David L. Marcum* and Kelly P. Gaither^NSF Engineering Research Center for Computational Field Simulation
Mississippi State University
Abstract
A procedure is presented for anisotropic adaptation ofinviscid type flow features with CFD simulation usingan unstructured grid. The basic grid generation schemeis an existing advancing-front, local-reconnection(AFLR) procedure. This procedure is modified to allowthe field point spacing and alignment to vary based ona flow field solution using adaptive grid regeneration.Multiple physically based feature detectors are used toisolate regions in the field which contain physicalfeatures of interest. Using pattern recognitiontechniques, these regions are reduced to groups ofsimple geometric entities representing the physicalfeatures. Features such as expansion or stagnationregions are reduced to singular points. Shock waves andcontact discontinuities are reduced to curves. In theadaptive grid regeneration procedure, singular pointsare treated as adaptive sources which modify theboundary and field point spacing. Curves are treated asembedded boundaries. These embedded dual-sidedboundaries provide a framework for generating alignedhigh-aspectnratio elements. A description of theprocedure is presented along with results fortwo-dimensional applications with multiple features.The results clearly demonstrate the capability of theoverall procedure for CFD applications. And, theyindicate that the method can be further developed intoa robust and useful procedure.
IntroductionUnstructured grid technology provides a powerfulcapability for computing complex flow fields aboutrealistic aerospace configurations. The geometricflexibility of the unstructured approach makes it ideallysuited for geometrically complex configurations and,when combined with solution adaptation, for complexphysics. The full potential of this approach, for manyapplications, can only be realized when it is trulycapable of handling complex geometries and adaptingto complex physics in an automatic and efficientmanner. Several unstructured grid generation and flowsolver procedures with isotropic solution adaptation
*Professor, Mechanical Engineering, Department,Senior Member AIAA.
^Research Assistant II, Engineering Research Center.
have been developed and successfully demonstrated forinviscid flow about realistic configurations1"7.Considerably less work has been done with unstructuredanisotropic solution adaptation for inviscid or viscousflow simulations8"15. Many physical features, such asshock waves, contact discontinuities, and shear-layershave property gradients that are primarily aligned in asingle direction and can be resolved more efficientlyusing anisotropic refinement. This is especially criticalin three-dimensional applications wherehigh-resolution with isotropic refinement is notpractical for realistic configurations. For viscous flowapplications, most unstructured procedures haveadopted a pseudo-structured type approach withinboundary-layer regions. Approaches to solutionadaptive anisotropic refinement that have beenimplemented use predominantly non-alignedhigh-aspect-ratio elements. While non-alignedelements can provide high levels of accuracy andresolution, they adversely effect the performance oftypical CFD solution algorithms. Geometric meshquality decreases with increasing aspect-ratio. Elementangles approaching 180° typically reduce stability andconvergence rates, resulting in significantly increasedCPU time required. Using an approach whereinanisotropic elements are generated such that they arehighly structured and aligned with the flow featuresoffers the advantage of improved resolution withoutadversely impacting solver performance. This approachhas been used successfully for adaptation of viscouswakes15. The objective of the present work is to developprocedures for anisotropic solution adaptation usinghigh-quality elements aligned with physical features.
The anisotropic approach used in the present workrequires development of methodologies for detectingphysical features, sorting the features into spacialgroups, recognizing and classifying features within agiven group, reducing the groups to representativegeometric entities, and utilization of the geometricentities to generate a solution adapted unstructured grid.Detection of physical features can be performed usingtechniques commonly used for standard h- orr-refinement solution adaptive grid generation.Typically a function based upon solution gradients isused. In the present work, the feature detection isaccomplished using previously developed, gradientbased, multiple feature detector functions7*16. These
functions can be used to identify spacial regions in aflow field which contain physical features of interest.The feature detector functions can be viewed as a datareduction tool which reduces the entire field to a smallersubset of only the most relevant physics (for solutionadaptation). These subset regions can be directly sortedinto groups using the underlying connectivity of thecomputational grid. Recognizing and classifyingfeatures within a given group is accomplished using aheuristic approach wherein knowledge of the physics(available from the feature detection function) and thelocal geometry is used to determine what geometricentity or entities (lines and points in 2D) best representsthe feature. After classification, the group can bereduced to a representative geometric entity usingsolution gradient information. For example, a shockwave should be classified as a line in 2D. The actual linecan be generated by tracking a maximum solutiongradient or iso-value in a manner similar to streamlinetracing. The creation of representative geometricentities can be viewed as a final step in an overall datareduction process. In this process the computed solutionis reduced to a simple set of geometric entities (pointsand lines) which best represent the relevant physics.Utilization of these geometric entities within a gridgeneration procedure can be accomplished by treatingthem as point and line sources or by including them asembedded boundaries within the computationaldomain. Using embedded boundaries, alignment withthe physical features occurs automatically. It also offersa convenient way to obtain very controlled directionalrefinement using marching techniques that advancefrom the embedded boundaries.
