ICAS 2000 CONGRESS

244.1

A SOLUTION ADAPTIVE TECHNIQUE USINGTETRAHEDRAL UNSTRUCTURED GRIDS

Shahyar Z. PirzadehNASA Langley Research Center

Hampton, Virginia, U.S.A.

Keywords: unstructured grids, grid adaptation, remeshing

Abstract

An adaptive unstructured grid refinementtechnique has been developed and successfullyapplied to several three dimensional inviscidflow test cases. The method is based on acombination of surface mesh subdivision andlocal remeshing of the volume grid. Simplefunctions of flow quantities are employed todetect dominant features of the flowfield. Themethod is designed for modular coupling withvarious error/feature analyzers and flowsolvers. Several steady-state, inviscid flowtest cases are presented to demonstrate theapplicability of the method for solvingpractical three-dimensional problems. In allcases, accurate solutions featuring complex,nonlinear flow phenomena such as shockwaves and vortices have been generatedautomatically and efficiently.

1 Introduction

Generation of efficient grids for thecomputational fluid dynamics (CFD)applications usually requires some priorknowledge of the flow behavior in order tomatch the grid resolution to the essentialfeatures of the problem. While suchinformation is usually unavailable in advance,a number of 'trial-and-error' iterations betweenthe solution and grid generation are oftenrequired to tailor the grid to the specific nature

of the problem at hand. Alternatively, an overlyfine grid is often generated to guarantee thedesired solution accuracy. In both cases, theamount of time, effort, and computationalresources may become excessively large forsolving complex problems.

The solution adaptive grid technology is apowerful tool in CFD, which provides threeimportant benefits: automation, improvedefficiency, and increased solution accuracy.Since the distribution of grid points areefficiently determined by an automatic process,an adapted grid contains far fewer number ofpoints than an initial fine grid having similarlocal resolution at the crucial regions. Thisimportant feature of grid adaptation results in asubstantial saving in computational time andmemory requirement.

In general, most adaptive methods fall intothree broad categories: grid movement (r-refinement), grid enrichment (h-refinement), andlocal solution enhancement (p-refinement).While the methods in the first two classes modifythe grid density to improve the solution accuracy(grid adaptation), those under the third categoryenhance the order of numerical approximation atlocations where the solution undergoes abruptvariations (solution adaptation). Most adaptivetechniques used in the CFD applications fall intothe first two classes.

In the grid movement approach, nodes areredistributed and moved towards regions where ahigher degree of accuracy is needed. Since thegrid topology remains unchanged throughout thegrid adaptation, the process of grid movement

Copyright © 2000 by the National Aeronautics and SpaceAdministration. Published by the International Council ofthe Aeronautical Sciences, with permission.

Report Documentation Page Form ApprovedOMB No. 0704-0188Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, ArlingtonVA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number.

1. REPORT DATE SEP 2000

2. REPORT TYPE N/A

3. DATES COVERED -

4. TITLE AND SUBTITLE A Solution Adaptive Technique Using Tetrahedral Unstructured Grids

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Research Center Hampton, Virginia, U.S.A.

8. PERFORMING ORGANIZATIONREPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR’S REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited

13. SUPPLEMENTARY NOTES See also ADM002333. International Congress of Aeronautical Sciences (ICAS) (22nd) Held in Harrogate,United Kingdom on August 27-September 1, 2000. U.S. Government or Federal Purpose Rights License,The original document contains color images.

14. ABSTRACT An adaptive unstructured grid refinement technique has been developed and successfully applied toseveral three dimensional inviscid flow test cases. The method is based on a combination of surface meshsubdivision and local remeshing of the volume grid. Simple functions of flow quantities are employed todetect dominant features of the flowfield. The method is designed for modular coupling with variouserror/feature analyzers and flow solvers. Several steady-state, inviscid flow test cases are presented todemonstrate the applicability of the method for solving practical three-dimensional problems. In all cases,accurate solutions featuring complex, nonlinear flow phenomena such as shock waves and vortices havebeen generated automatically and efficiently.

15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT

SAR

18. NUMBEROF PAGES

13

19a. NAME OFRESPONSIBLE PERSON

a. REPORT unclassified

b. ABSTRACT unclassified

c. THIS PAGE unclassified

Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

Shahyar Z. Pirzadeh

244.2

can be simply incorporated into the solver in amodular fashion. In addition, no data transfer(i.e., interpolation among different grids) isrequired since the grid structure remains intactduring the process. Therefore, no solutionaccuracy is lost from one adaptation cycle tothe next. The method is especiallyadvantageous for transient problems involvingmoving surfaces and unsteady solutions.However, since the number of grid nodesremains constant, transferring nodes from onepart of the grid to another may cause local'depletion' of grid elements, and thus severedistortion of the grid may be introduced [1].Adaptation by grid movement has primarilybeen applied to structured grids and 2Dtriangular meshes.

