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Algebraic grid generation
PART B
MESH GENERATION AND REZONING
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
64 (1987) 285-300
ALGEBRAIC GRID GENERATION Robert E. SMITH NASA Langley Research Center, Hampton, VA 23665, U.S.A.
Lars-Erik ERIKSSON* Old Dominion University, Norfolk, VA 23508, U.S.A. Received 22 September
1986
Algebraic grid generation is the direct expression of a physical coordinate system as a function of a uniform grid in a rectangular computational coordinate system. Algebraic grid generation is based on mathematical interpolation and is presented in general terms of multivariate transfinite interpolation. The multisu~ace method and the two-bounda~ technique are described as univa~ate procedures that can be applied within the context of transfinite interpolation. A technique for grid clustering is described. Problems that are commonly encountered in three-dimensional grid generation are discussed and approaches for dealing with complex physical domains using multiple computational grid blocks are presented.
1. Introduction The purpose of computational mechanics is to solve approximately the governing differential equations that describe the motion of a continuous medium [l-3]. This is accomplished by imposing a collection of points called a grid on the continuum domain, approximating the governing equations at the grid points with algebraic equations, and numerically solving the resulting system of algebraic equations. Defining a grid on a continuum domain is called grid generation, and unlike the governing differential equations there are no physical laws for grids to obey. Grid generation is a construction process which must accommodate the characteristics of the governing equations and the geometry of the physical domain [4-61. In general this leads to two highly desirable characteristics that a grid should possess: (1) A grid should conform to the boundaries of the region of interest. (2) A grid should be concentrated in regions where there are high gradients in the solution. There are two basic types of grids, which are (1) structured grids and (2) unstructured grids. A structured grid (Fig. 1) can be generated by a rule or correspondence between unifo~ly spaced points in an idealized computational domain and points in a physical domain. Consequently, the relative position between neighboring grid points in the physical domain are governed by mathematical expressions, and grid point coordinates can be completely identified by increasing sequences of integers (indices within arrays). Unstructured * Research Associate Professor, on leave from FFA, Sweden. 00457825/87/$3.50
0
1987, Elsevier Science Publishers B.V. (North-Holland)
286
R. E. Smith,
L.-E.
Eriksson.
Algebraic
grid generation
or
rl
J
I I r)
or
J
x(I,J) Y(I,J)
+i
or
Fig. 1. A structured
I grid.
grids (Fig. 2) consist of arbitrarily placed points in a physical region, and in general require an explicit directory indicating neighboring grid points that are connected. An unstructured grid can, in part, be generated by mathematical expressions, but there can be many exceptions that must be coordinated into the overall grid. Historically speaking, structured grids have been used in conjunction with finite difference techniques and finite volume techniques for solving the governing equations. Unstructured grids are more suitable for use with the finite element technique. The algebraic grid generation techniques described herein produce structured grids. Algebraic grid generation techniques are based on interpolating functions that transform an idealized rectangular computational domain into a physical domain (Fig. 3). A grid in the computational domain is created by partitioning the rectangular coordinates into uniform intervals, and under the transformation, the uniformly spaced coordinates map into the
Fig. 2. An unstructured
grid.
R.E.
Smith, L.-E.
Computational
Eriksson,
Algebraic grid generation Physical
Domain
Fig. 3. Computational
287
Domain
domain and physical domain.
physical domain (Fig. 4). Changing the number of grid points does not change the transformation or its characteristics. When the boundaries of the computational domain map onto the boundaries of the physical domain, the transformation is said to be boundary-fitted. Also, coordinate transformations imply that the governing equations must be transformed [7]. In this paper the basic concepts of interpolation are applied to create algebraic coordinate transformations, and grids are discrete evaluations of these transformations. First, a very general description of transfinite interpolation [8-111 is presented. Virtually all algebraic grid generation techniques can be described under the umbrella of transfinite interpolation, but individual applications can be quite different. An application of transfinite interpolation accredited to Eriksson [lo, 12,131, the two-boundary technique [7,11,14] described by Smith, and the multisurface method [15-181 described by Eiseman are used to demonstrate this point. Algebraic grid generation allows direct control of clustering grid points, and an approach to this important feature is described. Generally speaking, interpolation methods to express three-dimensional algebraic coordinate transformations are quite simple and straightforward. There are, however, many considerations that must go into their practical application. Two of these considerations are transformation singularities and geometric complexity of physical boundary surfaces. The singularity problem is discussed and a building block approach is advocated for dealing with complex geometry.
Computational
Grid
x
=
Fig. 4. Grids in the computational
UJl,r)
Physical
Grid
domain and in the physical domain.
R.E.
288
Smith, L.-E.
Eriksson.
Algebraic grid generation
2. Algebraic grid generation techniques The transformation function,
from a computational
where, without loss of generality, Os(sl,
OGndl,
domain to a physical domain is a vector-valued
we can assume that O<<
Multivariable coordinate transformations (1) can be described in a very general manner using transfinite interpolation. Transfinite interpolation in three dimensions is applied through a recursion of three univariate interpolations, where blending functions and associated parameters (specified point positions and/or derivatives) determine transformations that are explicit forms of (1). Individual applications of interpolation within the context of transfinite interpolation can be quite different. Two direct applications and one variation of transfinite interpolation are discussed. Eriksson has adapted the transfinite interpolation formulation to use only exterior boundary descriptions and certain normal derivatives at the boundaries. He has derived blending functions with exponential terms to concentrate a grid near boundaries. The two-boundary technique described by Smith is similar to the Eriksson application for the univariate case. The two-boundary technique is based on the description of two opposing boundary surfaces, tangential derivatives on the boundary surfaces which are used to compute normal derivatives, and hermite cubic blending functions. Intermediate variables that are functions of the computational coordinates are embedded in the blending functions to control grid spacing. The multisurface method developed by Eiseman is, in its original form, a univariate formula (multisurface transformation) for grid generation based on describing two boundary surfaces and an arbitrary number of interior control surfaces. Blending functions are implicitly derived from what Eiseman calls global and/or local interpolants which result from an expression of the tangential derivative spanning between exterior boundary surfaces. The multisurface transformation can, for the most part, be described in the context of transfinite interpolation, except that the conditions on the blending function are imposed on the interpolants that are used in the derivative expression.
