GRIDGEN S IMPLEMENTATION OF PARTIAL DIFFERENTIAL EQUATION BASED STRUCTURED GRID GENERATION METHODS. John P. Steinbrenner and John R.

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1 GRIDGEN S IMPLEMENTATION OF PARTIAL DIFFERENTIAL EQUATION BASED STRUCTURED GRID GENERATION METHODS John P. Steinbrenner and John R. Chawner Pointwise, Inc., Bedford, TX, USA. gridgen@pointwise.com ABSTRACT Elliptic and hyperbolic partial differential equation methods for generating both surface and volume structured grids have been implemented in the Gridgen software system. Their implementation provides the means by which very high quality quad and hex grids can be generated for computational fluid dynamics (CFD). The grids, in turn, provide the basis for highly accurate CFD simulations of viscous flow fields. The development and implementation of these methods resulted in several unique attributes: the ability to create grids that are topologically independent of the underlying geometry model, the ability to use a hyperbolic method to generate a surface grid spanning multiple trimmed B-Spline surfaces, and special boundary condition treatments that allow multi-block abutting grids to be generated using the hyperbolic method. Keywords: grid generation, partial differential equation methods, hyperbolic PDE, elliptic PDE, structured grid methods, quadrilateral, hexahedral 1. INTRODUCTION 1.1 Pre-Processing for CFD Computational fluid dynamics (CFD) requires discretization of the spatial region being analyzed. The discretization process, commonly referred to as grid generation, typically requires the vast majority of the labor for each analysis. The remaining labor is split between the numerical solution and post-processing. Therefore, the greatest gains in analysis productivity will be derived from grid generation advancements. There are several active areas of research[1][2] in the grid generation community including geometry model repair, automatic geometry feature detection and deletion, automatic topology creation, grid quality optimization, parametric grid re-generation, and grid generation methods. The subject of this paper belongs to the last area. 1.2 Grid Generation Methods Grid generation methodologies are typically divided into two classes: unstructured and structured. Unstructured meshes consist of quadrilateral or triangular (2D) and hexahedral or tetrahedral (3D) cells arranged in a manner that requires explicit ordering. Structured grids consist of quadrilateral (2D) and hexahedral (3D) cells arranged in a manner that has an implicit order. Although unstructured grids are growing in popularity among CFD practitioners, they currently are not widely used for analyses that require accurate simulation of viscous flows including boundary layers. For example, aircraft, marine, turbomachinery, and some automotive flow fields are still the realm of structured grids. The relative merits of structured grids for these applications include both the level of solution accuracy that may be achieved with quadrilateral and hexahedral cells and the low cell count that is made possible by a structured grid s tolerance of very high aspect ratio cells. 1.3 Structured Grids In a two dimensional structured grid quadrilateral cells are arranged in an I J array and in a three dimensional structured grid hexahedral cells are arranged in an I J K array. Furthermore, structured grids are typically implemented using a multi-block topology in which the overall domain to be discretized is subdivided into any number of subregions called blocks. Many different types of methods are available for generating structured grids including: interpolation methods such as transfinite interpolation (TFI)[3] partial differential equation (PDE) methods[4], including elliptic, parabolic, and hyperbolic methods

2 optimization methods conformal and harmonic mappings TFI and PDE based methods are the two most widely implemented structured grid methods. TFI s popularity is derived mainly from the speed with which a grid may be generated. However, the quality of a TFI grid in terms of smoothness, orthogonality, and clustering may be less than optimal. What PDE methods lack in speed they make up for in their ability to generate a high quality grid. Specifically, iterative application of elliptic PDE methods improves the quality of any structured grid. Hyperbolic PDE methods allow a high quality structured grid to be created initially. It it these two PDE methods which have been implemented into Gridgen and are described herein. 