In the grid enrichment technique, morenodes are added to regions where higheraccuracy of the solution is desired. Nodes canalso be removed from locations where thesolution is smooth and requires less gridresolution. Due to node addition or deletion,the topology (connectivity) of the grid changesfrom one adaptation cycle to another.Consequently, interpolation of data betweenconsecutive grids is required which curtailsthe applicability of these methods for unsteadyproblems. Adaptive methods by gridenrichment are particularly attractive for theirflexibility, especially when applied inconjunction with unstructured grids.

Among the adaptive grid methods byenrichment, two techniques are notable: gridsubdivision and grid remeshing. In gridsubdivision, 'parent' cells are divided intoseveral smaller cells. The method is efficientand fast. Once a systematic data structure isestablished prior to the adaptation cycles, bothgrid refinement and coarsening can easily beimplemented. The grid subdivision methodshave been best demonstrated on Cartesianmeshes [2] and can be implemented intriangular grids conveniently. However, theirapplication to tetrahedral grids involves

complex data structures and, in most cases,results in refinement complications and griddistortion [3].

Global and partial remeshing have also beenemployed for adaptive grid refinementsuccessfully [4,5]. Two significant advantagesof these methods are 1) flexibility for refinementand unconstrained coarsening (in subdivisionmethods, for example, grids cannot be coarsenedbeyond their initial resolutions) and 2) goodquality grids generated in each refinement cycle.On the other hand, the grid generation time andthe cost of solution interpolation are extensive inthese methods, especially in the globalremeshing.

As there is no single grid type or generationmethod (e.g., structured, unstructured, etc.) to fitall classes of CFD problems, neither is anindividual adaptive methodology which can beuniversally applied to a variety of problems.Different methods offer certain advantages todifferent classes of grids and problems [2,5].Therefore, it is beneficial to exploit theadvantages of several techniques in a hybridadaptive grid method for solving complexproblems [1]. In the present work, an attempthas been made to combine the efficiency of h-refinement and the flexibility of remeshing forsolution adaptive refinement. A grid movementtechnique has also been developed (not presentedin this paper) for geometric adaptation of volumegrids to moving or deforming surfaces (see Ref.6). The focus of this paper is on the refinementmechanism aspect of the solution adaptiveproblem as applied to realistic 3D problems.

2 Approach

The proposed grid adaptation strategy issummarized in the flowchart shown in Fig. 1. Inthis chart, each block represents an independentmodule readily exchangeable in the system.Starting with a reasonably coarse mesh and acorresponding flow solution, the grid adaptationprocess proceeds with an assessment (analysis)

A SOLUTION ADAPTIVE TECHNIQUE USING TETRAHEDRAL UNSTRUCTURED GRIDS

244.3

of the current flow solution to determinewhere in the field the solution needs furtherimprovement. For a successful adaptation, theinitial coarse grid should be adequatelyresolved in order to prevent the adaptivesolution from converging to a 'wrong' solution.Once the locations requiring solutionimprovement are identified, the correspondinggrid elements are flagged for resolutionenhancement. A local remeshing strategy isthen employed to refine the grid at theselocations. A new solution is next obtained onthe modified mesh followed by anothersolution assessment. The process continuesuntil a preset goal such as a desired level ofsolution accuracy, the optimization of anobjective function, or simply a certain numberof adaptation cycles is accomplished.

There are two main components in anygrid adaptation technique. First, a strategy isemployed to determine where in the field the

grid (solution) needs modification, e.g., bymeans of error estimation, flow feature detection,or any other type of solution analysis. Secondly,a mechanism is utilized to either change the griddensity or modify the solution method. Thefocus of this paper is mainly on the latter, i.e. agrid alteration procedure that is automaticallycontrolled by the flow solution characteristics.Although it is not the objective of this paper toelaborate on the former topic, which is thesubject of a separate paper, a brief discussion oferror estimation and flow feature detection ispresented for completeness.

