AIAA 2000-2245OVERSET STRUCTUREDHYPERBOLIC GRID GENERATIONON TRIANGULATED SURFACES

William M. ChanELORET, Army/NASA Rotorcraft Division,NASA Ames Research Center,Moffett Field, California 94035

Fluids 200019-22 June, 2000 / Denver, CO

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics1801 Alexander Bell Drive, Suite 500, Reston, VA 20191–4344

AIAA 2000-2245

OVERSET STRUCTURED HYPERBOLIC GRID GENERATION

ON TRIANGULATED SURFACES

William M. Chan�

ELORET, Army/NASA Rotorcraft Division,NASA Ames Research Center,Mo�ett Field, California 94035

Abstract

A scheme is presented for the creation of oversetstructured grids on a surface geometry described bytriangles. Two types of surface features are identi�edand preserved: seam curves and seam points. Seamcurves lie along surface discontinuities and high curva-ture contours while seam points are points with non-unique normals on the surface such as corner and apexpoints. Methods are described for extracting these sur-face features on a triangulated surface. Other surfacecurve creation schemes on a triangulation involvingcutting-plane intersection and stencil-walks are alsopresented. A hyperbolic marching method is then em-ployed to generate structured surface grids from thesurface curves and seam points over the triangulatedgeometry. Examples are shown for overset grids cre-ated for several complex surfaces including parts of arotorcraft, a space vehicle, and a triceratops. With theaid of a graphical interface, overset surface grid gen-eration time for a moderately complex geometry cantypically be accomplished in about half a day.

1. Introduction

The e�cient design and analysis of aerodynamic ve-hicles require rapid modelling of the surface geometryto create computational grids suitable for numericalsimulations. Ideally, surface grids should be gener-ated directly on the CAD representation of the geome-try. Unfortunately, such tools are not yet convenientlyavailable for creating structured surface grids that fol-low the Chimera overset gridding philosophy.1 In thepast, overset surface grid generators2;3 utilized a mul-tiple panel network representation of the geometry asinput, where each panel network is a rectangular arrayof points that de�nes a set of bilinear quadrilaterals.Conversion from CAD data to multiple panel networkformat requires manual e�ort and can be a time con-suming process. Tools that have been used for thisprocedure include GRIDGEN,4 ICEMCFD,5 and oth-

�Senior Research Scientist, ELORET, Senior Member AIAA

This paper is a work of the U.S. Government and is not subject to copy-

right protection in the United States.2000

ers. Clearly, a faster method to reach the CAD dataneeds to be found.The ideal approach is to have independent library

routines that any surface grid generator can access toperform tasks like point projection and surface nor-mal interrogations on the CAD model. While suchlibrary routines do exist as part of commercial CADpackages, independent interfaces of this kind are notreadily available. The CAPRI6 application program-ming interface (API) provides interrogation routinesfor solid models created from Pro-Engineer, CATIA,and a few other commercial CAD packages, but therespective commercial licences are required to use thesoftware. A next best option from direct CAD modelinterfaces is to employ a surface geometry format thatcan be derived quickly from CAD data, while keepinga high �delity representation of the true surface geom-etry. Surface triangulations are excellent candidatesfor this purpose.Many CAD packages already o�er output in various

surface triangulation formats. Moreover, a rapid andessentially automatic method to create high qualitysurface triangulations is available in the CART3D7;8

software utilizing CAPRI. The maximum distance be-tween a triangular face and the CAD model, as wellas certain grid quality constraints, are prescribable towithin a speci�ed tolerance.This paper describes a scheme that takes advantage

of this easily available geometry format for creatingstructured surface grids. While the multiple panel net-work surface description has the advantage that smallgaps and overlaps are allowed between neighboringpanel networks, this is also a drawback since automaticalgorithms for domain decomposition are di�cult toachieve.9 Since all face connectivities are known in asurface triangulation, construction of automatic algo-rithms may become a more tractable problem.The Chimera overset grid approach for ow simula-

tions using structured grids has been successfully ap-plied to many complex con�gurations.10�14 The over-set surface grid generators2;3 mentioned above employhyperbolic grid generation methods which have provedto be e�cient and convenient for the task. Result-ing grids are of high quality, nearly orthogonal, and

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each is created by marching from a single speci�edinitial curve. With a multiple panel network surfacedescription, such initial curves are usually producedby extracting a subset of a network in the structuredindex space. With an unstructured surface triangula-tion, more sophisticated surface curve creation meth-ods are needed. Moreover, such initial curves shouldalso capture important geometric features on the sur-face. These problems are addressed in Section 2 of thispaper. Hyperbolic grid generation methods from sur-face curves and corners are described in Section 3. Inparticular, a projection scheme for triangulated sur-faces is presented. Several test cases are shown inSection 4 to illustrate the capability of the new scheme.Concluding remarks are given in Section 5.

