Full exploitation of the sophisticated physical mod-eling capabilities of computational fluid dynamicshas been hampered by the inability to generate highquality grids in a timely manner. Complex geome-tries further exacerbate grid generation difficulties.Despite the potential for fully automatic grid gen-eration with the relatively new unstructured tech-niques, such a capability has, in large part, re-mained elusive. The principle difficulty with thegeneration of such grids lies within the surface dis-cretization, the operation for which, though diffi-cult in its own right, must be intimately coupledwith CAD data to retain geometric integrity andachieve the desired efficiency. This paper presentsa new method for generating high quality unstruc-tured surface grids directly on NURBS surfaces orany other surface for which a similar parameteriza-tion exists; surface distributions are automaticallyadapted to local curvature which renders a high fi-delity representation of the geometry with optimumuse of grid points. The method is applied to numer-ous surfaces possessing large variations in curvatureto demonstrate its effectiveness at producing highquality surface grids automatically.
Introduction
Computational fluid dynamics (CFD) has be-come an indispensible design tool in today's highlycompetitive engineering and manufacturing envi-ronment. Traditional means of product develop-ment, which have included prototype developmentand testing, have been supplemented in large partwith the sophisticated physical modeling capabili-ties available within a wide range of research andcommercial CFD products. Such codes have seenapplication to very complex problems and have pro-duced useful data for product design. Collectively,these codes represent a powerful analysis capabil-ity, however, full integration of these capabilitiesinto the design process has been stymied due to gridgeneration difficulties, user interface limitations and
solver complexities. Adaptive Research has beenengaged in a vigorous development program to ad-dress these problem areas with the CFD20001 soft-ware product incorporating developments in each ofthese areas. Despite these efforts, grid generationdifficulties continue to remain a significant obsta-cle inhibiting full exploitation of the technology asthe non-expert is often daunted by the large invest-ment of time required to produce grids for complexconfigurations.
Engineering and manufacturing organizationsconcerned with product development times and as-sociated costs continue to demand streamlined de-sign methods. Such organizations typically havea Computer Aided Design (CAD) system in placefor geometric modeling and insist that such a de-scription be used not only to speed disseminationof geometric data but also to maintain geometric in-tegrity. The Initial Graphics Exchange Standard2
(IGES) currently serves as the primary standard bywhich such data is communicated. This standardconsists of various geometric entities, the most flex-ible of these being the NURBS3 entity. The need forthe CFD analyst to translate CAD data into otherless rigorous forms in order to generate a suitableCFD grid is quickly becoming an unacceptable so-lution.
The development of unstructured grid meth-ods has spurred increased application of CFDacross many engineering disciplines. Though thetechnique is relatively new, it has already seenwidespread use as its inherent flexibility makes itvery attractive from the standpoint of grid genera-tion for complex geometries; the lack of an imposedstructure, while having definite implications withinthe flow solution algorithm, has significantly easedthe burden of grid generation. Difficulties remaineven with this approach, however, as initial surfacediscretizations, which play an important role in thequality of the volume grid, continue to be difficultand time consuming to generate as the user is typi-cally required to interact with the process on a fun-damental level. This is compounded by the inabil-ity to fully integrate available CAD data directlyinto the grid generation process. In order to renderCFD useful to a broad class of users and streamline
the grid generation process, a methodology whichpermits the automatic generation of high qualitytriangular meshes directly on NURBS surfaces isrequired.
Previous investigators such as Nakahasi andSharov4 have recognized the need for a higher levelof automation in the grid generation process. Theyhave devised a method to generate surface trian-gulations which are sensitive to the local curva-ture. Their method incorporates the AdvancingFront5 technique which links front propagation toestimates of surface curvature determined not froma parametric definition of the surface, but from adiscrete representation. Surface edge distributions,which are also computed as a function of the localcurvature, serve as the initial front.
