NURBS-based surface grid redistribution and remapping algorithms

COMPUTER AIDED GEOMETRIC DESIGN ELSEVIER

Computer Aided Geometric Design 12 (1995) 675-692

NURBS-based surface grid redistribution and remapping algorithms B h a r a t K. S o n i a, *, S h a o c h e n Y a n g b NSF Engineering Research Center, Mississippi State University, Mississippi State, MS 39762, USA b Department of Mathematics, Mississippi University for Women, Columbus, MS 39701, USA Received January 1994; revised January 1995

Abstract Algorithms to redistribute, refine, remap, and optimize structured surface grids utilized in computational fluid dynamics (CFD) simulations involving complex regimes are presented. The non-uniform rational B-spline (NURBS) representation is utilized for defining parametric surfaces. A semi-automatic remapping algorithm presented here results in a simplified surface grid generation process with significant reduction in generation labor time for surfaces involving holes, gaps and interior objects. A hermite transfinite interpolation method and elliptic generation systems are enhanced to optimize (in view of orthogonality and smoothness) surface grids. Computational examples are presented to demonstrate the success of these methodologies.

1. I n t r o d u c t i o n

In the last few years, structured grid generation has evolved as an essential tool in obtaining numerical solutions of the partial differential equations of fluid mechanics. The geometry-grid generation is considered as the most labor-intensive part of any CFD application. Also, 80-90% of the grid generation labor time is usually spent on the geometry processing and the surface grid generation. In most C F D applications these original surfaces are defined in the C A D / C A M system as a composition of explicit or implicit analytical a n d / o r sculptured definition. These surfaces are supplied in the grid system as a discretized network of points.

" Corresponding author. 0167-8396/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 8 3 9 6 ( 9 5 ) 0 0 0 1 2 - 7

B.K. Soni, S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

676

However, in most grid systems, the original definition of the surfaces is not available during the redistribution and refinement of grid points. The redistribution and refinement of grid points is usually accomplished by taking a discretized network of initial points as the geometry definition. Hence, the challenge in surface grid generation is to redistribute and refine surface grids while maintaining the fidelity of the geometrical description of the original surfaces. The algorithms to redistribute, refine, remap and optimize structured surface grids utilizing the NURBS representation are discussed in this paper. Redistribution and remapping procedures with appropriate examples are presented in the section followed by basic concepts and definitions. The optimization algorithms based on Hermite transfinite interpolation technique and elliptic surface generation scheme are presented in the final section. The computational examples demonstrate the success of these methodologies. For more details about CFD and NURBS, see the references.

2. Basic concepts Structured surface grid generation involves the establishment of a one-to-one correspondence between the non-uniformly distributed physical surface and the uniformly distributed computational plane. Let r = ( x ( u , v), y(u, v), z(u, v)) denote a parametric surface with Euclidean coordinates (x, y, z) and parameter values (u, v). The surface grid associated with the computational domain {(El, ~2)[ 1 ~ 1 4 n , 1 ~2~
~_~ ~ ( x u _ x

u_,)2+(y

_yu_ l)2+(z~_z~_l)

2

u =i I + 1

r i -~

i2

£ U

,

~(Xu--Xu-l)2+(y.--Yu-l)2+(Zu--Zu-l)

(1)

2

=ii+1

where i = i s + 1 , . . . , i 2, satisfies the definition of a distribution curve. Definition 1.2. L e t ( x i j , Yij, Zij)' i = i I .... ,i2' J =Jl .... 'J2 be the physical Cartesian coordinates of a set of points on a surface. Then a mesh (si], ti]), i = i 1, ij + 1 . . . . , i2 J =Jl, Jl + 1 , . . . , J2 is called a surface distribution mesh if (sid . . . . , si2j) for j =Jl . . . . , J2 represents the distribution curve for the curve ((xij , Yij, Zij), i = il,

V~:~X~

B.K. Soni, S. )fang / Computer Aided Geometric Design 12 (1995) 675-692

1--;+..2:.), h7"-,<

. ~

.f

7

"

,,

]

/_

, '

Illlllliillllllllllliilllllll~

~ ~

.v-c-.Lu,v.~,./~,< ,~ !

