,To appear: Proceedings, 12th CanadianCongress of Applied Hechanics, Hay 28-JunE! 2,- 1989

SMAllT ALGOIllTlIMS ANIl ADAPTIVE !\lETIIODS INCOMPUTATIONAL FLUID DYNAMICS

J. TINSI.EY ODEN

Tua., Institute for Computational Mechanic3The Uniltersity "r l'uas al Au31in

1 Introduction

The original title of this lecture. conwele.l some

months berore the manllscript was dlle, IIsed

the words " ... Artificial Illtelligence ... " instead

of "Smart Algorithms ... " After considera!>le reo

l1ection, I have decided to omit reference to "ar·

tificial intelligence" in the tille of this work. even

thollgh some wOllld argile that the lerms are

cerlaill'y appropriate in tl,e preselli. IIsa~e. III

partir.ular, I am lold hy experls ill the area, a

,Iistindioll for which I .10 1101. qualify, Ihat A I, in

its most primitive intNprelal.ion, merely refers

to the lise of 1\ computational devirr 10 perform

some fllnclion ordillarily requiring hlln1<11Iintel·ligence. III that .ensp., virtll;llIy all compulation

cOllld he rp.garded as a crllde form of AI, Dllt

others tell me that the terms nowadays slIggest

weak or nonexistent algorithmic conlelll. What

I have in mind is somewhat deeper than the

coding and implementation of a nllmerical al·

gorithm; it is the nse of a rat.ionally· based set

of crileria for automatic decision making in an

allemptto produce optima.l simulations of rom-

plex 1111iddYllamics phellomena. The informa·

tion needed to makl' these decisions is not kno\\'n

heforehand and evoh'es in structure and form

during the nnmerical solutioll of t he no\\' pro!>-

lem. Onc~ the code makl's a decision based on

the data at hand, much of lhe slrucillrl' of the

data may change, and criteria IIIlIst he lIpplied

anew to redirect the analysis toward an accl'pt-

able end. Thus, intelligent decisions arl' made

by processing vast amounts of data that evohe

in an unpredictable way .Iuring the calcnlation.

This is one kind of AI employell in modem

adaptive computational methods. Rl'ference to

"smart algorithms" as opposed to AI acknowl-

edges that mllny aspects of adaptive compllting

are "algorithmic" to a strong d('gree. There are

others that are not, and these non-algorithmic

components may prove to be l'ssential in even·

tllally !>ringing much of modem computational

fluid dynamics (eFD) into use in engineering

Ilcsign.

Th~ foclls here is aflaptive methods in CFD.

I look at tl1l'se lIIelhods M techniques designed

to tackle the most hasic isslles in computational

mechanics: how good arc the answers and what

can he done to improve them? The lint issue

can be treated algorithmically or not ,Iepending

on how one measures the l)uality of a solution, Ifthe mathelllaticnllllodcl is well established, the

issne reduces to one of accuracy, and Oll~ hopes

to estimate accuracy lhrough the use of rigor-

OilS aposteriori error esti mates. If lhe model is

not firmly seL an "expert" must view the results

and dl'terllline if they agree with his experience

as to how the s}'stl'm should hehave - ciSI' the

model itself must he changed - i.e., it must bl'

adapted. If one a.ssumes for present purposes

that Ihe model is adequate for the simulations

of int.erest, then one must proceed in the deter-

minatioll of error estimates and to modify the

strncture of the approximation to improve the

result. These ideas form the heart of reselHch

in modem adaptive methods in eFlJ and will

he ,Iisr.ussed in more detail in the pages which

follow.

The works and results I summarize in the

pages which fol\ow arc drawn from collaborative

work completed or in progrl'ss with several col·

leagnes. In partir.ular, I acknowledge significant

contributions by Leszek Demkowicz, Jon Dass,

Waldek Rachowicz, Theofanis SI.rouhoulis, and

Philippe Devloo. Also, Roger Chen helped in

using our AlJAPT™ code 10 produce exam·

pies on viscolls flow around a cylinder. C. Y.Huang hM also worked wil.h this tMm, hut his

work for us on three-dimensional Euler solvers

is not yet complete and lIlust await discussion

ill some lat.er commullication. Some of the ex·

amples presellted in the last sectioll are excerpts

from my survey on "Progress in Adaptive Meth·

ods in eFD," to be published by SIAM Publi·

cal ions.

The plan here is to review the basic compo-

nents of adaplive methods and I.heir applir.a·

tiou to very complex prohlems in fluid dYllam·

ics. These arc:

I. Dat.a St rnctures - how can one cbange the

structure of an approximat.ion 10 reduce N-

ror?

2. Error Estimation - what. techniques exist

for estimating till' "volution of error in IIeFD calculation?

3. Solvl'rs - what Illgoril.\lIns are avail"ble

that can function IlII changing meshes?

4. Examples - what numerical results are

available to dernollst.rate th" viability of

these approaches?

SCHne of the discussioll and results pwsented

here are also extracted from earlier papers and

reports (sec, e.g., [1-1 iI).

2 Data Struct \Ires for AdaptiveMethods

Assume that. the sit.ualion is lhis: we have com·

pleled a rough calclllatioll of the flow fields in

a numerical solution of the Navier-Stokes equa·

tions on some coarse inilial mesh. We perform

operations on lhis initial solution so as to obtain

an estimate of the computational error. Genl'r'

ally, if the illit.ial calculation is done on a fillite

difference or a I1nite element mesh, we will com-

pute local ~fTor indicators 0., e = 1,2,···. E(A)for each cell n. in a partition A of the flow do-

maill n C liN, N = 1.2 or 3. The llllllllll'rs

O. indicate the computational error eh over n.in some suitable norm. The global error is th ..

tolal error in I.he computalioll and is approxi·mated by II global indicator such M

We will briefly discuss methods for computing

O. in Seclioll 5.

Now we must decide what to do in order to

reduce the error in the fastest and most effective

way. There are some natural approaches that

sIIggest thelllsdves:

I. h-method.. (also including embedding meth-

ods). lIere one simply refines (divides into

smaller pieces) the mesh sizes h where the

error is "large" (or, equivalently, one em-

beds a 11ner mesh over or wit hin the coarse

mesh at places where the error is too largp).

2. r-mell.od.s (or node-redi...tribution method..or moving node methods). lIere nodal

points are relocated so that. their density is

greater around regions of high error, keep.

ing the number of grid points (and un·

knowns) constant.

3. p.metho~ (or spectral methods). In this

case, lhe number of grid r.ells and the num·

ber of grid points is held COllslant while

the local order of the approximation is in-

creMed - such as Ihe local spectral liniN

of a polynomial approximatilln of flow v~ri·

abies or the local degree of the polynomialshape lunctions in a finite element environ·

ment.