Though they present some impressive results,there remain issues over the formulation of theirtechnique. In the distribution of points along sur-face edges, the user must contend with 4 interde-pendent input parameters to control relative andabsolute spacing as well as point count. Further-more, as the front advancement is performed inphysical rather than parametric space, surface pro-jections, each involving an iterative procedure, arenecessarily performed to ensure that the points in-troduced during the triangulation lie on the basissurface. Another undesirable aspect of their ap-proach involves the construction of what appears tobe structured surface grids which serve as the basissurface. A fair amount of effort may be required togenerate grids which provide sufficient resolution.In the case of trimmed surfaces (those possessingan internal trimming curve), it becomes a somewhatmore user intensive process to produce the requiredset of surface grids. Lastly, though application ofthe technique did show that high curvature regionswere better resolved, numerous areas with minimalcurvature were over-resolved. We seek a simpler,more robust method for the generation of unstruc-tured grids directly on NURBS surfaces.
In an earlier paper, Abolhassani6 offers a methodfor computing triangulations for NURBS surfaces.His approach first involves representing the NURBSsurface only approximately through numerous bi-linear patches with an Advancing Front tech-nique used to discretize the individual patches.A Newton-Rhapson technique is then invoked toproject each of the points in the resulting trian-gulations onto the actual NURBS surface. The re-sulting grid may require smoothing due to irregu-larities which may be introduced during the projec-tion. Each smoothing operation must necessarily befollowed by a re-projection of the points since thesmoothing filter operates on physical rather thanparametric coordinates. Furthermore, the fidelityof the surface patching operation will have a signifi-
cant impact on the quality of the final triangulation.Such a technique appears rather cumbersome anddoes not possess the desired level of automation.
Of more recent appearance, a method presentedby Hufford, Mitchell and Harrand7 generates un-structured surface meshes on NURBS surfacesthough in an indirect manner. They too dis-cretize surface edges first using what appears to bea rather complicated process involving three userspecified parameters. For the surface discretiza-tion, they also employ the Advancing Front ap-proach which incorporates an iterative process tointroduce points in parameter space such that sur-face triangles with the desired physical character-istics are created. Their method offers additonalcontrol in that not only are estimates of surfacecurvature used to guide the creation of triangles,but other constraints available from the backgroundgrid may also be manifested in the surface triangu-lation. Though it is noteworthy that a completegrid generation system has been built around theirapproach, the example surface meshes they present,however, tend to exhibit a lack of smoothness inboth surface edge distributions as well as trianglearea variation.
In the current work, we seek an automaticmethod to generate curvature adapted surface gridsdirectly on NURBS surfaces using the Delaunay tri-angulation method. The Delaunay method8 hasbeen shown to generate planar meshes very effi-ciently with little user intervention. We supplementthis approach with the concept of equidistributionas postulated by Dwyer, Kee and Sanders9 to clus-ter points in response to surface curvature. Unlikethe previous methods, the current effort operatesdirectly in parametric space thereby providing thatall points introduced during the triangulation auto-matically lie on the NURBS surface, hence, no pro-jections or other iterative procedures are required.At the conclusion of the process, the mapping pro-vided by the NURBS definition is used to yield thephysical coordinates.
Though there are distinct advantages to gener-ating grids in parameteric space, Nakahashi andSharov present valid concerns which do not fa-vor such an approach. The principle difficultyinvolves the nature of the mapping from two-dimensional parametric space to three-dimensionalphysical space which, when highly and unequallystretched, can result in excessively skewed trianglesin the physical plane. Such problems are well un-derstood by anyone working in the field of grid gen-eration. We have successfully addressed this prob-lem in the current effort by incorporating withinthe grid generation process various links to physi-cal space. These constraints, which have been em-bedded within the two-dimensional Delaunay trian-
gulator, permit the generation of isotropic surfacegrids on complex NURBS surfaces while retainingthe efficiency of a two-dimensional operation. Fur-thermore, the method to be presented requires thespecification of at most two constants which controlthe degree of edge and surface resolution.
The remainder of this paper presents the newmethods for determination of both curve and sur-face distributions. These methods are then appliedto various curves and then to simple parametric andNURBS surfaces, each of which possess significantvariations in curvature.
Curve Distributions
The generation of point distributions along sur-face edges is the first step in the process of surfacegrid generation. In keeping with our goal of directoperation on NURBS surfaces, we consider curveswhich are defined parametrically. This will permitapplication not only on NURBS surface edge curvesbut on any curve for which a similar parameteriza-tion exists. For a curve defined by an arbitrary dis-tribution of points, such a parameterization mightbe constructed from the arc-length variation.