2. Surface Feature Capturing

The �rst step in surface grid generation is to de-compose the surface geometry into multiple domainssuitable for the creation of surface grids. One of themain advantages of the overset grid approach is thatsuch domains can overlap arbitrarily. The boundariesof these domains, and hence the boundaries of the sur-face grids, are allowed to oat freely, making it easierto produce high quality almost orthogonal grids. How-ever, constraints must be applied to the grid lines sothat surface features of interest are properly captured.Typically, such surface features are captured by oneor more of the grid boundaries. This suggests that theproblems of surface domain decomposition and surfacefeature capturing are closely connected.

2.1. Surface Triangulation Data Format

The schemes presented in this paper are based onan indexed surface triangulation format described be-low. This format can also be easily converted fromother triangulation formats. The triangulation is as-sumed to consist of a set of non-duplicated vertices.Connectivity between the vertices is prescribed by theindices of the three vertices that make up a triangularface. This information is stored in a face-vertex list.It is also assumed that none of the faces intersect eachother and that the surface geometry is trimmed, i.e.,only the outer mold line (wetted surface) is described.Each face of the triangulation also carries a tag whichis an identi�er for the component in which the facebelongs. For example, a face in an airplane geometrymay belong to the fuselage, wing, pylon or nacelle com-ponent. Such a triangulation is conveniently describedby the CART3D7 or FAST15 formats.From a given face-vertex list, the following data

structures are built so that geometric operationsneeded for surface feature extraction can be conve-niently performed.

Vertex-face list - the indices of the faces surroundingeach vertex.

sharp surface edges

intersection curves between componentshigh curvature contours

open boundary curves

Fig. 1: X-38 Crew Return Vehicle example showingvarious types of seam curves.

Vertex-edge list - the indices of the edges surroundingeach vertex.

Edge-vertex list - the indices of the two vertices con-nected to each edge.

Edge-face list - the indices of the two faces next toeach edge. For edges on an open boundary, thereis only one neighboring face.

The vertex-face list and vertex-edge list are storedas 1-D arrays. A pointer array is used to point to the�rst face and edge for each vertex on the vertex-faceand vertex-edge list arrays, respectively.

2.2. Surface Feature Extraction

For a geometry described by any format, two typesof surface features may be identi�ed: seam curvesand seam points. Seam curves can be classi�ed intothe following four types: (1) curves along surface dis-continuities (e.g., wing trailing edge), (2) intersectioncurves between components (e.g., wing/fuselage junc-tion), (3) high curvature contours (e.g., wing leadingedge), and (4) open boundary curves (symmetry planecurve for a half-body). Seam points are singular pointsof the surface with non-unique normals and can be oftwo types: (1) a corner which is an intersection pointof multiple seam curves, and (2) the apex point of alocally cone-like geometry with smooth cross section.Examples of these surface features are shown in Fig-ures 1 and 2.Detection and extraction methods for the above sur-

face features are given below for a surface geometrydescribed by a triangulation.

Surface Discontinuity Curves

Surface discontinuity curves are created by auto-matically concatenating a set of agged edges on thesurface triangulation. Using the edge-face list, an edgeis agged if the normals of the two adjacent faces de-viate by more than a user speci�ed threshold. Sucha scheme is appropriate for capturing sharp discon-tinuities on the surface but is unable to detect highcurvature regions with rounded turns.

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apex point

seam corner point

Fig. 2: Intersected teardrops example showing cornerand apex points.

Component Intersection Curves

Component intersection curves are easily extractedautomatically from the component tag information foreach face. Going through the edge-face list, an edgeis agged if the two adjacent faces belong to di�erentcomponents. Such an edge must lie on the intersectionbetween neighboring components. The set of aggededges are concatenated automatically to form intersec-tion curves.

