Elliptic grid generation methods have been used for many years to smooth and improve grids generated by algebraic interpolation schemes. However, the elliptic system that must be solved is nonlinear and convergence is generally very slow for large grids. In an attempt to make elliptic methods practical for large three-dimensional grids, a two-stage implementation is developed where the overall grid point locations are set using a coarse grid generated by the elliptic system. The coarse grid is then interpolated to generate a finer grid which is smoothed using only a few iterations of the elliptic system

Mastin, C. Wayne

English

NASA Technical Reports Server

N95- 28765
MULTILEVEL ELLIPTIC SMOOTHING OF
LARGE THREE-DIMENSIONAL GRIDS
C. Wayne Mastin*
Mail Stop 125
NASA Langley Research Center
Hampton, VA 23681
SUMMARY
Elliptic grid generation methods have been used for many years to smooth and improve grids
generated by algebraic interpolation schemes. However, the elliptic system that must be solved is
nonlinear and convergence is generally very slow for large grids. In an attempt to make elliptic
methods practical for large three-dimensional grids, a two-stage implementation is developed where
the overall grid point locations are set using a coarse grid generated by the elliptic system. The
coarse grid is then interpolated to generate a finer grid which is smoothed using only a few iterations
of the elliptic system.
INTRODUCTION
Elliptic grid generation methods have become less applicable to large scale problems due to the
time required to solve the elliptic system of partial differential equations. The equations themselves
are nonlinear and are difficult to solve efficiently even using the traditional multigrid methods.
Although there are some variations of the equations, this report assumes the elliptic system is of
the form
gll(r_ + P r_) + g22(rvv + Q rn) + g33(r_¢ + R r¢) + g12r_n + g13r_c + g23rn_ = 0
where r = (x, y, z), gij are the contravariant metric tensor components and the functions P, Q, and
R are used to control the distribution of grid points. The objective of this report is to demonstrate
that in many cases it is not necessary to solve the elliptic system to generate a smooth grid with the
required grid point distributions. If a coarse grid is first generated by solving the elliptic system,
then this grid can be interpolated to generate a finer grid and the fine grid can be smoothed with
only a few iterations of the elliptic difference equations. If this procedure is to work in practice, it is
essential that the interpolated grid be smooth and give a good approximation of the final solution
of the elliptic system on the fine grid. Thus, the fine grid iterations are primarily used to eliminate
interpolation errors which are local and of high frequency. The actual residual on the fine grid may
not be close to zero. This multilevel approach is efficient if only a few fine grid iterations are to
give a smooth grid. It is well known that if the initial grid deviates greatly from the final elliptic
grid, the first few iterations may generate large scale oscillations which decay very slowly. The
multilevel approach can also be used to generate grids with specified boundary orthogonality and
spacing. Coarse grid computations can be used to generate good initial approximations of control
functions which may be fixed or further adjusted during the fine grid iterations.
*This work was performed while the author held a National Research Council - NASA Langley Research Center
Research Associateship.
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https://ntrs.nasa.gov/search.jsp?R=19950022344 2020-06-16T07:36:56+00:00Z
Thetechniquesdescribedin this reportcanbeeasilyimplementedin existingsoftwarepackages.
Coarsegrids solutionshavebeenusedto generatestarting valuesfor the elliptic systemin the
GRIDGENcodeof Steinbrennerand Chawner(ref. 1) andthe 3DGRAPEcodeof Sorenson(ref.
2). Eisman(ref. 3) hasalsoemployeda similar conceptin usingelliptic systemsto distribute
controlpointsfor hisalgebraicgrid generationschemein the the GRIDPROcode.
It shouldbenotedthat the proposedschemeis not atruemultigridscheme.Theschemehereis
a simplegrid sequencingschemeprogressingfrom a coarsegrid to progressivelyfinergrids.There
is no cycling between coarse and fine grids with the objective of eventually achieving a higher rate
of convergence to the solution of the elliptic system of partial differential equations on the finest
grid. Multigrid methods have been shown to increase convergence rates, but there have been no
applications to large systems with moving boundary points and adaptive control functions. Some
difficulties that may arise are discussed in the paper by Stiiben and Linden (ref. 4).
