NASA Technical Reports Server (NTRS) 19980021427: Directional Agglomeration Multigrid Techniques for High Reynolds Number Viscous Flow Solvers

16 Pages·1.3 MB·English

by  NASA Technical Reports Server (NTRS)

AIAA-98-0612
Directional Agglomeration Multigrid
Techniques for High Reynolds
Number Viscous Flow Solvers
D. J. Mavriplis
Institute for Computer Applications in Science
and Engineering (ICASE)
NASA Langley Research Center
Hampton, VA
36th Aerospace Sciences
Meeting & Exhibit
January 12-15, 1998 / Reno, NV
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, Virginia 20191-4344

DIRECTIONAL AGGLOMERATION MULTIGRID TECHNIQUES FOR
HIGH REYNOLDS NUMBER VISCOUS FLOW SOLVERS
D. J. Mavriplis
Institute for Computer Applications in Science and Engineering
MS 403, NASA Langley Research Center
Hampton, VA 23681-0001
1. Abstract efficiently the thin boundary layers and wakes which
occur at high Reynolds numbers. Additional stiff-
A preconditioned directional-implicit agglomeration ness is induced in regions of low Mach number flow,
algorithm is developed for solving two- and three- due to the disparity in eigenvalues corresponding
dimensional viscous flows on highly anisotropic un- to the acoustic and convective wave speeds, as the
structured meshes of mixed-element types. The Mach number tends to zero.
multigrid smoother consists of a pre-conditioned The construction of an efficient solver requires
point- or line-implicit solver which operates on simultaneous treatment of these effects. Semi-
lines constructed in the unstructured mesh using a coarsening multigrid techniques as wcll as implicit
weighted graph algorithm. Directional coarsening or line-solvers can be used effectively on structured
agglomeration is achieved using a similar weighted grids to relieve the stiffness associated with highly
graph algorithm. A tight coupling of the line con- stretched meshes [2, 3]. The basic semi-coarsening
struction and directional agglomeration algorithms strategy consists of constructing coarser multigrid
enables the use of aggressive coarsening ratios in levels by coarsening the original grid in the coordi-
the multigrid algorithm, which in turn reduces the nate direction normal to the grid stretching, rather
cost of a multigrid cycle. Convergence rates which than in all directions simultaneously. When conflict-
are independent of the degree of grid stretching are ing stretching directions exist, multiple coarse grids
demonstrated in both two and three dimensions. must be constructed, each generated by a coarsen-
Further improvement of the three-dimensional con- ing in a particular coordinate direction [4]. However,
vergence rates through a GMRES technique is also when a single stretching direction can be identified,
demonstrated. only one family of directionally coarsened grids is
required [5].
Semi-coarsening techniques can be generalized to
2. Introduction unstructured meshes as directional coarsening meth-
ods [6, 7, 8, 9]. Graph algorithms can be constructed
The goal of this work is the development of an to remove mesh vertices based on the local degree
efficient solver for compressible steady-state high and direction of anisotropy in either the grid or
Reynolds number Navier-Stokes flows on unstruc- the discretized equations. This is achieved by bas-
tured meshes. The overall strategy is based on a ing point-removal decisions on the values of the dis-
multigrid approach. Multigrid methods form the crete stencil coefficients. This is the basis for alge-
basis of some of the most efficient available solvers braic multigrid methods [9], which operate on sparse
for such problems, both on structured and unstruc- matrices directly, rather than on geometric meshes.
tured grids. For inviscid transonic flow problems, These techniques are more general than those avail-
multigrid methods can deliver converged solutions able for structured meshes, since they can deal with
in under 100 cycles [1]. However, for high-Reynolds multiple regions of anisotropies in conflicting direc-
number Navier-Stokes problems, and for flows in- tions.
