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I 

NASA Technical Memorandum 101 069 

Generation of 

Three-Dimensional Body-Fitted 

Grids by Solving Hyberbolic 

Partial Differential Equations 

Joseph L. Steger 

(BASA-TH-IO 1069) 66YERITICII CF t189-I6763 

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January 1989 

National Aeronautics and 

Space Administration 
NASA Technical Memorandum 101 069 

Generation of 

Three-Dimensional Body-Fitted 

Grids by Solving Hyberbolic 

Partial Differential Equations 

Joseph L. Steger, Ames Research Center, Moffett Field, California 

January 1989 

NASA 

National Aeronautics and 

Space Administration 

Ames Research Center 

Moffett Field, California 94035 
SUMMARY 

Hyperbolic grid generation procedures are described which have been used in external flow simulations about complex configurations. For many practical applications a single well-ordered (Le., structured) 

grid can be used to mesh an entire configuration, in other problems, composite or unstructured grid procedures are needed. Although the hyperbolic partial differential equation grid generation procedure has 

mainly been utilized to generate structured grids, an extension of the procedure to semiunstructured grids 

is briefly described. Extensions of the methodology are also described using two-dimensional equations. 

INTRODUCTION 

In finite difference simulations it is often advantageous to generate a body-fitted grid about an arbitrary closed body. Such a grid can simplify the accurate application of boundary conditions, enhance 

computer vectorization, allow clustering to action areas of the field, and may allow the use of certain simplifications to be made to the partial differential governing equations. For a large class of problems (e.g., 

external aerodynamics), the grid must conform only to the body surface and does not necessarily have to 

coincide with a specified outer boundary. For such cases grids can often be efficiently generated by solving 

hyperbolic or parabolic partial differential grid generation equations. 

THREE-DIMENSIONAL HYPERBOLIC GRID GENERATION EQUATIONS 

An extension of the cell-volume hyperbolic grid generation scheme [ 13 to three dimensions has been 

described in [2,3]. In this approach, as in most other partial differential grid generation procedures, the 

equations are transformed to a uniform computational space, E, q, (. Here, the body surface was chosen 

to coincide with (( x, y, z) = 0 and the surface coordinates were taken as ( and q. The outer boundary 

(( x, y, z) = cannot be specified for a hyperbolic set of equations. 

Two orthogonality relations with respect to the (-outward ray and a user-specified cell-volume condition are used as the governing equations: 

1 
These equations comprise a system of nonlinear partial differential equations in which z, y, and z are 

specified as initial data at [ = 0. Local linearization of the equations about a nearby state, 0, results in the 

system of grid generation equations which can be written as 

with r'= (5, y, z)~, coefficient matrices A0 ,Bo, CO, and I= (0,O ,A V + AVO)^. Although the algebraic verification is not trivial, C;' A0 and C;' Bo are symmetric matrices so the linearized system Equation (2) is hyperbolic and can be marched with [ serving as the "time-like" direction. 

The nonlinear system of grid generation equations given by equation (2) has been solved with an 

approximate factorization noniterative implicit finite difference scheme using central differencing in 6 and 

7, with the nearby known state 0 taken from the previous (-step. Numerical dissipation terms are also 

appended, and if carefully chosen [2,4,5], the hyperbolic grid generation scheme can be quite robust. An 

unconditionally stable implicit scheme has the advantage that the marching step size in C can be arbitrarily 

selected based only on considerations of generating the grid. Applications are given in [3,6] while figure 1 illustrates a grid for the space shuttle orbiter taken from [6]. 

Composite Grids 

If only a single, body-fitted, structured grid is used to mesh a complex configuration, the resulting 

grid will be too skewed or poorly clustered. For this reason, body-fitted composite grid or unstructured 

grid strategies are generally used to discretize complex configurations. 

One composite mesh scheme is the chimera overset grid method [7,8] in which each body component 

of a complex configuration is meshed independently and the composite grid is made up of the superposition 

of the individual grids. Figure 2 shows body symmetry planes of a set of grids generated for the integrated 

solid rocket boosters. The individual grids are then connected by cutting out nonrealistic grid points (e.g., 

points of one grid that fall within another body) and setting up interpolation links between cut-out-hole 

and outer boundaries. An efficient structured grid flow solver can then be used to solve the flow equations 

on each grid and only a small segment of the code must deal with the unstructured boundary data supplied 

from other grids. Because the grids are superimposed in this approach, the outer boundary location of 

each individual grid is arbitrary. Thus, the use of a hyperbolic grid generation procedure is generally 

quite adequate, and all of the shuttle grids shown have been generated in this way. Computational results 

obtained with these grids can be found in [6]. 

I 

I space shuttle vehicle in which individual grids are formed for the shuttle orbiter, external tank, and the 

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Semiunstructured Grids 

A practical grid for high Reynolds number viscous flow applications could be formed as follows. 