2.1 Error and Flow Feature DetectionIn most grid adaptation techniques, the questionof where to modify the grid resolution or thesolution accuracy is addressed through theconcept of 'error equidistribution'. This principlestates that grid nodes should be clustered in sucha way that the computational errors are uniformlydistributed throughout the grid. In other words,the grid should be proportionally denser wherethe solution incurs larger error, e.g., where theflow undergoes rapid changes, and vice versa.The principle of error equidistribution is strictlyapplied in the methods by r-refinement and theglobal remeshing techniques to redistribute gridpoints in the field optimally. The magnitude ofthe computed errors directly determines the gridspacing parameters in these methods.

In the methods based on h-refinement,however, error estimation practically serves as ameans to locate the grid elements experiencinglarge computational errors [7]. A separatemechanism modifies the distribution of gridnodes at these locations without considering themagnitude of errors. Unlike the r-refinementwhich aims for an optimum grid with equalerrors at the nodes, the h-refinement fulfills theobjective of reducing the maximum error throughseveral preset refinement steps. Theoretically, asthe number of h-refinement cycles increases, thedistribution of errors approaches an equilibriumstate throughout the grid. In practice, however,

Figure 1. Flowchart of the proposed gridadaptation strategy.

Stop

Adaptationconvergence

?

New grid

Solution/erroranalysis

Local gridrefinement

yes

Initial grid

no

Gridsmoothing

Solutioninterpolation

Flowsolution

Shahyar Z. Pirzadeh

244.4

only a few cycles of refinement are usuallyperformed to adapt the grid. Therefore, therole of error estimation for h-refinementreduces to indication of computational errorsinduced by the dominant flow features.

Many error (feature) indicators in use arebased on some physical flow quantities suchas density, pressure, entropy, etc. Functionsof the first or second gradients of thesequantities are usually employed to estimateerrors or detect dominant flow characteristics.For h-refinement, even a crude indicator suchas a simple increment of a flow quantity issufficient as long as it correctly detects thedesired flow feature. In this work, thefollowing simple indicator is employed todetect expansion and compression waves.

ϑι = (1 + δi / δa) |∆pi| / pi (1)

where pi and ∆pi are the local static pressureand its increment associated with the ith gridelement, respectively, δi is the local gridspacing, and δa is an average spacing in thegrid. The inclusion of the grid-spacingcorrection factor in Eq. 1 results in a betterdetection of weak flow discontinuities inlarger grid cells that are away from thegeometry.

Functions based on vorticity or entropyhave been used as indicators for detectingvortices and adapting grids to vortical flows.In this work, the following simple measure ofentropy is used to capture vortices.

εi = (γ pi / ρiγ) -1 (2)

where ρ i is the local density and γ is thespecific heat ratio. It should be emphasizedthat the parameters defined by Equations (1)and (2) do not represent the magnitude ofcomputational errors but only indicate thelocation of dominant flow features inducing

errors. In practice, the flow feature indicatorsmeasured at each grid element are compared withsome threshold constants prescribed by the user.If the value of an indicator is greater than thethreshold, the corresponding grid element isflagged for refinement.

The challenge in the practicalimplementation of an adaptive method forsolving complex problems is the choice ofappropriate error or indicating functions. Whilea particular indicator may work well for certainclass of flow features, it may not be as effectivein recognizing other flow phenomena. Usually, aprior knowledge of the flow characteristics isneeded in order to select relevant functions.Since the information about the flow is generallynot available in advance, an automatic indicatorbased on a global objective (e.g., drag reduction)is desirable. Such a universal indicator should beable to capture all relevant flow features thatinfluence the objective function (e.g., shockwaves, vortices, etc.) and should even determinethe flow characteristics that contribute to theformation of these flow features. In addition, theindicator must be 'smart' enough to distinguishbetween the actual flow variations and thenumerical ‘noise’ present in the solution.Otherwise, the grid may be refined in the wronglocations. The development of an automaticuniversal indicator requires comprehensiveresearch, which is beyond the scope of thepresent project and this paper. Further in-depthstudy of the subject is planned for future work.Once such a capability becomes available, it canbe readily incorporated into the present modularadaptation system.

2.2 Adaptive Local Mesh RefinementThe inherent irregularity of unstructured gridsoffers two important benefits: 1) high degree offlexibility to handle complex shapes and 2) easeof mesh alteration. The lack of a regularstructure in tetrahedral grids provides arbitrarycell groupings which, in effect, makes every partof a grid independent of the rest. Consequently,

A SOLUTION ADAPTIVE TECHNIQUE USING TETRAHEDRAL UNSTRUCTURED GRIDS

244.5

any section of a tetrahedral grid can beremoved and locally remeshed withoutdisturbing the rest of the grid. Furthermore,the local resolution of a grid can be arbitrarilychanged when the grid is partially remeshed.This important property makes unstructuredgrids particularly suitable for adaptive localremeshing.