COARSE GRID ITERATIONS
The first step in the procedure is to coarsen the initial algebraic grid. This scheme that has
been used here is to remove every other grid point in each coordinate direction. This refinement
may proceed to several levels as long as the grid dimensions in each direction are odd integers.
For large grids, this reduces the size of the problem by approximately eight. A second coarsening
would reduce the original problem by a factor of approximately 64. Complementing the reduction in
problem size is an increase in the rate of convergence when going to a coarser grid. The coarse grid
iterations should be optimized for rapid convergence. For example, all coarse grid calculations here
have used locally optimal acceleration parameters. Any control function and boundary condition
options should be implemented so that the converged coarse grid has all of the desired characteristics
of the final grid.
INTERPOLATION
The interpolation scheme is a critical component in this procedure. Therefore, a tricubic Her-
mite interpolation procedure has been developed to generate a smooth grid. The slope information
is calculated using central differences on the coarse grid. There are two options for the bounding
surface grids. Either the original surface grids can be used or the surface grids can be redefined
using the same interpolation scheme used at the interior points. The choice can have a significant
effect on the success of the coarse grid solution in reducing the amount of work needed to obtain
a usable grid. If the coarse grid accurately resolves the surface so that there is little change is
curvature or spacings between grid points, then one can generally use the original fine surface grids
and still have a reasonably smooth grid to start the elliptic system on the fine grid. On the other
hand, if there are significant changes in surface or grid properties between the interpolated grid
and the original surface grids, then using the original grids on the boundary surfaces will result in
large changes in grid spacings and angles at the boundary, and possibly even some places where the
interpolated grid folds over the boundary. This would result in a poor starting grid for the elliptic
system and the main purpose in using the coarse grid solution would be lost.
The ability to prescribe grid distributions on the final grid is very dependent on the control
functions of the elliptic system. There are two options for calculating control functions which seem
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to workequallywell. The controlfunctionscaneither becalculatedbasedon the initial finegrid
andthenrestrictedto eachcoarsergrid, or thecontrolfunctionscanbecalculatedoneachgrid from
restrictionsof the initial grid to that level.It is generallyrecommendedthat the controlfunctions
notbecomputedon thecoarsestgrid andinterpolatedto thefinergrids.This will oftenresultin a
lossof distributionon thefinergrid. Theonly timethat controlfunctionshavebeeninterpolatedto
finergirdsis whenusingthe controlfunctionsto controlboundaryspacingan orthogonality.Even
then,the interpolatedcontrolfunctionsareblendedwith thecontrolfunctionsfrom theinitial grid
sothat the interpolatedvaluesareonly effectivenearthe boundary.
There is oneimportant fact that shouldbe emphasizedwhentransferringcontrol functions
betweenfineand coarsegrids. When the control functionsare restrictedto a coarsergrid, then
theyshouldalsobemultipliedbya factorof two. This scalingfactoris necessarysothat the same
ellipticsystemis approximatedonboth thecoarseandfinegrids. Conversely,if thecontrolfunctions
aretransferredfrom a coarseto a finergrid, thenthe controlfunctionsshouldbeinterpolatedand
dividedby two. This assumesthat the grid coarseningis doneby removingeveryotherpoint in
eachcoordinateddirection.Othercoarseningschemeswouldresultin differentscalingfactors.For
example,if onlyeverythird point wasretainedin generatingthe coarsegrid, thenthe factorof two
wouldbe replacedby three.
FINE GRID ITERATIONS
At thispoint it is assumethat thecoarsegrid iterationsandinterpolationprocedurehasresulted
in a smoothgrid that hasthe desireddistribution of grid points. In manycasesthis grid would
begoodenoughto computea CFDsolution.However,theremaybea fewripplesin thegrid due
to the Hermiteinterpolation. If the originalboundarysurfacesaremaintained,the grid mayneed
someadditionalsmoothingneartheboundary.Sincethe objectivehereis onlyto smooththegrid
and not to obtain convergenceof the elliptic system,thereshouldbe a changein the relaxation
parameterso that the iterationsareunderrelaxed.Another effectivewayof smoothingthe grid
whilemaintainingthe existingdistribution of grid points is to introducea time derivativeinto
the partial differentialequations,and solvethe resultingparabolicsystemusinga time marching
method.