,I. COllIbille<llllelhod.. (snch as !..r, r.p, or "~opmethods). lIere one simnltaneously moves

nodes anti refines the mesh or sinllllt.ane-

ollsly increases p anti refines the mesh, etc.

The degree of difficnlty in implementing t.hese

various possibilities varies, and the success of

any adaptive strategy strongly turns on howef·

ficiently these ideas are carried out. We shall

hriefly outlille a. few adaptive strategies that we

have used to treat certain now problems.

An h·RefinementfUllrefinement Method. One

h-procednre involves the following steps:

I. For a given domaiu n. such as that shown

in Fig. I, a coarse finite element mesh is

constructed which contains only a nUlllhl'r

of elements suflicient to model !>asic geo·

metrical features of the 11011' domain.

2. As our adaptive process will he designed to

handle groups of four elements at a. time(fOT the two-dimensional CMI'). we may

generatl' a finer starling grid h>' a bisectionprocess, indicated iu Fig. la, t.o obtain an

iuitial set of element grollps.

3. We initiate t.he nnmerical solution proc ..·

r

c.

<JUNREANE

(b

Fignre 1: (a) A coarse init.ial mesh consistingof four-element groups and ( b) refinement andunrefillcmcnt of 1\ four-·clenll'nt group.

dutcs on this initial cnar6c grid. and com.

pute crror indicators 0. oveJ' all At elemcntsin the grid. Let

4. Next, we scan groups of a fixed nllml>er Pof e\cmcnls allli COlli pute

p

O~Jf\our = I:Oc.k=1

with eight I>rick elements constituting a group.

One possihle adaptive scheme for time-dependent problems is:

1. Advance the solution N time steps !it using

an appropriate time-marching scheme.

2. Calculale error eslimates.

3. Refine the mesh.

4. Hedo the N time-step calculations using

the new refined mesh.

5. Redo the error est.imation.

G. Unrefine the mesh.

7. Go to 1.

There are several rather ohvious alternative ver-

sions of this algorithm, but this is the approach

used in the sample calculations presented laterin this paper.

As all example of our mesh refillcment strat-

egy, consider the uniform grid of four elements

shown in Fig. 2a a/lll suppose that the error

estimators dictat.e that element II is to be re·

fincd. Thus, II is divided into four elemcnts, I,

2, :1, .1. as shown, alHlthe solution valnes at the

junction nodcs, shown circled in the figurc. arc

constrained to coincide with thc averaged values

between those marked X.

Next, assnme that an additional refinement is

required. and that we must next refine element

wherc Ck is the e\emenl for group k. We

take P = 4 in our current. code.

5. Error t.olerances are defined by two real

numbers, 0 < 0, /3 < ]. If

O. ~ /30MAX

A

c

c'

B

D

• 381 B

Z 1

CI

C D

(b

we rcfine element Oc. This is done I>y bi-

sect.illg O. into four nclV subelements. If

O~nour ~ OOMAX

• 3 B 7

I 2 5 6

±0[1

Fignre 2: Seqnenc!' of refinements of a nniformmesh.

we unrefine the group k by replacing this

group with a single new elemenl with nodes

coincident with the corner nodes of thegroup.

This general process can be followecl for any

choice or an error indinltor. Moreover, it can

also be implemented at each time step. Three-

dimensional generalizations are straightforward

(e (d

x - AcnVE NODE

o . CONSTRAINED NODE

, I

where

Xi = polynomial of degree $ I' in ~ E [-I, II

Lagrange .hope lunClions:

llie'8lchicol shape fuoclions X(~):

Fignre ·1: Concept of hierarchical shape func·tions.

lIeB Ui, i = 1,2,3, ,t, the tangential derivatives

akll/{Jrk, k = 1,2, ... ./, at nodes 5,6,7,8 and

mixed derivat.ives {J'"ll/D{'a,{, l+r = 01 =1,2, .. "p at node 9.

To fix ideas, consider first the 1-1) ca.~e. In the

c1l15sical FEM (e.g., Lagrange interpolation),

shape functions for various order of approxi.

mation are constructed independently. For ex-

ample, passing from a linear element with two

linear shape functions to II qnadratic element,

we construct the lI,ree qnadratic shape func-

tions independently of the shape functions for

the linl'ar element. An alternative way to con·

struct the same second order approximation i.

to r.omplete the set of two linear shape functions

by including a third, quadratic shape function.

At the moment the definition of lhis third shapefunction and a rorrespondillg degree of freedom

is somewhat arbit.rary. the only restriction be·

ing that the set of shape functions must form a

dual basis to til" set. of degrees of freedom, i.e.,

Jbhn- .;SndIIIII~ ~ Alf.'Ei> - --=="

.. =-- A_ --A-

~......~

37~

4

These polynomials have hierarchical st ructure

in the ~ense indicated in Fig. ·1, which ('nsmes

the property that the element matrices corre·

sponding to an approximation of degree p con-

tain as proper submalrices all of those element

matrices corresponding to approximations of de-

gree less than p. For the 2D elem('nt shown in

Fig. 3, the degrees of freedom lire t.he nodal val·

'Pij(~, II) = LXi(OXj(fj)i,i

3. We impose the restriction that each element

~ide clln have no lIIore than lwo elemcnts con-

nected to it. Thus, before :l can be refined. ele·

ment 8 mllst he refine,l. The constrained Node

81 in Fig. 21> now I>ecomes aclive, while node

Cl remains a constrained node. With B hi·

sectt'd, we proceed to refine 3 into subelements

0, (J, -y, 6 and new constrained nodI'S, again cir·

cled in Fig. 2 .. , arc prodllccd. In this case, only

clement B had t.o he refined first in order to reo

fine 3, but, in general. the number of elements

that must be refined in order t.o refin" a partic·

ular clement cannot be slwcified.

A p.mdlaod. The idea of increasing the ofller

of an approximation while keeping mesh sizes

fixed is II natural one in the elISe of problems

with thin boundar)' layers o~ singularities. In

results to be outlined later, we employ a hier·

archical p-version of the finite element method.

For two-dinwnsiollal problems, the type or ele·

ment shown in Fig. 3 is used. The idea is to

choose element shape functions of the form

Figure 3: Degrees or rreedom for a spectralelcmCflI wilh hiernrchical shape functions

8~

-\- 9

5

t

2

6

'Pi(Xj) = 0 i,j = 0, I.2

where 'Pi, i = 0, 1,2 denote the degrees of free·

dam and Xj, j = 0, 1,2 the corresponding shape

functions. Since the two degrees of freedom

associated with the linear shape functions are

function values at. the endpoinb, this implies

that lhe added qua.lratic shape function must

vanish at both endpoints.