677

H~I+J~++4+J-HIII~!II+t+Ht ][j]l[l~q}lllllllll~lllllllI] I:~llllllllll~ IIIIIIII

.9

!!!!]iii!ii!!iiiiiiiiii!!iiii!

.4.. ' / / / ~ •

[!II[lli::::::::~!!ili~::::l: [ l l l l ~ l l l l l l l l l IIIIII1

•

I[lllI!! l l l l I l l i l l l l lil lil , iiii!iiiiiiiii ,,,: ......... ]]ll]lli~llllllllllliilllIllll

[[llJlIi~lllllllllll;:[llll]ll

i;~llIllI[IIIIIIIlJ U~JI]'" ....................

C

A

B

Physical Space

Distribution Mesh

A

•

B

Computational Space

~

C

Fig. 1. Relationship between physical space, distribution mesh and computational space.

i1+ 1,...,i2) for J=Jl,...,J2 and (tih..... t%) for i = i 1. . . . . i 2 represents the distribution curve for the curve ((xii, Yiy, zij), J =J], Jl + 1,...,J2), for i = i l, il + 1 , . . . , i 2. Also, there exists a one-to-one correspondence between the physical space and the surface distribution mesh, and between the surface distribution mesh and the computational space. These relations are demonstrated in Fig. 1. The surface definition using NURBS representation (Farin, 1990; Mortenson, 1985; de Boor, 1978; Bartels et al., 1987) is widely used in C A D / C A M systems and scientific visualization. The NURBS definitions for the curve and surface are stated as follows.

Curve: C(t) = ~ W(i)d(i)Ni~(t)/~ W(i)Nik(t), i

O

i

(2)

O

(t-T(i))Nik-'(t) (V(i+k)-t)Ni+k , l (t) N~k(t)= T ( i + k - 1 ) - T ( i ) + T ( i + k ) - T ( i + l )

'

(3)

where k is the order of the curve, T is the knot sequence, Nil(t) = (10

ifT(i)~t
W is the weight function, d are the control points, N is the basis function, and n is the last index of the control points.

Surface: ~ W( i, j)d(i, j)Nik~(s)Njk~-(t) i = 0 j=(} =

r ( s , t)

~

kW(i,j)Nk,(s)Nff_(t)

,

i-o j=o

where the basis functions are defined in the same manner as Eqs. (2) and (3).

(4)

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B.K. Soni, S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

The relationship between order k, number of knots n k n o t , and the number of control points (n + 1) can be expressed by n k n o t = (n + 1) + k. The convex hull, local support, shape preserving forms, and variation diminishing properties of B-spline functions (Farin, 1990) contribute to the generation of the well-distributed smooth grid. Also, widely utilized geometrical entities can be exactly or approximately (with desired tolerance) converted to NURBS presentation (Yu and Soni 1994). The derivatives rs, r,, rss, rst, r t t , . . , etc. can be readily evaluated from the expressions (2)-(4). The algorithm to evaluate NURBS representation for curves and surfaces from a given sculptured data, originally developed by de Boor (1978) and then enhanced by Yu (1992) and Yu and Soni (1995) is utilized in this work.

3. Redistribution and remapping The parametric space associated with a NURBS surface is transformed as the surface distribution (normalized chord length based) mesh. The redistributed and remapped surface grid is obtained by evaluating the NURBS surface at the respective parameter associated with the redistributed and remapped distribution mesh. For example, the original sculptured surface and the associated distribution

r !TI i'!rr ! i" ! .

(b)

la)

Ic)

(d)

Fig. 2. (a) Originalsculpturedsurface.(b) Distribution mesh associatedwith the surface(a). (c) Desired distribution mesh.(d) Redistributed surface grid.