In this work, an unstructured tetrahedralgrid generation system, VGRIDns [8], is usedto generate initial grids. The grid generationmethod is based on the Advancing-Front [9]and the Advancing-Layers [8] techniques.Both techniques are based on marchingprocesses in which tetrahedral cells grow onan initial front (triangular boundary mesh) andgradually accumulate in the field around thesubject geometry. The front, made of theexposed triangular faces of the tetrahedrons,continuously evolves and marches outward asnew cells are created and added in thecomputational domain. The process continuesuntil the entire domain is filled withtetrahedral cells when no triangular faceremains in the grid. The grid characteristics,used during the marching process, areprescribed through a set of source elementsincluded in a 'transparent' Cartesianbackground-grid [10]. The information is firstdistributed smoothly from the sources onto thebackground grid nodes by solving a Poissonequation and then interpolated in the field todistribute unstructured grid points during themarching process.

An important feature of the advancingfront technique, like any other marchingmethod, is that the solution process can berestarted at any time. Since a grid segment,once constructed, does not influence the restof the mesh yet to be generated, the processcan be stopped and restarted without"carrying" the grid portion already generatedin the previous run(s). The only data required

to restart the generation process are thosedefining the current front. The flexibility ofunstructured grids for local remeshing, alongwith the restart feature of the advancing frontmethod, offers an excellent opportunity for gridadaptation. An efficient grid restart capabilityand a local remeshing technique have previouslybeen developed and incorporated into theVGRIDns system for post-processing of thegenerated grids [11]. In this work, the existingcapabilities are extended for adaptive gridrefinement.

The process of local grid refinement isdemonstrated on a simple triangular grid in Fig.2. In this example, a transonic flow field arounda simple airfoil is assumed. An initial coarsegrid (Fig. 2a), along with a corresponding flowsolution, is supplied to the adaptive refinementscheme. An appropriate flow analyzer, such asthat given by Eq. (1), is used to detect thedominant flow features. For example, a diffusedshock wave and a large pressure gradient at theleading edge of the airfoil are assumed in thiscase. The grid cells experiencing large variationsin the flow properties, along with additionallayers of cells, are identified for removal (shadedtriangles in Fig. 2b). In the next step, the flaggedelements are deleted to create voids (emptypockets) in the mesh (Fig. 2c). The remaininggrid points and cells are then renumbered, andthe faces exposed in the pockets are grouped toform a new front in the grid. If any portion ofthe geometry is exposed in the voids, thecorresponding faces on the surface are h-refined,and the newly inserted nodes are projected ontothe geometry model. The rest of the front facesin the field remain unrefined to maintain acontiguous connectivity between the elements ofthe new grid segment (being generated in thepockets) and the original grid. Finally, the griddensity is readjusted in the pockets, and the voidsare remeshed by the Advancing-Front method asin a regular restart run (Fig. 2d).

Shahyar Z. Pirzadeh

244.6

The process of local h-refinement isdepicted in Fig. 3 for a portion of a 3Dtriangular surface mesh. In this figure, theshaded surface triangles are assumed coveredwith tetrahedral volume grid (Figs. 3a and b).The unshaded area represents a portion of thesurface mesh exposed in a void after asegment of the volume grid is removed at thelocation of a shock wave (Fig. 3c). To refinethe exposed triangles, new grid points are firstadded to the edges of the triangles (Fig. 3d).Each interior triangle is then divided into foursmaller triangles by connecting the new nodes(Fig. 3e). The "buffer" triangles (thoseadjacent to the unrefined region) are dividedinto two or three triangles, depending on theirnumber of edges on the pocket boundary.

Finally, the void is remeshed and filled withsmaller tetrahedrons (Fig. 3f).

Since the length of each surface mesh edgeis cut in half by the h-refinement, the spacingparameters defined by the background grid arealso reduced by 50% for regenerating the volumegrid in the voids. Therefore, every time a pocketis opened for remeshing, the newly generatedgrid portion becomes twice as fine as thesurrounding parent grid. The modified gridspacing provides the required compatibilitybetween the h-refined surface and locallyremeshed volume grids. To ensure a smoothvariation of grid resolution between the refinedcells and the surrounding parent grid, an averagegrid spacing is employed to generate the firstlayer of tetrahedral cells on the pocket walls.

refined gridopen pocket

flagged cellsflow discontinuity

(a) (b)

(c) (d)Figure 2. Adaptive refinement steps by local remeshing: (a) initial grid, (b) flagged cells in regions of rapid flowgradients, (c) removal of the flagged cells, and (d) locally refined grid.