3. Transfinite interpolation Transfinite
interpolation
in a general form expresses the vector function F( t,n,
6) as
R.E. Smith, L.-E. Eriksson, Algebraic grid generation
where the parameters
289
are the position and derivatives described by
s(&n,I)=A;(n,i),
l=l,2
,...,
L, n=O,l,...,
P,
$(W)=B;(&i),
l=l,2
,...,
M,n=O,l,...,
Q,
5
1=1,2 ,...,
N, n=O,l,...,
R,
(5, q, 5J = C;t( 5777) ,
and the blending functions are &j”‘(t),
1=1,2 ,...,
L, n=O,l,...,
P,
&“‘(T/),
Z=1,2 ,...,
M,n=O,l,...,
Q,
y?‘(c),
1=1,2 ,...,
N, n=O,l,...,
R,
subject to the conditions
4. E&won’s
grid generation
Eriksson has adapted transfinite interpolation in the form of (2) to create grids about aerodynamic configurations where the governing equations are for inviscid fluid flow. In his applications of (2) he uses known parameter information only at the outer boundaries (Fig. 5). The information contains outer boundary surface descriptions and outward derivatives on certain boundary surfaces. The parameters L, M, and N in (2) are set equal to 2 .and P, Q, and R vary depending on the application, but are usually between 1 and 3. The outward derivatives, which control grid orthogonality and outward spacing at the boundaries, are derived from the surface descriptions and Eriksson has explicitly defined blending functions to achieve a desirable exponential stretching of grid points between boundary surfaces. An example of the first univariate interpolation is:
R. E. Smith, L.-E.
290
Eriksson. Algebraic grid generation
a<”
Back
Surface
n-0, l...
a“G$,r), Left
Surface
a<” Right
Fig. 5. Specified parameters
‘y;o)( 5) =
1 -
eK5 1; 1 ;t
cp([)=
e;;r;-;6,
eK
Similar blending functions
,
and parameters
Bottom
Surface
on exterior boundaries
cy;l)(+-
a;“(
Surface
f-
arl”
5. The two-boundary
1)
0
eK’+;t, eK = 0 .
are used in the second and third recursion
steps.
technique
As in Erisksson’s application of the first step in transfinite interpolation, the two-boundary technique uses two opposing boundary surface definitions and outward derivatives on the surfaces. The blending functions are cubic polynomials which satisfy the conditions in (2) and in effect provide hermite cubic polynomial interpolation between the two boundaries. An intermediate variable is used in the blending functions and is itself made a function of a computational coordinate. The one-to-one functional relation between the computational
R.E.
Smith, L.-E.
Eriksson,
291
Algebraic grid generation
coordinate and the intermediate variable provides a convenient means of controlling the spacing between physical points, when the computational coordinate is uniformly descritized. A general discussion of this method of grid spacing control is presented in a later section. The outward derivatives in the two-boundary technique are normal derivatives that are computed from the cross product of tangential derivatives on the surfaces (Fig. 6). The first step of transfinite interpolation for the two-boundary technique is,
(4) where
and 4%)
47
=W~)13
- 3m>1’ + 17
5) = 4Lf(~>1’ + 3Lf(81” 7
4”(t> = M~>l” - w(5)1’+.&9
7
‘yY( s> = v(s>1” - M 81’ .
The function f( 6) must be monotonically increasing in the interval 0 s 5 s 1. Where f( 6) has a small first derivative, there will be a relative high concentration of grid points relative to the &computational coordinate, and where f( 6) has a large first derivative, there will be a dispersion of grid points relative to the &computational coordinate. A function suitable for the concentration of a grid toward a boundary is
where K is a constant. Outward
Derivative from
Cubic
Blending
functiois
Fig. 6. Two-boundary
technique.
Determined
R. E. Smith. L.-E.
292
Eriksson.
Algebraic grid generation
The second and third steps of transfinite interpolation have been applied with the two-boundary technique [14,19] using the remaining two pairs of boundary surfaces and blending functions which have been derived to control the effect of the boundaries on the interior grid. The expressions are,
(5)
The functions g(q), g(q), h( l). and h( 6) must increase monotonically and vary between 0 and 1. They control how the boundaries corresponding to 77 and 5 affect the interior grid.
6. The multisurface method The multisurface method is a procedure for generating grid points between two exterior boundary surfaces F(0, n, 5) and F(l,n, 6). An arbitrary number of intermediate surfaces (F( 5,,77, C>, I= 2, . . . 3 L - 1) are introduced to control the grid. The blending functions are derived indirectly by integrating the tangential derivative in the direction of the interpolation (Fig. 7) which is expressed as a linear combination of position differences on neighboring surfaces and interpolation functions. The interpolation functions have the same delta function conditions as the blending functions have in (2). The blending functions that result from the multisurface method satisfy delta function conditions at the exterior boundaries and not necessarily at the interior control surfaces. An abbreviated derivation of the univariate multisurface method is presented. The multisurface derivative expression is
(6) where G-i)
= 4i 3
l&=0,
1#i,
6,,=1,
l=i.
Integrating (6) and applying boundary the general multisurface transformation:
conditions
yields an equation
which Eiseman calls
R. E. Smith, L.-E. Eriksson, Algebraic grid generation
ential
293
Derivatives in
the
Direction
Fig. 7. Multisurface
of
Interpolation
method.
The basic ingredients of the multisurface method are a partition of the computational coordinates in the direction of interpolation (ii, i = 1, . . . . , L - l), the interpolation functions, and the surface definitions. Choosing ql( 6) to be polynomials of degree L - 1, a curve connecting the boundary surfaces is of degree L. For instance, if there are four surfaces and the interpolation functions are
where
Then the multisurface
transformation
is
F,( 6,~~ f) = F(O, 7, 6) + (4t3 - 9t2 + WNF(629
+ et3
+ 352)[w3,
+ (4t3
- 352)[F(L
77, 0
-
?), 5) - F(
~3 l) - F(O, 7,
m27
537
77
77
01
01
01 *
Rearranging the terms in (8) such that the surface positions expression in the form of the first step in (2). That is
are coefficients
(8) results in an
R. E. Smith. L.-E.
294
Eriksson,
Algebraic grid generation
where (Y;“)(5) = 45j - 95’ - 65 + 1 ,
CY;“( 5) =
a:)(
CXy:o’( 5) = 45j - 35? .