We seek to determine a point distribution whichis sensitive to the local curvature such that, inregions of large curvature, smaller spacing resultsand vice versa. This suggests that the concept ofequidistribution, which typically serves as the ba-sis for many structured grid adaption techniques,might be useful for determining such a distribu-tion. A variation of this theme, the equidistributionspring analogy, is used by Nakahashi and Sharov.
We propose another approach which requiresa solution to a differential form of the one-dimensional equidistribution law in which a secondtransformation from parametric space, «, to com-putational space, f, has been made. This equationis given as
0 (1)which is solved subject to the boundary conditions
U(0) = 0, «(£„,«*) = «o (2)
where the parameterization is defined on the inter-val 0 < « < «0. In its most basic form, 7 is thelocal curvature, K. This equation is to be appliedon each of the surface edges in what amounts to one-dimensional parametric space. An iterative processis required for the solution to equation (1) sincethe local curvature must be recomputed at the newnode points for each cycle.
A fundamental problem exists with the applica-tion of equation (1), however, as it satisfys equidis-tribution in parametric rather than physical space.
Thus, even for the surface edge distributions we canexpect that highly stretched distributions in physi-cal space may result due to the nature of the map-ping. To demonstrate this, a distribution has beencomputed for the parametric curve defined by
x = 12«y = 5e~usm(47r«) (3)
with 0 < u < 1. Given the solution, «(£), from theequidistribution equation, the physical coordinatesare then calculated from equation (3) to yield thedistribution shown in the following figure.
Figure 1 Nominal distribution on damped sine wave.
The resulting distribution is indicative of the draw-backs inherent within methods which compute dis-tributions in parametric space. With 52 pointsspecified on the curve, the maximum to minimumphysical spacing ratio is approximately 27:1 andsuch a highly stretched distribution on a surfaceedge would undoubtedly translate into a poor sur-face discretization. To circumvent this problem, wemodify 7 in equation equation (1) such that it be-comes a function not only of the local curvature butof additional physical constraints and a user definedconstant, <j>. This formulation introduces the phys-ical variation into the solution and also permits usto control the extent of the clustering through ad-justment of <j> which is constrained to take on valuesin the range 0 < <f> < 1 where <f> = 0 corresponds toa uniform distribution in physical space. The distri-bution for the same curve is now regenerated withthe modified form for 7 and <j> = 0.8, the results ofwhich are shown in figure 2.
Figure 2 Distribution on damped sine wave,modified 7, <j> = 0.8.
Such a formulation yields a much more reasonabledistribution.
Equation (1) is also applied to a more complexcurve taken from the cross-section of a generic re-entry vehicle. This curve features a blended wingsection with high curvature tip. In this applica-tion, the cross-section is parameterized using thedimensionless arc-length. The distribution, whichconsists of 55 points, is computed with <j> = 0.9 andis shown in the following figure.
Figure 3 Distribution on re-entry Vehicle cross-section,</> = 0.9.
The clustering ratio achieved is quite reasonableeven in the region of the wing tip where the highcurvature tip is in close proximity to the flat wingsection. The distribution was also recomputed with0 = 0.75, with the resulting distribution shown inthe figure below.
Figure 4 Distribution on re-entry vehicle cross-section,<t> = 0.75.
It can be seen that a reduction in (j> has reduced theamount of stretching within the distribution though
the higher curvature areas continue to remain bet-ter resolved. This approach represents a highly ef-ficient and robust method for the automatic dis-cretization of edge curves as it requires that, for agiven point count, the user only need specify <f>.
Two-Dimensional Triangulation
The surface triangulation methodology used inthis effort is based upon earlier development effortsof the current authors. We employ the Delaunaytriangulation method, the specific implementationbased in large part on the technique presented byHolmes and Snyder10. This technique first com-putes a boundary triangulation using the prede-fined edge points only. As the boundary pointsare sequentially introduced into the triangulation,a circumcircle defined using the triangle vertices isconstructed to ascertain whether an edge swappingoperation must be employed to ensure that no ver-ticies of any triangle fall within the circumcircle.This edge swapping operation propagates through-out the domain to satisfy this constraint. In thisway the scheme maximizes the minimum angleswithin each of the triangles yielding a good qual-ity mesh essentially automatically.
