarizes the development of a new class of algorithms using paralleladaptive hp finite elements for the analysis of Stokesian flows. Adaptive strategies, meshpartitioning algorithms and a domain decomposition solver for such problems are discussed.

1 Introduction

In this paper. we describe some new algorithms and early experiences with them, in combin-ing adaptive hp finite element methods with parallel computing for the analysis of Stokesianflows. Adaptive hTI finite element methods. in which grid size and local polynomial order ofapproximation are both independently adapted. are capable of delivering super-algebraic ande\,('n exponential rates of convergence. as seen in the work of Babuska. Oden. lJemkowicz andothers [1. 2). With para.llel computing. these methods have the potential to dramatically reduc('

~ computational costs associated with realistic finite element approximations.

-I' The development of several good (I posteriori estimators [3. 4] has removed one of the" principal difficulties in implementing hp adaptive methods. However mallY other difficulties

must be surmounted before such performance can be achieved for real simulations. Majordifficulties in doing so a.re: non-conventional adaptive strategies that produce good hp meshesmust be developed: the linear systems arising out of non-uniform hp meshes are difficult topartition into load halanced sub-domains and are very poorly conditioned for efficient paralleliterative solution. Tlw subsequent sections describe algorithms designed to overcome each ofthese difficulties. Section 2 introduces t he Stokes problem its' finite elemeut formulation and

•NSF Post-doctoral Fellow. TICAMICockrell Famil)' Regents Chair no. ? in Engineering

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the appropriate function spaces necessary for a dl'srription of the various algort.hms used in thesolution process. Section 3 describes a simple adaptive strategy for producing good hp meshes.We also discuss here a construction of compatible approximation spaces for the velocity andpressure spaces. Section 4 reviews a recursive load hased bisection type mesh partitioningstrategy. Section 5 discusses a domaiu decomposition type solver for such problems.

2 The Stokes Problem

The mixed finite dement approximation of the Stokes probleIll is classically given by:

Find Ill; E V", Ph E Hr"

such that

== -b(ii.ql;)

(1)

(2)

whereul; + It is the approximate vl'locity field and PI; is t he approximate pressure lielt! of anincompressihle viscolls fluid flowing through a given domain!! C lld with imposed velocity ii atthe boundary an of n. The domain n is partitioned into sub-domains !!; and finite elelll('ntsWI.;. It is assumed that sub-domain boundaries coillcidl' with element houndaries. The finiteelement spaces \/1; and H'l;o are conforming finite dimensional approximations of (HJ(I1))2 andL5(n).

\II; == {l'h E VI;P,vl;lwK E 1/~(wI.;),VK.vl; == (] on al1}

IFh == {'II. E L2(11).qhlwK E IVp(WK).VK}.

WhO == {'II, E Who fo qhdx == O}

\lp(wrd and H'p(w/() are tensor product polynomial spaces whose precise construction is de-scribed in the next section.

3 3 Step Adaptive Strategy

This type of adaptive strategy was first proposed in Oden and Patra [6]. We briefly reviewthe underlying ideas. Most conventional adaptive strategies propose an incremental type ofr('finement wherehy a certain heuristically det.ermined fraction of the mesh is refined/enriched ,..to the next level. While this strategy ultimately leads to a good mesh. it causes a large nnmber ~intermediate solution steps on non-optimal meshes. The cost of these intermediate solutionsmight negate all advantages of adaptivity. Clearly a goml adaptive strategy must deliver a meshfor a desired level of error in one or two iterations. Moreover. in the context of lip adaptivity.we also need a wa.y of choosing bel\Vl'en II and p adaptivity.

....

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3.1 Basic Pt'inciples and Strategy

We now describe the basic ideas underlying the adaptive strategy.

• Optimal meshes equidislribute erl'Or over the whole domain.

• Asympt.otic a priori error bounds arc treated as equalities e.g .

where u is the exact solut.ion. e is the error. II. and ]I are mesh parameters.

• Use of a good (/ postel'ior; error estimate to compute the const.ants A/,' (mesh parameterindependent) in the above II priori error estimate from a coa.rse mesh solution. Conver·gence rates II and II, if unknown, can be estimated from two coarse mesh solutions.

• ;"[esh parameters II and p, required for a desired error. are estimated locally from the apriori estimate and the need to equidistribl/le till' error.