High Curvature Contours

A completely automated scheme to detect high cur-vature contours such as those along leading edges ofwings and nacelles is di�cult since tests based on a sin-gle edge of a triangle alone are not su�cient. A schemehas to be designed to examine the total turning angleover a local surface region. Currently, a robust methodto de�ne such a local region is not available. Untilsuch a scheme is developed, an alternative approachused in this paper is to utilize a graphical interfacecalled OVERGRID16 and a stencil-walk scheme. In-side OVERGRID, the user selects two vertices on ahigh curvature contour by visual inspection. A curvebetween the two vertices is automatically constructedby a stencil walk from one vertex to the other. Thewalk is constrained to lie on existing edges of the trian-gulation and utilizes the vertex-edge list to walk fromone vertex to the next. In the edge selection process forthe path, a higher preference is given to an edge thatseparates two faces with higher normal vector devia-tion than those from alternative edges. Curved leadingedges, such as those found on nacelles, can thus becaptured by this method (see Figure 3).

Open Boundary Curves

Open boundary curves are easily extracted from theedge-face list by automatically concatenating edgeswith just one adjacent face.

Corner Points

The seam curves identi�ed by the above four meth-ods are all extracted from existing edges of the surface

A

B

Fig. 3: Surface curve created along nacelle leading edgeby connecting existing edges between vertices A andB.

triangulation. Identi�cation of point coincidence canthus be performed in index space. A seam curve issplit at an interior point if such a point coincides withthe end point of another seam curve. After the split,a point where three or more end points of seam curvescoincide is agged and extracted. The degree of a cor-ner point is de�ned to be the number of seam curvesconnected to the point.

Apex Points

An apex point on a surface is topologically equiva-lent to the apex of a cone. Since the surface is locallysmooth other than at the apex point, there are no seamcurves connected to the point. In order to determine ifa vertex on a triangulation is an apex point, a vertexnormal is �rst computed by averaging the surroundingface normals using the vertex-face list. The point is anapex point if the angles between vertex normal and allconnected edges are larger than some threshold (takento be 90 degrees in this paper).

2.3. Other Surface Curve Creation Schemes

For reasons explained below, it is convenient to con-struct two other methods for creating surface curveson triangulations. These are described in the follow-ing two subsections. With the surface curve creationschemes presented in Section 2.2, points on the re-sulting curves are all derived from vertices on thetriangulation. This allows for storage of these curvesin terms of the index space of the triangulation ver-tices. For the surface curve creation schemes describedin this section, points on the curves can be arbitrarilylocated and do not admit index space storage.

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cutting plane

Fig. 4: Surface curves created on a business jet geom-etry by intersection with a single Cartesian plane.

Sectional-Plane Cut

For geometries with no surface features (e.g., a com-pletely smooth closed surface), or where no smoothseam curves are available (e.g., a poorly resolved sur-face with many faceted faces or many minute surfacefeatures), surface curves are more easily created byintersection with a cutting plane. Currently, only in-tersection with a Cartesian x, y, or z cutting planehas been implemented. The set of faces that intersectsthe speci�ed Cartesian cutting-plane can be easily ex-tracted by examining the bounding box of each face.Each such face may intersect the cutting plane at 2distinct points (2 edge intersections), along an entireedge (edge is coplanar with cutting plane), at 1 dis-tinct point (vertex intersection), or along the entireface (face is coplanar with cutting plane). The lasttwo cases do not produce line segments and are disre-garded. The remaining two cases produce a line seg-ment for each face/cutting-plane intersection. Theseline segments are automatically concatenated to createone or more sectional curves for a single cut. Figure 4shows four curves created by a single Cartesian planeintersection through a business jet geometry.

Direct Path Between Two Vertices

After surface grids are created from the seam curves,gaps are sometimes left uncovered in the smooth re-gions of the surface. An algorithm to automaticallycover these gaps for a multiple panel network sur-face description is given in Ref. 9. Unfortunately, thescheme cannot be easily extended to cover gaps on atriangulated surface description. The remedy in thiscase is to construct surface curves along one or moreboundaries of the gaps and then cover the gaps usinghyperbolic or algebraic schemes.