The overall concept for the procedure presented in thispaper is derived from a previously developed solutionadaptive procedure for viscous wakes15. In thatprocedure the features of interest are wakes. Detectionand generation of the representative lines is obtained bytracing streamlines within detached viscous regions.The wake lines are then used as embedded boundariesto obtain a solution adapted grid with alignedhigh-aspect-ratio elements within boundary layers andwake regions. Embedded wake boundaries and theresulting solution adapted grid for a multi-elementairfoil case is shown in Figs. 1 and 2. The viscous wakeadaptation procedure can be extended for the presentwork by incorporating appropriate feature detection anddeveloping pattern recognition processes for inviscidfeatures.
A brief overview of the basic grid generation proceduresand details of the solution-adaptation process ispresented in this paper. Also presented are examplecases for two-dimensional flow fields with multiplefeatures which demonstrate the capabilities of thepresent approach.
Fig. 1 Initial triangulation for multi-elementairfoil case with multiple wakes after insertion ofembedded wake boundary points.
Fig. 2 Final solution adapted unstructured grid formulti-element airfoil case with multiple wakes.
The overall procedure utilizes solution adaptivegrid-regeneration to improve the resolution of acomputed solution. The basic steps in the overallprocess are listed below.1) Generate a coarse initial grid with resolution
sufficient to represent the geometry and at leastweakly resolve relevant flow features.
2) Compute an initial solution.3) Reduce the solution data to geometric entities that
represent the relevant physical features in thecomputed flow field.
4) Include the geometric entities which represent thephysical features within the computational domainas embedded boundaries and adaptation pointsources.
5) Generate an anisotropic solution adapted grid usinghigh-aspect-ratio elements near the embeddedboundaries.
6) As an alternative to step 5, an isotropic solutionadapted grid can be generated by treating theembedded boundaries as point sources in step 4.Low—level isotropic refinement can be moreeffective for generation of the first adapted grid ifthe initial grid is very coarse.
7) Generate a new solution using the adapted grid.8) If more resolution is required return to step 3 with
the grid and solution from steps 6 and 7.
Feature DetectionFor the present approach, a feature detector is requiredthat detects and locates appropriate features in acomputed flow field. Multiple or split feature detectorsare used to provide flexibility in isolating varyingfeatures of differing type and magnitude. Each detectorcan isolate a particular type of feature. The featuredetectors used have been successfully utilized to reducethe flow field to a set of adaptation sources in anisotropic adaptation procedure7'16. The featuredetectors used are set to the negative and positivecomponents of a flow property gradient given by
fi=-min[(V/\V\)'Vp, 0]f2=max[(V/\V\)'Vp, 0]
and the magnitude of the gradient in all directionsnormal to the velocity vector given by
f3 =\ Vp - (V/ \V\) [(V/ \V\) »Vp7 Iwhere f i , / 2 , and./} are the feature detector functions, pis any suitable fluid property (density is used for allresults presented in this paper), and V is the velocityvector. The feature detectors represent expansions andcompressions in the flow direction and shear gradientsnormal to the flow. Each feature detector is computedat all solution points and treated independently. A lowerlimit for each is determined statistically and detectorvalues below the limit are zeroed. A given node isflagged as being part of a relevant feature if any of thefeature detectors are non-zero. An example of a typicalcollection of feature nodes and elements is shown in Fig.3 for a transonic airfoil case. The elements which areshaded contain nodes which have been flagged. Theregions identified include upper and lower surfaceshock waves as well as leading and trailing edgecompression and expansion regions. At the completionof the feature detection process the grid nodes identifiedare flagged and labelled by originating feature detectionfunction or functions. With isotropic adaptation theflagged nodes can be used as is for adaptation pointsources (along with a source strength dependent uponthe gradient magnitude). In the present anisotropic
method, this information is used in the subsequentpattern recognition and geometric representationprocess.