From a body boundary surface grid ( = 0, lines can be radiated outward by marching hyperbolic grid 

generation equations from one A(-plane to the next so as to form a three-dimensional grid. All of the 

interface logic, link lists, etc. set up and specified on the body surface - a two dimensional (2-D) problem 

- would be maintained for each (=constant plane. Because a ray-like structure along which ordered grid 

points in the outward direction exists, it is easy to tightly cluster grid points to the surface to resolve thin 

viscous flow layers and it is also easy to use an implicit operator in this direction. Away from the body a 

fully unstructured grid could be used if the rays coalesce. 

The hyperbolic grid generation procedure has been extended to generate this kind of mesh by treating 

the semi-unstructured data in a c=constant plane as follows. The AD1 implicit algorithm used to solve 

the governing equations at each (=constant plane was discarded and an explicit marching scheme was 

substituted. However, an explicit method places limitations on the marching step size that are not always 

compatible with how the grid should be varied in the outward direction. Because the governing equations 

have been formulated with respect to a curvilinear coordinate system which no longer exists on the unstructured grid, a local coordinate system was defined for each point. Finally, differencing schemes and 

link-lists for the semiunstructured grid and pointwise local coordinates were coded. 

A choice was made to difference the equations at node points (not cell centers), and the points were 

linked together as sketched in figure 3. On a (=constant surface, a local (, q coordinate for each grid point 

was defined and derivatives in f and rj were formed using the four linked points. First-order-accurate, centered, difference approximations were used, and for a derivative with respect to f formed as (see notation, 

fig. 3) 

with A (*I = ( 

q, numerical dissipation is added by using some averaging of the four linked points. 

- FO ) 2t where is the unit vector in (. Because central differencing is used in ( and 

An explicit method composed of a forward Euler predictor and followed by two applications of either 

backward Euler or trapezoidal rule correctors was used to difference and march the equations in (. This 

scheme is conditionally stable for central differencing of and q terms for ordered grids, but the impact 

on stability due to the use of local coordinates has not been analyzed or severely tested. 

Although all of the logic for treating unstructured grids was coded, ordered patched grids were used to 

test the grid generation scheme because general procedures to set up node connectivities and link lists were 

not available. Test calculations were carried out using either an ordered grid or, to stress the unstructured 

logic, patched grids were set up on a sphere by rotating segments of the grid and/or locally halving the 

3 
grid spacing. The two axes and the disjoint patched boundaries required the general link logic. Control 

volumes were chosen to simply radiate the spherical surface shell outward so an affine grid was generated. 

TWO-DIMENSIONAL ELLIPTIC-HYPERBOLIC GRID GENERATION EQUATIONS 

The hyperbolic grid generation equations represent conditions to control mesh size and orthogonality 

(skewness), while the conventional elliptic grid generation equations [9,10] represent conditions of averaging (or smoothing) an interior distribution of grid points to specified boundary-point distributions. A 

combination or blend of the two equation sets should give good control of the natural constraints usually 

imposed on a grid. 

The elliptic and hyperbolic 2-D grid generation equations can be combined as 

22 where CY = x,, + y,, , p = xtx,, + yty,, 7 = xz + yi, and 

If the parameter p = 0 the hyperbolic equations alone are recovered for marching in 7; otherwise, the 

equations are elliptic. 

Iterative solution algorithms such as Gauss Seidel and certain implicit approximate factorization 

schemes can be built to solve equation (3) such that when p = 0 a single-sweep hyperbolic scheme is 

recovered. For example a Gauss Seidel scheme ( for marching outwards, v > 0 ) is given by 

where a, p, 7, B and A are evaluated using c-', (if p = 0) and/or c. The small positive parameter E is 

used to maintain a smoothing term if p = 0, and A 7 = 1 so 77 = k. 

The hyperbolic terms (and perhaps the fortified approach discussed below) can give an efficient alternative to the usual control-function approach used with elliptic equations [9,10]. Figure 4 shows an 

example (developed by William R. Gutierrez) of using this kind of scheme where p was set to zero near 

the body and rapidly increased so that away from the body the usual elliptic equation solely dominates. 

4 
FORTIFIED HYPERBOLIC GRID GENERATION IN TWO DIMENSIONS 

As a marching scheme, hyperbolic grid generation methods can break down if unrealistic conditions 

are imposed. For example, consider using a hyperbolic grid generation scheme to generate the interior of a 

cylindrical grid. As the method marches inward, it will simply break down as the singularity is approached. 

The hyperbolic grid generation scheme has sometimes broken down at airfoil sharp trailing edges because 

the constraints of controlled volume and orthogonality can be meaningless there. In each case, though, the 

singular grid solution is known and could be specified, and this information can be imposed on the solution 

algorithm by a simple fortified [ll] approach. 