A

B

Fig. 5: Surface curve created by connecting line seg-ments along direct path between A and B.

An arbitrary surface curve can be constructed be-tween two existing vertices, A and B, on a surfacetriangulation. Starting at vertex A, the vertex-facelist is used to �nd the face F1 that contains the mostdirect path to vertex B. The resulting path throughF1 either (1) follows an edge of F1 or (2) intersects theedge of F1 that is opposite to vertex A. In case (1),the end point of the path in F1 is another vertex andthe same procedure can be followed to �nd the nextsegment of the path. In case (2), the end point of thepath in F1 is a point on an edge. The edge-face listis used to locate the neighboring face F2. The mostdirect path to vertex B through F2 is then computed.Again the end point of this path through F2 could landon another edge or at a vertex. The above procedurecan then be repeated until vertex B is reached (seeFigure 5).

2.4. Surface Domain Decomposition

In this paper, the automatic domain decompositionscheme described in Ref. 9 is extended. Previously,only the corner points are identi�ed and a polar gridtopology is placed around all such points. In this pa-per, apex points as described above are also identi�ed.It is proposed that any singular point of a surfacecan be classi�ed as a corner point or an apex point.Furthermore, the surface region around such singularpoints can be covered using a surface grid with anO-type or H-type topology. The O-type topology isidentical to those found in the polar grids describedin Ref. 9. This topology can be used on any singularpoint and is the only option for corner points of degree4 or higher, and at apex points. For a corner point ofdegree 3 or 4, either the O-type or H-type topologymay be applicable as shown in Figure 6.

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O−type

H−type

Fig. 6: O- and H-type grid topologies at a corner point.The three seam curves connected to the corner pointare also shown.

The H-type topology is preferable when the seamcurves are approximately orthogonal at a corner. Someheuristic rules are given below to automatically deter-mine the choice of topology for a corner of degree 3.Similar criteria can be established for corners of degree4.

(1) When two of the three seam curves lie along anopen boundary of the geometry, there are onlytwo angles separating the three seam curves. AnH-type topology is selected if the ratio of the twoangles is less than R where R is chosen to be 1.3for this paper.

(2) When the corner is in the geometry interior, thereare three angles a; b; c separating the three seamcurves. Let Rab be the ratio between angles aand b and let similar de�nitions holds for Rbc andRca. An H-type topology is selected if (a) theratio of one angle with � is less than R and theratio between the remaining two angles is also lessthan R, or (b) Rab, Rbc and Rca are all less thanR.

After the seam points are classi�ed, the procedureproceeds as described in Ref. 9 to retract seam curvesfrom the O-type singular points, and to redistributegrid points on the retracted seam curves. At H-typecorners, two of the seam curves have to be reconnected(see Section 3.2). The result is a decomposition of thesurface domain around the surface features describedin Section 2.2.

2.5. Grid Point Redistribution

An automatic scheme is used to redistribute gridpoints on all surface curves such that the followingcriteria are satis�ed.

(1) The maximum grid spacing is no larger than agiven value.

(2) Grid points on the redistributed curve lie on thepiece-wise linear segments of the original curve;and grid points on the original curve are no morethan a speci�ed distance from the piece-wise lin-ear segments of the redistributed curve. Thisensures that grid spacings are automatically re-duced at high curvature regions.

(3) Sharp turns on the original curve are preserved.(4) Local grid spacings are reduced at sharp turns.(5) Grid spacings vary smoothly along the redis-

tributed curve.

3. Grid Generation Scheme

A marching scheme is used to create a surface gridfrom an initial curve or an initial point. For surfacecurves not connected to an H-type corner, a basichyperbolic method2;3 can be used (see Section 3.1).For surface curves connected to an H-type corner, thebasic hyperbolic method is still employed but a correc-tor smoothing scheme is added (see Section 3.2). Foran initial point (O-type corners), a special algebraicscheme is used to generate the �rst grid layer awayfrom the point, and a hyperbolic method is used onthe subsequent layers (see Section 3.3).