B.K. SonL S. Yang / Cornputer Aided Geometric Design 12 (1995) 675-692

679

(b) {a) Fig. 3. (a) C-type grid on a surface. (b) Remapped O-type grid.

mesh are presented in Figs. 2a-b. The NURBS representation (control points, weights, knots,...) is evaluated by utilizing the enhanced de Boor's algorithm (de Boor, 1978; Yu, 1992; Yu and Soni, 1995). A candidate desired distribution mesh demonstrated in Fig. 2c is then evaluated using the NURBS representation. The resulting distribution mesh is demonstrated in Fig. 2c. The construction of the distribution mesh (Fig. 2c) is application dependent. In this example, the distribution mesh is constructed on a 0-1 square by using the transfinite interpolation scheme. The distribution on the outer four boundaries is computed by using hyperbolic tangent stretching with user supplied spacings. The arclength associated with outer four boundaries of the NURBS surface is utilized in the specification of these spacings. The resulting surface grid is smooth and well distributed as demonstrated in Fig. 2d. Now, consider the sculptured surface presented in Fig. 3a. The surface grid is mapped as a C-type grid (Thomson et al., 1985) on the surface. However, if the user is interested in mapping an O-type grid (Thomson et al., 1985) on the surface, a remapping process is accomplished by essentially creating a distribution mesh which when evaluated results in a surface grid of Fig. 3b. The associated parametric (distribution) mesh are presented in Figs. 4a-b. The creation of the distribution mesh for remapping is intuitive and highly application dependent. Here, an interactive program which specifies x, y, z coordinates and associated parameters (u, u) of the NURBS definition is utilized in determining the parameters of interest. The four boundaries associated with the desired distribution mesh for

t

i

i~

[!ll

(a)

.

.

.

.

;:[i:',::

:1;

H :: ; ; ;

.......... " " 1

, i ti

]

,lII;;;;:;:i]!lll

Iit

(b)

Fig. 4. (a) Distribution mesh associated with C-type grid (b) Desired distribution mesh for O-type grid.

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B,K. SonL S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

(a) (o)

(b)

(d)

Fig. 5. (a) Initial O-type grid. (b) Distribution mesh associated with initial O-type grid. (c) Remapped desired distribution mesh to accomplish H-type grid. (d) Resulting H-type grid.

O-type grid is generated and then a combination of transfinite interpolation and elliptic grid generation scheme (Soni, 1985, 1992) is applied to produce a smooth distribution mesh (Fig. 4b). The second example describing the process of remapping an O-type grid into an H-type grid (Thomson, 1985) is presented in Figs. 5a-d. The remapped parametric surface is created by re-establishing the associated transformation between the physical space and the computational space. The generation of H-type grid (Fig. 5d) is accomplished by evaluating parametric distribution presented in Fig. 5c at the NURBS surface resulting for the sculptured O-type grid presented in Fig. 5a and the associated distribution mesh Fig. 5b. The remapped desired distribution mesh is obtained by considering the physical to computational space mapping criteria applied in the O-type and H-type grid respectively. The pictorial views of the physical to computational mapping associated with O-type and H-type grid are provided in Figs. 6a-c. The remapped desired distribution is generated using weighted transfinite interpolation by mapping the configuration in Fig. 6b onto the one in Fig. 6c. Remapping of the surface grid is also required when the interior object(s) is (are) to be kept fixed as part of the interior surface grid. For example, consider the sculptured surface and the interior object presented in Fig. 7a. The desired distribution mesh and the resulting redistributed surface along with the interior object are presented in Figs. 7b-c. The remapping process needs to blend an interior object as a part of the surface grid. An interpolation-search algorithm based on the NURBS presentation is developed to evaluate parameters associated with the interior object. This evaluation is demonstrated in Fig. 8. The evaluation of parameters is accomplished by minimizing ]Jr - F JJ where F is the point on the interior object for which parametric values s, t are to be determined, and r is the closest point from F resides on the surface. The cell location is determined by evaluating the sign of ( r - F ) . r ~ and ( r - F ) . r t where s, t are the parametric

681

B. 14. Soni, S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

E

F

A

Y~L~

_

B

D

F

C[

B

A

(b)