5) =
4
- 65’ ,
65” - 125’ + 6[ ,
In this case, however, the blending functions CX~“‘( 5) d o not satisfy the delta function conditions, but the interpolants $J,(6) d o satisfy similar conditions. The second and third steps as described in (2) could be applied for multivariate interpolation or the blending functions for the remaining two coordinate directions could be derived with the multisurface transformation and Boolean sums performed. An interesting corollary [ 181 to the four-surface multisurface transformation described above is obtained by differentiating (8) with respect to the computational coordinate, expressing the position on the second and third surfaces in terms of the derivative evaluation at the exterior boundaries and resubstituting these expressions into (8). This yields the hermite cubic blending functions used in the two-boundary technique.
7. Grid spacing control The algebraic techniques previously discussed are transformations that map a rectangular computational domain to a physical domain with irregular boundaries. A uniform grid in the computational domain is obtained by partitioning each computational coordinate into equal increments, and under the transformation the discrete points in the computational domain map into irregularly spaced points in the physical domain creating a grid. The spacing between grid points in the physical domain is primarily controlled by the blending functions in the interpolation equations. Simple blending functions that produce the desired shape of a grid (i.e. relative orientation between points) may not produce the desired spacing between points. An example is hermite cubic polynomial blending functions on the occasions where the boundary derivatives are relatively small. These blending functions then produce a grid which is concentrated at the two boundaries. In order to obtain grid concentration somewhere else, additional information must be provided. One approach is to design or modify the blending functions to exactly produce the desired concentrations. Another effective approach is to define an intermediate control domain between the computational domain and the physical domain. An intermediate control domain is defined to be a rectangular domain where each coordinate is related to a corresponding computational coordinate by single-valued functions. For instance,
u =fM
7
u = g(T)
2
w=h(().
Under the application of these functions uniformly spaced points domain map to rectangular nonuniformly spaced points in the control blending functions are redefined with the intermediate coordinates as Following this logic, the hermite cubic polynomial blending functions
in the computational domain (Fig. 8). The independent variables. could be written as
R. E. Smith, L.-E.
Eriksson,
Intermediate
Algebraic grid generation Domain
Physical
Computational
295
grid
Grid
Fig. 8. Intermediate
control domain.
u3-2u2+14,
‘yY)( 5) = 2u3 - 3z.Z + 1 )
iq)(
aF)( 6) = -2u3 + 3u2 )
c$)( 6) = u3 - u3 )
LJ) =
where u=f(S). This is equivalent to the way the blending functions are written in (4). From a practical point of view, it is advantageous to make the intermediate coordinates proportional to the arclengths in the physical domain corresponding to the computational coordinate directions.
8. Grid singularities A singularity occurs when the Jacobian of the transformation between the computational domain and the physical domain is zero. Three-dimensional transformations that map the six-sided computational domain about a closed body will always have singularities. There are different types of singularities and their effect on the solution of the governing equations depends on the type of singularity, the solution technique, and where the singularity is relative to high solution gradients in the physical domain. Two common types of singularities that can occur when mapping a six-sided rectangular domain into a closed three-dimensional region are polar singularities and parabolic singularities (Fig. 9). For polar singularities an entire side of the computational domain maps to a singular line in the physical domain resulting in all grid points for that surface having a zero Jacobian. Using a finite difference technique to solve the governing equations and clustering
296
R. E. Smith, L.-E.
Polar
Eriksson,
Algebraic grid generation
Singularity
Fig. 9. Polar and parabolic
singularities.
grid points near a polar singularity can lead to computational difficulty because of the large elements in the Jacobian matrix of the transformation and the relatively large differences in the elements between neighboring points. If the phenomenon of interest is near a polar singularity, it is best to determine another transformation where the singularity is somewhere else or a transformation with another type of singularity. Parabolic singularities are less severe than polar singularities. For a parabolic singularity only edges of the computational cube map into singular curves in the physical domain. When they can be applied, such as at the tip of a wing, parabolic singularities are preferable to polar singularities. The importance of singularities is dependent on the technique that is used to solve the governing equations. Generally speaking, finite difference techniques are very sensitive to singularities but finite volume techniques are not as sensitive. In finite difference techniques the Jacobian matrix appears explicitly in the formulation, and special provisions must be taken near a singularity. On the other hand, finite volume techniques use areas of the faces of volume elements formed from neighboring grid points to compute fluxes across the faces. These techniques can tolerate singularities as long as the areas approach zero smoothly.
Fig. 10. Grid-crossing
singularity.
R.E.
Smith, L.-E.
Eriksson,
Algebraic grid generation
297
Another type of singularity that can happen in algebraic transformations rendering the transformation useless occurs when a grid curve in a computational coordinate direction becomes double valued, as shown in Fig. 10. This type of singularity should be avoided.
9. Complex boundary topology bodies is a primary goal of The computation of grids about closed three-dimensional algebraic grid generation. Unfortunately, closed bodies can have very complex topologies containing sharp corners and deep cavities (Fig. 11). Generating structured grids about complex geometries that map into a single rectangular computational block is, for all practical purposes, impossible. Considering this dilemma, there are two directions that can be taken for creating multiple block structured grids. They are: (1) create multiple grids that adjoin each other, or (2) create multiple grids that overlap each other. In many cases, multiple-block grids can be created where adjoining grid points and derivatives match. Examples of this approach are a dual-block grid [13] (Fig. 12) for the configuration shown in Fig. 11, a nineteen-block grid (Fig. 13) described in [20], and a 324-block grid described in [21]. Another variation of matching adjoining grid blocks is to match adjoining surfaces with interpolation to individual grid points. To the authors’ knowledge there are no examples that have been done with complex three-dimensional geometries using this approach, but, using some of the techniques to maintain conservation of fluxes [22], it would seem to be feasible. Multiple overlapping grids [23,24] where individual grid blocks conform to particular regions of the physical domain without concern about where one grid ends and another begins seems like the ideal approach. Unfortunately, the complexity of the interpolation process between grids is extremely high. Also, where grids overlay and the relative spacing in the overlap region is of considerable concern [25]. Investigators have applied unstructured grids in conjunction with the finite element technique to solve nonlinear continuum problems [26-281. Unstructured grids about geometries such as aircraft surfaces, however, are highly complex, require a great deal of effort to generate, and require considerable computer storage and computer time to maintain and
Vertical
Tail
ards
Iniet Fig. 11. Fighter aircraft configuration.