3.1. Marching Scheme from Surface Curves

In hyperbolic grid generation, a grid is created bymarching from an initial curve over a speci�ed dis-tance via a sequence of grid spacings. The sequence ofgrid spacings is conveniently prescribed using a hy-perbolic tangent stretching function which providescontrol of grid spacings at the initial curve and at theouter boundary. At each step, starting from the ini-tial curve, the marching scheme produces a new layerof points. This new layer of points is then projectedonto the reference surface. The marching and projec-tion procedures are repeated until the �nal distance isreached.In Ref. 2, a projection algorithm was described for a

reference surface consisting of multiple panel networks.In this paper, the same marching scheme is used butan algorithm is presented below for projection onto atriangulated surface. Prior to generating any grids, apre-processing step is performed once to create a setof face bounding boxes and a layer of prisms over thefaces of the surface triangulation.Each face bounding box is simply the Cartesian-

axes-aligned bounding box for the face with a tolerance

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(a) (b)

Fig. 7: Entities used by the projection algorithm fora surface triangle. (a) Extended bounding box. (b)Bilinear prism.

added to all 3 directions (see Figure 7a). This ensuresa �nite volume for the face bounding box even if theface is coplanar with a Cartesian plane. Furthermore,any arbitrary point close to the surface should be con-tained inside at least one of the face bounding boxes(see below for a mechanism that guarantees this con-dition).The prisms are formed by extruding each vertex by a

distance D along the vertex normal which is computedby averaging the normals of the surrounding faces (seeFigure 7b). As will be shown later, the exact value ofD is not important as long as it is small relative tothe size of the entire geometry. For this paper, D ischosen to be 0.001 times the geometry's bounding boxdiagonal.

Let a point P be created by the marching schemeduring hyperbolic grid generation. The projection pro-cedure onto the surface is divided into several steps.

(1) A list of candidate faces whose bounding boxescontain P is determined. If P is found to be out-side of all face bounding boxes, each face bound-ing box is automatically expanded in all 3 direc-tions by a small fraction of its original dimensions.The process is repeated until at least one facebounding box is found for P. Note that the facebounding box expansion is only needed when P isnot close to the surface. Such situations will onlyoccur if the marching step is large relative to thelocal surface curvature. In other words, the gridspacing is too large for accurate surface resolu-tion - a situation that should be avoided. Hence,a warning is issued to the user if the automaticbounding box expansion extends beyond severallevels.

(2) For each face in the candidate list, a Newton iter-ation is performed to compute interpolation coef-�cients in a bilinear representation of the prism17

over the face. The interpolation coe�cient in thenormal direction is disregarded in testing for con-vergence of the iteration. In other words, theexact height D of each prism is irrelevant in de-termining whether the point belongs to the prism.

This procedure generates a list of prisms whereeach prism contains the point.

(3) For most cases, only one prism will be found inthe list from step 2. The interpolation coe�-cients in the tangential direction are then usedto project the point to the surface triangle. Ifmultiple prisms are found, the one that gives theshortest distance between P and the surface tri-angle is chosen.

The projection scheme described above is found tobe robust and reasonably fast. Numerical experimentsshow that it took 3 CPU seconds on an SGI R10000175 MHz workstation to generate a surface grid with2436 points on a triangulation containing 4005 verticesand 8006 faces. Another surface grid with 950 pointswas created in 6.5 CPU seconds on a triangulationcontaining 37700 vertices and 75000 faces.

3.2. Marching Scheme at H-type Corners

In this section, a scheme for treating an H-type cor-ner of degree 3 is described. Simple extensions canbe made to handle H-type corners of degree 4. Af-ter the heuristic rules given in Section 2.4 are usedto determine that a corner is of H-type, two of theseam curves connected to the corner are reconnectedautomatically to form an initial curve for hyperbolicmarching. The remaining curve becomes a referencecurve that an interior grid line of the hyperbolic gridmust follow during the marching procedure. In Fig-ure 8a, curves C1, C2 and C3 are reconnected to forman initial curve running through H-type corners at H1and H2. Curves C4 and C5 become interior referencecurves for the resulting initial curve.

Given an initial curve, occurrence of an H-type cor-ner in an interior point Jc is automatically detectedby searching for end points of reference curves coin-ciding with Jc. After each hyperbolic marching andprojection step, the predicted point along the gridline emanating from Jc is projected onto the referencecurve. Several smoothing steps are performed locallyand the smoothed points are projected back onto thereference surface. Figure 8b shows the surface gridproduced from an initial curve with two interior H-type corners.