(a) @ H-Type ~

g

kF

~2

C

A

B

D

(c)

Fig. 6. (a) Physicalto computationalmapping.(b) Computationalspace(O-type).(c) Computational space(H-type). values associated with point r. An automatic algorithm based on weighted transfinite interpolation (Soni, 1985, 1992) in two dimensions is developed which blends the parameters associated with the interior objects into an overall distribution

L[_IIfI~II111!|!'ILL!j

(a) (b) Fig.7. (a) Initialsculpturedsurfacewithinteriorobject.(b) Desireddistributionmesh.(c) Redistributed surfacewith interiorobject.

682

B.K. Soni, S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

(c) Fig. 7 (continued).

I lllllllllllllllf I I IlJllll|HlllllJl [ I I

i i IllfilHl~l

i

" iiiii~--iiii~![[

f r

P i iiiiifi~.fiiiiiii : ,

i .................. ,

, ,...m,.,,,

I ]

[

[

!

i

i

i

I

I

I

iii .............. II 1 I Illflilllll

I I

I Illl[IH|mlllfll I I I

I IIIIIIIIlllllll I I I Fig. 8. Desired distribution mesh with associated parametric values of the interior object.

mesh. The resulting re-parameterized distribution mesh is demonstrated in Fig. 9. The surface grid is then evaluated with respect to this new distribution mesh. The geometry associated with the interior object is kept precisely on the surface grid. The resulting remapped surface grid is presented in Fig. 10.

Fig. 9. Reparametized distribution mesh.

Fig. 10. Resulting surface grid.

B.K. Soni, S. Yang/ Computer Aided Geometric Design 12 (1995) 675-692

683

(a)

i!! ! !i !!i! ! !!!i i!! ! ~! ! !i i~ !!i F - ~ ~ t J~'T'T'~f J ' "

i iii il iliiiiii

'miN,LI3Ad"~

(b)

(c)

(d) Fig. 11. (a) Sculptured surface with interior objects. (b) Desired distribution mesh with associated

parametric values on the interior objects. (c) Reparameterized distribution mesh. (d) Resulting surface grid.

An example involving three interior objects is presented in Figs. l l a - d . The sculptured surface and the associated parametric space are presented in Figs. l l a - b . The interior objects on the surface are shaded for clarity. The resulting

684

B.K. Soni, S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

re-parametrized distribution mesh is demonstrated in Fig. 11c. The surface grid evaluated at the new distribution mesh is displayed in Fig. lld.

4. Surface grid optimization There are two main categories in grid generation methodologies: direct methods, where algebraic interpolation techniques are used, and indirect methods, where a set of partial differential equations is solved. Two distinct approaches for surface grid optimization will be discussed here. The first approach is based on a transfinite interpolation (TFI) technique which is a widely utilized algebraic grid generation method. The second approach is to solve an elliptic system with appropriate control functions, a commonly used partial differential equations method (Thomson, 1987; Soni, 1985, 1992).

4.1. Hermite transfinite interpolation on distribution mesh The grid generated by TFI technique is developed by applying a Boolean sum of a selected one dimensional interpolation operator in each direction. The interpolation operators allowing linear, quadratic, cubic lagrange, cubic hermite, quintic and Bezier polynomials, piecewise polynomials-spline, B-spline, NURBS schemes are utilized in grid generation. In general, the TFI technique can be presented as follows. Let LI

P~'(rii) = E

It

E @.,t(sii)?~71}

(5)

l=1 m=l

and L2

Jt

P~2(rij ) = Y'~ Y'. llJml(tij)l'(ilm -1)

(6)