298
R. E. Smith, L.-E.
Cross
Eriksson,
Algebraic grid generation
Sections
Fig. 12. Dual-block
grid for a fighter
configuration
apply a connectivity relation between grid points. On the other hand, unstructured grids solve many of the problems associated with complex geometries. It is possible that a combination of structured and unstructured grids will evolve and, in part, be generated by algebraic techniques. A very important aspect of generating three-dimensional grids about complex geometries such as about aircraft surfaces is the grid generation on the body surface. This can be the most tedious and time consuming part of the entire grid generation process [19]. The surface geometry must be defined, intersections of components determined, and the grid topology chosen relative to the defining topology. There are surface generating programs available from commercial and government sources. Such programs may or may not provide the information that is needed for a particular grid
Fig. 13. Multiple-block
grids (many grids).
R. E. Smith, L.-E.
Eriksson,
Algebraic grid generation
299
generation project. Consequently, the use of these programs in conjunction with the overall grid generation procedure should be thoroughly investigated before embarking on a complex grid generation project.
10. Grid computation Designing, viewing, and modifying grid boundaries and controls require a sophisticated interactive environment. The environment should be built around a workstation with color graphics, dynamic rotations and translation, interactive devices such as tablets, light pens, and dials. Three-dimensional grid generation has many common features with computer-aided design (CAD) and should be approached in a similar manner [29].
11. Conclusions Algebraic grid generation is an application of mathematical interpolation. Multistep transfinite interpolation or the multisurface variation are highly suitable for three-dimensional grid generation, and there are many variations for applying each univariate step. A convenient approach for grid spacing control is to create an intermediate rectangular domain between the computational domain and the physical domain. Construction of monotonic functions between the computational domain and an intermediate domain is an effective means to concentrate and disperse grid points in the physical domain while maintaining simple blending functions that control the relative orientation of neighboring grid points. Certain singularities are unavoidable in creating grids in closed three-dimensional regions. The effect of singularities is dependent on the type of singularity, where they are located, and the solution technique. Multiple adjoining or overlapping blocks are required for a structured grid about complex geometries, and the use of interactive computer graphics is essential in the algebraic generation of complex three-dimensional grids.
References [l] P.C. Roache, Computational Fluid Dynamic (Hermosa, Albuquerque, NM, 1972). [2] D.A. Anderson, J.C. Tannehill and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer (Hemisphere, Washington, DC, 1984). [3] R.W. MacCormack, The numerical solution of the compressible viscous flow field about a complete aircraft in flight, in: Computational Methods in Viscous Flow (Pineridge, Swansea, U.K., 1984). [4] J.F. Thompson, Z.U.A. Warsi and C.W. Mastin, Numerical Grid Generation Foundations and Applications (North-Holland, Amsterdam, 1985). [5] J.F. Thompson, Z.U.A. Warsi and C.W. Mastin, Boundary-fitted coordinate systems for numerical solution of partial differential equations-a review, J. Comput. Phys. 47 (1982) l-94. [6] J.F. Thompson, ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982). [7] R.E. Smith, Two-boundary grid generation for the solution of the three-dimensional compressible NavierStokes equation, NASA TM 83123, 1981. [8] W.J. Gordon and C. Hall, Construction of curvilinear coordinate systems and applications to mesh generation, Internat. J. Numer. Meths. Engrg. 7 (1973) 461-477.
R. E. Smith, L. - E. Eriksson,
300 [9] W.J.
[lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23] [24] [25] [26] [27] [28] [29]
Algebraic grid generation
Gordon and L.C. Thiel, Transfinite mappings and their application to grid generation, in: J.F. Thompson, ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982) 171-192. Lars-Erik Eriksson, Transfinite mesh generation and computer-aided analysis of mesh effects, Doctoral Dissertation, Department of Computer Science, Uppsala University. Uppsala. Sweden, 1984. R.E. Smith, Algebraic grid generation, in: J.F. Thompson, ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982) 137-176. Lars-Erik Eriksson. Three-dimensional spline-generated coordinate transformations for grids around wingbody configurations. Numerical grid generation techniques, NASA CP 2166. Lars-Erik Eriksson, Flow solution on a dual-block grid around an airplane, Comput. Meths. Appl. Mech. Engrg. 64 (1987) 79-93 (this volume). R.E. Smith and M.R. Wiese. Interactive algebraic grid-generation, NASA TP 2533, 1986. P.R. Eiseman. Coordinate generation with precise controls over mesh properties, J. Comput. Phys. 47 (3) (1982) 341-351. P.R. Eiseman, Automatic algebraic coordinate generation. in: J.F. Thompson. ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982) 447-465. P.R. Eiseman, Grid generation for fluid mechanics computation, Ann. Rev. Fluid Mech. 17 (1985) 487-520. P.R. Eiseman and R.E. Smith, Mesh generation using algebraic techniques. Numerical grid generation, NASA CP 2166, 1980. R.E. Smith, Algebraic grid generation about wing-fuselage bodies, in: Proceedings 15th Congress of the International Council of the Aeronautical Sciences, 1986. S.L. Karman, J.P. Steinbrenner and K.M. Kisielewski. Analysis of the F-16 flow field by a block grid Euler approach, in: Proceedings 58th Meeting of the Fluid Dynamics Panel Symposium on Applications of Computational Fluid Dynamics in Aeronautics, 1986. W. Fritz and S. Leicher, Numerical solution of 3-D inviscid flow field around complete aircraft configurations. in: Proceedings 15th Congress International Council of the Aeronautical Sciences, 1986. M. Rai, A conservation treatment of zonal boundaries for Euler equation calculations, AIAA Paper 84-0164, 1984. CF. Dougherty, J.A. Benek and J.L. Steger. On applications of Chimera grid schemes to store separation. NASA TM 88193, 1985. J.A. Benek. P.G. Buning and J.L. Steger, A 3-D Chimera grid embedding technique, AIAA Paper 85-1523. 1985. C.W. Mastin, Interface procedures for overlapping grids, in: Proceedings First International Conference on Numerical Grid Generation in Computational Fluid Dynamics, 1986. A.J. Baker and M.O. Soliman, An Accurate and efficient finite element Euler equation algorithm. in: Proceedings International Conference on Numerical Methods in Fluid Dynamics, 1982. A.J. Baker, Finite Element Computational Fluid Mechanics (Hemisphere. Washington, DC. 1983). A. Jameson, T.J. Baker and N.P. Weatherill. Calculation of inviscid transonic flow over a complete aircraft, AIAA Paper 86-0103, 1986. R.E. Smith, Three-dimensional algebraic grid generation, AIAA Paper 83-1904, 1983.
