,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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