1=1 m = l

be the interpolation operators applied in the ~:1 and s¢2 coordinate directions respectively, where,

q~,m( ),

m=1,2,...,I1,1=1,2

..... Li

denotes the blending function for the operator in the ~1 direction,

~,m( ),

m=l,2,...,Jl,

l=l,2,...,L2

denotes the blending function for the operator in the sc2 direction,

?m (l-1)~-( (0~1),_ ~(/--1)~, j

, mj

m

=

1, 2 . . . . . I,

B.K Soni, S. Yang / Computer Aided Geometric Design 12 (1995) 675-692

685

at all points j = 1, 2 . . . . . M and 0(t-l)f -:(t-l)=

(0~2) 1-1

tim

im

,

rn= 1,2,...,J

1

at all points i = 1, 2 . . . . . N r e p r e s e n t available b o u n d a r y and g e o m e t r y information, and (s,j, tgj), i = 1, 2 . . . . . N, j = 1, 2 . . . . . M represent the distribution m e s h of the surface grid of size N X M. T h e Boolean sum (Pg' *

Pg2)(rij)

results in the grid g e n e r a t e d by T F I scheme. For example: Taking L 1 = L 2 = 2, I~ = J~ = 2 in the o p e r a t o r s (5)-(6) results in the grid g e n e r a t e d by cubic h e r m i t e interpolation, such that P~t( rij ) = CI)ll( S i j ) r l j .4- (/)12r2j Jr" CI)21( Sij)l'tlj q- (I)22( Sij)~'t2j,

P~.i(rij)

= cl) , , ( g i j ) r i , -}- ~ 1 2 ( tij)ri2 q- (P21( tij)r[1 q- dlD22( tij)r;2 ,

wherc

ru= rlj,

r2j = rNj,

r'lj =

"

,

r2~ =

lj ril = ril,

ri2 = r i m ,

ril =

-0~2 -

,

Nj ri2 = iM

and qS,,(w) = (1 + 2 w ) ( 1 - w) 2,

(/)21(W) = (1 -

w)2w,

(iOI2(W) = (3 -- 2W)W 2,

1~)22(W) = (W -- 1)2W 2.

T h e redistributed surface grid is usually smooth and well distributed. T o ensure orthogonality at the b o u n d a r i e s

must be satisfied. T h e a p p r o a c h taken here is to refine the p a r a m e t r i c distribution m e s h to account for orthogonality on the boundaries. T h e p a r a m e t r i c distribution m e s h is refined by applying weighted H e r m i t e transfinite interpolation utilizing a cubic h e r m i t e interpolation technique. T h e following equations are solved to obtain the slopes at the {:2 = I and ~:2 = {:max 2 boundaries, respectively, ~ 1 • ~a2 = 0

and I] r~ × r& I[ = A

(desired cell area off the b o u n d a r y in the {:2 direction).

T h e s e equations can be t r a n s f o r m e d to the p a r a m e t r i c mesh as (?~s~, + ~,t~,)' (~s~_, + ~,t~2) = 0

(7)

686

B.K. Soni, S. Yang/ ComputerAided Geometric Design 12 (1995) 675-692

and

ll(7-sS~, + rtQi) x (rsS~2 +

?,te2)II

(8)

=A.

Eqs. (7)-(8) can be rewritten as

(~, "?,)se,se2 + (?, "Ft)te,Qz + (?s "?,)(Se,Q2 + s~2t~,) = 0

(9)

and

II f, x ~, II(se,te~-se2te, ) = A .

(10)

Eqs. (9) and (10) are then solved for se2 and te2 at ~:2= 1 and s~2= ~max 2 boundaries. Notice that at ~:2= 1 and ~:2_ ~2ax boundaries tex = 0, and hence Eqs. (9)-(10) result in A

tu

II fs x ~, II s e, II~s x~, II ~=0,

and

se2

(r,'fs)

se~4=O, and

t#,

(11)

IIGxGII ~=0

can be assumed without loss of generality for standard grid generation applications. Similarly the slopes s# and t¢2 are evaluated at scl = 1 and scl =SCmax boundaries. The surface grid is then re-evaluated at the new refined distribution mesh.