3.3. Marching Scheme at O-type Corners

For an O-type corner or an apex point, a polar topol-ogy is used for the surface grid wrapping around theinitial point. This subsection outlines a special alge-braic scheme to generate the �rst layer of points awayfrom initial point. The hyperbolic marching methoddescribed in Section 3.1 is employed for the subsequentlayers. A mechanism similar to that presented in Sec-tion 3.2 is used to guarantee that each seam curve

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C1

C2

C3

C4

C5

C6

C7

C8

H1

H2

O1

H3

(a) (b)

(c) (d)

C9

C10

C11

Fig. 8: Curves and surface grids at H-type and O-typecorners. Curves are labelled Ci with with i = 1; 11.Corners are labelled Hi with i = 1; 3 and O1. (a)Initial and reference curves. (b) Surface grid from H-type corners. (c) Surface grid from O-type corner. (d)Surface grids from nearby retracted seam curves.

connected to the initial point is followed by a radialsurface grid line during hyperbolic marching.Since all seam points are assumed to coincide with

a vertex on the surface triangulation, the vertex-edgelist is used to �nd all edges connected to the initialpoint. Then the ordering of the edges around the ini-tial point and the angle between each pair of adjacentedges are determined. The circumferential directionis then divided into sectors where each sector is sep-arated from the next by a seam curve connected tothe initial point. For an apex point, there is just onesector. The angle subtended by each sector is thencomputed by summing the angles between the corre-sponding ordered edges connected to the initial point.Next, the number of points in the circumferential

direction in each sector is automatically determined.Based on the polar topology of the grid, the arc lengthalong the outer boundary of each sector can be es-timated from the marching distance and the anglesubtended by the sector. The number of points in thecircumferential direction in the sector is computed byassuming a uniform spacing along the outer boundaryno larger than the end grid spacing in the marchingdirection.9

Grid points on the �rst layer are created by extrud-ing radially on the surface from the initial point to adistance equal to the prescribed initial spacing for thissurface grid. The angular positions of points in the

mast mounted site

Fig. 9: Rotorcraft surface geometry.

�rst layer in each sector are made to form a uniformdistribution over the angle subtended by the sector.In Figure 8c, radial grid lines of the surface grid

grown from O1 are made to follow C5, C7 and C8.Curves C9, C10 and C11 are the retracted versions ofthe connected seam curves. Other surface grids aregrown from these curves as shown in Figure 8d.

4. Test Cases

Three test cases are presented below to illustruatethe schemes described in the previous sections. Theseinclude a rotorcraft, a space vehicle and a triceratops.

4.1. Rotorcraft

The �rst test case is a rotorcraft shown in Figure 9.For a complex con�guration such as this, it is usu-ally easier to treat the problem in several parts. A�ne surface triangulation for the mast mounted sitecontaining 61000 faces and 30000 vertices is shown inFigure 10a.Seam curves are automatically extracted around the

surface discontinuities and intersection curves of theattachments. Several other surface curves are ex-tracted by sectional cuts to create more initial curvesfor hyperbolic marching so that the surface can be cov-ered completely (see Figure 10b). The hyperbolic gridsshown in Figure 11 are generated via the OVERGRIDgraphical interface in less than 3 hours of manual ef-fort. A total of 17 surface grids are created with28660 points. About two thirds of the time is usedin adjusting marching distances so that the surface iscompletely covered with proper overlap between neigh-boring grids. Similarly, 23 surface grids for the fuselageare created in about 5 hours of wall clock time (see Fig-ure 12). The fuselage reference triangulation contains226000 faces and 114000 vertices while the total num-ber of points in the resulting surface grids is about18000.

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(b)(a)

Fig. 10: Rotorcraft mast mounted site. (a) Surfacegeometry. (b) Seam and sectional cut curves.

Fig. 11: Hyperbolic surface grids for rotorcraft mastmounted site. Not all grids are shown to improve dis-play quality. Grid boundaries are depicted by darklines.

4.2. Space Vehicle

The second test case is an experimental space vehicleshown in Figure 13. Only the left half of the symmet-ric vehicle is considered here. The surface geometryis highly resolved with about 75000 faces and 38000vertices. Both �ns have �nite thickness trailing edges.Seam curves are automatically extracted around allsharp surface edges. The stencil-walk scheme is neededto create the surface curves over the high curvaturecontours along the leading edges of the �ns, and thelower side walls of the body ap (see Figure 14).

