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 (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, extension of the procedure to semiunstructured grids is briefly described. Extensions of the methodology are also described using two-dimensional equations

Steger, Joseph L.

English

NASA Technical Reports Server

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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) 6 6 Y E R I T I C I I  CF t189-I6763 
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L I k P E R E I I I A L  EQtAqIoIS ( b A S 1 )  12 pCSCL O l A  Unclas 
G3/02 0 19 1236 
January 1989 
National Aeronautics and 
Space Administration 
https://ntrs.nasa.gov/search.jsp?R=19890007392 2020-03-20T03:22:24+00:00Z
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 simula- 
tions 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 pro- 
cedures 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 ar- 
bitrary 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 sim- 
plifications 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 con- 
dition 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 alge- 
braic verification is not trivial, C;' A0 and C;' Bo are symmetric matrices so the linearized system Equa- 
tion (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 fig- 
ure 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 
I 
~ 
2 
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 un- 
structured 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, cen- 
tered, 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 aver- 
aging (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 
2 2  where CY = x,, + y,, , p = xtx,, + yty,, 7 = xz + y i ,  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 al- 
ternative 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, S U M  J. Sci. Stat. Comput. 1, 1980. 
2. Steger, J. L. and Rizk, Y. M. Generation of Three-Dimensional Body-Fitted Coordinates Using Hy- 
perbolic 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 Simu- 
lating 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 Work- 
shop on Computational Fluid Dynamics, Univ. of California, Davis, June 17-18, 1986. 
6 
Figure 1.- Various computational planes of the orbiter grid. 
- 
Figure 2.- Symmetry planes of all grids. 
7 
A&* = (71, -70) 8~ 
Figure 3.- Local differencing notation. 
Figure 4.- Elliptic hyperbolic grid. 
8 
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 simu- 
l 
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 unstruc- 
tured 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 semi- 
unstructured grids is briefly described. Extensions of the methodology are also described using two- 
dimensional equations. 
I 
1 17. Key Words (Suggested by Authods)) I 18. Distribution Statement 
1 Grid generation 
' Numerical methods 
I ~ o d y  coordinates 
Unclassified-Unlimited 
Subject Category - 02 
I I 1 
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Generation of three-dimensional body-fitted grids by solving hyperbolic and parabolic partial differential equations

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