volving large regions of low velocity fluid, multigrid One of the drawbacks of semi- or directional-
convergence rates degrade seriously. This degrada- coarsening techniques is that they result in coarse
tion is due partly to the stiffness induced by the grids of higher complexity. While a full-coarsening
highly stretched grids which are required to resolve approach reduces grid complexity between succes-
sively coarser levels by a factor of 4 in 2D, and 8 in
Copyright _)1998 by the American Institute of Aeronautics
and Astronautics, Inc. All rights reserved. 3D, semi-coarsening techniques only achieve a grid
complexityreductionof2,inboth2Dand3D.This ual of a standard upwind scheme as the sum of a
increasethsecostofamultigridV-cyclea,ndmakes convective term and dissipation term:
theuseofW-cyclesimpracticalP. erhapms oreim-
portantlyfor unstructuredmeshcalculationsth,e
amounotfmemoryrequiredtostorethecoarsleevels
neighbors 1
isdramaticalliyncreasedp,articularlyin3D. 5(F(wi) + F(wk)).nik -- 1IAikI(WL __WR) (1)
An alternativeto semi-coarseniinsgto usean k=l
implicitline-solveirn thedirectionnormaltothe
gridstretchingcoupledwitharegularfullcoarsen- where the convective fluxes are denoted by F(w), nik
ingmultigridalgorithm.Althoughpredetermined represents the normal vector of the control volume
gridlinesdonotexistinanunstructuremdeshs,uch face separating the neighboring vertices i and k, and
linescanbeconstructebdyidentifyingandgrouping Aik is the flux Jacobian evaluated in the direction
togethenreighborinmgeshedgeussingagraphalgo- normal to this face. WL and wR represent extrap-
rithm[10,11].Byusingaweightedgraphalgorithm olated flow values at the left and right hand sides
with edgeweightswhichreflectthedegreeofcou- of the control volume face respectively. For a first
plinginthediscretizatiobnetweenneighboringgrid order-scheme, these are taken as the values at the
pointss,etsoflineswhichpropagatienthedirection vertices to the left and right of the control volume
ofstronggridcouplingcanbeconstructe[d12]. interface, whereas for a second-order scheme, these
Thesolutionstrategydescribedin this paper are extrapolated from the corresponding vertex val-
addressethseanisotropy-inducsetdiffnessproblem ues using solution gradients pre-computed at these
vertices.
throughacombinatioonfimplicitlinesolverscou-
pledwithdirectionacloarseninmgultigrid.Thiscou-
pledalgorithmpermitsfastecroarseninrgateswhich
resultinmoreoptimalcoarsgeridcomplexitiesT.he I
low Mach number stiffness problem is addressed us-
ing preconditioning techniques [13, 14, 15, 16], which
are integrated into the overall directional implicit
multigrid algorithm. The combination of these three
techniques into a single solver has previously been
demonstrated in the context of geometric multigrid
for two-dimensional problems [12]. The current work
represents an extension of this strategy to the more FIG. 1. Median control-volumes for stretched
practical agglomeration or algebraic multigrid ap- quadrilateral and triangular elements
proach for unstructured meshes, as well as the ex-
tension to three dimensions.
In this work, a matrix artificial dissipation is em-
ployed. The matrix-based artificial dissipation
3. Discretization
scheme is obtained by utilizing the same transfor-
mation matrix [Aik[ as the upwind scheme, but
The governing equations are discretized using a
using this to multiply a difference of blended first
finite-volume approach. Flow variables are stored
and second differences (i.e. blended Iaplacian and
at the vertices of the mesh, and control volumes
biharmonic operator) rather than a difference of
are formed by the median-dual graph of the origi-
reconstructed states at control-volume boundaries.
nal mesh, as shown in Figure 1. A control-volume
The traditional scalar artificial dissipation scheme
flux balance is computed by summing fluxes evalu-
[17, 18, 19] is obtained by replacing the four eigen-
ated along the control volume faces, using the av-
values u, u, u+c, u-c in the IAik[ matrix of the ma-
erage values of the flow variables on either side of
trix dissipation model by the maximum eigenvalue
the face in the flux computation. This construction
[u]+ c, where u and c denote local fluid velocity and
of the convective terms corresponds to a central dif-
speed of sound, respectively. This matrix dissipa-
ference scheme which requires additional dissipation
tion construction has been found to deliver accuracy
terms for stability. These may either be constructed
comparable to an upwind scheme, while eliminating
explicitly as a blend of a Laplacian and biharmonic
the need to compute and store flow gradients at mesh
operator, or may be obtained by writing the resid-
vertices.
Thethin-layerformoftheNavier-Stokeesqua- in efficiency can be obtained by using a Jacobi
tionsisemployeidnallcasesa,ndtheviscoutserms preconditioning approach in conjunction with the
arediscretizedtosecond-ordaecrcuracybyfinite- multi-stage scheme [22, 23, 24, 25]. The (q + 1)th
differenceapproximation.Formultigridcalcula- stage of a Jacobi preconditioned multi-stage scheme
tions,afirst-ordedriscretizatioinsemployefdorthe can be written as:
convectivteermsonthecoarsegridlevels.