Proceeding in t.hi. manner, we define the se-

quence of so-called hierarchical sllape functions

of increasing order of Fig. 4.

An r-mctllOd. As we will show later, if the

· .

mesh size h and the polynomial degree pis fixf'd,

one can show thalthe optima.! mesh is that for

whir.h the noell'S are positioned so that the error

is e'luidistributed over the mesh; i.e., the error

over each element is lhe slime. Di~z, Kikuchi,

and Taylor (18) have nsed this fact to produce II

simple algorithm for r'lIdaptivity:

1. For fixed hand p on a mesh of quadrilateral

elements, compllte error estimators 0, for

all eleml'nts in a mesh Hh C R2.

2. For each 4-elenH'ut group (reclIlI Fig. I)

calculate the error per uuit area O,/A, nf

rach element in the group.

3. Computr the Ucentrnid of error"

4 O.LTj X+"3;=1 IIJ

11k = 4 ' k = group no.

L~j=1 A;

where r; is the p"sitioll vector of the cen·

troid of element j in group k.

4. Ilelocale the central nod.· in the gronp at

!Jh so as to (approximately) e'luilli3trihnh'

error over the 'I-element group.

5. Continue t his process over all groups uutil

the error is e'luidistributed over the entin'

mesh.

This simple procedure is easy to implemellt

and i. effective in many classes of prohlems.

There are, of course, more sophisticated mov·

ing mesh methods. The most popular are thosl'

introduced by Miller (19, 20]. There the FEM

approximations is defined in trrms of nndal co·

ordinates which themselves arc functions of po·

sition lind time and the actuallocalion of nodes

is determined (for thl' I'rohlem III = L( u)) so as

to minimize the L2.residual.

/l = lIu/- f,(u)lIhll)

(See (19, 20, 21) for more details on these meth·

ods.)An h.p (Ia.spectral) Method for Un..trucillred

Meshes. If a mesh is rrgular (meaning here that

node points a.nd degrees of freedom are appro·

priately matched) ami aojacellt elements are of

the same order, then (in t.he ca~e of conform·

ing finite elements) the resulting ~pproximatiollacross the interelement boundary is easily made

to be r.ontinuous. This situation changl's if we

have to match elements of different order, even

on a regular grid.

A typical situation is shown in Fig. 5. Dif·

ferent orders of approximalion in two aojacent

elements produce an undesired discontinuous

approximlltion along the interelement bonno·

ary. Enforcemelll of the continuity condition

has led initially to applications of Lagrange mul·

tiplier technique or penalty methods; however,

the most altractive and practical solution to the

prohlem has beeu offered by meaJls of the so-

called hierarchical shape functions.

Boolean Conslraint MaLt-ices. The key to

thp. h·p scheme is the so-calleo conslrailled-

I\pproximation: the imposition of constraints on

discoJltinllous shape functions of dillerent poly.

nomial degree so that continuity across element

interfaces is maintained. The issne is cOlJlpli.

cale,1 by the fact that elements of different size

may share a boundary, such as is the case in the

mesh in Fig. 2. The prohlem is solved ml\lh·

ematically by thl' construction of Boolean con-

straint matrices 122). The basic slel)s are as fol·lows:

I. A space Xhp of discontinuous shape fllnc-

tion6 is defined 011 all irrglliar mesh, ~lIch

as that shown in Figs, 2 or Ii. DilTerent

polynomial degrees p. lIlay reside in differ·

cnt elcments !l•.

2. The .hape functions {xi} for .'ach element

e are lensor prodllcts of hierarchical poly.

nomials as in the I" method discussed ear·

lier. At this slagI', thc degrees of free-

dom arc nodal values II; (i = 1,2,:1,4) at

corners of a quadrilateral element, tangen·

tial derivativcs ErU/aT', ~ = 2,3, ... ,p a.t

Figurc 5: Example of IInconstrainrd, discontin·1I0llS approximatioll on all irregular'mesh.

I •

the midpoints of each side (or aninI' maps

of snch derivatives at midsides of a mas.

tel' square clement) and mixed derivatives

D"u/D{iD,f, i + j = 2,3, ... , p, at the cen.troid.

3. A subspace Xhp C Xhp is sought in which

all global basis functions are continuous.

This space is constructed by developing

constraint matrices n which relate degrees

of freedom to he left active with those that

are to be linearly constrained. Thus. if

{.pj}f=1 = I>asis of Xhp

we seek n such that.

(a)

(b)

Here s(k) is the set of labels of constrained

degrees of freedom associated wilh (de-

pending on) the active degree of freedom

lal>eled A,. If, for example, k is a stiffness

matrix for an element computed using dis.

continuous ba.~is functions. the constrained

matrix actually used in the h-p calculationis

where il is the Doolean matrix redncillg kto its active degrees of freedom.

4. A criterion for l' selection must be chosen

to determine n. One which we USI' is this:

if any clement with shape functions of poly.

nomial degree p has a neighl>or with poly-

nomials of degree q < p, additional shape

functions of degree ~ pare ad,led 10 the

neighbor sullicient to allow the global ap.

proximations to be cont.inuous across the

int.erface. This decision makes it possibleto uniquely determine n.

5. 11 is sufficient to consider a "master" ele-

ment K = [-1,1] X [-1,0] with boundary

DR = 1-1,IJ inlerfacing t~o smaller cle-

ments tJKidl = [-1.01 and tJK,i/IJlt = [0.1].By merely demanding that the approxima.tion Von K of degree p match exactly those

of Kidl of degree Pi on aKidl and those of

Krighl on 8K,ight of degree p" explicit for.mulae for Rij are obtainable if the "max.

imum rule" is enforced: only the highest

degree polynomial survives on oK. Thus,

for the situation shown in Fig. 5, if thelarge element has shape functions of degree

(say) 5, the small element along AS has

those of clegree 3, and the small element

with side BC has polynomial shape func.

tions of degree 4, lhe p = 5 dominates, and

the smaller elements are enriched by the ad-

dition of shape functions of degree 5 struc.

tured by the choice of n to provide conti-

nuity along AC. On the interface shared hy

the small elements, however, P = 4 domi-

nates, so the bollom small clement is again

enriched by the acldition of functions of de-

gree " to achieve continuity along that in.terface as well.

Full details on these types of constraints aregiven in our paper [22].

The Optimal h.p Mesh. One of lhe basic is-

sues that must be resolved in an h.p calculation

is to determine the best possible change in the

mesh structure t.o most efficiently and quickly

rednce local error. In other words, for a given

local error measllre, what should be done to im.

prove the solution, decrease h or increase 1', or

I>oth? While this "optimal path" question is

still open, we have employed one technique that.

is reasonably elfect.ive. 11 is based on lhe follow.ing analysis: Let

0, = error indicator density for clement ne Ina regular 2D linite element mesh.

h = piecewise constant mesh size function;

h(x, y) = h. = dia (ne) for (x,g) E lie

p = piecewise constant polynomial degree func·lions,

1'(X, Y) = 1'- = hif!:IIl'st pnl)'nomial degree ofshape fUlldiolls supported by element H,for (x,y) Ene.