MESH GENERATION AND REZONING
COMPUTER METHODS NORTH-HOLLAND
IN APPLIED
MECHANICS
AND ENGINEERING
64 (1987) 285-300
ALGEBRAIC GRID GENERATION Robert E. SMITH NASA Langley Research Center, Hampton, VA 23665, U.S.A.
Lars-Erik ERIKSSON* Old Dominion University, Norfolk, VA 23508, U.S.A. Received 22 September
1986
Algebraic grid generation is the direct expression of a physical coordinate system as a function of a uniform grid in a rectangular computational coordinate system. Algebraic grid generation is based on mathematical interpolation and is presented in general terms of multivariate transfinite interpolation. The multisu~ace method and the two-bounda~ technique are described as univa~ate procedures that can be applied within the context of transfinite interpolation. A technique for grid clustering is described. Problems that are commonly encountered in three-dimensional grid generation are discussed and approaches for dealing with complex physical domains using multiple computational grid blocks are presented.
1. Introduction The purpose of computational mechanics is to solve approximately the governing differential equations that describe the motion of a continuous medium [l-3]. This is accomplished by imposing a collection of points called a grid on the continuum domain, approximating the governing equations at the grid points with algebraic equations, and numerically solving the resulting system of algebraic equations. Defining a grid on a continuum domain is called grid generation, and unlike the governing differential equations there are no physical laws for grids to obey. Grid generation is a construction process which must accommodate the characteristics of the governing equations and the geometry of the physical domain [4-61. In general this leads to two highly desirable characteristics that a grid should possess: (1) A grid should conform to the boundaries of the region of interest. (2) A grid should be concentrated in regions where there are high gradients in the solution. There are two basic types of grids, which are (1) structured grids and (2) unstructured grids. A structured grid (Fig. 1) can be generated by a rule or correspondence between unifo~ly spaced points in an idealized computational domain and points in a physical domain. Consequently, the relative position between neighboring grid points in the physical domain are governed by mathematical expressions, and grid point coordinates can be completely identified by increasing sequences of integers (indices within arrays). Unstructured * Research Associate Professor, on leave from FFA, Sweden. 00457825/87/$3.50
0
1987, Elsevier Science Publishers B.V. (North-Holland)
286
R. E. Smith,
L.-E.
Eriksson.
Algebraic
grid generation
or
rl
J
I I r)
or
J
x(I,J) Y(I,J)
+i
or
Fig. 1. A structured
I grid.
grids (Fig. 2) consist of arbitrarily placed points in a physical region, and in general require an explicit directory indicating neighboring grid points that are connected. An unstructured grid can, in part, be generated by mathematical expressions, but there can be many exceptions that must be coordinated into the overall grid. Historically speaking, structured grids have been used in conjunction with finite difference techniques and finite volume techniques for solving the governing equations. Unstructured grids are more suitable for use with the finite element technique. The algebraic grid generation techniques described herein produce structured grids. Algebraic grid generation techniques are based on interpolating functions that transform an idealized rectangular computational domain into a physical domain (Fig. 3). A grid in the computational domain is created by partitioning the rectangular coordinates into uniform intervals, and under the transformation, the uniformly spaced coordinates map into the
Fig. 2. An unstructured
grid.
R.E.
Smith, L.-E.
Computational
Eriksson,
Algebraic grid generation Physical
Domain
Fig. 3. Computational
287
Domain
domain and physical domain.
physical domain (Fig. 4). Changing the number of grid points does not change the transformation or its characteristics. When the boundaries of the computational domain map onto the boundaries of the physical domain, the transformation is said to be boundary-fitted. Also, coordinate transformations imply that the governing equations must be transformed [7]. In this paper the basic concepts of interpolation are applied to create algebraic coordinate transformations, and grids are discrete evaluations of these transformations. First, a very general description of transfinite interpolation [8-111 is presented. Virtually all algebraic grid generation techniques can be described under the umbrella of transfinite interpolation, but individual applications can be quite different. An application of transfinite interpolation accredited to Eriksson [lo, 12,131, the two-boundary technique [7,11,14] described by Smith, and the multisurface method [15-181 described by Eiseman are used to demonstrate this point. Algebraic grid generation allows direct control of clustering grid points, and an approach to this important feature is described. Generally speaking, interpolation methods to express three-dimensional algebraic coordinate transformations are quite simple and straightforward. There are, however, many considerations that must go into their practical application. Two of these considerations are transformation singularities and geometric complexity of physical boundary surfaces. The singularity problem is discussed and a building block approach is advocated for dealing with complex geometry.
Computational
Grid
x
=
Fig. 4. Grids in the computational
UJl,r)
Physical
Grid
domain and in the physical domain.
R.E.
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Smith, L.-E.
Eriksson.
Algebraic grid generation
2. Algebraic grid generation techniques The transformation function,
from a computational
where, without loss of generality, Os(sl,
OGndl,
domain to a physical domain is a vector-valued
we can assume that O<<
Multivariable coordinate transformations (1) can be described in a very general manner using transfinite interpolation. Transfinite interpolation in three dimensions is applied through a recursion of three univariate interpolations, where blending functions and associated parameters (specified point positions and/or derivatives) determine transformations that are explicit forms of (1). Individual applications of interpolation within the context of transfinite interpolation can be quite different. Two direct applications and one variation of transfinite interpolation are discussed. Eriksson has adapted the transfinite interpolation formulation to use only exterior boundary descriptions and certain normal derivatives at the boundaries. He has derived blending functions with exponential terms to concentrate a grid near boundaries. The two-boundary technique described by Smith is similar to the Eriksson application for the univariate case. The two-boundary technique is based on the description of two opposing boundary surfaces, tangential derivatives on the boundary surfaces which are used to compute normal derivatives, and hermite cubic blending functions. Intermediate variables that are functions of the computational coordinates are embedded in the blending functions to control grid spacing. The multisurface method developed by Eiseman is, in its original form, a univariate formula (multisurface transformation) for grid generation based on describing two boundary surfaces and an arbitrary number of interior control surfaces. Blending functions are implicitly derived from what Eiseman calls global and/or local interpolants which result from an expression of the tangential derivative spanning between exterior boundary surfaces. The multisurface transformation can, for the most part, be described in the context of transfinite interpolation, except that the conditions on the blending function are imposed on the interpolants that are used in the derivative expression.