Thesingleequationturbulencemodeol fSpalart w (q+l) ----w_°) ÷ [Di] -1 x l_(w (q)) (3)
i
andAllmaras[20]isutilizedtoaccounftorturbu-
lenceeffectsT.hisequationisdiscretizeadndsolved where the scalar time step At from equation (2) is
inamannecrompletelaynalogoutostheflowequa- replaced by the matrix time step given by the inverse
tions,withtheexceptionthattheconvectivteerms of the matrix
areonlydiscretizetdofirst-ordearccuracy.
neighbors
Thisparticuladriscretizatioinsdesignetdoenable OIL(w) 1]Aik]) (4)
theuseofmixedelemenmt esheisntwodimensions [D,]- _ ----( Z
k:l
(quadrilateraalsndtriangles)andthree-dimensions
(tetrahedrap,rismsp, yramidsh,exahedraM).eshes which is a 5 x 5 matrix (4 x 4 in two dimensions)
ofdifferingelementtypesarehandledbyemploy- corresponding to the pointwise Jacobian of the resid-
ingasingleedge-baseddatastructuretoassemble ual. Note that for a scalar dissipation scheme, this
thefluxesacrossallelementtypes[21].In twodi- matrix becomes diagonal, and the scalar time-step
mensionqs,uadrilateraellementasreemployeidnthe estimate is recovered, thus reducing the scheme to
regionosfhighmeshstretchingw,hiletriangulaerle- the standard explicit multi-stage scheme.
mentsareemployeidnisotropicregionosfthemesh.
Additional preconditioning of the type described
In threedimensionsh,exahedroar prismsareem-
in [13, 14, 15, 16] must be implemented in order
ployedinregionnsearthewall,whiletetrahedraare
to address the stiffness problems induced by re-
generallyemployeedlsewhereT.heuseofdifferent
gions of low Mach number flows. Traditionally,
elementtypesinregionsofhighmeshstretchingen-
such preconditioners are described as a matrix mul-
ablesamorecompletedecouplinogfthediscretiza-
tiplying an explicit updating scheme, and a similar
tioninthestretchingandnormaldirectionsa,sdis-
matrix-based modification to the dissipation terms,
cusseidnsection5.
which improves the accuracy at low Mach numbers.
Thus, the (q÷ 1)th stage of the standard multi-stage
4.
Preconditioned Smoothing scheme (c.f. equation (2)) is rewritten as:
Once the governing equations are discretized, they w(q+l) -- (0) +PAti ×
must be integrated in time to obtain the steady-state
neighbors
solution. This is achieved using a preconditioned ( (F(wiq)+F(wkq))nik
multi-stage time-stepping scheme. An explicit k-
k=l
stage scheme can be written as:
1p-11pA,kI(WLq--WR q) ) (5)
2
w_°) _-- w._n)
In the present work, we wish to implement
this type of "preconditioner" in the context of
a point-implicit (Jacobi-preconditioned) or line-
w(q) = w_°)+ At, ×1%(w(q-l))
i implicit scheme. Since the low Mach number pre-
(2) conditioning matrix is a point-wise matrix, its imple-
Wi(qTi) = W} 0) ÷ Ati x Ri(w (q))
mentation for point-implicit schemes is similar as for
line-implicit, or any implicit scheme. The approach
taken, which was originally described in [26, 21], is
to modify the dissipation terms in the discretization,
as per equation (5) ,and then simply take this mod-
where At represents the scalar time step estimate. ification into account in the point-wise linearization
While such a scheme is commonly used for scalar ar- that is required for the point-implicit Jacobi scheme.
tificial dissipation discretizations, for upwind or ma- Thus, the (q + 1)th stage of the low Mach number
trix dissipation discretizations substantial increases
preconditioneJdacobmi ulti-stagsechembeecomes:
the line length becomes one (i.e. 1 vertex and zero
edges), as is the case in isotropic regions of the mesh.
w(q+l) _ (0)
In summary, the final scheme, which is used as
i = w i -F
a smoother for multigrid on all levels, results in
a point-implicit low-Mach number preconditioned
×
neighbors - 1 multi-stage scheme in isotropic regions of the mesh,
and a line-implicit low-Mach number preconditioned
neighbors
multi-stage scheme in regions of high mesh stretch-
ing. A three-stage multistage scheme with stage
k=l
] coefficients optimized for high frequency damping
--2P-I[PAikI(WL a- wRq) ) (6) properties [28], and a CFL number of 1.8 is used
in all computations.