The total error is given by the functional,

J(h,p) = L 1 O,(h,p)dn(' Of':

The total number of degrees of freedom active

in such an h·p approximatioll is roughly givenby the functional

· .

( p2N = in n(p,h)dn, n(p,!.) ~ 1,2

The optimization problem is to lind a particlliar

hop distribution, (h.,p.) slIch that J is a min-

imlllll snbject to the constraint the N = con·

stant:

A simple Lagrange.multiplier argument shows

that the optimal mesh ocr.nrs whenever

dO. = constdn.

that is, when the rate of change of error per de·

gree of freedom is eqllidistrihllted over lhe mesh.

We ha\'(~ developed a simple one-step opti.

mization algorithm to apply this crilerion to I D

and 2D elliptic problellls. Some results are gi\'en

later.

Figure 6: Example of a mesh produced hy anadaptive implicit-explicit solver a a given timeinstant; the time step is constant and stabilitycriteri .. determine if all implicit method (blackelement) or an explicit method (white element)is used.

5. Go to 2

for some appropriate lIorm on the discrete

solution tt").

1. The situation changes each time step, the

choice of implicil or explicit depends upon

the cllrrent element Cnurallt nnmber.

thesecalculal.ions,sample3. In

2. The choice of h is detl'rmine.1 from accuracyconsiderations and t:.t is hdd fixed. If h is

selected to mect error "ontrol and tot is not

fixed, purely explicit schemes may force t:.t10 almost zero to prescrve stal>ility in some

4. Corrector (For fixed tol and II. estimate if

the predictor step is stahle; if not, switch to

an im plicit solver to ensure stability with·

ont the necessity of reducing tot).

3. Test Stability (Check the local Con rant

(CFL) number vto determine stahility con-

straints on the choice of time step tot or

mesh size I" e.g., for lin h·method,

In this way, a spl'ckled mesh such as that

shown in Fig. II is ohtained in which implicit

methods are used on the shaded elements and

explicit methods are used elsewhere. Note the

following:

1. Initialize (Compnte initial conditions, ele-

ment mat.rices, boundary condit.ions)

Dllring a r.alclliation on an adaptive IIlesh, hoII'

can one be sllre lhatthe algorithm or 11011'wlver

selected to solve the discrete problem he still

appropriate for the condition prevailing at that

time in the hehavior of the solntioll? Beller

still, why should one p.xpecl a sillgle 11011' solver

to he aITer.live everywhere, at II gil'en time in-

stant, thronghont the cllrrenl mesh? Answers:

one cannot expect a single algorithm to remain

elTeclive and it may 1>1' neCl.'ssary to nse dilTNent

algorithms at. different portions of the mesh at

diITerl.'nt times.

One example of a family of adaptive algo-

rithms is the adaptive implicit.-explicit schemes

which use a mixt.ure of implicit allli explicit

algorithms lhroughont the mesh (51'1' Tezdn-

yar and Liou 123]). Thesl' methods are hased

on corrector-predictor schl'mes which contain a

"switch" enabling thl' 1151' of eit.her an explicit

or an explicit corrector depending on stahility

conditions prevailing at each cell.

For example, consider an h-adapt.ed mesh

such a.s that shown in Fig. 6, and consider thesmart predictor-corrector algorithm:

3 Adaptive and Automatic Al-gorithm Selection

2. Predictor (Use anl.'xplicit solver to advance

the soilltioll inti me for each elemellt in the

current mesh)

mixed schemes have proved 10 be suhslan.

tially more efficient thall a Pllrely illl plicit

scheme 123]. But using such methods in

an adaptive rEM strategy, suhstant.ial im-

provements in computing efficiency can beachieved.

which h·p strur.ture, algorithm, and model were

ultimately used to complete the analysis.

5 Aposteriori Error Estima-tion, General

It is clear that the use of the algorilhms outlined

above equip a CFD code with substantial "in.

telligence" for making critical der.isiolls during

the evolution of a 110w calculation:

1. The user supplies a very crude mesh - per.

haps only roughly defining t.he now domain

of inlerest, and he specifies tolerances hewill accept in the error (or in the cos I of

the calculation).

4 The IntelligentCode

Adaptive

We shall discuss here some general properties

of the evolution and distribution of error in fi·

nite element meshes and the use of so-called

interpolation crror eslimates. These easily im·

plementable local estimalors sometimes provide

only a rather crude estimate of the actual local

error but can be devised to give a correct indica·

tion of relative error between successive meshes

or approximation orders and, thus, correrlly di·

rect an adaptive strategy to syslematically reo

d uce local error.

Interpolation Errors. Let tl he a smooth

function defined over a regular domain t.wo-

dimensional fl. The Wr,P(fl) Sobolev semi-

norm of u is defined by

2. Thereafter, I.he computer code computes an

initial solntion, checks error alld slahility

lolerances, and dp-cides if an adaptat.ion is

nC'eded t.o improve the solution; if so:

3. The code chooses an optimal approxima'

tion structure - an algorithm u_ for most

crrectivl!ly reducing the error. The mcsh,

spectral order, and properties of the solu·

tion algorithm are changed accordingly and

a new solution is produced. The adapted

mesh also adapts itself more closely 10 fill

up the actual now domain.

4. Post-processing routines enhance the solu·

tion, prest'nt it to the user, plot est.imat.ed

errors, all wit.hout int.erference from the an·

alyst.

5. The simulaled physical behavior is tested

against, a data base underlying an expert

system package. If the behavior is not

viewed as physically reasonable, eilher the

mathematical lIIodel of t.he now is changed

or the general model is ret.ained hut paramo

eters are changed so as t.o produce now fea·

tures deemed more desirahle by t.he "ex·

perl."

'" [ai+iu]p }l/P~ ~ df!

i+i=' ax! ax~id~o

where I $ " $ 00 and r is a non-negative inte·ger. For the special case of p = 00, which is also

of interest, the W'OO(H) Sobolev semi-norm is

given hy

I ai+iu Imax ----;---""""rEO aXlax~i+i=r

i,i~O

With these definit.ions of the semi-norm, the

Sobolev norm of u is then

{

m }l/P1II1I1IVm,r(ll) = E lul\v"'(O)

Let G he an arbitrary convex subdomain (a

finite element.) of fl over which II is interpolatedby a function Un which contains complete piece-

wise polynomials of degree k, Then it can be

shown that the local interpolatioll cr1'Or in the

IVm,P( G)·semi-norm is

hk+l !l!l

lu - unlw""(G) $ C ..,m . h. - p lullV'+J"(G)

where

h = the diameter of the dOlllain GThe engineer need not know at I.his point how

the code chose to compllte the solut.ion in hand.