1.4 Grid Quality The use of elliptic and hyperbolic PDE grid methods is based on the high quality grids they produce. Indeed, the use of structured grids for CFD, as described above, is predicated based on their quality for use in analyses of viscous flows. Despite this, quantitative guidelines for grid quality are still lacking. Theoretically, truncation and discretization error estimates could be derived for each CFD solution algorithm and the grid constructed so that they are minimized. In practice, however, qualitative guidelines are used to direct the grid generation process. smoothness: The gradient of grid cell size and shape should be minimized. clustering: The grid should be clustered in regions where high solution gradients will appear. orthogonality: Grid lines should be mutually orthogonal. However, the fact that the guidelines are qualitative does not diminish their importance. Consider the following: A high quality grid created using the methods described herein and a grid created using another method were both used in the computation of a turbomachinery flow field at various flow conditions. Not all of the CFD solutions computed on the other grid converged, and where they did converge the solutions computed on the higher quality grid more closely matched the experimental data. Two grids were created for computation of the flow field around a high speed projectile. The CFD solution on the high quality grid converged six times faster than the solution on the lower quality grid. There are other practical benefits to the ability to employ a high quality grid method. Without such a method one must usually resort to grid topology as a means of improving the grid quality. Since topology is the current bane of structured grid generators, this last resort usually slows down the grid generation process. 1.5 Preface The following sections of this paper describe the implementation of elliptic and hyperbolic structured grid methods into the Gridgen software system. In the interest of brevity, basic mathematics about the methods are included by reference only. Both surface grid (including NURB constrained) and volume grid implementations are described as are supporting methods for diagnosing grid quality. 2.1 Overview 2. ELLIPTIC PDE METHODS The use of elliptic PDEs for structured grid generation began in 1977[5]. An elliptic PDE method is a boundary value problem that involves iterative solution of Poisson s equation with control functions (right hand side terms) that affect the grid s smoothness, clustering, and orthogonality. Because the PDEs are elliptic in nature, an iterative solution method is required. 2.2 Elliptic PDEs Employing the tensor notation used in Reference [4], Poisson s equation for a volume grid may be written as 2 ξ i = ( ξ i ξ i )ϕ i i = 1, 3 (1) where ϕ i is the control function. For numerical solution the roles of the dependent ( ξ i ) and independent (x,y,z) variables are reversed which results in ( g v g ij )r ξ i (2) ξ j + ( g v g kk )ϕ k r ξ k = 0 i j k Poisson s equation for a 3D surface may be extracted from Equation (2) by assuming that lines transverse to the surface are orthogonal and have zero curvature. The resulting equation is ( g s g ij )r ξ i (3) ξ j + ( g s g kk )ϕ k r ξ k = i j k 2 2 g s g ij ( )( r ξ i ξ j nˆ ) nˆ i j The unit normal vector nˆ is obtained at each grid point by evaluating the underlying surface shape definition. Furthermore, consider the case in which the 3D surface shape is defined by some parametric representation F( u, v). In this case the elliptic PDEs are recast entirely in terms of the parametric coordinates w = ( u, v). However, keep in mind that in each of these three variations is still the same basic Poisson equation. 2.3 Numerical Solution and 3D Surface Shape The initial conditions for applying the elliptic PDE method consist of an existing grid r ijk. Finite differences are used to approximate the elliptic PDEs s derivative terms and a pointwise successive over relaxation (SOR) numerical algorithm is used to solve for a new grid r' ijk. The relaxation procedure uses optimal relaxation factors[6] which greatly accelerates convergence. The iterative solution procedure is repeated until certain grid criteria are met. However, con-