In regionswhere the Maeh number isrelatively
large,the low Mach number preconditioningma-
5. Directional Agglomeration
trixP becomes the identitymatrix, and effectof
and Line Construction
the preconditionervanishes.Inthiscase,theabove
scheme revertsto the3acobipreconditionedscheme
The stiffness due to grid anisotropy is addressed by a
ofequations(3).Likewise,forscalardissipationdis-
directional agglomeration multigrid strategy coupled
cretizations (i.e. when [PAik[ is approximated as
with a line-implicit smoother. The combination of
a diagonal matrix), this scheme reverts to the low
these two strategies into a single algorithm has been
Mach number preconditioned schemes characterized
found to result in a more robust and efficient solution
by equation (5) and described in [13, 14, 16]. The
method than the use of either strategy alone [29, 12].
particular form of the preconditioning matrix P em-
ployed is that described in [27]. The implementa-
tion described therein is attractive because it can In regions of high grid stretching, standard direc-
be achieved without any change of variables in the tional agglomeration (i.e. coarsening) results in the
original discrctization. removal of one grid point for every retained coarse
grid point. This produces a sequence of coarse grid
Equation (6) represents the scheme used in
levels for which the complexity between successive
isotropic regions of the mesh. In regions of large
levels decreases by a factor of 2. Isotropic agglom-
mesh stretching, this pointwise scheme is replaced
eration, on the other hand, produces a coarse grid
by a line implicit scheme, operating on grid lines
complexity reduction of 4:l in 2D and 8:1 in 3D. The
which are pre-constructed in the grid. The im-
higher complexity of the directionally coarsened lev-
plicit system generated by the set of lines can be
els greatly increases memory overheads, particularly
viewed as a simplification of the general Jacobian
in three dimensions, and makes the use of the multi-
obtained from a linearization of a backwards Euler
grid W-cycle impractical, since the operation count
time discretization, where the Jacobian is that ob-
of the W-cycle becomes unbounded in such cases as
tained from a first-order discretization, assuming a
the number of grid levels is increased.
constant Roe matrix in the linearization. For block-
diagonal preconditioning, all off-diagonal block en- The implicit-line solver achieves superior smooth-
tries are deleted, while in the line-implicit method, ing of error components along the direction of the
the block entries corresponding to the edges which implicit lines, as compared to a regular explicit
constitute the lines are preserved. The line-implicit scheme. This in-turn permits the use of an ac-
solver is introduced into the current solution strat- celerated coarsening schedule by the agglomeration
egy as an extension of the Jacobi preconditioner. multigrid algorithm. However, since the implicit
At each stage in the multi-stage scheme, the_orr6c- line-solver is only effective at smoothing error com-
tions previously obtained by multiplying the resid- ponents along the implicit lines, multigrid coarsen-
ual vector by the inverted block-diagonal matrix are ing must proceed precisely along the direction of
replaced by corrections obtained by solving the im- these lines. This requires a close coupling between
plicit system of bl0ck-tridiagonal matrices generated the directional agglomeration algorithm and the line
from the set of lincs. This implementation has the construction algorithm. Both techniques are based
desirable feature that it reduces exactly to the block- on weighted graph algorithms, and must employ the
diagonal preconditioned multi-stage scheme when same definition of the graph weights.