The intelligent engineer, however, would be ex·

pected to interrogate the solution to determine n

= the diameter of the largest sphere

that can he inscribed inside G

= the dimension of the dOTllain n

, .

I', q = posilive numbers, 1 ~ p, q ~ 00

1. II = 2, III = 0, k = 1. I' = 1, t.hen

If -y is proportional to It, and if it remains pro-

portional in rl'fin('ments of fI defined by para.

metrically redncing It, we have

with c independent of u, 1', or It.

Evolution Equalioll for Error. Consider the

continuity eqllation for the evolution of mass

density through a domain n with known 110w

velocity u. A weak form of the continuity equa·tion is

{ t/>p,dn = - { V· (up)t/>dH for all t/>E IVill ioA semidiscrete approximation of the above

equatioa consists of seekillg all approximate

density ph snch that, over sOllie suital>le finite-

dimensional space of test functions IY",

for r ~ O. For an hop scheme, we expect

I' = min(", p + 1)

= a constanl. independent of It, -y,and II

C

with '·lm.9.0 = 1·I\I'm.tlll)' etc., and Eh = u- til..

Such estimates can be used to devise adap-

tive schemes. Snppose that u on the right side

of the above eqnation is replaced hy a finile

element approximation Uh and that IUhIHI,p

= Iulktl,p + O(h). Then the equation indiocates thaI the local error in the Wm,r(G) semi-

norm is a proportional to the error indicator,

h"!q-"!p+HI-mlu!k+',p' Some choices for 71, III

k, p and q of int.eresl are:

In this case, one mllst approximate till'

11'1,1 send-norm of II over Gi i.e., the I}.

norm of second partial derivatives of II.

The error indicalor O. is then set equal to

IE"I"'IIl,) for finite "I.'ment H•.

2. II = 2, p = 00, q = I, !. = 0, III = 0 then

If;"I,.'(G) = CIt'IE~"'A8.1

~ Ch3

1ukoo,G

< Clt3 max IV. tt(r)1- rEG

Then we have for the error indicator O. over

an element G = fl.,

{ t/>hP~dn = - ( V.(tlp")t/>"dn for all t/>hE W"ill in(".1)

If the spar.e or test functions IF/. is such that

W" C IV, we ma.y choose t/>= ,p", suhtrar.t the

above eqllation and ohtain the fllllowillg condi·

tioll on the error e"(x, t) = p(x, I) - p"(x, I):

in t/>"c~rln = in -V'(IIch)t/>"rlll for all t/>" ElY"(5.2)

The exact and approxi.natr solutions are re-

lated according to p = p" + e" where eh is the

approximation error. Thus, the error satisfies

lhe evolution equation.

in(t/>e~+v'"cht/»rln =< r",t/> > for all t/> E W(5.3)

IE"I average over n, ~ o. = h. max IV'"(r)1rEO,

In all of these ca...es, it is also possible 10 estimate

the constant C. While we shall not describe how

this is done in the present study, our experience

is that it is a worthwhile cornput.a.tion that can

lead to schemes with good "erfectivity indir.es"

(i.e. ratios or exact error Ill''' II to approximate

error IIE"II close to unity).

p- flflli/I-l' fll'pmxillllliioll. For the p. or h·p

versions of th .. finite eh'ment method, differentinterpolation estimates hold heclIuse the mesh

parameter is no longer just the m ..sh size It but

is also the spectral order I' or both It and p. rn

these cases, if II E H""(H), t.hen one can show

(24) that for th .. p·method (11 = const.)

where < r",t/> > is the residual function,

<r",t/»=- in(p~t/>+V'"P"t/»d!l (5.4)

If we rrplace t/> by <Ph, < r",t/>" >= 0 hy (5.1)and the e\'olul ion equation reduces to merely

the orthogonalit.y condition (5.2), which auto-

matically sal.isfied hy tloe error.

An approximal.e evolution eqllation for the er·ror may be oht.ained as follows: ).<It E" denotp.

a fine-grid approximation or eh, i.... ,

I. " "" N (e (x,I)",E (x.t)=~E (t)I)IN(:r) 5.5)N

where 1)1 N(r) denotes a polynomial basis func-

tion defined on a subgrid of finer ml'sh size thanthat used to calculate p". Then, introduction of

..

where

1st Older EOE ocheme

interpola\ion method

2nd order EOE scheme IExact I

ELEMENT NO. X

= 3IG X 10-3

= II X 10-3

,N,.J

::~ -

II e lIexaeL

compared with the error computed using inter.

polation methods. Obviously, the interpolation

mel hod is completely inadequate in lhis exam.

pie. Indeed, the actual global L2.error and the

estimated global errors are:

Figure 7: Compnrisnn of Error Bstimatiou:EOE method vs. interpolation method.

II c IIEOElllelh.,,1 = 303 X 10-3 (using Ist·order 10·caltesl functions)

II c 11I"'[>lIIelh",1 = 311i.67 X 10-3 (using 2nd-

order lor.al test functions)

The iuterpolation I>ound is 30 times less than

the actual error which t.he new estimation tech.

nique agrees with the exact to wilhiu lhree sig.

uificant figures. Figure 8 shows contours of the

actual error comparelj with those of the esti·mated error.

Further details aud results can be fonnd in

the recent note by Oden, Strouboulis, and Dass

[261·o in nau(Jf + a· VII

o N=1,2, .. ·,N

(5.5) into (5.3) and replacing'" hy 'It N gives

LM( mNl., EM + k( It )N1o!EM) - '-N(t) =(5.6)

fIlNM

rN = <r".Wh>

Many possible wa)'s for implementing (5.6)

present themselves. These equations, for ex-

ample, need not he global in the sense that

an element.hy.element or patch of elemenls in

a fine mesh, obtained through a mesh refine.

ment, may produce suflicient accnracy 10 allow

for an adequate indication of the evolution of er·

ror. The local velocities fI and residual rN can

be interpolated using Q.·approximations on a

fine mesh level.