3 vergence of the numerical solution is not required. Consider now solution of the 3D surface form of the elliptic PDE (Equation (3)). In practice many variations of surface shape constraints may occur: unconstrained - No particular 3D shape is desired so the surface is free to assume the shape arrived at by numerical solution of the PDE. parametrically constrained - The grid is to be generated on a surface defined by the parametric function F. geometry model constrained - The grid is to be generated on a geometry model comprised of an arbitrary collection of trimmed and untrimmed surfaces, usually NURBs. The last two cases are detected automatically by Gridgen s implementation of the elliptic PDE solution algorithm. Consider a geometry model consisting of a collection of trimmed NURBs, untrimmed NURBs, and wire frame surfaces (for example, the front of the space shuttle in Figure 1). Figure 2. A surface grid spanning multiple NURB surfaces. Figure 1. NURB model of the space shuttle. Furthermore, consider a surface grid (Figure 2) that is to be generated so that it spans any portion of the geometry model. Begin by projecting the grid that onto the model so that each grid point knows what model surface it is on and the corresponding (u,v) parametric coordinate. As the elliptic PDE SOR algorithm sweeps through each grid point, the point s neighbors in the finite difference stencil are examined to determine whether they all lie on the same NURB surface. If they do, that NURB s parametric coordinates become the definition of the shape function F. The 2D parametric form of the elliptic PDE is solved at that grid point and the model is evaluated to obtain the new parametric grid point location w' and F is evaluated to arrive at the new grid point r'. If the grid point s neighbors are not all on the same NURB surface, the 3D surface form of the PDE is solved and projection (ray cast or closest point) is used to ensure that the new grid point is on the model. Again, keep in mind that these are simply two variations of the same Poisson equation so that neighboring points feel the same control function influence regardless of how the 3D shape is determined. Figure 3. A NURB constrained surface grid after application of the elliptic PDE method. Note from Figure 3 that this algorithm permits grid points to migrate from model surface to model surface based on the PDE solution without requiring assembly of the model into a higher order solid. Furthermore, in cases where the projection fails (for example, the projection ray is cast into a gap between two surfaces) the grid point remains at the location computed by the elliptic PDE. Given the smoothness properties of the PDE, the grid lines across the gap will be smooth, provided of course that the gap is small relative to the grid spacing. 2.4 Control Functions The control functions ϕ provide control over the grid s smoothness, clustering, and orthogonality. They may be classified according to whether they primarily influence the grid points on the interior of the region (background) or near its boundaries (foreground). The foreground and background control functions are then blended together to obtain

4 the best qualities of each method. The properties of each method are illustrated by showing their effect when applied to the grid in Figure 4 that was generated by TFI. enforcement of the angle and spacing constraints (Figure 8). This stringent enforcement of grid constraints near the boundary is vitally important to accurate prediction of heat transfer, boundary layers, and other fluid phenomena. Figure 4. Baseline grid (generated using TFI) for illustration of control functions. Thomas-Middlecoff[7] background control functions force the clustering of grid points on the interior to match the clustering of boundary grid points (Figure 5). They are computed on the grid s boundaries from ( r ϕ i ξ i ξ i r ξ i) = (4) r ξ i r ξ i and is then interpolated onto the interior. Figure 6. Effect of LaPlace control functions on the baseline grid. Figure 7. Effect of fixed grid control functions on the baseline grid. Figure 5. Effect of Thomas-Middlecoff control functions on the baseline grid. LaPlace background control functions simply set ϕ i = 0 and result is a smooth variation of cell size across the grid (Figure 6). So-called fixed grid background control functions are computed by smoothing the control functions obtained by solving Poisson s equation for ϕ i on an existing grid. The subtle effects of fixed grid control functions are the elimination of slope discontinuities from grid lines (Figure 7). Hilgenstock-White[8][9] (H-W) foreground control functions provide a means of controlling the orthogonality of transverse grid lines at the boundary (angle constraint) and the cell size immediately adjacent to the boundary (spacing constraint). H-W control functions are adjusted each iteration of the elliptic PDE solution to provide nearly exact Figure 8. Effect of Hilgenstock-White control functions on the baseline grid. Steger-Sorenson[10] (S-S) foreground control functions are a more mathematically elegant method for enforcing angle and spacing constraints but sacrifice exact enforcement relative to H-W in order to provide an added degree of smooth-