• Orthogonality of error to finite element space leads to a good a.pproximation of the normof the exact solution norm i.e. Ill/lli.n;:::: IIuhplIi.n + llellr.n

The adaptive strategy comprises of :3 steps: 1) selecting an intermediate error level be-tween the initial lIlesh error and the final t.arget mesh. and estimating different parametersusing a coarse mesh solution, 2) keeping polynomial orders Ph" constant adapting the grid size(change hg) to achieve the int.ermediate error while equidistributing the.error.:I) keeping gridsize constant and changing the local polynomial orders Ph' to achieve the target. error whileequidistributing the error.

3.2 Compatible approximations of velocity and pressure

vVewill now specify the exact polynomial spaces IIp(WK) and Wp(WK) that avoirl the commonlyknown phenomena of "mesh locking" while preserving approximation properties in all adaptivelip mesh. In mathematical terms. these spaces satisfy the requirement that the LBB constantis bounded away from zero. Our ideas here are motivated by the work of Stenberg and Suri [5]on the p version.

Let Ui(X) = f~, Li(t)dt. where L;{x) is the Legendre Jlolynomial of degree i and let Pp(S)denote a polynomial of degree totaling ]I defined on the unit square S = [-1.1]2. Now theapproximation over each clement W]{ may he defined hy the sum of int.ernal fUllctions Jp(S)anti external functions Ep( 8) defined as

./p(S') = {lIll1 = Lf:i~l llijUi(X)Uj(y).llij Ellp 2 2}

F.'p(S) = Pdix)Pp(i,,) U Pp(ix)Pdi,,)

50X

EEJ-------· -.· ., ., .· .· .· .· .

. .- - -- ";. . -.,- -. L. ";

t . .' •Figure I: Basic space filling curves.

where Ix and ly Me the unit intervals [-1. L. Thus in two dimensions /';p is made up of fOllrsets of polynomials associated with the fOllr edges of S. In an adaptive lip mesh these ilia.)' aIJhe different. Now let Pm = "IIUlX{PI. {l2,/J3. f!4}. Compatible spa.ces Fp(S) and "'p(S) CiUl hedefined as

Vp(S) =

where it, i2 etc. denote the sides of S The standard finite element mapping process cau heused to obtain Vp(wJ<) and Wp(wJ()'

4 RLBBO - Mesh Partitioning Algorithnl

An essential part of applying parallel computation to these problems is thc partitioniug ofthe mcsh into load balanced pieces with rninimal interfaces. Partitioning adaptive hp meshes,however. poses special difficulties since i)the load distribution is irregular and localized ii) agood choice of an a priori measure of computational load is difficult.

We use a. recursive bisection of an ordering of the elements created using a space lillingcnrve (see Fig. 1 for an illustratiou) passing through the element centroids. The curve ishisected using a composite load measure cODlprisin~ of the load estimates on each element ina partition and the engendered interface. As a load measure we use the degrccs of freedomin each element, the error in a coarser mesh and the degrees of freedom on iuterfaces. Suchalgorithms are discussed in Patra and Odell 17].

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5 Domain Decomposition Solver

The solver described here is an extension of the domain decomposition solver proposed foradaptive hp methods for elliptic problems in Oden. Patra and Feng (8). The primary concernswith this type of solver are: I) is the preconditioner good enough to guarantee convergencewith increasing p. and 2) is the solver efficient for paraUel computing. The solution process forStokesian flows poses two additional difficulties: 1) obtaining solution in divergence fITe space.2) the solution of indefinite lincar systems. The basic solution procedure for elliptic problemscovered by the following scheme:

Apply partial orthogonalization first at the element level to eliminate, the interior functions andthen at the sub-domain level to obtain reduced system on the interface ,W, = F . where S = L:vv.':j,.and S,. = L M;r 1\·..."1,.and F = L Fi = L MT Ii

• RO = F, pO = 0

• Iteration in n

- Precondition Gn = C-I Rn- Compute direction of descent

pll = Gil + < {{".G" > pn-l< Un-I. (;"-1 >

Compute

NoZ" = ~ S' p., (I. _ < R", en >L...J I I II - -

i:l < Zrl1

• end loop on n

• Add correction term to interior unknowns u,. = u? + M;r fi

5.1 Solution in divergence free space

It is clear fron the above algorithm that to construct divergence free velocities by the aboveprocedure, one must ensure that each of the search direction P" is a divergence free vector.This is aCCOml)lished by modifying the preconditioning step C Gn = J?" to

CG"+ jj'/"p = 9fiG" = 0

where

SIO

and J! is a vector of average preSSllr" per sub·domaill. This computatioll reduces to one coarsesoh'eper iteration of a problem of dilJlension equal to the number of suh-domains. and the initialcost ofsetling up and factoring Be-1BT.