A SOLUTION ADAPTIVE TECHNIQUE USING TETRAHEDRAL UNSTRUCTURED GRIDS

244.7

The average spacing is based on the actualsize of the interior front faces (those which arenot h-refined) and the modified backgroundgrid spacing. These “buffer” cells provide agradual transition from the coarse to the finecells generated in the pockets as shown in Fig.2d.

3. ResultsAdaptive solutions are presented in this paperfor three steady-state, 3D test cases: 1) anONERA M6 wing at transonic speed, 2) anexperimental high performance fighter aircraftat subsonic speed, and 3) an experimentalaerospace vehicle at supersonic speed. Eachof these flow cases represents a distinctaerodynamic feature suitable for the adaptivesolution. The examples clearly demonstratethe three benefits that grid adaptationprovides, i.e. accuracy, automation, andefficiency. All inviscid flow computations,presented in this paper, were performed using

the upwind, cell-centered, finite-volume,unstructured grid solver USM3D [12].

3.1 ONERA M6 WingAn ONERA M6 wing configuration has beenemployed to demonstrate the transonic shockcapturing capability of the present solutionadaptive grid method. The flow condition is at afree stream Mach number of 0.84 and anincidence of 3.06 degree.

A reasonably coarse grid with a nearlyuniform point distribution chordwise wasgenerated to serve as the initial grid foradaptation. The grid, shown in Fig. 4a, contains2,615 boundary nodes, 15,432 total nodes, and83,356 tetrahedral cells. An inviscid flowcomputation on this grid reveals the presence ofa weak "λ" shock wave on the upper surface ofthe wing. The surface pressure contours are alsoillustrated in Fig. 4a. As expected, the flow isunder-expanded on the upper surface at theleading edge, and the shock wave is diffused dueto coarseness of the grid.

(e)(d) (f)

(a) (b) (c)

Figure 3. Process of local surface mesh subdivision in three-dimension: (a) initial coarse grid, (b) footprint of flowdiscontinuity on the surface, (c) exposed triangles after a portion of the volume grid is removed, (d) insertion of newpoints on the edges of exposed triangles, (e) subdivision of the exposed triangles, and (f) final adapted grid.

Shahyar Z. Pirzadeh

244.8

Using the surface grid subdivision andlocal remeshing procedures described earlier,three levels of adaptive refinement wereperformed for this case. The indicator givenby Eq. (1) was employed to detect regions oflarge pressure variation. In each refinementcycle, cells indicating 20% or larger incrementin the pressure parameters were deleted andremeshed. The final grid contains 9,739boundary nodes, 54,385 total nodes, and288,739 tetrahedrons. Figure 4b shows theadapted surface grid and the correspondingpressure contours. As evident, the grid is

efficiently refined at the shock location and theleading edge of the wing at the so-called "suctionpeak" region. The effect of the automatic gridrefinement is clearly reflected on the surfacepressure contours, which show a sharp shockdefinition including minute details of thepressure gradient at the wing tip.

To investigate the effect of partial gridrefinement on accuracy of the adapted solutions,a globally fine grid with a resolution similar tothat of the adaptively refined grid was generated.This large grid contains 40,424 boundary nodes,394,155 total nodes, and 2,217,001 tetrahedrons.The surface grid and the corresponding pressurecontours are shown in Fig. 4c. A comparison ofthis solution with that of the adapted gridindicates that the differences between the two arenegligible, and that the grid adaptation hasproduced an identical result with about an orderof magnitude smaller grid size. Figure 5illustrates several chordwise distributions of thesurface pressure coefficient (Cp) for the initialcoarse, adapted, and unadapted fine grids ascompared with the experimental data at sixdifferent spanwise stations. As expected, thereare insignificant differences between the adaptedand the fine grid results. From the Cpdistributions, it appears that the result of thecoarse grid is in satisfactory agreement with theexperimental data at the shock locations.However, it is well known that inviscid solutionspredict stronger shocks further downstream asindicated by both the fine and adapted gridcurves in Fig. 5. Addition of viscous effectsusually weakens and moves shock wavesupstream to the correct locations. As illustratedin Fig. 5, the automatic refinement of the grid atthe leading edge has corrected the solution at thesuction peak area. Also, note that both segmentsof the λ shock wave are captured by the adaptiveand the fine grid solutions at the span stationy/b=0.8. Accurate computation of the flow atthis particular station is difficult due to itsproximity to the coalescing shock waves.