4.2. Elliptic surface generation A general method to generate three-dimensional grids by elliptic systems was initially developed by Thompson (1987). Much work has been done on the surface grid generation since then. In particular, Thomas (1982) proposed a method of generating a quasi-two-dimensional grid on a curved surface specified by the equation z = f ( x , y), where the function f is single-valued and twice differentiable. However, the equation z = f ( x , y) for a surface is not always known. This limits the flexibility of curved surfaces. Then Takagi and Miki (1985) and Whitney and Thomas (1983) described a curved surface by two independent parameters, and hence a new elliptic system is derived for such surface representation. Another important role in elliptic system is the control functions. A procedure to compute these control functions developed by Soni (1985) is used. Let ? = (x, y, z) and ,O = (~:1, sc2, ~:3) denote physical and computational space respectively. Define covariant and contravariant vectors (Thomson et al., 1985) as follows: a i = covariant vectors ?~i (i = 1, 2, 3), a i -- contravariant vectors Vsci

gij =ai "aj=gji gii=ai.aJ=gji

(i = 1, 2, 3),

(i = 1, 2, 3, j = 1, 2, 3), ( i = 1, 2, 3, j = 1, 2, 3),

g = det I gul = [a 1 - ( a 2 X a3)] 2.

B.K. Soni,S. )rang/ ComputerAided GeometricDesign12 (1995)675-692

687

Now

(gij)~,-~ (derivative of gi~ with respect to ~:k) - ?~i~k • ?~ + ?~- ?~i~,

i, j, k := 1, 2, 3.

(12)

Using (12), the following statement can be obtained ?~/~-?~k=

( gik)U- ( gii)~* + (gjk)~,

, i,j, k=1,2,3. (13) 2 Consider a three-dimensional elliptic grid generation system (Thompson et al., 1985) 3

Z

3

3

Egij?¢'~ ' + Ethkgkk?~ k = 0

i=1 j = l

(14)

k=l

where 1

gil= 7(gjmgkn -gj, gkm),

i, l= 1, 2, 3, (i, j, k) and (l, m, n) cyclic.

Our analysis for evaluation of control functions 4'k, k = 1, 2, 3 is as follows. Eq. (14) can be rewritten by taking the dot product with ?e,, q = 1, 2, 3 as 3

3

3

E EgijT,~ie"r~ ',q- E~Pkgkkr~ k'r('=O, i=1 j = l

q = 1, 2, 3.

(15)

k=l

Using Eqs. (12)-(14), Eq. (15) can be written in terms of metric terms and their derivatives as 3

3

3

3

3

E Egi'i(giq)~J + E~kgkqg k k - Z Zgij(f,~i'r~j,~q) =0, i=1 j = l

k=l

q=1,2,3.

i=l j=l

(16) Also

gii = ?e~ "?el = II ~e, II 2

(17)

represents an increment of arc length on a coordinate line along which ~i varies and g i J = ~ e " ~ ¢ ' = II~e, ll" II~¢Jll'cos 0,

i4=j

(18)

represents a measure of orthogonality between grid lines along which ~i and ~:J varies. These quantities can be evaluated if the desired increment in the arc length and desired angles between grid lines are known. Looking at the "precise control of spacing" property of the algebraic grid (Soni, 1992), the quantities gsj can be evaluated from the well-defined algebraic grid, and using

B.K. Soni, S. Yang/ Computer Aided Geometric Design 12 (1995) 675-692

688

where 0 is the desired angle between sci, scy grid lines, the quantities gii, i ¢ j can be evaluated. Once all gij'S a r e known, then Eq. (16) can be solved for the forcing functions ~k, k = 1, 2, 3. In particular when orthogonality is assumed i.e., 0 = 90 ° or gii = 0 for i 4:j, then q5k, k = 1, 2, 3 can be formulated as

gkk ) ,

1 d ( 4~, = -~ dsC-----/ In - -

(i, j, k ) cyclic, k = 1, 2, 3

(20)

gii gjj

and

~b, =

g

,

(i, j, k ) cyclic, k = 1, 2, 3;

(21)

we assume that gii -# 0 for i = 1, 2, 3. These control functions will be utilized in surface grid refinement.