EXAMPLES
Threesamplegridswill beconsideredto test the conceptsof this report. All threeinitial grids
weregeneratedusingtransfiniteinterpolation. All threealsoareobviouslynot suited for CFD
computationsbecauseof negativeJacobiansor extremelyskewedcells.After applyingthe elliptic
smoothing,in twoof thethreecasesthefinal grid wasfreeof negativeJacobians.In theothercase,
a fewnegativeJacobiansremainedafter usingthe elliptic methods,evenwhenconvergenceof the
elliptic systemwasattemptedon theoriginalfinegrid.
The first exampleis thegrid in the interior of aduct. Theduct is plotted in Figurel(a). The
crosssectionis very irregular.An initial grid wasconstructedwith dimensionsof 33by 33by 65.
Therearealargenumberof grid pointswhichfall outsideof the ductascanbeseemin the plot of
an interior grid surfacein Figurel(b). Thisgrid canbe improvedusingelliptic methods,sothat
nonegativeJacobiansappear.In fact, negativeJacobianscanbeeliminatedwithout resortingto
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grid coarseningin only seveniterations.However,it takesmanymoreiterationsto givea smooth
grid. The negativeJacobiansin the initial grid causelargevaluesof the residualin the solution
of the elliptic system.Nonellipticphenomenalike ripplesandwavescandevelopin the numerical
solution.Theseperturbationsmaybeoflargemagnitudeanddecayveryslowlyduringtheiterative
solution.Forexample,the plot in Figurel(c) is the sameinterior grid surfaceafter ten iterations
of the elliptic system.Ripplesin the grid appearastriangular shapedcellsin the lowerpart of
the surfacegrid. Nowconsiderthe casewherethe originalgrid is coarsenedto givean 17by 17
by 33grid. Theelliptic systemcanbeconvergedon this grid in lessthan sixtyiterations,whichis
equivalento lessthat eightiterationson thefinal grid. Thiscoarsegrid is interpolatedandfurther
smoothedto eliminateanyroughspotsnearthe boundary.For comparisonwith Figurel(c), ten
iterationson the finegrid wasperformedafter convergenceon the coarsegrid, and the resulting
grid plotted in Figure l(d). As a further comparison,the originalgrid wasusedto calculatea
convergedsolutionof the elliptic systemandit tookabouttwiceasmanyiterationsto convergeon
thefinegrid asit did on thecoarsegrid. Thus,whenconsideringboth thesizeof thegrid andthe
convergencerate,it canbeconcludedthat the useof coarsegrid iterationsresultedin a reduction
in workby afactorof fiveoverwhatwouldbeneededto generateaconvergedellipticallygenerated
grid. Zerocontrolfunctionswereusedin all of thesecalculations.Thegrid points werefixedon
the wallsof the duct,but wereallowedto slidealongthetwo circularendcaps.
The secondexampleis a grid for a regionabout an aircraftwing. The initial grid is a C-grid
constructedusingtransfiniteinterpolation.The grid wasconstructedasa four blockgrid eachof
whichwas33by 53by 33for a total of slightly over230thousandgrid points. Theedgesof the
blocksareillustrated in Figure2(a) Thereareno negativeJacobiansin this grid, but the grid is
highlyskewedin regionsnearthewingtip ascanbeseenin Figure2(b). Severaloptionshavebeen
exercisedin thisexamplewhichtend to reducetherateof convergencefor the elliptic system.The
controlfunctionshavebeeninterpolatedfromtheblockboundaries(usingtheThomas- Middlecoff
technique)to maintainthe interiorgrid point distributionduringthesolutionof theelliptic system.
Thecontrolfunctionshavebeenallowedto adjustnearthe surfaceof thewing (asin theGRAPE
code)to generateorthogonalgrid linesat the wing surface.Finally, the grip pointshavebeen
allowedto float alongthe two planarboundarysurfacesintersectingthe endsof the wing. An
indicationof the slowrate of convergenceis evidentin Figure2(c). After 10 iteration, starting
with the original algebraicgrid, there is little differencebetweenthe grid generatedusingthe
elliptic difference quationsandthe originalalgebraicgrid. Thereis someindicationof boundary
orthogonality. However,thereis considerabledifferencewhenexaminingFigure 2(d) whichwas
generatingusing100coarsegrid iterationsfollowedby 10finegriditerations.Thecontrolfunctions
weretreatedasdescribedabove.Thecontrolfunctionswerefirst interpolatedfrom the coarsegrid
to thefinegrid anddividedbytwo,sincethecoarsegrid wasgeneratedby takingeveryotherpoint
of the finegrid. Thesecontrolfunctionswerethenblendedwith the controlfunctionscomputed
from the initial interpolatedgrid. Forthis example,the originalsurfaceswereusedwith the fine
grid. It wasthereforenecessaryto continueto adjustthe controlfunctionsduring the finegrid
iterationsto correcttheslight skewnessat the boundarywhichresultedfrom theinterpolation.