5 ness (Figure 9). boundary is computed from the elliptic PDE solution. 2.6 Implementation Figure 9. Effect of Steger-Sorenson control functions on the baseline grid.. Figure 10. Effect of combining Thomas-Middlecoff and Hilgenstock-White control functions on the baseline grid. Gridgen s implementation of the elliptic control functions allows the background and foreground control functions to be blended together using an exponential decay function that reduces the foreground control function s boundary effect to zero over a user-specified number of grid points. Figure 10 illustrates the effect of using a hybrid control function that applies Hilgenstock-White foreground control functions near the boundaries to obtain orthogonality and Thomas-Middlecoff background control functions on the interior to obtain clustering control. This combination is, in fact, the default control function formulation used in Gridgen. 2.5 Boundary Conditions Three types of boundary conditions may be used with the default. The default is a Dirichlet (fixed grid point) condition. One may also employ a von Neumann type of boundary condition in which points will move along the boundary shape so that transverse grid lines will be orthogonal to the boundary. Finally, a floating boundary condition may be applied so that two or more adjacent grids can be run through the solver as though they were one. The common The implementation of the elliptic PDE methods is as important to their success as is their mathematics. In Gridgen, the following features are available. Default setting of elliptic PDE solver attributes. Singularities on the grid s boundaries are automatically detected and the appropriate adjustments made to the control functions. Any number of grids may be run in the elliptic PDE solver at a time, each grid with its own unique combination of attributes. The solver can by run interactively by stepping one iteration at a time, a preset number of iterations, or run until stopped by the user. For geometry model constrained surface grids, the specific surface entities and projection method for each grid may be selected. Each boundary of each surface grid may have a unique formulation of the foreground control functions (off, H- W, or S-S). Spacing constraints for the foreground control functions may be computed by interpolating from the boundaries end points, by keying in a specific value, by using whatever the existing spacing is, and by extracting the spacing from an adjacent grid. Angle constraints for the foreground control functions may be computed to be orthogonal or by any of the methods listed above for the spacing constraint. 3.1 Overview 3. HYPERBOLIC PDE METHODS A hyperbolic PDE method for generating a structured grid involves propagating grid fronts outward from an initial grid subject to constraints on a. the trajectory of propagation and b. the grid cell sizes in the propagation direction. The partial differential equations that result from the constraint equations are hyperbolic in nature such that a marching solution via finite differencing is possible. The use of hyperbolic PDEs for structured grid generation began in 1980[11]. While many researchers have contributed to the advancement of hyperbolic methods, the maturation of hyperbolic grid methodology since 1991 has been due largely to Chan[12]. 3.2 Surface Constraint Equations Three constraint equations are needed to solve for the x, y, and z components of the grid points r i, j. First, we specify that the η-directed grid lines (the marching direction) must be orthogonal to the ξ-directed grid lines (the transverse direction). r ξ r η = 0 (5)

6 The second constraint is obtained by introducing a surface, represented by its normal vector nˆ, to which the trajectory of the η-directed lines will be tangent. (6) The surface normal vector nˆ is computed in either of two ways. For surface grids that are not NURB constrained, nˆ is the normal vector of a plane. Note that this does not constrain the resulting surface grid to be planar; rather nˆ is constant for all points on the front. For NURB constrained surfaces, nˆ is computed locally at each grid point by evaluating the underlying NURB surface. With the trajectory of grid lines in the marching direction specified, the third and final constraint equation specifies the cell area. 3.3 Surface Cell Area nˆ r η = 0 V i, j = nˆ ( r ξ r η ) The value used for the cell area in the last constraint equation is computed