3. Transfinite interpolation Transfinite
interpolation
in a general form expresses the vector function F( t,n,
6) as
R.E. Smith, L.-E. Eriksson, Algebraic grid generation
where the parameters
289
are the position and derivatives described by
s(&n,I)=A;(n,i),
l=l,2
,...,
L, n=O,l,...,
P,
$(W)=B;(&i),
l=l,2
,...,
M,n=O,l,...,
Q,
5
1=1,2 ,...,
N, n=O,l,...,
R,
(5, q, 5J = C;t( 5777) ,
and the blending functions are &j”‘(t),
1=1,2 ,...,
L, n=O,l,...,
P,
&“‘(T/),
Z=1,2 ,...,
M,n=O,l,...,
Q,
y?‘(c),
1=1,2 ,...,
N, n=O,l,...,
R,
subject to the conditions
4. E&won’s
grid generation
Eriksson has adapted transfinite interpolation in the form of (2) to create grids about aerodynamic configurations where the governing equations are for inviscid fluid flow. In his applications of (2) he uses known parameter information only at the outer boundaries (Fig. 5). The information contains outer boundary surface descriptions and outward derivatives on certain boundary surfaces. The parameters L, M, and N in (2) are set equal to 2 .and P, Q, and R vary depending on the application, but are usually between 1 and 3. The outward derivatives, which control grid orthogonality and outward spacing at the boundaries, are derived from the surface descriptions and Eriksson has explicitly defined blending functions to achieve a desirable exponential stretching of grid points between boundary surfaces. An example of the first univariate interpolation is:
R. E. Smith, L.-E.
290
Eriksson. Algebraic grid generation
a<”
Back
Surface
n-0, l...
a“G$,r), Left
Surface
a<” Right
Fig. 5. Specified parameters
‘y;o)( 5) =
1 -
eK5 1; 1 ;t
cp([)=
e;;r;-;6,
eK
Similar blending functions
,
and parameters
Bottom
Surface
on exterior boundaries
cy;l)(+-
a;“(
Surface
f-
arl”
5. The two-boundary
1)
0
eK’+;t, eK = 0 .
are used in the second and third recursion
steps.
technique
As in Erisksson’s application of the first step in transfinite interpolation, the two-boundary technique uses two opposing boundary surface definitions and outward derivatives on the surfaces. The blending functions are cubic polynomials which satisfy the conditions in (2) and in effect provide hermite cubic polynomial interpolation between the two boundaries. An intermediate variable is used in the blending functions and is itself made a function of a computational coordinate. The one-to-one functional relation between the computational
R.E.
Smith, L.-E.
Eriksson,
291
Algebraic grid generation
coordinate and the intermediate variable provides a convenient means of controlling the spacing between physical points, when the computational coordinate is uniformly descritized. A general discussion of this method of grid spacing control is presented in a later section. The outward derivatives in the two-boundary technique are normal derivatives that are computed from the cross product of tangential derivatives on the surfaces (Fig. 6). The first step of transfinite interpolation for the two-boundary technique is,
(4) where
and 4%)
47
=W~)13
- 3m>1’ + 17
5) = 4Lf(~>1’ + 3Lf(81” 7
4”(t> = M~>l” - w(5)1’+.&9
7
‘yY( s> = v(s>1” - M 81’ .
The function f( 6) must be monotonically increasing in the interval 0 s 5 s 1. Where f( 6) has a small first derivative, there will be a relative high concentration of grid points relative to the &computational coordinate, and where f( 6) has a large first derivative, there will be a dispersion of grid points relative to the &computational coordinate. A function suitable for the concentration of a grid toward a boundary is
where K is a constant. Outward
Derivative from
Cubic
Blending
functiois
Fig. 6. Two-boundary
technique.
Determined
R. E. Smith. L.-E.
292
Eriksson.
Algebraic grid generation
The second and third steps of transfinite interpolation have been applied with the two-boundary technique [14,19] using the remaining two pairs of boundary surfaces and blending functions which have been derived to control the effect of the boundaries on the interior grid. The expressions are,
(5)
The functions g(q), g(q), h( l). and h( 6) must increase monotonically and vary between 0 and 1. They control how the boundaries corresponding to 77 and 5 affect the interior grid.
6. The multisurface method The multisurface method is a procedure for generating grid points between two exterior boundary surfaces F(0, n, 5) and F(l,n, 6). An arbitrary number of intermediate surfaces (F( 5,,77, C>, I= 2, . . . 3 L - 1) are introduced to control the grid. The blending functions are derived indirectly by integrating the tangential derivative in the direction of the interpolation (Fig. 7) which is expressed as a linear combination of position differences on neighboring surfaces and interpolation functions. The interpolation functions have the same delta function conditions as the blending functions have in (2). The blending functions that result from the multisurface method satisfy delta function conditions at the exterior boundaries and not necessarily at the interior control surfaces. An abbreviated derivation of the univariate multisurface method is presented. The multisurface derivative expression is
(6) where G-i)
= 4i 3
l&=0,
1#i,
6,,=1,
l=i.
Integrating (6) and applying boundary the general multisurface transformation:
conditions
yields an equation
which Eiseman calls
R. E. Smith, L.-E. Eriksson, Algebraic grid generation
ential
293
Derivatives in
the
Direction
Fig. 7. Multisurface
of
Interpolation
method.
The basic ingredients of the multisurface method are a partition of the computational coordinates in the direction of interpolation (ii, i = 1, . . . . , L - l), the interpolation functions, and the surface definitions. Choosing ql( 6) to be polynomials of degree L - 1, a curve connecting the boundary surfaces is of degree L. For instance, if there are four surfaces and the interpolation functions are
where
Then the multisurface
transformation
is
F,( 6,~~ f) = F(O, 7, 6) + (4t3 - 9t2 + WNF(629
+ et3
+ 352)[w3,
+ (4t3
- 352)[F(L
77, 0
-
?), 5) - F(
~3 l) - F(O, 7,
m27
537
77
77
01
01
01 *
Rearranging the terms in (8) such that the surface positions expression in the form of the first step in (2). That is
are coefficients
(8) results in an
R. E. Smith. L.-E.
294
Eriksson,
Algebraic grid generation
where (Y;“)(5) = 45j - 95’ - 65 + 1 ,
CY;“( 5) =
a:)(
CXy:o’( 5) = 45j - 35? .