Grids around most of the sharp surface features ofthe space vehicle, other than ones on the outboard �n,are shown in Figure 15. Generation of the 16 gridsdisplayed took about 4 hours using the OVERGRIDinterface. The di�culty with this geometry lies in themoderately complex arrangement of seam curves and

Fig. 12: Hyperbolic surface grids for rotorcraft fuse-lage. Not all grids are shown to improve display qual-ity. Grid boundaries are depicted by dark lines.

Fig. 13: Space vehicle geometry with about 75000 facesand 38000 vertices.

Fig. 14: Seam curves for the space vehicle viewed fromthe back.

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Fig. 15: Surface grids around seam curves in the baseregion of the space vehicle (grid boundaries depictedby dark lines).

Fig. 16: Triceratops surface geometry with 5660 facesand 2832 vertices.

seam points in the base region that need to be cap-tured. Also, time is needed to create surface curvesalong the high curvature contours in several places.It is anticipated that creating grids for the outboard�n and �lling the remaining smooth parts of the sur-face geometry would require approximately another 4hours of user time.

4.3. Triceratops

In the previous two test cases, the surface geometryis essentially smooth in most places except for cer-tain local regions, and accurate representation of localdiscontinuities and high curvature regions are of highimportance. For third test case, a mostly faceted ir-regular surface is chosen to illustrate the capability ofthe current scheme applied to a di�erent class of geom-etry. A triangulation over a triceratops (Figure 16) ischosen for this purpose. Here it is not critical that allfacets of the surface be captured accurately. The sur-face geometry contains 5660 faces and 2832 vertices.

Since the surface consists of many facets, initial

Fig. 17: Triceratops surface curves.

Fig. 18: Surface grids for triceratops. Not all grids areshown to improve display quality. Grid boundaries aredepicted by dark lines.

curves for the hyperbolic marching scheme are createdby various sectional-plane cuts on the body, and bythe stencil-walk method along the crown of the headand the base of the feet (Figure 17).

All surface grids are obtained by hyperbolic march-ing, a sample of which are shown in Figure 18. Aspecial option in the grid generator is used to close theouter boundary of the marching scheme to a point forgrids around the two sharp horns and four feet. Thecomplete con�guration consists of 17 surface grids andare created with about 14000 grid points. Startingfrom the surface triangulation, the process of surfacecurve creation and surface grid generation took aboutone and a half hours of user time using the OVER-GRID graphical interface.

5. Concluding Remarks

A scheme is presented for generating structuredgrids over a triangulated surface geometry de�nition.Surface features are classi�ed into seam curves andseam points and are captured under an overset do-main decomposition procedure. Various methods forcreating surface curves for triangulated surfaces arealso described. A hyperbolic marching scheme is usedto generate grids from initial curves and points. Spe-cial treatments are needed at the initial points whichcan be of O- or H-type. A robust and e�cient surface

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projection scheme that is coupled to the hyperbolicgrid generator is also presented. It is shown that gridsfor moderately complex con�gurations can be gener-ated in a reasonable amount of time. The human e�ortneeded to model CAD geometry data with computa-tional grids, and thus the design cycle time, is reducedwith this new capability.A surface triangulation is a popular geometry repre-

sentation that can be easily derived from CAD modelsas well as scanned objects. While this opens up thepotential for tackling a wide range of problems, thelack of a standard format can be annoying to theuser. In addition to the CART3D and FAST formatsmentioned in this paper, other commonly used for-mats include stereolithography (STL), �nite elementneutral (FNF), and Virtual Reality Markup Language(VRML). Varying amounts of information are presentin the di�erent formats while some also allow dupli-cated vertices. Degeneracies such as self-intersectionsoccur quite frequently and require pre-processing workto remove. In order to truely streamline the geometryhandling and grid generation process, the user com-munity has to agree on a robust standard format forsurface triangulations.

Acknowledgements

The author is grateful to Michael J. Aftosmis andDr. Shishir Pandya for many valuable ideas and dis-cussions on surface triangulations. This work wasperformed when the author was employed by MCAT,Inc. under NASA contract number NAS2-14109, Task23. Funding for the work was provided by the DODHPCMP CHSSI CFD-4 program.