In recent weeks, we have developed a nell'

error estimation technique based on the EOE

(evolution of error) technique embodied ill a

variant of (5.6) and the discontinuous finite ell"

ment method of l..esaint and Raviartl25j for hy·

perholic partial differential equations. The key

to our approach is the use of disr.ontinnous 10'cal shape functions and an amendmentlo a local

(elementwise) version of (.'i.G) thaI contains the

jump term,

1'0+1 L hIfI . 1/-.1 [E ).,"NdfldtIn 80;-

where I[Eh]1 is the jnmp in the error approxima.

tion Eh across the innow boundary of element

ne.Figure 7 contains results of a sample calcula·

tion of steady-state solutions of the pure con·

vection problem,

II = 9 on an-where

n = (0.1) x (0,1) C R2

6 Flow SolversSchemes 011

Meshes

for AdaptiveU nstruct ured

a = (I, 1)

Results of the estimated error versus actual

error as a function of distance along the diago·

nal of the square domain are shown in this figure

Most of the applications we have in mind involve

solutions of the unstread)'. compressible Navier·

Stokes equations. In three dimensions, without

body forces or external heat addition, these canbe wril!en,

9 = Ion- = {(a',y) : (x,O)U(y,O)}

sin (~nlanh G(y - 32)) on x = 0

o on y = °8U 8E OF 8G d' S-+-+-+-= IV8/ Ox 8y 8: (6.7)

..

o

Ilere p is the total mass density, ll, tl alld ware

the velocity components, p is lhe f1nid pressure,

r,j are the component.s of the viscous stresses,

e is the total energy defined by c = e + Hu2 +v1 + w1) where e is the thermodynamic internal

energy per unit mass, and q is the heat nuxvector.

The constitutive relation used to evaluate theviscous stress r;j is given hy

Figure 8(a): Contours of error in a 2Dconvection problcm. Exact error.

S=

Tr.z,% + TrY,1I + Tx%,%

rry,r + Tyy.y + Tyr,%

Trz,r + Tyt,y + Tzz,z3 3 3

~~T"ll"+~q"L-, ~ I) ),' L 1,1

i=1 j=1 1:1

(6.11)

and an equation which relates the temJleratureto the internal energy

where µ and .x are the first, a.nd second coeffi.

cients of viscosity, ll;,j are the comJlonents of

the velocity gradient (1I'.j = {)ll;/Dxj, XI = X,

X2 = V, X3 = z), and 6;j is the I{ronecker

delta. For the sample problems of Section 7,

the Stokes' relation.x = -5/1 has been assumed,

which makes the hulk viscosity l.NO.

In addition to tlw partial dilferenti,,1 equa-

tions "hove, two thermodynamic relations are

also ueeded to close the system of equations.

These relations are lhe ideal-gas state e(luation

(6.12)l' = h-1)peFigure R(b). Estimatcd error obtained usingbilinear test functions.

where U, E, F, G and S are vectors given hy

lIere -y is the specific heat ratio and Cv is the

specific heat at constant volume. With these

two additional equat.ions we now have a com-

plete system which cau be solved for the vector

of unknown quantities (p, ll, V, 1lJ, C, p, 1').

[

p 1 r pu ]pu pu2 + I'

U = pv E = PUl'

pw PU1lJ

pe (pe+p)u

(6.8)

c = cu'r (6.13)

[

pv ]pill'

F = p,,2 + /'pvw(pc + p)v

lpw ]puw

G = PVIV

pw2 + p(pe + p)w

(6.9)

(6.10)

For the class of prohlems considered here,

a weak formulation is defined in terms of two

classes of functions: V, the class of trial func-tions, to which the solution U belongs, and lV,

the class of test (or weight) functions which are

integrated against the residual of the governing

equations. The resulting weak form is:

Find U in a class V such that

, ."

for all test functions .p = [4>1,4>2"" ,4>~J in nT:

where [0, T] is lhe time interval of interest, fl is

the region through which the nuid moves, ao is

the boulldary of the flow re~ion fl, and" is the

vector of bonndary l1uxe~. II. is underst.ood that

the viscons stress terms on the right~ltand side

of (6.14) also appear in the integrated form.

surfaces of discontinnity where the jump condi·

tions hold.In a strictly formal way, the finite element

approximatioll of the now is obtaillcd from the

weak statement of the conservalion laws, hy in·

terpreting 0 a.< a 'JU1"lrilateral or I>rick element,

replacing U by the discrete approximation Uh

and replacing the lesl fnnctions .p by the dis·

crete functions .ph. Then (6.15) holds over each

element Oe(I), e = 1,2··· E. Choosing an ap-

propriate numerical illtegration rule to evaluate

the time integral yields a local fillite element

model of the conservation laws.

We shall now give examples of a few explicit

algorithms We have nsed successfully on unsl.ruc·

tured meshes. These are by no means olTered

with any strong endorsement; perhaps no other

component of the adaptive technology is more

in need of furl.her work and improvement.so that differentiabilit), of rij in L' is notneces·

sarily required.

Our numerical approximation of lhe now

problem will begin with a ,Iiscrele npproxima-

tion of a slightly different weak form of the prob.

lem. for a time intervnl [II, /2] 1I.~:

find U = t/(r,l) E V such that

- 1.'2 ( UT .p,dUdl" 10(1)

6.1 Two-Step Taylor-Galerkin/Lax·\Vendro(f AIgOl'ithm5

A two-step Taylor·Galerkin I l.ax- Wendroff

scheme is derived from the discrete version of

(6.15) hy using the midpoillt intgration rule:

Partitioning the lime interval of interest into a

finite number of steps,

()= 10 < II < '2 < .. , < IN = T

for all .p E tv

one arrives at the followiug two-step scheme:

First Step: For each clement fle, calculate an+1conslant element vector Un,_ 1 from

Second Step: for each node, calculate ut·n+1

Ity solving the following system of equations:

A~+I u~~t=t{(1" \}lidO) u~,n.=1 n,

~I A~+I (f aW; dfl) Qi,n }-2 A;+t In:'+~axp n,(J

where A~ is the area of lhe elemenl. at time tn,lI~·n is the a-component of vector U at time

tn at node i, V'i is a piecewise bilinear shape

function which has a value of unity at node i and

is zero at every olher node, an,] 0 = 1,2" .. ,5.

(6.15)= 1.'2 ( Q(U): V.pdfldt

" 10(1)

+ 1.'2 f S(U): V.pdfldlI, In(1)

1.'21 T+ 0 .pd.,dl" ao(,)

Here, .p = (<P •• 4>2·· ·4>s}T, dO == dr. 4>n(z,t).x E 0(1), a == 1.2 .. ·5, <P, == a<pI{}t, and" isan outward unit normal vector to the I>oundary.

Also, here Q is the full convective flux Q( U) =(E,F,G).