Fig. 3 Elements and points identified by featuredetection for a transonic airfoil case.
Pattern Recognition and Geometric RepresentationPattern recognition is used to determine how to furtherreduce the previously described feature points tosimpler geometric entities. Before starting this processthe points identified by feature detection are firstgrouped into logical sets. A valid group of points isdefined to be ones which originate from compatible orthe same feature detection function and that arephysically connected to one another through the gridconnectivity. Isolated groups of small physical size notconnected to another or to a boundary are discarded asnoise. Rules based upon known physics can be used tosplit points in overlap regions or remove points whichmay be generated solely by errors in the solution. Forexample, expansion points are removed from a groupwith primarily compression points. After the sets ofpoints are created during this initial grouping process,classification and pattern recognition are then used toreduce them to simple geometric entities.During classification, each group is dealt withindividually and only the points of a given group areused. To classify a group, a regular grid is first imposedon the unstructured elements contained in the group.The row and column size are determined by the elementspacing within the group. This regular grid is thencolored or marked according to whether or not a cellcontains a feature point of the given group. The purpose
of the regular grid is solely to assist in classification bypattern recognition.Extremes of a given group can be found by traversingthrough the regular grid along columns and rows. Cellsare labeled as extremes if they are on the boundary or donot have a marked neighbor cell. Physical informationcan also be used to help identify extreme cells.Iso-values or vectors orthogonal to the gradient traversethrough extremes in line type features, such as shockwaves or contact discontinuities. Once the extreme cellshave been identified, a group can be classified andcompressed into either a point source, a set of pointsources, a line, or a set of lines. The classification of thegroup is made in the following manner:1) Compression Point Source (Compression or
Stagnation Region)- If the group contains onlycompression features and it passes a specified fillratio and a specified shape constraint, then theregion is classified as a point source. Compressionregions which are entirely subsonic are alsoclassified as a point source. Emeging very weak orpoorly resolved shock waves are initiallyrepresented as point sources. The point source isspecified as the vertex in the group that has themaximum gradient magnitude.
2) Compression Line (Shock Wave) - If the groupcontains compression features, has supersonic flow,and it contains only two extreme values, the regionis compressed to a single line. The line isconstructed as a trace from one extreme region to theother extreme region using the tangent to thegradient vector. Both end-points of the trace arefound by obtaining the maximum gradientmagnitude in both marked regions.
3) Compression Lines (Multiple Shock Waves) - If thegroup contains compression features, hassupersonic flow, and it contains more than twoextremes, the region is compressed as a set of lines.Each region on the regular grid marked as anextreme can be represented as a single point byfinding the maximum gradient magnitude in theextreme region. All extreme points are tested todetermine whether there are ambiguous regionswhere lines connect. This is done by generating atrace on the unstructured grid which is tangent to thegradient vector until the trace is no longerconsidered to be valid. A trace is no longer valid ifit intersects another trace, enters an element whichhas been visited by another trace, violates somecurvature over a defined set of trace points, orabruptly exits the group. Each ambiguous regioncan be represented as a single point by computingthe regions centroid and searching for a localmaximum gradient magnitude. Traces areconstructed from each extreme point to other
extreme points or to the center of its ambiguousregion.
4) Expansion Point Source (Expansion Region) - If thegroup contains expansion features and it is subsonic,then the region is compressed to a single pointsource. The source is the maximum gradientmagnitude in the group.
5) Expansion Fan - If the group contains expansionfeatures and it is supersonic, then the region iscompressed to a fan marked by three point sources.The first point source is the apex of the fan and isfound as the maximum gradient magnitude. Theother two points represent each edge of the fan.
6) Shear Line (Contact Discontinuity) - If the groupcontains shear features, then the region iscompressed as a line or set of lines in the samemanner as a compression line.
At the completion of classification and geometricrepresentation the feature points are reduced to a set ofgeometric entities (points and lines) representing thephysical features. Lines generated are smoothed tominimize the effects of low resolution or errors in thesolution. An example of the geometric entities createdfrom feature points is shown in Fig. 4 for a transonicairfoil. The upper and lower surface shock waves havebeen correctly identified and represented as lines. Theleading edge and trailing edge compression andexpansion regions are represented as individual pointsources (not visible in figure) located on the airfoilboundary.
Fig. 4 Representative lines and source pointscreated by pattern recognition and geometricrepresentation for a transonic airfoil case.