For example, to fortify the 2-D hyperbolic difference equations, they are modified as 

where the forcing term x( r' - Fk+l) is appended. Here r) is the specified solution near the singularity or 

breakdown, and x 2 0 is a user-specified function of and 7. For x >> 1, ?k+k+l -+ F', while if x is set 

to 0 the algorithm is unaltered. 

By specifying the grid-line points emanating out from the trailing edge and letting x be large nonzero 

just along this line, the regular grid generation solution is neatly overwritten at this line with the specified 

solution. As the solution is marched out, the singularity disappears and x can be allowed to approach zero. 

A grid generated in this way is shown in figure 5. 

5 
REFERENCES 

1. Steger, J. L. and Chaussee, D. S. Generation of Body-Fitted Coordinates Using Hyperbolic Partial 

Differential Equations, SUM J. Sci. Stat. Comput. 1, 1980. 

2. Steger, J. L. and Rizk, Y. M. Generation of Three-Dimensional Body-Fitted Coordinates Using Hyperbolic Partial Differential Equations, NASA TM 86753, 1985. 

3. Rizk, Y. M., Steger, J. L. and Chaussee, D., Use of a Hyperbolic Grid Generation Scheme in Simulating Supersonic Viscous Flow About Three-Dimensional Winged Configurations , NASA TM 

86776,1985. 

4. Kinsey, D. W. and Barth, T. J. Description of a Hyperbolic Grid Generation Procedure for Arbitrary 

Two-Dimensional Bodies, AFWAL TM 84-191-FIMM, 1984. 

5. Cordova, J. Q. and Barth, T. J., Grid Generation for General 2-D Regions using Hyperbolic Equations. 

AIAA Paper No. 88-0520, 1988. 

6. Buning, P. G., Chiu, I. T., Obayashi, S., Rizk, Y M., and Steger, J. L. Numerical Simulation of the 

Integrated Space Shuttle Vehicle in Ascent, AIAA-88-4359-CP, 1988. 

7. Steger, J. L., Dougherty, E C. and Benek, J. A.: A Chimera Grid Scheme. In Advances in Grid 

Generation, K.N. Ghia and U. Ghia, Eds., ASME FED-5, 1983. 

8. Benek, J. A., Buning, P. G., and Steger, J. L.: A 3-D Grid Embedding Technique, AIAA Paper No. 

85-1523, July 1985. 

9. Thompson, J. F., Thames, F. C., and Mastin, C. W. Automatic Numerical Generation of Body-Fitted 

Curvilinear Coordinate System for Field Containing Any Number of Arbitrary Two-Dimensional 

Bodies, J. Comp. Phys., 15, 3, July 1974. 

10. Thompson, J. F., Warsi, Z. U. A., and Mastin, C. W . Numerical Grid Generation - Foundations and 

Applications, North-Holland, New York, 1985 

11. Van Dalsem, W. R. and Steger, J. L., The Fortified Navier-Stokes Approach, proceedings of the Workshop on Computational Fluid Dynamics, Univ. of California, Davis, June 17-18, 1986. 

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Figure 1.- Various computational planes of the orbiter grid. 

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Figure 2.- Symmetry planes of all grids. 

7 
A&* = (71, -70) 8~ 

Figure 3.- Local differencing notation. 

Figure 4.- Elliptic hyperbolic grid. 

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Figure 5.- Fortified hyperbolic grid (periodic points not connected). 

9 
1. Report No. 2. Government Accession No. 

NASA TM-101069 

4. Title and Subtitle 

Generation of Three-Dimensional Body-Fitted Grids by 

Solving Hyperbolic Partial Differential Equations 

3. Recipient's Catalog No. 

5. Report Date 

January 1989 

7. Author(s1 8. Petforming Organization Report No. 

Joseph L. Steger 

9. Performing Organization Name and Address 

Ames Research Center 

Moffett Field, CA 94035 

I A-89022 

505-60 

11. Contract or Grant No. 

10. Work Unit No. 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages 

Unclassified Unclassified 10 

22. Price 

A02 

12. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

4 13. Type of Report and Period Covered 

Technical Memorandum 

I 

15. Supplementary Notes 

Point of Contact: Joseph L. Steger, Ames Research Center, MS 258-1, Moffett Field, CA 94035 

(415) 694-6459 or FTS 464-6459 

16. Abstract 

I Hyperbolic grid generation procedures are described which have been used in external flow simul 

I lations about complex configurations. For many practical applications a single well-ordered (i.e., 

structured) grid can be used to mesh an entire configuration, in other problems, composite or unstructured grid procedures are needed. Although the hyperbolic partial differential equation grid generation 

procedure has mainly been utilized to generate structured grids, an extension of the procedure to semiunstructured grids is briefly described. Extensions of the methodology are also described using twodimensional equations. 

I 

1 17. Key Words (Suggested by Authods)) I 18. Distribution Statement 

1 Grid generation 

' Numerical methods I ~ody coordinates 

Unclassified-Unlimited 

Subject Category - 02 

I I 1 

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