Agglomeratiomnultigridmaybeviewedasasim- thisorderedlististhenpickedasthestartingpoint
plifiedalgebraicmultigridstrategy.Coarselevel foraline.Thelineisbuiltbyaddingtotheoriginal
gridsareconstructebdyfusingtogetheorragglomer- vertextheneighborinvgertexwhichismoststrongly
atingneighborincgontrovlolumetsoformacoarser connectetdothecurrentvertex,providedthisver-
setoflargerbutmorecomplexcontrolvolumesI.n texdoesnotalreadybelongtoaline,andprovided
thealgebraiicnterpretationofagglomeratiomnulti- theratioofmaximumtominimumedgeweightsfor
grid,thecoarselevelsarenolongergeometrigcrids, thecurrentvertexisgreaterthan
a, (using a = 4
butrepresengtroupingosffinegridequationwshich
in all cases). The line terminates when no addi-
aresummetdogethetroformthecoarsegridequa- tional vertex can be found. If the originating vertex
tionssets[6,30].Thereforeit,isimportanttobase is not a boundary point, then the procedure must be
thedirectionaalgglomeratioanndlineconstruction repeated beginning at the original vertex, and pro-
graphweightosnalgebraiqcuantitiessuchasstencil ceeding with the second strongest connection to this
coefficientrsa,therthangeometriqcuantitiessuchas point. When the entire line is completed, a new line
edgelengthsw, hichmaybeill-definedonthecoarse is initiated by proceeding to the next available vertex
levels.Howevera, one-to-onceorrespondenbcee- in the ordered list. Ordering of the initial vertex list
tweenstencicloefficientasndgridedgeosnlyexists in this manner ensures that lines originate in regions
forscalaerquationasndisnotpossiblfeorsystemosf
of maximum anisotropy, and terminate in isotropic
equationsF.orthisreasont,heedgeweightsforthe
regions of the mesh. The algorithm results in a
line-constructioanlgorithmandthedirectionaalg-
set of lines of variable length. In isotropic regions,
glomeratioanlgorithmaretakenasthestencicloef-
lines containing only one point are obtained, and
ficientsofascalacronvectioenquationdiscretizeodn
the point-implicit or Jacobi pre-conditioned scheme
thefinegridusingthefinite-volumaepproachO. n is recovered.
thefinelevel,thesecorrespontodthearea-weighted
normalsofthecontrolvolumefacesdelimitingtwo The agglomeration algorithm consists of choosing
neighborinvgerticesO. nthecoarselervelst,heseare a seed point (i.e. a control-volume) which initiates
a local agglomeration, and then agglomerating the
constructebdysummingtheconstituenftinelevel
facenormals. neighboring control volumes to the seed point. The
isotropic version of this algorithm [32, 33, 18] consti-
Forhighlystretchedquadrilateraclells,thisre-
tutes an unweighted graph algorithm. In this version
sultsin largeweightsbeingassociatewdith grid
of the algorithm, each time a seed point is chosen, all
edgesnormalto thedirectionof stretchinga, nd
neighboring points are agglomerated to this point.
smalledgeweightsin thedirectionparalletlo the
The directional agglomeration algorithm is based on
stretchinga,scanbeinferredfromtherelativesizes
a weighted graph technique. The edge weights are
ofthecontrovlolumefacesinFigure1.Howevefro,r defined in the same manner as for the line construc-
stretchedtriangulacrellst,hediagonaglridedgerse-
tion algorithm. Once a seed point is chosen, only
sultinweightswhichmaybecomparabilnethetwo
those neighboring points that are connected to the
directions.Thisweakerdecouplinogfthenormal
seed point through an edge of weight greater than
andstretchingdirectionfsortriangularelementisn
/3 × maxweight are agglomerated, where maxweigh_
twodimensioncsanproduceundesirablreesultsin
denotes the maximum edge weight incident to the
thelineandagglomeratioanlgorithms.Therefore,
seed point. Taking j3 -- 0.5 reproduces the isotropic
weemployquadrilateraellementinstwodimensions
agglomeration algorithm in regions were all edge
inregionosfhighmeshstretchinga,ndprismatic(or
weights are close in size. Howcver, in regions where
hexahedrael)lementisnhighlystretchedregionsfor
one edge weight is much larger than the others, a
threedimensionamleshesA. n alternateapproach
directional coarsening is achieved. This results in
wouldbetoemployadifferenctontrovlolumedefi-
a 2:1 coarsening ratio in such regions. In order to
nition,suchasacontainment-dubaalsedcontrovlol-
obtain a 4:1 coarsening ratio, the process must be
ume[31],andretainsimpliciaellementisnthesere- repeated. This will result in the agglomeration of
gionsa,lthoughthishasnotbeenattemptedtodate.
points or control volumes which were not originally
neighbors of the initial seed point. This type of ag-
Thelineconstructionalgorithmbeginsbypre- gressive coarsening can only be tolerated in regions
computintgheratioofmaximumtoaveragaedjacent where the implicit line solver is used as a smoother.
edgeweightforeachvertex.Theverticesarethen Therefore, the coarsening process is repeated only
sortedaccordingtothisratio. Thefirstvertexin if the agglomerated control volume is joined to the
currentseedpointbyanedgewhichispartofan
implicitline.Theprocesissrepeateduntilfourcon-
trolvolumesareagglomeratetodgethero,runtilno
lineedgescanbefound.