Once the boundary triangulation is complete,Weatherill's11 approach is employed to sequentiallyintroduce points at the centroids of the triangles un-til a locally defined size threshold has been reached.As these points are introduced, the current triangleis subdivided with the edge swapping operation in-voked as required. After extraneous triangles havebeen removed, the resulting mesh is then smoothed.It is worth noting that careful attention to the de-sign of data structures and searching algorithmswithin the computer code play an extremely impor-tant role in the resulting efficiency of the code. Theauthors have constructured a computer code whichruns extremely fast and requires minimal memorymaking it ideal for applications on personal com-puters and workstations.
Surface Triangulation
Since our approach involves the generation ofthree-dimensional surface grids in two-dimensionalparametric space, we make substantial use of thetriangulator described in the previous section. Thisis an obvious advantage of the approach, but wemust also contend with the difficulties introducedby the mapping as previously described. Modifi-cations have been incorporated into the triangula-tor which introduce various physical constraints inorder to compensate for any stretching within themapping. The resulting surface grids are thus, in
this respect, independent of the nature of the pa-rameterization used for the NURBS surface.
The generation of the surface mesh begins with atriangulation performed using the boundary pointsonly. These distributions are computed in paramet-ric space using equation (1) applied to the surfaceedges. The points comprising these distributionsare then sequentially introduced into the triangula-tion.
The placement of points within the interior of thesurface is to be a function of the local surface curva-ture. We seek to satisfy, at least approximately, therequirement of equidistribution which is expressedas
< a (4)
where A is the local cell area. This will provide forsmaller triangles in regions of high curvature andvice versa. The exponent on the area provides amechanism to introduce additional stretching intothe mesh thereby permitting a larger variation incell area distribution. The surface curvature is com-posed of estimates of the principal curvatures inboth parametric directions and is given as
K = (5)
where «„ and KV are the curvatures which corre-spond to the « and v parametric directions, andc is a small number to prevent surface curvaturesfrom becoming identically zero. The equidistribu-tion constant, a, is denned from the edge distri-butions. As in the case of the computations forthe edge curve distributions, we introduce a fur-ther constraint such that K(«, v) — » K(U, v, V>) where0 < i> < 1 which provides for global control of theresolution across the surface. Although it is not re-quired, it is permissible and indeed intuitive that
rather than the physical coordinates ensures thatthe points remain on the surface and no projectionoperation need be invoked. Embedded physical cri-teria within the filter ensure smooth grids in thephysical plane even though parametric coordinatesare used.
Applications
Exponentially damped sine wave
The methodology described is first applied toan exponentially damped sine wave surface definedparametrically as
x = 10«y = 12wz = 3e-tue-2"sin(47rt))
(6)
where 0 < u,v < 1. The surface distribution whichresults for ip = 0.8 contains 2692 triangles and 1398nodes and is shown in the following figure.
The surface triangulation is now constructed bysequentially introducing points into the existing tri-angulation directly in u, v space; no iterative processis required in this operation. These points are notintroduced according to simple cell size constraintsas was done in the baseline triangulator, but areintroduced to satisfy equation (4). Simply stated,when candidate triangles are examined for point ad-dition and possible subdivision, a for that triangleis compared against the product of the triangle areaand the surface curvature. If this quantity exceedsthe former, the surface has not been sufficiently re-solved and the point is introduced. This processcontinues until sufficient points have been intro-duced and triangles subdivided until equation (4) issatisfied for every triangle on the surface. The para-metric mesh coordinates are then passed througha smoothing filter. The use of these coordinates
Figure 5 Surface distribution for exponentiallydamped sine wave surface, ip = 0.8.
An exceptionally smooth distribution has been ob-tained with noticeable enhancement of the resolu-tion at the peaks and valleys of the surface. Cellsize variation is very well behaved in the vicinity ofthe boundaries even in the high curvature areas ofthe surface.
With the edge distributions fixed, surface trian-gulations were also generated for values of i}> cor-responding to 0.9,0.95, and 1.0 resulting in meshesconsisting of 3146/1625, 3670/1887 and 5390/2747triangles and nodes, respectively. The resulting sur-faces for each of these cases are shown in the figuresbelow.