References

1. Steger, J. L., Dougherty, F. C. and Benek, J. A.,\A Chimera Grid Scheme," Advances in Grid Genera-tion, K. N. Ghia and U. Ghia, eds., ASME FED-Vol. 5,June, 1983.

2. Chan, W. M. and Buning, P. G., \Surface GridGeneration Methods for Overset Grids," Computersand Fluids, 24, No. 5, 509{522, 1995.

3. Chan, W. M., \Hyperbolic Methods for Sur-face and Field Grid Generation," CRC Handbook ofGrid Generation, Eds. Thompson, Soni and Weather-ill, CRC Press, 1998.

4. Chawner, J. R. and Steinbrenner, J. P., \Au-tomatic Structured Grid Generation Using GRID-GEN (Some Restrictions Apply)," Surface Modeling,Grid Generation, and Related Issues in ComputationalFluid Dynamic (CFD) Solutions, NASA Lewis Re-search Center, NASA CP-3291, 463{476, 1995.

5. Wulf, A. and Akdag, V., \Tuned Grid Genera-tion with ICEMCFD," Surface Modeling, Grid Gener-

ation, and Related Issues in Computational Fluid Dy-namic (CFD) Solutions, NASA Lewis Research Cen-ter, NASA CP-3291, 477{488, 1995.

6. Haimes, R. and Follen, G., \Computational Anal-ysis PRogramming Interface," Proceedings of the 6thInternational Conference on Numerical Grid Genera-tion in Computational Field Simulations, Eds. Cross,Eiseman, Hauser, Soni and Thompson, July, 1998.

7. Aftosmis, M. J., Berger, M. J. and Melton, J. E.,\Adaptive Cartesian Mesh Generation," CRC Hand-book of Grid Generation, Eds. Thompson, Soni andWeatherill, CRC Press, 1998.

8. Aftosmis, M. J., Delanaye, M., and Haimes, R.,\Automatic Generation of CFD-Ready Surface Trian-gulations from CAD Geometry," AIAA Paper 99-0776,1999.

9. Chan, W. M. and Gomez, R. J., \Advances inAutomatic Overset Grid Generation Around SurfaceDiscontinuities," AIAA Paper 99-3303, Proceedings ofthe AIAA 14th Computational Fluid Dynamics Con-ference, Norfolk, Virginia, 1999.

10. Meakin, R. L., \Moving Body Overset GridMethods for Complete Aircraft Tiltrotor Simulations,"AIAA Paper 93-3350, Proceedings of the 11th AIAAComputational Fluid Dynamics Conference, Orlando,Florida, 1993.

11. Atwood, C. A. and Van Dalsem, W. R., \Flow-�eld Simulation About the Stratospheric ObservatoryFor Infrared Astronomy," J. of Aircraft, 30, No. 5,719{727, 1993.

12. Duque, E. P. N. and Dimanlig, A. C. B.,\Navier-Stokes Simulation of the RH-66 ComancheHelicopter," Proceedings of the 1994 American Heli-copter Society Aeromechanics Specialist Meeting, SanFrancisco, California, 1994.

13. Slotnick, J. P., Kandula, M. and Buning,P. G., \Navier-Stokes Simulation of the Space Shut-tle Launch Vehicle Flight Transonic Flow�eld Using aLarge Scale Chimera Grid System," AIAA Paper 94-1860, Proceedings of the 12th AIAA Applied Aerody-namics Conference, Colorado Springs, Colorado, 1994.

14. Rogers, S. E., Cao, H. V. and Su, T. Y.,\Grid Generation for Complex High-Lift Con�gura-tions," AIAA Paper 98-3011, June, 1998.

15. Walatka, P. P., Clucas, J., McCabe, R. K.,Plessel, T. and Potter, R, \FAST User Guide," RND-93-010, NASA Ames Research Center, 1994.

16. Chan, W. M., \OVERGRID { A Uni�ed OversetGrid Generation Graphical Interface," submitted toJournal of Grid Generation, 1999.

17. Dhatt, G. and Touzot, G., The Finite ElementMethod Displayed, Wiley, 1984.

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