as the product of the local marching step size and the local effective chord length along the front. V i, j = s j l i, j (7) (8) The marching step size s j is computed from an exponential function called a geometric progression. s j = s 1 ( 1 + ε) j 1 (9) The user specifies the size of the first marching step size s 1 and the constant value of ε, the ratio of successive step sizes in the marching direction. For step sizes that increase away from the initial front, typical values of ε are in the range 1.0 < ε < 1.2. Gridgen s implementation (see below) also allows the step sizes to mimic any other curve in the grid from which s j = 1, J is obtained. In this case, if the hyperbolic marching continues after the curve s J is exceeded, the marching step size remains constant at sj. The computation of the local effective chord length is a bit more complicated. First, a raw local chord length is computed at each point on the front. 1 c i, j = -- ( r (10) 2 i, j r r i, j 1 + i + 1, r j i, j ) These local chords are summed along the initial front yielding C j and an average local chord c j is computed. The effective local chord is then C j l i, j = [( 1 f )c (11) C 1 i, 1 + f c 1 ] The first term above is simply the ratio of the current front length to the initial front length. By scaling the local chord by the changing front size we ensure that the step sizes specified by the geometric progression will be enforced. Without this scaling, as the front propagated outward and became larger, the step size would shrink in order to meet the cell area constraint. The second term above (the one between the brackets) provides a mechanism to dissipate the non-uniform distribution of grid points on the initial front as the marching progresses as it provides a linear blending between the true local chord and the average local chord on the initial front. The scale factor f in the range 0 f 1 is computed from (12) The term g is a user definable constant in the range 0 g 1 that provides control over the rate at which the initial front s clustering will dissipate. The term j is the marching index at which the average cell will have an aspect ratio of 1. It is computed from the average cell on the initial front and the geometric progression growth rate. log( c 1 ) log( s 1 ) j = (13) log( 1 + ε) Practical limits on j are 10 j Volume Constraint Equations Three constraint equations are needed to compute the unknowns x, y, and z for each grid point in the marching (z,k) direction. The first two constraints specify that the trajectory of the z- directed grid lines is to be orthogonal to the front. The third constraint is the cell volume specification 3.5 Cell Volume f = min[ g - j, j 1 ] r ξ r ς = 0 r η r ς = 0 V ijk = r ξ ( r η r ς ) (14) (15) (16) The value used for the cell volume in the last constraint equation is computed as the product of the local marching step size and the local effective patch area along the front. V ijk (17) The local raw cell patch area on the current front is computed using finite differences. 1 p ijk = (18) 4 -- ( r r r ijk i 1, j, + k i + 1, r jk ijk ) Marching step size and local effective cell patch area are computed in the same manner as for surface grids. 3.6 Numerical Solution s k p ijk ( r ijk r r i, j 1, k + i, j + 1, r k ijk ) The constraint equations are linearized about a known state (the current grid front) and resulting matrix is solved via a 3 x 3 block tridiagonal inversion. 3.7 Numerical Smoothing = Practical application of hyperbolic methods requires the addition of smoothing terms to the governing equation. The

7 smoothing terms prevent slope discontinuities in the initial data from being propagated into the grid. Smoothing also attenuates the effects of odd-even grid point decoupling that may result from the use of central differences in the transverse direction. Smoothing is added both implicitly and explicitly in the transverse direction. Implicit smoothing[13] is also added to the marching direction. 