Figure 6 Surface distribution for exponentiallydamped sine wave surface, i/> = 0.9.
Figure 7 Surface distribution for exponentiallydamped sine wave surface, i/> — 0.95.
The implicit dependence between the degree ofglobal resolution and $ is obvious. Despite the highlevel of refinement prescribed for the last surface,the grid exhibits very smooth behavior even withthe proximity of high curvature and low curvatureregions.
Exponential surface
The second application involves an exponentialsurface which is denned parametrically as
x = (2« - I)x0
y = (2v-l)y0
z = 2e-*/2(7)
where R = x2 + y2 and the physical surface is de-nned within the region — x0 < x < x0 and — y0 <y <y0 where x0 = y0 — 3.5 and 0 < u, v < 1. Themaximum curvature for this surface is seen to existwithin the interior rather than on the edge as wasthe case for the previous application. In the nextseries of figures, we demonstrate the effect of chang-ing the point count along the edges with ip fixed at0.95. At present, the number of points along eachedge must be prescribed, though it is hoped thatat some point in the future, the process for deter-mining such could be automated as well. Figures 9through 12 correspond to 10, 14, 17 and 21 pointson each edge, respectively.
Figure 9 Surface distribution for exponential surface,10 points per edge.
Figure 8 Surface distribution for exponentiallydamped sine wave surface, ijj — 1.0.
Figure 10 Surface distribution for exponential surface,14 points per edge.
Figure 11 Surface distribution for exponential surface,17 points per edge.
28 points. The resulting surface mesh which corre-sponds to V" = 0.95 is shown in the following figure.
Figure 12 Surface distribution for exponential surface,21 points per edge.
Note that the extent of the clustering in the regionof the peak has changed in tandem with the resolu-tion local to the surface edges. This highlights thegoverning influence of the edge distribution on thesurface resolution and why such is so critical to theinterior distribution.
Ripple surface
The next surface to be examined is the "ripple"surface which is defined parametrically by the equa-tions
x - (2« - I)x0
y = (2t>-1)2/0
Figure 13 Ripple surface, V = 0.95.
The mesh is surprisingly good even for this com-plex surface. Note the differences in the resolutionaround the circumference of the surface. This isdue to differences in K which, for the parameter-ization employed, varies about the circumferenceeven though the surface exhibits axial symmetry.A reparameterization capitalizing on this would un-doubtedly produce a more uniform distribution.
First NURBS surface
The next case involves an actual NURBS surfacerepresenting a section of a generic re-entry vehicle.This surface features an integrated wing at high di-hedral angle, the cross-section for which was pre-sented in figure 3. The curvature variation of thissurface presents some difficult challenges for sur-face grid generation as regions of very high curva-ture are in close proximity to regions of zero cur-vature. In addition, due to the placement of theNURBS control points, the mapping is highly andunequally stretched as is indicated by the struc-tured grid generated for this surface as shown inFigure 14. This grid was generated using a uniformdistribution in both parametric directions. Note theunequal stretching in the spanwise direction.
(8)
defined within the interval x0 = y0 = 3.5 and0 < u,v < 1. This surface presents a more sig-nificant challenge as there is not only considerablevariation in surface curvature, but surface slopesattain large values. Schemes which operate exclu-sively in parameter space should see severe degre-dation in the resulting surface meshes. The edgedistributions for this surface were discretized using
To qualify the extent of mesh degradation due tothe mapping for this surface, a surface mesh wasgenerated without consideration given to the na-ture of the mapping. Selected constraints withinthe modified triangulator were disabled which thenproduced the mesh shown in the following figure.
ratio have been generated in regions where the map-ping is locally near-uniform with the new pointaddition scheme in equation (4) providing the in-creased resolution.
The modified triangulator is now employed withall constraints active to produce the surface gridshown in figure 16.
Figure 15 Surface triangulation for NURBS surfacedue to stretched mapping.
Figure 16 Surface triangulation for NURBS surface,ij> = 0.95.