Fromtheabovedescriptionit,isevidentthatthe
lineconstructioanndcoarseninpgrocesasreclosely
coupledandmustbecarriedout simultaneously.
Theedgeweightso,ncedefinedonthefinestlevel,
arecomputedonthefly foreachcoarselrevelas
theyarecreatedT. hewholeprocesissperformeidn
apreprocessipnhgasea,ndtheoutput,consistinogf
setsoflinesforeachlevelandcoarsgeridgroupings,
ispassedtotheflowsolver.
FIG.3.
Directional Implicit Lines Constructed on
i,' ._ .... _w_:-,_:-- .-_.,_ ........ _.-_i_-4 1 Grid of Figure 2 by Weighted Graph Algorithm
F.,>.>_" -+;_,_1
FIG. 2. Unstructured Grid Used for Computation of
Transonic Flow Over RAE _822 Airfoil. Number of
Points = 16167, Wall Resolution = 10 -6 chords
As an example, the directional implicit agglomer-
ation multigrid algorithm has been applied to the FIG. 4. First Agglomerated Multigrid Level Con-
grid of Figure 2. The lines created on the finest structed on Grid of Figure 2 Illustrating _:1 Direc-
grid level are depicted in Figure 3. The first coarse tional Coarsening in Boundary Layer Region
agglomerated level is illustrated in Figure 4, depict-
ing the agglomerated cells in the boundary-layer re-
gion near the leading edge, where a 4:1 directional
coarsening is observed. Table 1 documents the com-
plexity of the coarse grid levels using the isotropic
agglomeration algorithm of [18], as well as the coarse
grid complexity achieved using the current direc-
tional agglomeration multigrid algorithm. The re-
sulting complexity for a multigrid W-cycle is just
15 % larger for the directionally agglomerated grids
than for the isotropically agglomerated grids.
'. i ; ,' / '...'\ ', _ / /
Regular AMG Directional AMG
" i _/ " :: _\ \ k _ / I /
" _- \ !:///i : i_'_X\ i I p
Mesh Level Nnode Ratio Nnode Ratio
_'_:: "it',,:_',/:?/)/;'/.--C"|P"l_\ _ ! _,,
i )ri, _......... :-:-:'.-_-_'_
16167 1 16167 1
4074 3.99 4476 3.61
'- _-',_'k\',"_J / : x
1038 3.92 1383 3.24 --_/:"!} ,.",..._/ ,/ ! "\ ,/
" " : ' ..... ' I ' /
268 3.87 585 2.36
'.
'y :' '\
W-Cycle
Complexity 1.89 2.18
FIG. 5. Computed Mach Contours on Grid of Figure
2. Mach= 0.73, Incidence = 2.31 degrees, Re = 6.5
Table 1: Comparison of Coarse Grid Complexity and
million
Resulting W-cycle Complexity for Regular Isotropic
Agglomeration and Directional Agglomeration Multi-
grid
6. Two Dimensional Results __ l,e-05 NORMAL SPACING
__ _ I.e-06NORMAL SPACING
The combined directional-implicit agglomeration ....... !.e-07 NORMAL SPACING
multigrid algorithm produces convergence rates in-
dependent of the degree of grid anisotropy. This is
_1.. EXPLICIT FULL COARSENING MG
demonstrated in two dimensions by solving the tran- "__ _::'"--:........................................
sonic flow over an RAE 2822 airfoil on three different
grids. All three grids contain the same distribution _.
of boundary points, but different resolutions in the
direction normal to the boundary and wake regions. _
The first grid contains a normal wall spacing of 10-5
chords, and a total of 12,568 points, while the second
grid contains a normal wall spacing of 10 -6 chords,
and 16,167 points, and the third grid a normal wall
spacing of 10-7 chords, and 19,784 points. The cells
in the boundary layer and wake regions are gener- I ' . _ ] "-it,
100 20_ 300 400 _0 6(]0
ated using a geometric progression of 1.2 for all three
Number of MG Cycles
grids. The second grid, depicted in Figure 2, con-
tains what is generally regarded as suitable normal FIG. 6. Comparison of Explicit Isotropic and
and streamwise resolution for accurate computation Directional-Implicit Agglomeration Multigrid Algo-
of this type of problem, while the first and third grids rithm Convergence Rates on 3 Grids of Varying Nor-
mal Resolution
are most likely under-resolved and over-resolved in
the direction normal to the boundary layer, respec-
tively.