5) =
4
- 65’ ,
65” - 125’ + 6[ ,
In this case, however, the blending functions CX~“‘( 5) d o not satisfy the delta function conditions, but the interpolants $J,(6) d o satisfy similar conditions. The second and third steps as described in (2) could be applied for multivariate interpolation or the blending functions for the remaining two coordinate directions could be derived with the multisurface transformation and Boolean sums performed. An interesting corollary [ 181 to the four-surface multisurface transformation described above is obtained by differentiating (8) with respect to the computational coordinate, expressing the position on the second and third surfaces in terms of the derivative evaluation at the exterior boundaries and resubstituting these expressions into (8). This yields the hermite cubic blending functions used in the two-boundary technique.
7. Grid spacing control The algebraic techniques previously discussed are transformations that map a rectangular computational domain to a physical domain with irregular boundaries. A uniform grid in the computational domain is obtained by partitioning each computational coordinate into equal increments, and under the transformation the discrete points in the computational domain map into irregularly spaced points in the physical domain creating a grid. The spacing between grid points in the physical domain is primarily controlled by the blending functions in the interpolation equations. Simple blending functions that produce the desired shape of a grid (i.e. relative orientation between points) may not produce the desired spacing between points. An example is hermite cubic polynomial blending functions on the occasions where the boundary derivatives are relatively small. These blending functions then produce a grid which is concentrated at the two boundaries. In order to obtain grid concentration somewhere else, additional information must be provided. One approach is to design or modify the blending functions to exactly produce the desired concentrations. Another effective approach is to define an intermediate control domain between the computational domain and the physical domain. An intermediate control domain is defined to be a rectangular domain where each coordinate is related to a corresponding computational coordinate by single-valued functions. For instance,
u =fM
7
u = g(T)
2
w=h(().
Under the application of these functions uniformly spaced points domain map to rectangular nonuniformly spaced points in the control blending functions are redefined with the intermediate coordinates as Following this logic, the hermite cubic polynomial blending functions
in the computational domain (Fig. 8). The independent variables. could be written as
R. E. Smith, L.-E.
Eriksson,
Intermediate
Algebraic grid generation Domain
Physical
Computational
295
grid
Grid
Fig. 8. Intermediate
control domain.
u3-2u2+14,
‘yY)( 5) = 2u3 - 3z.Z + 1 )
iq)(
aF)( 6) = -2u3 + 3u2 )
c$)( 6) = u3 - u3 )
LJ) =
where u=f(S). This is equivalent to the way the blending functions are written in (4). From a practical point of view, it is advantageous to make the intermediate coordinates proportional to the arclengths in the physical domain corresponding to the computational coordinate directions.
8. Grid singularities A singularity occurs when the Jacobian of the transformation between the computational domain and the physical domain is zero. Three-dimensional transformations that map the six-sided computational domain about a closed body will always have singularities. There are different types of singularities and their effect on the solution of the governing equations depends on the type of singularity, the solution technique, and where the singularity is relative to high solution gradients in the physical domain. Two common types of singularities that can occur when mapping a six-sided rectangular domain into a closed three-dimensional region are polar singularities and parabolic singularities (Fig. 9). For polar singularities an entire side of the computational domain maps to a singular line in the physical domain resulting in all grid points for that surface having a zero Jacobian. Using a finite difference technique to solve the governing equations and clustering
296
R. E. Smith, L.-E.
Polar
Eriksson,
Algebraic grid generation
Singularity
Fig. 9. Polar and parabolic
singularities.
grid points near a polar singularity can lead to computational difficulty because of the large elements in the Jacobian matrix of the transformation and the relatively large differences in the elements between neighboring points. If the phenomenon of interest is near a polar singularity, it is best to determine another transformation where the singularity is somewhere else or a transformation with another type of singularity. Parabolic singularities are less severe than polar singularities. For a parabolic singularity only edges of the computational cube map into singular curves in the physical domain. When they can be applied, such as at the tip of a wing, parabolic singularities are preferable to polar singularities. The importance of singularities is dependent on the technique that is used to solve the governing equations. Generally speaking, finite difference techniques are very sensitive to singularities but finite volume techniques are not as sensitive. In finite difference techniques the Jacobian matrix appears explicitly in the formulation, and special provisions must be taken near a singularity. On the other hand, finite volume techniques use areas of the faces of volume elements formed from neighboring grid points to compute fluxes across the faces. These techniques can tolerate singularities as long as the areas approach zero smoothly.
Fig. 10. Grid-crossing
singularity.
R.E.
Smith, L.-E.
Eriksson,
Algebraic grid generation
297
Another type of singularity that can happen in algebraic transformations rendering the transformation useless occurs when a grid curve in a computational coordinate direction becomes double valued, as shown in Fig. 10. This type of singularity should be avoided.
9. Complex boundary topology bodies is a primary goal of The computation of grids about closed three-dimensional algebraic grid generation. Unfortunately, closed bodies can have very complex topologies containing sharp corners and deep cavities (Fig. 11). Generating structured grids about complex geometries that map into a single rectangular computational block is, for all practical purposes, impossible. Considering this dilemma, there are two directions that can be taken for creating multiple block structured grids. They are: (1) create multiple grids that adjoin each other, or (2) create multiple grids that overlap each other. In many cases, multiple-block grids can be created where adjoining grid points and derivatives match. Examples of this approach are a dual-block grid [13] (Fig. 12) for the configuration shown in Fig. 11, a nineteen-block grid (Fig. 13) described in [20], and a 324-block grid described in [21]. Another variation of matching adjoining grid blocks is to match adjoining surfaces with interpolation to individual grid points. To the authors’ knowledge there are no examples that have been done with complex three-dimensional geometries using this approach, but, using some of the techniques to maintain conservation of fluxes [22], it would seem to be feasible. Multiple overlapping grids [23,24] where individual grid blocks conform to particular regions of the physical domain without concern about where one grid ends and another begins seems like the ideal approach. Unfortunately, the complexity of the interpolation process between grids is extremely high. Also, where grids overlay and the relative spacing in the overlap region is of considerable concern [25]. Investigators have applied unstructured grids in conjunction with the finite element technique to solve nonlinear continuum problems [26-281. Unstructured grids about geometries such as aircraft surfaces, however, are highly complex, require a great deal of effort to generate, and require considerable computer storage and computer time to maintain and
Vertical
Tail
ards
Iniet Fig. 11. Fighter aircraft configuration.