Figure 4. ONERA M6 wing surface grids andpressure contours at M∞=0.84 and α=3.06

o: (a)initial, (b) adapted, and (c) fine unadapted.

(a)

(b)

(c)

83,356 cells

288,739 cells

2,217,001 cells

A SOLUTION ADAPTIVE TECHNIQUE USING TETRAHEDRAL UNSTRUCTURED GRIDS

244.9

Figure 5. Chordwise distributions of surfacepressure coefficient for ONERA M6 wing at M∞=0.84and α=3.06o.

As mentioned earlier, a salient feature ofthe adaptive grid methods by remeshing is thequality of the refined grids they produce.Every time a portion of the grid is removed forlocal remeshing, new cells are regenerated inthe pockets without affecting the quality ofgrid elsewhere (unlike adaptation by gridmovement.) The accuracy of the presentedadaptive flow solutions and their ease ofgeneration substantiate the viability andquality of the adapted grids generated with thepresent method. Furthermore, this exampleunderscores the advantage of grid adaptation

for providing more accurate flow solutionseconomically.

The initial grid/solution as well as the adaptedresults were all generated using a SiliconGraphics Octane workstation with a R10000processor. While the mesh for the fine grid wasalso generated on the same workstation, thecorresponding flow computation was performedon a CRAY C90 supercomputer due to its largememory requirement. A converged solution onthe fine grid took about 36,548 CPU seconds onthe CRAY C90. A sum of 40,335 CPU secondsof the SGI workstation was spent to obtain theinitial as well as three levels of adaptivesolutions. For the cases presented in this paper,all adapted solutions were started from thefreestream condition at each adaptation cycle.Interpolation of solutions from one grid onto thenext adapted grid (planned for future work)would expedite the overall solution convergence,resulting in a substantial saving in the adaptivesolution time.

3.2 Modular Transonic Vortex InteractionConfigurationTo demonstrate the effectiveness of the presentsolution adaptive method for predicting vorticalflows, a generic fighter model referred to as theModular Transonic Vortex Interaction (MTVI)has been employed. The configuration features achine forebody with an included angle of 30degrees, sixty-degree cropped delta wings,partially deflected wing leading-edge flaps, andtwin vertical tails. All edges of the geometry aresharp inducing flow separations and vortices,which are independent of viscous effects.Inviscid solutions were obtained at a free streamMach number of 0.4 and an incidence of 20degrees.

An initial grid generated for this geometrycontains 31,565 nodes and 163,619 tetrahedrons.As in the previous case, the grid density ismarginally adequate to resolve the main featuresof the flow. No attempt has been made to clustergrid points at locations where vortices are

coarse gridfine grid

adapted gridexperiment

.0 .2 .4 .6 .8 1.x/c x/c

y/b=0.20 y/b=0.44

y/b=0.65 y/b=0.80

y/b=0.90 y/b=0.95

Cp

Cp

Cp

.0 .2 .4 .6 .8 1.1.0

0.5

0.0

-0.5

-1.0

-1.5

1.0

0.5

0.0

-0.5

-1.0

-1.51.0

0.5

0.0

-0.5

-1.0

-1.5

Shahyar Z. Pirzadeh

244.10

expected. Three levels of adaptation refine thegrid at the critical locations, producing a finalsize of 108,014 nodes and 564,727 cells.Figure 6a illustrates the surface triangulationfor the initial coarse grid (port) and afteradaptation (starboard). Cross-sections of theinitial and adapted volume grids are shown ata streamwise station ahead of the vertical tailsin Fig. 6b. The automatic refinement of thesurface and volume grids, as adapted to thechine and wing vortices, is clearly indicated inthese figures.