It is easily verified that (6.15) is eqnivalent

lo the !!ntire syslem of NaviN-St.okes equations,

Rankine-I\ugoniot jnmp cOllditions (when S =0). and iuit.ial conditions on U (at. I = til when-

ever U is a C1·function everywhere except at

· ,

where -0 + {J = 1,

£flu _ 8 {~A al.' + k0{fk Oil +lif' - CJij Ot k ox:; J I (Jij;

+AjAk~ (ildr) }

ixe(~)cPxe (AtWf)

&(~)o (A 811)(Jl jl1xj

The time-derivative of the Jacol>ian of the nux

~ is ralculaled as follows:

6.2 Taylor-Galerkill Algorithms

lIere, Q~JJ denol.es an dementwise averaged

value of the nux. and N is t.he lolal number

of nodes in the discretizalion.

In order to advance the solution over a lypical

lime intl'rval [t",lntt!, we employ an explicit

Taylor-series in lime 10 obtain all explidt algo-

rithm or an implicit Taylor-series t.o oblain an

implicit algorilhm, namdy:

FOM/lard- Taylor-series expansion.

Using this formula we obtain

(8 )"lI"tl = u" + t:..t 8';t:..1

1 (8111)"+2 8/1

t:../3 (/flu)" + O(M1)+6 lJ/3

Dacl.."lJUlrd-1'aylor-scI'ic., eXIJRllsion.

u" = 11,,+1 _ t:..l (OU)"tlOt

+ t:..t1 (('Pu)"+12 8/2

t:../3 (If' )"tl-6 8t~ +O(M1)

Followillg the idea of I.ax and WendrorT, we

use the governing partial din'erl'ntial ellnation

in order to trade time-derivatives with spa<:e-

derivatives in the Taylor-expansion as follows:

an 8Fj A 8118t = OXj = - j aXj

where..., - {)= I.

where Aj = 8Fj8u ' Inserting the .•e expressions ror the derivatives

into the Taylor-series expansions and laking the

convex combination of the explicit and implicitformula, we have:

"

]"8 8u+ AjAk8xk (a1)

+ 06[(<'JAtAj + Aj <'JAt) (Ak~){)1't DXt DXk

8 ( 8U))n+AjAk- At-aXk DXt

l(oAt OAt) ( 8U)+ (1 - 0)6 -Aj + Aj- Ak-

UX( ux, aJ'k

{J ( 8 )]n+J}+ AjAkfu:; At a;:Ilere 0 E [0,1]. -o+t1 = 1, ...,-/J = 1. In order 10

obtai II a finite-element algorithm IIII' Uul>nov.

Galerkin method is used to discretize the above

equatioll in space. Baker and Kim 127] have

shown that with proper choice of parameters the

abo\'e formula can reproduce most of the known

one-step schemes.

0.2.1 3.2.3 Runge-Kulln Algorithms.

Let us write the semi-discrete version of the COli·

servation law obtai lied by nsing the Bul>nov.

Galerkin method on a finite-dimensional sub·

space defined by the finite element discretization

of the spatial domain, as follows:

lIere R denotes the "load" vector with length

equal to the numher of nodes in the spatial dis-

cretization; the component correspolllling t.o the

J·th node is:

1 alpJ t .llJ = Fj-u 1m - FjVjlpJtls.n Xj all

Here 'PJ denot.es the hasis function correspolld·

ing to node J, Vj dellotes the j-th compollent

of the boundary normal. Moreover, M is the

mabs-matrix with components of the form

where I is t.he (4x4) identity matrix. If one uses

an m-stage Ruuge.J(utta method to integrate

in time the system of ordinary differential equa-

tions

one employs the m + 1 steps:

u~O) = 1t7Itll) = Il~O) + 0 I tiIL~O)

7 Sample Problems

We shall now cil.e results of adaptive calcula·

tions on several sam pie prohlems selectl'd from

recent and forthcoming papers.

7.1 Snmple Problem 1: h-AdnptiveHesults for Subsonic Viscous FlowAbout n Cylinder

As a first example. we consider an h.adapti\'e so-

lution of the unsteally Navier·St.okes e'luations

for the problem of viscous compressible now past

a rigid cylinder. The 110wfield, shown in Fig.

9, is a circle of 8-times the cylinder diamet.er.

The calculatiou was run at a Reynold's num·

her of 1.1 x 10-5• Mach numher 0.6·1. and free-

stream temperat.ure of 530 degrees Rankine. In·

now conditions involved sper.ification or a uni·

form It velocity with II and w set to zero.

Typical computl'd results are shown in Figs.

8-12. Figure 9 shows the instantaneous mesh

after 1000 time st.eps and the calculated den·

sity r.ontonrs, Mach lIulnber contours, and ve-

locity \'eclors. The mesh is continually chang-ing, with the mesh and the solut.ions at 1500

time steps shown in Fig. 10. Note the l'volution

of vortices shedding off of the cylinder bound·

ary. We estimate t.hat the use of the adaptive

algorithm here saved 60-100 percent in compll-

tationaltime compared to that needed hy a uni·

rorm mesh 10 obta.in similar resolutions of the

1101\' properties.

, ."

7.2 Sample Problem 2:

The problem of supl'rsonic now over a nat. plate

was solved in order to verify the calculation of

the viscous lerms. This problem has been solved

by several investigators slarling with Carter

(NASA TR R.385) who ohtained numerical so·

lutions usi ng the Brailovskaya scheme. The

problem was Sf:t up ill the compntational box

shown in Fig_ 11. The supersonic free-stream

conditions (llfoo = 3, 0 = 0", -y = 1.4, 1'00 =390" R) are specified on sides AD and DE while

(.,

(1l)' -- -- .... _- ... -- ,~ .• _-

(d)

(e)

(d)

....- - -.,,-, ..- .--. ..-- .

Figure 9: Viscous cylilHIl'f problem with !If =0.6-1. now perturbed resulls after 1000 timesteps. The adaptive grid captures the gf:nera-tions and shedding of vortices: (a) the instan-taneolls adapted grid, (h) dellsity contours, (c)Mach number contours, and (d) velocilies.

Figure 10: Viscous cylinder problem with M =O.(j,l, now perturbed results after 1500 timesteps. The adaptive grid captures the genera·tions and shedding of vortices: (a) the instan-taneous adapted ~rid, (b) density contours, (c)Mach numbl'r conl.oms and (c) velocities.

(a)

o

A ::r=='i. L C.... 1" B -'_1 ,'0. 1._.11 _, ••• J.ft··tj \.ttJ!...... ".,ocot_

(a)

(b) (b)

(e)

NON-DIMENSIONAL, Y·VELOCITY

Figure 12: Error calculation; (a) contours oferror inclicator 03• (b) error indicator 02, and(c) el'£or indicator Ot.