(a)

(b)

If) Fig. 12. Surface grid optimization. (a) Original surface grid (view 1). (b) Optimized surface grid (view 1). (c) Original surface grid (view 2). (d) Optimized surface grid (view 2). (e) Original surface grid (view 3). (f) Optimized surface grid (view 3).

689

B.K, Soni, S. } T a n g / C o m p u t e r A i d e d Geometric Design 12 (1995) 675-692

(el

(b)

-.,

(d)

1'(\,', i ~,_~-.

U.i/,, t !

~. ~, ~---~-.t I lI-,---L~, .~-ZU"I"--~ l t. l j ,

¥ ~ % ~ , . / / i / i,' ~ .

(e)

(f)

Fig. 13. (a) Original surface. (b) Original surface (close-up near upper, left hand corner). (c) Optimized surface. (d) Optimized surface (close-up near upper left hand corner). (e) Distribution mesh (original surface). (f) Optimized distribution mesh.

A quasi two-dimensional elliptic system for surface generation can be formulated (Thomas, 1982) as , g 2 2 ( r f L ~ i - q~lrf) -- 2g12?¢,¢z + gll(?sc2f2- (~lrf2) = H I?¢, × ?t21(?¢, × ?,~2) (22) where H = gzzree "( fe × ~ ) - 29127~ '( ?~ × ~n) + g'lr~n "( ~ × ?n)

I~e × ~,

I3

(23)

Here, the surface is represented as a coordinate surface sc3 = constant, the sc~ and sc2 lines are assumed to be perpendicular to the sc3 lines (g13 = 0 and g23 = 0), and the principal curvatures of the ~3 lines vanish on the surface.

B.K. SonL S. Yang/ Computer Aided Geometric Design 12 (1995) 675-692

690

Eqs. (22)-(23) can be transformed to the parametric mesh (s, t) as follows (Whitney and Thomas, 1983) g22(s~I~,-(bla~,)-2gm2S,l~z+gll(s~z~2-dP2s,2)+

: 2 A " (OL1Fs--~1F,) iFs×Ftj2 =0,

(24) g22(t~z~l - qblt~ ) - 2g12Q,~2 + g11( Q2~ 2 - ~b2Q2) +

J2A'(3/ll't-~lrs)

=0,

(25) where Jz=s¢,t¢2-s~2Q~,

A =O~l?ss--2[31?st+3/l?tt ,

o~l = x2t + Yt2 + Z 2 '

[31= XsXt + ysYt + ZsZt,

3/1__ Xs2 + y2 + Zs2,

I ~s × ~, 12 = ( X s y , - y s X , ) 2 + ( X s Z , - z s x , ) 2 + ( y s z , - z s z , ) 2

Three distinct views of the surface grid on a cylinder are presented in Fig. 12. The optimized surfacegrid presented in Figs. 12b, 12d and 12f shows near orthogonality and smoothness of grid lines. The control functions are computed using Eqs. (20)-(21). The metric terms are evaluated directly from the algebraic (original surface) grid. Eqs. (24)-(25) are solved to obtain the optimized distribution mesh and then the associated NURBS surface is evaluated to obtain the optimized surface grid. The next example presenting an arbitrary surface grid optimization is presented in Figs. 13a-f. Two distinct views of the original surface (Figs. 13a-b) and the optimized surface (Figs. 13c-d) are presented. The distribution mesh associated with the original surface and optimized surface are presented in Figs. 13e-f. The optimized surface is nearly orthogonal and smooth.

5. Conclusion

The algorithms to redistribute, refine, remap and optimize structured surface grids utilizing the NURBS representation are presented with examples. The NURBS representation offers the common data structure and preservation of geometric fidelity which is of utmost importance in addressing CFD applications. The redistribution algorithm is readily applicable to the grid adaption scheme (Yang and Soni, 1993; Yang, 1993) where the desired distribution mesh can be evaluated directly by applying the equidistribution principle to selected solution features. The methodologies presented in this paper have been cast into an interactive computer code, CAGI (Computer Aided Grid Interface), under development at the NSF Engineering Research Center.

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