3.8 Boundary Conditions When the initial front forms a closed loop a special periodic form of the block tridiagonal solution algorithm is used so that the grid produced by the hyperbolic marching is a fully closed surface mesh with common and continuous grid points along the seam. In cases where the initial front does not form a closed loop a technique called splay[14] is used. Splay influences the boundaries so that they open outward by using an extrapolation from the interior of the grid. η r 1, j = ( 1 s) η r 2, j + s( 2 η r 2, j η r 3, j ) (19) grid in the near surface region is tightly clustered and orthogonal to the airfoil surface NURB Constrained Surface Grids In case where a hyperbolic PDE method is to be used to generate a surface grid spanning a geometry model consisting of wireframes and trimmed and untrimmed NURB surfaces, the normal vector nˆ is evaluated locally at each grid point by querying the underlying geometry model via Gridgen s NURB kernel. After each marching step is completed, the new grid front is projected onto the geometry model using a closest point algorithm. Like the Gridgen s elliptic PDE implementation, the hyperbolic grid method allows the marching front to cross over gaps and overlaps in the geometry model. The NURB-constrained formulation of the hyperbolic grid method is demonstrated for the X-38 Crew Return Vehicle (geometry supplied courtesy of NASA Johnson). Figure 12 shows the 133 trimmed NURB surfaces that comprise the shape definition. Valid values for the user-specified parameter s are in the range 0 s 1 but the upper bound in practical applications is 0.2. Splay is very useful for overlapping grid applications since it helps ensure that adjacent grids overlap and allows the user to influence the amount of overlap. Splay is disabled when abutting grids are being generated. 3.9 EXAMPLES D Multi-Element Airfoil The 2D grid for a 3-element airfoil in Figure 11 demonstrates the robustness of the current hyperbolic method for a complex domain. The initial front consists of grid points around the circumference of each component airfoil plus two small branch segments connecting the slat to the airfoil and the airfoil to the flap, respectively. The branch segments were created in Gridgen while the airfoil shapes were imported from a geometry file. Figure 11. A single structured surface grid was created by extruding outward from the airfoil surfaces. Note in Figure 11 that the grid lines are able to negotiate the sharp corners and concave regions as they propagate outward toward the far field boundary. At the same time, the Figure 12. X-38 trimmed NURB model. The initial front for the grid generation consists of the definition of grid points around the perimeter of the vehicle s base region. The surface grid, shown in yellow in Figure 13, was then marched along the surfaces up to the nose of the vehicle in approximately 60 seconds Multi-block Abutting Volume Grids As mentioned previously, internal boundaries that are shared between adjacent components of the initial front are detected automatically so that, as the grid is marched outward, common boundaries match exactly. The result is a non-overlapping (abutting) multi-block grid. Consider the geometry model in Figure 14 with each NURB surface given a different color.

8 Figure 13. Hyperbolically generated surface grid on the X-38. Figure 15. Initial conditions for hyperbolic multiblock volume extrusion. Figure 14. NURB geometry for multi-block hyperbolic extrusion. Figure 15 shows surface grids on this model. The green surfaces were created using an elliptic PDE method but all other surface grids were hyperbolically extruded on the NURB model. The hyperbolic volume method was used to march outward 11 steps, creating a thin layer of hex cells around the model (Figure 16). Each surface grid was extruded independently but a special interface boundary condition allowed the extruded boundary between each surface to be created as part of the hyperbolic solution resulting in the abutting multi-block grid shown below. The purple grids are faces of the extruded hex blocks. Figure 16. Surfaces of multi-block grid extruded hyperbolically from NURB surfaces. This technique is very useful for creating layers of hex cells in the viscous layer around a body. The remainder of the hybrid mesh will be filled with tetrahedral cells. Of course this method may be used to generate the entire volume grid out to the outer boundary. 4. GRID QUALITY DIAGNOSTICS In order to monitor the quality of grids created by the PDE (and other) methods, Gridgen has implemented a graphical tool to quantify grid quality measures such as cell size, cell size variation, Jacobian, skewness, and aspect ratio. This diagnostic tool may be operated in either of two modes. Criteria may be established and only grid cells exceeding