5.2 Solution of Indefinite Systems

If the pressures are discontinuous across inter-element houndaries: then they illay be eliminatedat the element level using one /Ilor!' step of partial orthogonalization as shown below. COllsiderthe elellH'nt matrix

Apply partial orthogonalization

Bel'

.s-O

Following up with one more level of partial orthogonalization

Element average pressures Pe call he recovered hy postprocessing the original equationswith the velocities. Thus the final pressure i, formed as P = Pi + Pe + P.

5.3 Choice of Preconditioner

As described in Odell. Patra. and Feng [8] matrix S is naturally blocked into a small portion(IV IV) corresponding to the linear on the interface and the larger Jlortion corresponding to theunknowns associated with the higher order polynomials( E E) and their interactions N E andEN. As a precollditioner C, we explore two choices. denoted IIPP] alld HPP2. respectively:1) the IV IV block and the diagonals of EE 2) the N N block and the hlock diagonals of EEcorresponding to a particular edge. For these choices of preconditioller. Letallec and Patra[9] have established firm theoretical hounds on the wnditioning of till' syst.em. gurant.ceingconvergence in a reasonable number of iterations.

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6 Numerical Results

Example 1. The well known problem of driveu cavity flow is used as the first numerical example.The domain. boundary conditions and mesh are shown in Fig. 2. Figure :1 shows performanceof the domain decomposition solver with respect to increasing polynomial order and problemsize. Convergence with respect to both appears to be robust and correspond to the theon'ticalhounds. Page limitations prevent inclusion of further results hNe.

u=O

v=o

y~

<>AIv=o ]

1 Iu::u, \-'=0 --~ ~ ,,-

1.0

Figure 2. a)Driven cavity flow - boundary conditions and domain b )Sample adaptive /'p mesh...._ ...~ ......,--.

i! ~,

.,.. J H • C U "* lOIJOll

Figure ;l.a)Convl'rg;;;;;-~·7r iterative solver with HP I'1 prec~',~~iitioner for different elementorders. Mesh of li-t elements with ll/h=4. Numbers in parentheses are the total number ofunknowns. h)Scalability of iterative solver with HPPI preconditioner for diff<'l"l'nt elementorders. Effect of problem size keeping number of elements per processor constant for differentelement orders.

Acknow ledgements

The sUJlport of this work by All PA under contmct no. VA DT63-92-C-OO.f2 is gratefullyacknowledged.

References

[1] 1. /labuska and M. Suri. "Thl' p and h-p versions of thl' finite element IIlI'thod. BasicPrinciples and Properties'. SIAM Rwi(w. Vol. :W. NumbPr'1. Decl'mber 19!1·1.

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[2] J. T. (>den and L. Demkowicz ... h-p adaptive linte element methods in computationalfluid dynamics". Computer ldethods in Applied Mechanics and Engineering. Vol. 89. 1991.

[3] ~1. Ainsworth. J. T. Oden. "A Unified Approach to a-posteriori Error Estimation UsingElemcnt Residual Methods" .Numerische Malhemillik, vol. 65 (1993) 1)1).23·50.

[4] H. I':. Bank. R. K. Smith. "II posteriori Error estimates based on hieran:hical ba.ses".SIAM Journal on Numerical Analysis. vol ;~ono. '1. pp. 92]·935.

[5) H. Stenberg and M. Suri. "1\lixed hp finite elellwnt methods for problems in elasticity andStokes /low". preprint. February 1994.

[6] J.T. Odell and Abani Patra. "A Parallel Adaptive Strategy For hp Finite Elements",inCamp. Meth. in App. Mech. and Engg., vol 121. March 1995. pp. 449-HO.

[71 Abani Patra and J. T. Oden "Problem Decomposition Strategies for Adaptive hp FiniteE1Plllent Methods'. to a.ppear in Computing Systems in Engineering.

[8) J. T. Oden. Ahani PatrOl. Y. S. I-eng ... ··Domain D!'composition for Adaptivc lip Finite El-ements", in J. Xu and D. Keyes ed. Procedings of VII th International Conferenceon Domain Decomposition Methods. State College. Pennsylvania.

[9] P. Lctallcc and A. K. Patra. "Non-overlapping Domain decomposition Methods for StokesProblems with Discontinuous pressure fields". TICA M Report (in press).