Thefinal exampleis includedto demonstratethat this methodcanbeappliedto a grid with
highlynonuniformspacing.The initial grid wasagainconstructedusingtransfiniteinterpolation.
Thegrid wasto beusedto computeviscousflowaboutanHSCTconfiguration.Thewing/fuselage
configurationisplotted in Figure3(a). Sincethe HSCTconfigurationwassymmetric,only half of
the body wasusedto generatethe grid. The grid wasconstructedin two blocks,eachof which
was177by 81by 61. Thusthe total grid consistsof nearlyoneandthree-quartermillion points.
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Therearea largenumberof negativeJacobians. A comparison of the initial grid and the elliptic
grid after 100 coarse grid iterations appears in Figures 3(b) and 3(c). For clarity, only the coarse
grid points are plotted. The elliptic method was able remove most of the negative Jacobians and
still maintain the spacing at the boundary. However, it could not remove all negative Jacobians
and generate a grid which would be suitable for CFD calculations. While this may be considered
a failure of elliptic methods, it is a successful application of the multilevel method. Knowing when
and where elliptic methods fail to generate a suitable grid will allow the user to proceed with his
efforts in redefining the topology or redistributing points.
CONCLUSIONS
The discussion and examples contained in this report should give the grid generator a guide for
using coarse grid iterations for smoothing and improving computational grids for CFD applications.
The main point is that the coarse grid solution should be converged to some specified tolerance.
After that, only a few fine grid iterations are needed. It is also important to treat the control
functions correctly to generate the desired distribution of grid points along boundary surfaces.
Simply interpolating these functions from the coarse grid to the fine grid is generally not sufficient.
There is one area where further study is needed. There should be some way of projecting an in-
terpolated volume grid onto the original boundary surfaces without effecting the overall smoothness
and orthogonality of the grid. One possible approach would be to include boundary information
into the interpolation scheme.
REFERENCES
.
.
°
,
Steinbrenner, J. P.; and Chawner, J. R.: " Recent Enhancements to the GRIDGEN Structured
Grid Generation System," in Software Systems for Surface Modeling and Grid Generation,
NASA CP-3143, NASA Langley Research Center, April 1992, pp. 253-271.
Sorenson, R. L.: The 3DGRAPE Book: Theory, Users' Manual, Examples, NASA TM-
102224, NASA Ames Research Center, July, 1989.
Eisman, P. R.; Lu, N.; Jiang, M.; and Thompson, J. F.: "Algebraic - Elliptic Grid Genera-
tion", in Numerical Grid Generation in Computational Fluid Dynamics and Related Fields,
Weatherill, N. P.; Eisman, P. R.; H_user, J.; and Thompson, J. F. (eds.), Pineridge Press,
Swansea, 1994, pp. 37-48.
Stiiben, K.; and Linden, J.: "Multigrid Methods: An Overview with Emphasis on Grid
Generation Processes", in Numerical Grid Generation in Computational Fluid Dynamics,
H£user, J.; and Taylor, C. (eds.), Pineridge Press, Swansea, 1986, pp. 483-509.
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dFigure1. (a) Duct geometry,(b) initial surfacegrid, andsurfacegridsafter 10iterations
startingwith (c) initial grid and(d) convergedcoarsegrid solution.
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a b
d
Figure 2. (a) Region about wing, (b) initial surface grid at wing tip, and surface grids after
10 iterations starting with (c) initial grid and (d) converged coarse grid solution.
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bFigure3. (a) HSCTconfiguration,and(b) initial and(c) elliptic coarsegrid
at wingfuselagejunction.
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Multilevel elliptic smoothing of large three-dimensional grids

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