9 the criteria will be displayed and tabulated. For example, the red cells toward the rear of Figure 17 show hexes with aspect ratios exceeding the criteria. Alternatively, one may obtain high level of detail about the quality measure anywhere in the grid. Figure 17 also shows two cutting planes of constant computational coordinate on which the aspect ratio of each cell is displayed according to the color bar. Histograms and cell hiliters are also available. Figure 17. Two grid volume grid planes colored by cell aspect ratio. Cells drawn in red exceed the established aspect ratio criteria. 5. CLOSING The current work accomplished the following goals. Elliptic and hyperbolic PDE grid methods have been implemented in order to support the accurate application of CFD to viscous flow fields. Elliptic control function formulations, developed independently and by several researchers, have been combined in a manner that preserves their best properties so that smoothness, orthogonality, and clustering may be obtained simultaneously. A 3D surface formulation of the elliptic PDE method has been implemented that allows grids to be generated on 3D geometry models regardless of the model s topology and quality (gaps and overlaps are handled without resorting to repair or stitching). Hyperbolic grid methods were implemented in Gridgen, including enhancements to the basic formulation that improve the quality of the resulting grids. The hyperbolic surface grid method was extended to allow grids to be marched along complex surface shapes defined by trimmed NURBs. New boundary condition formulations allow the hyperbolic method to be used to created non-overlapping (abutting) multi-block grids. 6. REFERENCES [1] Cross, M., et al, Numerical Grid Generation in Computational Field Simulations, proceedings of the 6th International Conference, held at the University of Greenwich, July 1998, ISBN [2] 7th International Meshing Roundtable 98, proceedings of the conference held in Dearborn, MI on October 1998, published by Sandia National Laboratories, Sandia Report SAND [3] Chawner, J.R., and Anderson, D.A., Development of an Algebraic Grid Generation Method with Orthogonality and Clustering Control, from Numerical Grid Generation in Computational Fluid Dynamics and Related Fields, ed. by -Arcilla, A.S., et al, proceedings of the 3rd International Conference, held in Barcelona, Spain, June 1991, pp [4] Thompson, Joe F., Warsi, Z.U.A., and Mastin, C. Wayne, Numerical Grid Generation, Foundations and Applications, Elsevier Science Publishing, [5] Thompson, Joe F., Thames, Frank C., and Mastin C. Wayne, Boundary Fitted Curvilinear Coordinate Systems For Solution of Partial Differential Equations on Fields Containing Any Number of Arbitrary Two- Dimensional Bodies, NASA CR-2729, [6] Ehrlich, Louis J., An Ad Hoc SOR Method, Journal of Computational Physics, Vol. 44, pp , [7] Thomas, P.D., and Middlecoff, J.F., Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations, AIAA Journal, Vol. 18, 1979, pp [8] Hilgenstock, A., A Fast Method for the Elliptic Generation of Three-Dimensional Grids With Full Boundary Control, Numerical Grid Generation in Computational Fluid Mechanics 88, ed. by Sengupta, S., et al, Pineridge Press ltd., Swansea, UK 1988, pp [9] White, Jeff A., Elliptic Grid Generation With Orthogonality and Spacing Control On An Arbitrary Number of Boundaries, AIAA paper no , June [10] Sorenson, R.L, and Steger, J.L, Grid Generation in Three Dimensions by Poisson Equations with Control of Cell Size and Skewness at Boundary Surfaces, Advances in Grid Generation, FED-Vol. 5, Ed. K.N. Ghia and U. Ghia, ASME Applied Mechanics, Bioengineering, and Fluids Engineering Conference, Houston, [11] Steger, Joseph L., and Chaussee, Denny S., Generation of Body-Fitted Coordinates Using Hyperbolic Partial Differential Equations, SIAM Journal of

10 Scientific and Statistical Computing, Vol. 1, No. 4, December 1980, pp [12] Chan, William H., Hyperbolic Methods for Surface and Field Grid Generation, from Handbook of Grid Generation, edited by Thompson, Joe F., et al, 1999 by CRC Press LLC, ISBN , pp [13] Kinsey, Don W., Barth, Timothy J., Description of a Hyperbolic Grid Generating Procedure for Arbitrary Two-Dimensional Bodies, AFWAL-TM FIMM, Flight Dynamics Lab at Wright-Patterson AFB, OH, July [14] Chan, William H., Steger, Joseph L., A Generalized Scheme for Three-Dimensional Hyperbolic Grid Generation, AIAA paper CP, LIST OF SYMBOLS c - raw chord along surface hyperbolic front l - effective chord along hyperbolic front g - metrics of the transformation from (x,y,z) to (ξ,η,ζ) i,j,k - grid point indices in the ξ, η, and ζ computational directions, respectively p - raw patch area along volume hyperbolic front nˆ - unit normal vector r - grid point coordinate (x,y,z) s - cell spacing constraint V - hyperbolic cell area/volume constraint x,y,z - components of the grid point coordinate r ε - geometric progression factor ξ,η,ζ - grid point coordinates in the transformed computational space ϕ - control function (RHS term of Poisson s equation)