This surface consists of 3500 triangles and 1800nodes. Despite the highly stretched mapping, themethod has performed very well producing trian-gles of very high quality in all regions of the surfacewith no evidence of the stretching due to the map-ping apparent. The transition from the smallesttriangles on the wing tip region to the flat portionof the wing is quite smooth.
Though the high curvature areas of the surfaceare well resolved, a fair number of elements havebeen introduced to achieve this. This can be ame-liorated by the use of the stretching factor built intoequation (4) as it provides a means to modify thenature of the area transition between surface trian-gles. To demonstrate this, the edge curves for theprevious surface were first recomputed using only80% of the points in the original distributions. Thesurface mesh was then regenerated with V = 0.80,the results of which are shown in the following fig-ure.
Such results are indicative of the difficulties in gen-erating grids in parametric space. Despite the gen-erally poor mesh quality, cells of reasonable aspect
Figure 17 Surface triangulation for NURBS surface,V> = 0.80, n = 1.
This surface contains 1600 fewer cells than the pre-vious surface but, as was expected, a significantamount of resolution has been lost in the wing tipand other regions. By increasing the exponent inequation (4), we can introduce additional stretch-ing into the mesh and achieve increased resolutiononly in these critical areas. With all other factorsunchanged, the mesh was regenerated with n = 2.5and is shown in Figure 18.
Using one-third fewer triangles, the resolution onthe wing tip as well as in other regions has beenenhanced to achieve nearly the same degree of res-olution in the critical areas as that for the meshshown in Figure 16. With suitable adjustment of if>and n, stretched grids having the desired resolutionmay be easily generated.
NURBS re-entry vehicle
The final case to be examined involves a completere-entry vehicle surface which includes the completewing section and main body. The wing root orig-inates from a near-elliptical body and exhibits aslightly curved tip at the trailing edge. There arenumerous areas which possess significant curvatureon this complex surface. The grid obtained withi> = 0.95 is comprised of 5435 triangles and 2787nodes and is shown in the following figure.
Figure 19 Surface triangulation for NURBS re-entryvehicle, $ = 0.95.
This NURBS surface is defined by a much morecomplex control net and, despite the highlystretched mapping, the method has performed ex-ceptionally well providing increased resolution inthe high curvature regions of the surface.
A coarse surface grid has also been generated hav-ing only 70% of the points along each edge. The gridshown in Figure 20 was generated with if> = 0.8 andis composed of 1997 triangles and 1047 points.
Figure 18 Surface triangulation for NURBS surface,V> = 0.80, n = 2.5.
Figure 20 Surface triangulation for NURBS re-entryvehicle, V = °.8.
It is evident that the method provides sufficient con-trol of surface grid density and resolution to pro-duce high quality grids even on very complex sur-faces.
Conclusions
An extremely versatile and robust methodologyfor the automatic generation of triangular gridsdirectly on NURBS surfaces has been presented.This methodology is ideal for computing initial dis-cretizations as they are automatically adapted tosurface curvature resulting in a high fidelity repre-sentation of the geometry. The method in large partretains the advantages of performing the triangula-tion in two-dimensional parameter space with thetypical problems caused by highly stretched map-pings having been eliminated to yield a methodwhich produces high quality triangulations very effi-ciently. Furthermore, a simple mechanism has beenbuilt into the method to permit global control ofmesh stretching. Except for user specification ofthe level of refinement and stretching, generationof these triangulations is completely automatic; noadditional user intervention is required.
References
1) CFDSOOO Version 3.0 User's Manual, AdaptiveResearch, Huntsville, AL, May, 1997.
2) The Initial Graphics Exchange Specification(IGES) Version 5.0, U.S. Department of Com-merce, National Institute of Standards and Tech-nology, September, 1990.
3) Piegl, L. and Tiller, W., "Curve and surface con-structions using rational B-splines," CAD, Vol.19, No. 9, pp. 485-498, 1987.
4) Nakahashi, K. and Sharov, D., "Direct Sur-face Triangulation Using the Advancing FrontMethod," AIAA Paper 95-1686, 1995.
5) Lohner, R. and Parikh, P, "Three-DimensionalGrid Generation by the Advancing FrontMethod," Int. Journal of Numerical Methods inFluids, Vol. 8, pp. 1135-1149, 1988.
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