cretization. The use of more accurate matrix dissi-
pation with the explicit full-coarsening multigrid al-
__ MACH NUMBER =0.73
gorithm produces slower and less robust convergence
___ MACH NUMBER =0.I rates. The directional implicit agglomeration algo-
....... MACH NUMBER =0.01
rithm operates on the matrix dissipation discretiza-
tion and uses a three stage time-stepping scheme
_ NON-PRBC_NDIT[ONED IMPLICrr MG with no residual smoothing but with point- or line-
preconditioning where the jacobians are evaluated
_" 'x .............'...."-:,
and inverted only at the first stage of the scheme and
then frozen for the remaining stages. Although Fig-
.....\_._ .. ......."."............ure 6compares the two schemes in terms of multigrid
cycles, the cost per cycle of both schemes is relatively
close, the directional implicit agglomeration scheme
being about 15% more expensive per cycle, which is
mainly due to the added work for the evaluation of
the matrix dissipation.
The benefits of low Mach-number preconditioning
w lOO 2{)0 3OO 4OO 5O0 6OO
Number of MG Cycles are demonstrated in Figure 7, where the flow over
an RAE 2822 airfoil at varying Mach numbers has
FIG. 7. Comparison of Low-Mach Number Precon-
been computed on the grid of Figure 2 using the di-
ditioned and Unpreeonditioned Directional-Implicit
rectional implicit agglomeration algorithm with and
Agglomeration Multigrid Algorithm Convergence
without the low Mach number preconditioner. For
Rates for Various .freestream Mach Numbers
the transonic case, the preconditioner is not active,
and both cases give identical convergence. However,
as the Mach number is lowered, the convergence rate
The Mach number for this case is 0.73, the incidence
degrades substantially for the cases run with no pre-
is 2.31 degrees, and the Reynolds number is 6.5 mil-
conditioning, while the preconditioned cases all con-
lion. The computed solution on the grid with nor-
verge to machine zero in approximately 300 cycles.
mal wall spacing of 10-6 chords is depicted in Figure
This example demonstrates the importance of em-
5. The flow is transonic and the low Mach number
ploying both techniques simultaneously (low-Mach
preconditioning matrix reverts to the identity ma-
number preconditioning and directional implicit ag-
trix for this case. With the effect of this precondi-
glomeration) in order to obtain rapid convergence
tioning removed, a more direct comparison between
rates for subsonic Navier-Stokes flows.
the directional implicit multigrid and the previously
developed unpreconditioned full coarsening multi-
The computation of high-lift flows simultaneously
grid method [18, 19] is possible. The convergence
involves regions of low velocity fluid and high grid
rates of both methods on all three grids are shown
anisotropy, therefore providing a good demonstra-
in Figure 6. The explicit full coarsening multigrid
tion of the current algorithm. Figure 8 depicts an
solver produces convcrgcnce rates which decay sub-
unstructured grid about a three-element airfoil high-
stantially as the grid stretching is increased. In
lift configuration. The grid contains a total of 61,104
fact, the asymptotic rate of this scheme for the most
points and a normal spacing of 10-6 chords at the
highly stretched grid is almost two orders of magni-
surface of each airfoil element. The implicit lines
tude slower than that achieved on the least stretched
generated by the graph algorithm for this case are
grid. On the other hand, the directional-implicit
depicted in Figure 9, and a qualitative view of the
agglomeration scheme produces convergence to ma-
solution as a set of Mach contours is given in Fig-
chine zero in under 600 cycles and is essentially un-
ure 10. The freestream Mach number for this case
affected by the degree of grid anisotropy. This com-
is 0.2, the incidence is 16 degrees, and the Reynolds
parison represents the best possible performance for
number is 9 million. The convergence rates of the
each scheme. The explicit full-coarsening multigrid
directional-implicit agglomeration scheme and the
algorithm employs a five stage time-stepping scheme
explicit full-coarsening agglomeration scheme are
which is augmented with implicit residual smooth-
compared on the basis of CPU time in Figure 11.
ing and is used to solve a scalar dissipation dis-
The explicit full-coarsening scheme employs a five