The basic unstructured grid generation procedure usedin the present work is the existing advancing-frontlocal-reconnection (AFLR) triangular/tetrahedral gridgeneration procedure. Complete details can be found inRefs. 15,17 and 18. This procedure is a combination ofautomatic point creation, advancing-normal pointplacement, advancing-front point placement andconnectivity optimization schemes. A valid grid ismaintained throughout the grid generation process. Thisprovides a framework for implementing efficient localsearch operations using a simple data structure. It alsoprovides a means for smoothly distributing the desiredpoint spacing in the field using a point distributionfunction. This function is propagated through the fieldby interpolation from the boundary point spacing or bya specified growth normal to the boundaries. Points aregenerated using either advancing-front type placementfor isotropic elements or advancing-normal type pointplacement for high-aspect ratio elements. Theconnectivity for new points is initially obtained fromdirect subdivision and then improved by iterativelyusing local-reconnection subject to a quality criterion.A min-max type (minimize the maximum angle)criterion is used. The overall procedure is appliedrepetitively until a complete field grid is obtained.
For the present work, the capability to generatehigh-aspect-Hratio elements, utilize embeddedboundaries, and include point sources are essential. Theuse of these features is identical to that for the viscouswake adaptation procedure. Details of theimplementation of these features is given in Ref. 19. Thelines representing physical features (from patternrecognition and geometric representation process) areincluded in the computational domain as embeddedboundaries. As such they can be used to advance fromin a very controlled manner and generate alignedhigh-aspect-^atio elements. Point sources are alsoincluded in the computational domain for localizedisotropic refinement. The normal spacing to the linesand point sources is set proportional to the local gradientmagnitude in the solution. Lines representing featuresmust be discretized prior to using them as embeddedboundaries within the grid generation process. The linesare discretized just as any other boundary curve.Typically, desired tangential point spacing is specifiedat the end points. With the present procedure the gridthat is generated is adapted to the solution withhigh-aspect-H-atio elements aligned with the physicalfeatures.
Application ExamplesTwo-dimensional application examples are presentedto demonstrate the overall procedures. The flow fieldsolutions presented here were obtained using an explicit
finite-element solver7-20. This solver uses asecond-order accurate spacial discretization obtainedfrom a Taylor-Galerkin weighted residualapproximation. The solution is advanced in time usingan explicit two-step Lax-Wendroff scheme. Thepresent adaptive procedure could be used with most anyunstructured CFD solver for compressible flow.For each case, an initial coarse grid and solution wasobtained. One low-level solution adapted grid was thenobtained from the initial coarse solution using isotropicadaptation. As relatively coarse initial grids were used,the anisotropic refinement at the first level offers noadvantage. High-aspect-ratios are inappropriate in thiscase as the shock locations may vary considerably withrefinement. In all cases, the initial grid resolution wassufficient to detect all relevant features. As is true withany adaptive procedure, the initial coarse grid must besufficient to at least weakly resolve all features ofinterest. A feature not identified in the initial grid maynever be resolved. The present approach with multiplefeature detectors minimizes this problem by allowingfeatures of varying type and magnitude to be identified.Using the isotropic adapted grid and solution, two moreanisotropic solution adapted grids and solutions wereobtained. As the resolution of the flow field is improvedwith adaptation, the aspect ratio of the anisotropicelements was allowed to increase along with a decreasein minimum point spacing.
Transonic AirfoilThe first case presented is for inviscid flow over aNACA0012 airfoil with a free stream Mach number of0.8 and an angle of attack of 1.25°. The initial grid,isotropic adapted grid, and final two anisotropic adaptedgrids are shown in Fig. 5. For the last grid the maximumaspect-ratio in the adapted regions was about 100:1.Computed pressure contours for this case are shown inFig. 6 and pressure coefficient distributions are shownin Fig. 7. For this case there is a relatively strong uppersurface shock and a weak lower surface shock. Thefeature detectors were able to identify the weak lowersurface shock starting with a low resolution solution.The final anisotropic grid provides very good resolutionfor all of the flow field features. An equivalentresolution adapted grid obtained using point sources(from the same feature lines) for isotropic refinementcontains 108,610 points compared to 6,655 points forthe anisotropic case.Supersonic Converging DuctThe next case presented is for inviscid supersonic flowthrough a converging duct with an inlet Mach numberof 3. The initial grid, isotropic adapted grid, and finaltwo anisotropic adapted grids are shown in Fig. 8. Forthe last grid the maximum aspect-ratio in the adaptedregions was about 200:1. Computed pressure contours
Fig. 5 Initial, isotropic adapted, and final twoanisotropic adapted grids with 1,426,1,889, 3,904,and 6,655 points, respectively, for transonic airfoil.