1

:c.----__- __-~S.... ...1" L'" •.'~ ~ .!.. I!~ ,.'.. ..'. 1._

...l •.....>-

IJ 2f~-=--.(d) L: .-l

Fignre 11: Supersonic l10w ovp.r a l1at plate.(a) Finite element mesh including elements upto level ,\ and contours of density obtainedwith the one-step Lax· WendruIT algorithm withMcCormack-Baldwin artificial dissipation, (h)contours of density, (c) contours of pressure, "lid(d) contours of llIach number.

Figure 1:1: SuperRonic now over a l1at plate.Exit V.velocity profile calculated using adaptiveTG2-scheme compared with 13,OOO-cell fixed-grid finite dirrerence solutions.

·.

outflow conditions are applied on the bound-

ary side CEo TIte nat plate is represented by

an isothermal no-slip bonndary along IIC; the

termperal.ure atoug the isothermal wall wa.. setequal to

TWALL=1"oo(I+ 1;1A[;')The boundary side AB in frout of the nat plate

was treated as a no-now hou ndary.

The ReYllolds nllml>er, Re = 103 was hased

on the length BC = ] and the Prandtl lHlml>erfor air was taken equal t.o (J.72.

Solutions for Carters flat plate problem were

obtained using several algorithms and some pre.

liminary comparisons indicate t.hat our results

are very close to Carter's sol u t.ion.

Three error indicators are used in these cal·culations.

(i) An error indicalor based on I.he maximum

component of the density gradiellt. normal.

ized I>y the densil.y, namely

I {ah Oh}01 = :h. II max -DP ,.L ,pr n. x ay

where p~ denoles an average \'alue of the

densit.y over elelnent n•.(ii) An error indir.ator based on the 1/ 2.

seminoml of the density normlilized hy thedensit.y gradient,

O• _ I/' !2,2,n,2 ----r--Ip II.2,n. '

lIere

I {(D2P)2 ( D2p )2IpI2,2,O. = \110• Bx2 + 2 DxBy

10. {(;=f+ (;~f}dn

(iii) An estimat.e of the normalized error in thedensity gradient

-I,, IVph - Vp 10,2,.

03 = 1,1IV f) 0,2 ••

- h •Here V p denotes the "ex traded" or

smoothed gradient alllll·lo,2.o. denotes theelement mean-square norm (L 2.norm) de.fined by:

Ivlo,2,0. = j foe v2df!

Shown in Fig. l1a is the finite ele-

ment mesh including adaptive elements up

to level 4 and conlours of density obtained

with the one-step Lax-Wendroff algorithm with

McCormack-Baldwin artificial dissipation. The

contours of density, pressure, and Mach num.

her are shown in Figs. II through lid. Figure

12 shows the contours of error indicators .. Fig-

ure 13 shows a comparison of computed veloc-

ity profiles with those of Carter obtained using

around 13,000 grid points. The savings provided

1»' adaptivity is clear.

7.3 Sample Problem 3: Intl'mal Flowin the Space Shuttle Main Engine

An instantaneous mesh ol>tained in an analysis

of viscous compressihle 110w past moving tur-

bille rotor hlades in a turbine engine is shown in

Fig. 14 with computed Mach number aud den·

sity coulours. As a general rille, the use of an

adal't.ive mesh rl'lluces lhe computational effort

for this class of problems around 60-70 percent

of that needed hy an esselltinlly uniform mesh

for delivering the sanle aCI:uracy.

7.4 Sample PI'oblclll 1: An hop Anal-ysis of Pot.ent.ial Flow

Figures 15 and 16 show a typical hop calculation

for two-dimensional and three-dimensional po·

tential now problems. In Fig. 15 we see an

optimal hop mesh in which the mesh appears

as an irregular pat.tern of colors (shown here as

different shades of grey). each color represent.

ing a different. polynomial degree, from p = 1 to

-p = 9. This Illesh was completely generated

I>ythe computer and wa.. attained using the hop

optimization scheme and hop constraint approx·

imation mentioned earlier. Fignre 16 shows a

similar three-dimensional calculation 011 a cu·

bic domain. Shown here is the mesh with wit.h

sOllie elements peeled away to reveal the actual

interna.l spectral orders chosen by the algorithm,

eact. identified by a diff.'rcnl. color (or shadehere).

7.5 Sample Problem 5: An h-rScheme for Shock Calculations inlnviscicl Compressible Flows

As a final example, some resllits of an h-r adap-

tive calculat.ion of shucks in inviscid compress-

ible flow prol>lems are presented. These results

,.

(,) (d)

(h)

(c)

p ~ I 2 4 5 6 1

Figure 14: (a) Instantaneous adapted gridaround moving rolor in it turboengine, (b) pres·sur .. contours, (c) Mach number distributions,and (d) velocilies.

Figure 1.5: An optimal hop grid wit.h clementsof different size alld spectral order-indicatedby different shades of elements.

...

Figurc 16: An examplc of three-dimcnsional h-p mcsh with different. shadcs indicating dirrercnlspectral orders.

Figure 17: An optimal h-r calculation ':'fill\'iscidsupersonic now pa..~t.a cylinder: (,,) t.he h-r meshand (h) densit.y contours.

Figure 18: Su~crsonic inviscid now over awedge: (a) lI.n h-r mcsh and (b) dcnsity con·t.ours; (c) a finc 6-level h·adaptive mesh with(d) density contours.

• • •

, ,

are taken from the forthcol11i ng paprr of Ed·

wards, Oden, and Demkowicz [28). Two resnlts

are noted. First, a distorted h·r mesh for calcn·

lating the bow shock on a. hlunt body is shown

in Fig. 17 with computed densit.)· cOlllours. A

similar calculation for lIow over a wedge is il·

lustrated in Fig. 18. To obtain a comparable

resolution of the shock using only h-adaptive, a

6-level h·mesh, shown in Fig. 18c must I>e used.

Acknowledgement.

The help of J. M. Dass, T. Stronhoulis, C. Y.

Huang, and C. W. Berry on the AD'\!'T/2IJTJIt

and ADAPT/3D™ codes, of R. CI1('n on tur·

bulence modeling, and of M. Edwards on the

h·r schemes is gratefully a.cknowledgNI. Thanks

are also given to L. Demkowicz who parl.i(:ipated

in many of I.he adaptive projects oV<'r the last

four years alullo \V. Hachowicz who continues

to contribute many original ideas to onr work

on h·p schemes. T. Westermann and O. Hardy

contributed to ongoing work on h.p methods.

The initiall\'ork Oil h·p data structures and h·

p methods for compressihle flow ....n., supported

hy the Ollicl1 of Na\'alltcscarch. SIIh5I"III"lIt .11"

velopmen!. in this area are under snppor! of the

Aerothermal Loads Branch of the NASA Lallg.

ley Research Center. Ollr adaptive work on h·

methods for turbomachinery calculations is sup·

ported by thl' NASA Lewis Research Center.

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• , .,

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