The accurate flow solution obtained inthis example highlights the benefit of

increased automation provided by gridadaptation. The solution has not only predictedall the details of the vortical flow structureaccurately, it has even revealed the onset of avortex breakdown phenomenon at the properlocation automatically. Figure 7 shows localrefinement of the volume grid (open pockets) atthe vortex locations at two different stages ofadaptation. A refinement of the initial grid,triggered by the first solution, indicates a chinevortex extending beyond the aircraft tail (Fig.7a). The final refined grid correctly predicts achine vortex burst ahead of the vertical tail asindicated in Fig. 7b. The vortex breakdownphenomenon has also been observed on thisgeometry experimentally. Figure 8 depicts awind tunnel visualization of the flow at a slightlydifferent condition (α=30o, undeflected flaps).Similar formation of the chine vortex and itsburst in front of the vertical tail is clearly visiblein this picture. The importance of the automation

Figure 7. Local refinement of the MTVI volumegrid: (a) initial grid and (b) adapted grid indicatingthe chine vortex burst.

unburst chine vortex

chine vortex breakdown

wing vortex

(b) adapted

(a) unadapted

Figure 6. Initial and adapted unstructured gridson the MTVI configuration: (a) surface mesh and(b) surface/volume grid.

Adapted

Unadapted

Adapted Unadapted

(b)

(a)

A SOLUTION ADAPTIVE TECHNIQUE USING TETRAHEDRAL UNSTRUCTURED GRIDS

244.11

aspect of grid adaptation for producingaccurate solutions is well demonstrated in thisexample, as the initial unadapted grid actuallyyields a misleading solution to the problem.

Figure 9a compares surface pressuredistributions before and after grid adaptationon the port and starboard sides of the aircraft,respectively. The adapted solution indicatescrisper footprints of the wing vortices and achine vortex, which does not extend as fardownstream as that of the unadapted solution(indicative of the vortex burst phenomenon.)The pressure distributions in the field,showing the vortices in a cross-sectionalplane, and on the surface are portrayed in Fig.9b. As evident, a well-defined vortexgenerated by the sharp leading-edge of thedeflected flap and even a smaller vortexemanating from the wing snag have beencaptured with grid adaptation (Fig. 9b,starboard). Figure 9b also shows a chinevortex in the field as predicted by theunadapted solution incorrectly (port). Theabsence of the chine vortex in the starboardside of the image indicates the breakdownphenomenon captured by the adapted solutionat this location.

To detect vortices, the feature indicatorgiven by Eq. (2) was employed in this

example. Grid cells experiencing entropyproduction levels of higher than a threshold (asmall fraction of the maximum entropy producedin the field) were flagged for removal at eachadaptation cycle. In the present example, athreshold value of 0.01 has been used. All thein i t ia l and adapt ive computa t ions(grids/solutions) were performed using the SGIworkstation described earlier.

3.3 X-38 Forebody ConfigurationThe last test case is presented to emphasize theefficiency of the adaptive method asdemonstrated on a supersonic flow featuring a

Figure 9. Initial and adapted static pressuredistributions on the MTVI configuration: (a) surfaceand (b) surface/volume.

Adapted

Unadapted

wing vortices

chine vortex

Adapted Unadapted

chine vortex burst

wing vortices

unburst chine vortex

(b)

(a)

Figure 8. Wind tunnel visualization of flow aroundthe MTVI configuration showing chine vortex burst.

Shahyar Z. Pirzadeh

244.12

detached shock wave. The configurationselected for this purpose is the front portion ofan experimental aerospace vehicle referred toas X-38. The flow condition is at Mach 2 anda zero incidence angle. This case represents aclassic example for which the generation of anefficient unadapted grid is challenging due tothe presence of a conical detached shock waveextending far into the field. Even with a priorknowledge of the shock location, it is difficultto control the distribution of grid points at acurved surface in the 3D space. Usually, thegenerated grids are either too coarse awayfrom the geometry, which fail to capture flowdiscontinuities accurately, or globally too fine,which make the computational cost excessive.The economic advantage of solution adaptivegridding becomes more tangible for suchapplications.

In Fig. 10, two separate cross-sections ofthe field grids are illustrated around thegeometry. The initial grid, shown on the left-hand side of the figure, contains 87,806 cells.This grid represents a typical unadapted grid,which is adequately clustered around thegeometry but is too coarse in the field toresolve the flow accurately. The grid afterthree levels of adaptive refinement (shown onthe right-hand side of the figure) contains

840,135 cells. The adapted grid is efficientlyrefined in the field at the 3D conical shock wavestructure as clearly indicated in Fig. 10. Even aweaker shock wave in front of the canopy isautomatically detected, and the grid is refinedaccordingly. Also, note the gradual transition ofthe grid spacing from the original coarse grid tothe core of the refined sections where the shockwaves are formed.