for this case are shown in Fig. 9. For this case there aretwo intersecting shock waves. This structure wascorrectly identified by the feature detection process asmultiple shock waves with a geometric representationof four lines that intersect at a point. The finalanisotropic grid provides a very well resolved solution.An equivalent resolution adapted grid obtained using
Fig. 6 Computed pressure contours with initial,isotropic adapted, and final two anisotropicadapted grids for transonic airfoil.
point sources (from the same feature lines) for isotropicrefinement contains 431,977 points compared to 9,219points for the anisotropic case.
Supersonic Double RampThe final case presented is for inviscid supersonic flowover a double ramp with 10° and 24° ramps and a free
—— Initial—— Source Adapted—— 1 si Directionally Adapted
2nd Directionally Adapted
0.0 0.2 0.4 0.6X/C
0.8 1.0
Fig. 7 Computed surface pressure coefficientdistributions for transonic airfoil.
stream Mach number of 3. The initial grid, isotropicadapted grid, and final two anisotropic adapted grids areshown in Fig. 10. For the last grid the maximumaspect-ratio in the adapted regions was about 25:1.Computed density contours for this case are shown inFig. 11. Computed Mach number distributions areshown in Fig. 12 along two different horizontal lines.For this case there are two shock waves that intersectand result in a single outer shock wave and contactdiscontinuity. This structure was correctly identified bythe feature detection process as multiple shock wavesand a contact discontinuity with a geometricrepresentation of four lines that intersect at a point.Also, the feature detection process was able to identifyall of the features, which vary significantly inmagnitude, starting with a low-resolution solution. Thefinal anisotropic grid provides a very well resolvedsolution for all of the relevant features. An equivalentresolution adapted grid obtained using point sources(from the same feature lines) for isotropic refinementcontains 77,554 points compared to 10,030 points forthe anisotropic case.
Performance ImprovementsThere are few key areas where the present procedurecould be improved upon for efficiency and usefulness.At present, the entire grid is regenerated and there is noenforced coupling between an adapted grid and theprevious grid. For efficiency, the existing grid could belocally de-refined and locally regenerated in regionswith embedded boundaries. R-refinement (pointmovement) could also be used to realign the featurelines and eliminate or reduce further regeneration.
Fig. 8 Initial, isotropic adapted, and final twoanisotropic adapted grids with 513, 1,362, 3,746,and 9,219 points, respectively, for supersonic duct.
These improvements would significantly benefit anunsteady application.
Three-Dimensional ExtensionThe most demanding future research issue for thepresent methodology is extension to three-dimensions.Many of the procedures used in the overall method canreadily be extended to three-dimensions. Featuredetection, feature grouping, along with embeddedsurfaces and point sources within the grid generationprocess have been or are readily extendable tothree-dimensions. However, a serious effort is requiredto develop a working 3D extension for patternrecognition and extraction of surfaces from featurepoints. While a 3D extension is unproven, surfaceextraction may be possible using networks of linesgenerated much like those in 2D.
SummaryA procedure has been presented for anisotropicadaptation of inviscid type flow features with CFD
Fig. 9 Computed pressure contours with initial,isotropic adapted, and final two anisotropicadapted grids for supersonic duct
simulation using an unstructured grid. Multiplephysically based feature detectors were used to isolateregions in the field which contain physical features ofinterest. Using pattern recognition techniques, theseregions were reduced to groups of simple geometricentities (points and lines) representing the physicalfeatures. These lines were treated as embeddeddual—sided boundaries which provided a framework forgenerating aligned high-aspect-ratio elements. Resultspresented demonstrate mat the present featuredetection, pattern recognition and classificationprocedures can automatically reduce a flow field to a setof simple geometric entities. The results alsodemonstrate that these entities can be utilized ingeneration of an anisotropic adapted grid for improvingthe resolution of a computed flow field.
AcknowledgementsThe authors would like to acknowledge support for thiswork through grants from the Air Force Office ofScientific Research, Dr. Leonidas Sakell, ProgramManager, and the Ford Motor Company, UniversityResearch Program, Dr. Thomas P. Gielda, TechnicalMonitor.
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