GRID GENERATION ON SURFACES 

IN THREE DIMENSIONS 

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Final Scientific Report 

(October 1, 1985 - September 30, 1986) 

Peter R. Eiseman 

Columbia University 

New York, NY 10027 

Support by NAVSEA 

Administered by NASA 

Grant Number NAG-1-617 

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(&AS&-CR-180559) GRID GENEBA41CY OH 107-27424 

ZSOSF&CES 18 3 DIHEEISICIJS Einal Scientific 

Geport, 1 oct, 19€5 - 30 SeF. 1SE6 

(CclumEia Univ,) 9 p Avail: NlIS HC Unclas 

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TECHNICAL CONTE~~T 

Under the NAVSEA project entitled "Grid Generations on Surfaces in 

Three-Dimensions" and administered by NASA as grant NAG-1-617, the 

development of a surface grid generation algorithm was initiated. In the 

initial research, the basic adaptive movement technique of 

mean-value-relaxation (1) was extended from the viewpoint of a single 

coordinate grid over a surface described by a simple scalar function to 

that of a surface more generally defined by vector functions and covered by 

a collection of smoothly connected grids. Within the multiconnected 

assemblage, the application of control was examined in several instances. 

In the first case, a smooth brick-like object was formed by an exponent 5 

superellipsoid and was covered by 6 coordinate patches. From an 

algebraically generated initial arid system, a uniforn; and then a curvature 

clustered system was generated. As a second and more complex case, the 

same experiment was repeated for the abrupt intersection of two such 

brick-like objects. The abrupt intersection was successfully treated by 

employing a numerical filter for tine weights. This was essential when 

strong curvature attraction was exercised. The reason is that the grid 

movement molecules straddling the intersection have weights that can 

suddenly acquire large values from small displacements of the points: 

this problem comes from the appearance of an infinite curvature along a 

curve. The weight filtering is the resonable cure and caused coordinate 

curves to smoothly follow the intersection rather than to smoothly weave 

back and forth through the intersection in an almost random fashion. The 

weaving effect had been witnessed prior to the filtering. In a similar 

manner, a stability restriction on the clustering intensity was witnessed 

and was then virtually eliminated by employing a new interpolation 

procedure (2) for the mean value relaxation process. Moreover, under 

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strong curvature attraction, an interplay was observed between the number 

of grid points in the patch grids and the pattern of high curvature 

regions. The result was a loose constraint on the number of points. 

To examine the effect of the generatin process a little more directly, 

we return to the second case where two of the brick-like objects 

intersected. As mentioned earlier, each such object is defined by a 

superellipsoid. This was done for convenience so that a simple geometry 

description could be given for surfaces with local high curvature regions. 

The superellipsoid retains the essential simplicity of the usual ellipsoid 

by only replacing the exponents of 2 with an arbitrary number. As that 

number increases in size from 2, the so called superellipsoid converges 

towards a rigid Cartesian box. As a consequence, high curvature regions 

arise and increase in the limit. Ir: our case, the exponent of 5 is already 

quite curved. Altogether, the specification of two such superellipsoids 

that also intersect is readily given and does not involve the often typical 

meticulous accounting that comes from the more commonly employed patch by 

patch approach (eg. Coons patches). As a consequence, a ready format was 

provided to test the basic grid generation procedure without the peripheral 

problems which could arise from a more elaborate geometry discription. 

As a starting point for the process, an initial grid was obtained by 

inward projections from a Cartesian box that entirely contained the object. 

Accordingly, the Cartesian grids on the six faces of the box were sent onto 

six corresponding coordinate patches on the object surface. This grid 

system inherits the grid topology of the surrounding box and is displayed 

in Figure 1. Because of the choice of topology, the grid has larger cells 

at the bulge in the center. Because of the constructional rigidity, the 

grid contains slope discontinuities and abrupt changes in cell size across 

the junctures between distinct coordinate systems. 

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From the initial grid, the mean value relaxation algorithm (i) with 

local barycentric interpolation (2) was employed within the general surface 

context to move the grids points into new positions on the basis of unit 

weights. The result was a push towards an equalization of cell areas 

subject to the constraint imposed by our choice of grip topology and the 

concurrent choice of the number of points in each coordinate system. The 

latter choice is, of course, restricted by the continuity conditions 

betwsen distinct coordinate systems. The associate grid is displayed in 

Figure 2. The push towards an equal distribution of cell areas is readily 

observed to occur within the grid. The opposing force arising from the 

choice of topology is also witnessed at the bulge where larger cells still 

occur. However, in a local sense, the area equalization is evident about 

each point in the entire grid. Also evident in the entire grid is a level 

of smoothness that was not present in the initial grid. 

To come closer to an area equalization in a global sense, the 

constraining topological force must be relaxed. This can be done by 

inserting new coordinate patches along the fiat portions of the buige. 

Looking down upon the flat portion, we roughly put a uniform 4 x 4 

Cartesian array of macro cells upon it. This gives three macro cells in 

each direction. The center cell is, however, consumed by the other 

intersecting superellipsoid. This then leaves 8 surrounding macro cells 

which now represent distinct coordinate systems. It is the corner 

coordinate patches that then provide a relaxation of the topological 

constraining force. 

Relative to the uniform conditions provided by the topologically 

constrained area equalization, the next step is to prosride a means to 

automatically obtain curvature clustering. In our development, this occurs 

directly from the choice of weight functions employed in the 

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mean-value-relaxation procedure. By adding a curvature term to the 

previous weight of unity, we get a new weight for curvature clustering 

relative to the unity term which represents the cell area equalization 

force. The result from an application of curvature attraction is given in 

Figure 3. There, the high curvature regions are seen to be resolved. 

Those regions correspond to the curve of intersection and the 

superellipsoid edges which in the limit of infinity exponent would become 

the edges of corresponding certain boxes. 

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REFERENCES 

(1) Eiseman, P.R., "Adaptive Grid Generation by Mean-Value Relaxation," 

ASME J. of Fluids Engr., Vol. 107, 1985, pp. 477-483. 

(2) Eiseman, P.R., "Adaptive Grid Generation," to appear in Computer Meth. 

in Appl. Mechanics and Engr., the text of an invited presentation at 

the First World Congress on Computational Mechanics, The University of 

Texas at Austin, Sept. 1986, 86 pgs. 

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Figure 1: Initial grid constructed by projecting Cartesian grids from the faces 

of a surrounding Cartesian box onto the combination of intersecting superellipsoids. 

For simplicity, only half of the object is displayed. 
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Figure 2: A uniform grid €or the chosen grid topology 
Figure 3: A curvature clustered grid -