An unadapted globally fine mesh, with aresolution similar to that of the adapted grid atthe shock locations, was also generated forcomparison. The fine grid (not shown) contains11,786,137 cells, which is a typical grid size forhypersonic flow computations. The largedifference (more than an order of magnitude)between the size of the two grids illustrates thehigh degree of efficiency offered by gridadaptation.

The indicator given by Eq. (1) was also usedto detect the detached shock waves in thisexample. The effect of the grid-spacingcorrection factor in Eq. (1) has resulted in abetter detection of small pressure differencesaway from the geometry. Consequently, thelarger cells, which have hardly experienced flowdiscontinuities in the initial solution, are alsoflagged and refined. Figure 11 shows staticpressure contours on the surface and in the field.

Figure 10. Initial and adapted tetrahedral gridsaround the X-38 forebody configuration.

Unadapted Adapted

Figure 11. Comparison of the initial and adaptedstatic pressure contours on the X-38 forebodyconfiguration.

Unadapted Adapted

A SOLUTION ADAPTIVE TECHNIQUE USING TETRAHEDRAL UNSTRUCTURED GRIDS

244.13

The adapted solution, on the right, exhibits asharp bow shock extending farther out to theoutflow boundary. A secondary shock wavein front of the canopy is also well predicted bygrid adaptation. The unadapted solution,exhibiting weak compression waves, is shownon the left-hand side of Fig. 11.

5 Concluding Remarks

A practical adaptive unstructured gridapproach has been developed and tested onseveral three-dimensional cases. The methodis based on the proven existing techniquesextended for grid adaptation. The "pilot"technology has demonstrated potential forsolving complex aerodynamic problems in anefficient and practical fashion. The presentedsample results have clearly shown thataccurate solutions can be generatedautomatically with substantially less amountof computational time and cost. Additionalwork is required to mature the pilottechnology and to extend its capabilities.Further developments planned for future workinclude the implementation of bettererror/feature indicators for accurate adaptationof solutions involving multiple dominant flowfeatures, solution interpolation betweenadaptation cycles, and extension of the methodfor the Navier-Stokes solution adaptivegridding.

6 Acknowledgements

The support of the High Performance AircraftOffice, the Configuration AerodynamicsBranch, and the Subsonic AerodynamicBranch at the NASA Langley Research Centerduring the course of this study is gratefullyacknowledged.

7 References

[1] Soni BK, Weatherill NP, and Thompson JF. GridAdaptive Strategies in CFD. Invited Paper,International Conference on Hydro Sciences &Engineering, Washington, D.C., June 7-11, 1993.

[2] Coirier WJ and Powell KG. A Cartesian, Cell-basedApproach for Adaptively-refined Solutions of the theEuler and Navier-Stokes Equations. Proceeding of theSurface Modeling, Grid Generation and Related Issuesin Computational Fluid Dynamics Workshop, NASAConference Publication 3291, May 1995.

[3] Mavriplis DJ. Adaptive Meshing Techniques forViscous Flow Calculations on Mixed ElementUnstructured Meshes. AIAA Paper 97-0857, January1997.

[4] Peraire J, Peiro J, and Morgan K. Adaptive Remeshingfor Three-Dimensional Compressible FlowComputations. Journal of Computational Physics,103, pp 269-285, 1992.

[5] Baum JD, Luo H, and Lohner R. A New ALEAdaptive Unstructured Methodology for theSimulation of Moving Bodies. AIAA Paper 94-0414,January 1994.

[6] Pirzadeh SZ. An Adaptive Unstructured Grid Methodby Grid Subdivision, Local Remeshing, and GridMovement. AIAA Paper 99-3255, June 1999.

[7] Baker TJ. private communications.[8] Pirzadeh S. Three-Dimensional Unstructured Viscous

Grids by the Advancing-Layers Method. A I A AJournal, Vol. 34, No. 1, pp 43-49, 1996.

[9] Lohner R and Parikh P. Three-Dimensional GridGeneration by the Advancing-Front Method.International Journal of Numerical Methods in Fluids,8, pp 1135-1149, 1988.

[10] Pirzadeh S. Structured Background Grids forGeneration of Unstructured Grids by Advancing FrontMethod. AIAA Journal, Vol. 31, No. 2, pp 257-265,1993.

[11] Pirzadeh S. Recent Progress in Unstructured GridGeneration. AIAA Paper 92-0445, January 1992.

[12] Frink NT. Tetrahedral Unstructured Navier-StokesMethod for Turbulent Flows. AIAA Journal, Vol. 36,No. 11, pp 1975-1982, 1998.