298
R. E. Smith, L.-E.
Cross
Eriksson,
Algebraic grid generation
Sections
Fig. 12. Dual-block
grid for a fighter
configuration
apply a connectivity relation between grid points. On the other hand, unstructured grids solve many of the problems associated with complex geometries. It is possible that a combination of structured and unstructured grids will evolve and, in part, be generated by algebraic techniques. A very important aspect of generating three-dimensional grids about complex geometries such as about aircraft surfaces is the grid generation on the body surface. This can be the most tedious and time consuming part of the entire grid generation process [19]. The surface geometry must be defined, intersections of components determined, and the grid topology chosen relative to the defining topology. There are surface generating programs available from commercial and government sources. Such programs may or may not provide the information that is needed for a particular grid
Fig. 13. Multiple-block
grids (many grids).
R. E. Smith, L.-E.
Eriksson,
Algebraic grid generation
299
generation project. Consequently, the use of these programs in conjunction with the overall grid generation procedure should be thoroughly investigated before embarking on a complex grid generation project.
10. Grid computation Designing, viewing, and modifying grid boundaries and controls require a sophisticated interactive environment. The environment should be built around a workstation with color graphics, dynamic rotations and translation, interactive devices such as tablets, light pens, and dials. Three-dimensional grid generation has many common features with computer-aided design (CAD) and should be approached in a similar manner [29].
11. Conclusions Algebraic grid generation is an application of mathematical interpolation. Multistep transfinite interpolation or the multisurface variation are highly suitable for three-dimensional grid generation, and there are many variations for applying each univariate step. A convenient approach for grid spacing control is to create an intermediate rectangular domain between the computational domain and the physical domain. Construction of monotonic functions between the computational domain and an intermediate domain is an effective means to concentrate and disperse grid points in the physical domain while maintaining simple blending functions that control the relative orientation of neighboring grid points. Certain singularities are unavoidable in creating grids in closed three-dimensional regions. The effect of singularities is dependent on the type of singularity, where they are located, and the solution technique. Multiple adjoining or overlapping blocks are required for a structured grid about complex geometries, and the use of interactive computer graphics is essential in the algebraic generation of complex three-dimensional grids.
References [l] P.C. Roache, Computational Fluid Dynamic (Hermosa, Albuquerque, NM, 1972). [2] D.A. Anderson, J.C. Tannehill and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer (Hemisphere, Washington, DC, 1984). [3] R.W. MacCormack, The numerical solution of the compressible viscous flow field about a complete aircraft in flight, in: Computational Methods in Viscous Flow (Pineridge, Swansea, U.K., 1984). [4] J.F. Thompson, Z.U.A. Warsi and C.W. Mastin, Numerical Grid Generation Foundations and Applications (North-Holland, Amsterdam, 1985). [5] J.F. Thompson, Z.U.A. Warsi and C.W. Mastin, Boundary-fitted coordinate systems for numerical solution of partial differential equations-a review, J. Comput. Phys. 47 (1982) l-94. [6] J.F. Thompson, ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982). [7] R.E. Smith, Two-boundary grid generation for the solution of the three-dimensional compressible NavierStokes equation, NASA TM 83123, 1981. [8] W.J. Gordon and C. Hall, Construction of curvilinear coordinate systems and applications to mesh generation, Internat. J. Numer. Meths. Engrg. 7 (1973) 461-477.
R. E. Smith, L. - E. Eriksson,
300 [9] W.J.
[lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23] [24] [25] [26] [27] [28] [29]
Algebraic grid generation
Gordon and L.C. Thiel, Transfinite mappings and their application to grid generation, in: J.F. Thompson, ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982) 171-192. Lars-Erik Eriksson, Transfinite mesh generation and computer-aided analysis of mesh effects, Doctoral Dissertation, Department of Computer Science, Uppsala University. Uppsala. Sweden, 1984. R.E. Smith, Algebraic grid generation, in: J.F. Thompson, ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982) 137-176. Lars-Erik Eriksson. Three-dimensional spline-generated coordinate transformations for grids around wingbody configurations. Numerical grid generation techniques, NASA CP 2166. Lars-Erik Eriksson, Flow solution on a dual-block grid around an airplane, Comput. Meths. Appl. Mech. Engrg. 64 (1987) 79-93 (this volume). R.E. Smith and M.R. Wiese. Interactive algebraic grid-generation, NASA TP 2533, 1986. P.R. Eiseman. Coordinate generation with precise controls over mesh properties, J. Comput. Phys. 47 (3) (1982) 341-351. P.R. Eiseman, Automatic algebraic coordinate generation. in: J.F. Thompson. ed., Numerical Grid Generation (North-Holland, Amsterdam, 1982) 447-465. P.R. Eiseman, Grid generation for fluid mechanics computation, Ann. Rev. Fluid Mech. 17 (1985) 487-520. P.R. Eiseman and R.E. Smith, Mesh generation using algebraic techniques. Numerical grid generation, NASA CP 2166, 1980. R.E. Smith, Algebraic grid generation about wing-fuselage bodies, in: Proceedings 15th Congress of the International Council of the Aeronautical Sciences, 1986. S.L. Karman, J.P. Steinbrenner and K.M. Kisielewski. Analysis of the F-16 flow field by a block grid Euler approach, in: Proceedings 58th Meeting of the Fluid Dynamics Panel Symposium on Applications of Computational Fluid Dynamics in Aeronautics, 1986. W. Fritz and S. Leicher, Numerical solution of 3-D inviscid flow field around complete aircraft configurations. in: Proceedings 15th Congress International Council of the Aeronautical Sciences, 1986. M. Rai, A conservation treatment of zonal boundaries for Euler equation calculations, AIAA Paper 84-0164, 1984. CF. Dougherty, J.A. Benek and J.L. Steger. On applications of Chimera grid schemes to store separation. NASA TM 88193, 1985. J.A. Benek. P.G. Buning and J.L. Steger, A 3-D Chimera grid embedding technique, AIAA Paper 85-1523. 1985. C.W. Mastin, Interface procedures for overlapping grids, in: Proceedings First International Conference on Numerical Grid Generation in Computational Fluid Dynamics, 1986. A.J. Baker and M.O. Soliman, An Accurate and efficient finite element Euler equation algorithm. in: Proceedings International Conference on Numerical Methods in Fluid Dynamics, 1982. A.J. Baker, Finite Element Computational Fluid Mechanics (Hemisphere. Washington, DC. 1983). A. Jameson, T.J. Baker and N.P. Weatherill. Calculation of inviscid transonic flow over a complete aircraft, AIAA Paper 86-0103, 1986. R.E. Smith, Three-dimensional algebraic grid generation, AIAA Paper 83-1904, 1983.