GENERATION OF STRUCTURED BLOCK BOUNDARY GRIDS

Transcription

1 Congreso de Métodos Numéricos en Ingeniería 2005 Granada, 4 a 7 de julio, 2005 c SEMNI, España, 2005 GENERATION OF STRUCTURED BLOCK BOUNDARY GRIDS Eça L. 1, Hoekstra M. 2 and Windt J. 2 1: Mechanical Engineering Department Instituto Superior Técnico Universidade Técnica de Lisboa Av. Rovisco Pais 1, Lisboa eca@marine.ist.utl.pt 2: R & D Projects Department Maritime Research Institute Netherlands P.O. Box 28, 6700 AA Wageningen, The Netherlands M.Hoekstra@marin.nl, J.Windt@marin.nl Key words: Structured Grid, Block Connections, Algebraic, Hyperbolic and Elliptic methods Abstract. In external flow hydrodynamics, the computation domain is typically bounded internally by a given surface shape, but externally the domain border can be conveniently chosen. After subdivision of the domain into several blocks, only a few block faces are located on the specified surface shape; the other block faces are specified by their edges only. The generation of a grid on a block face, given by its edges only, is therefore a frequently encountered task. In this paper, we present a grid generation package for structured grids on such block faces, which are arbitrarily positioned in space but only moderately curved usually. For generating a grid on such faces a quasi 2D grid approach is not only more efficient but also gives better results than a general 3D surface grid generation technique. One of the grid point coordinates is obtained with an algebraic interpolation technique applied in a rotated coordinate system that minimizes its value. The two remaining coordinates are obtained with algebraic, hyperbolic or elliptic methods of 2-D grid generation, which include the influence of the third coordinate. 1 INTRODUCTION In numerical ship hydrodynamics structured grids are widely used to obtain accurate viscous flow predictions. In many cases the complexity of the geometries requires a multiblock structured grid. The set-up of the grid and the ease with which it can be constructed play a key role in the success of such flow calculations. In external flow hydrodynamics, the computation domain is typically bounded internally by a given surface shape, but externally the domain border can be conveniently 1

2 chosen, provided that suitable boundary conditions can be imposed. After subdivision of the domain into several blocks, only a few block faces are located on the specified surface shape; the other block faces are initially specified by their edges only. The generation of a grid on a block face, given by its edges only, is therefore a frequently encountered task. In this paper, we present a grid generation package for structured grids on such block faces, which are arbitrarily positioned in space but typically only moderately curved. The idea is to reduce the problem so that it can be dealt with by a quasi 2-D, (x, y, grid generation method. Quasi means that the influence of the third coordinate, z, which is obtained with an algebraic interpolation technique, is still, approximately, taken into account. The basic procedure is as follows: Carry out a coordinate transformation so that the four edges of the block face are approximately in the (x, y plane. Generate an initial grid from given grid node locations on the four edges using an algebraic interpolation technique Perform a 2-D grid generation for (x, y using algebraic, hyperbolic or elliptic grid generators, [1], which are extended to incorporate the influence of the known z coordinate. One of the interesting properties of the present grid generation package is that it allows to combine the different techniques available. Moreover, for any of the methods available, it is possible to freeze part of a grid and update and improve only the remaining part. This paper is organized in the following way: section 2 describes the grid generation procedure and the methods available for 2-D grid generation. Section 3 presents two examples of application of the present grid generation package and the final remarks are summarized in section 4. 2 GRID GENERATION PROCEDURE The purpose of the present grid generation package is to generate structured grids on moderately curved surfaces defined initially by four edges only. This is the typical situation of block-connections of multi-block grids where only the four edges of the faces have to match a specific location. Starting from the grid nodes distributions at the four edges in a Cartesian coordinate system (X, Y, Z, the procedure to generate a (nx ny grid includes the following steps: 1. Transformation to a new coordinate system (x, y, z, so that z is minimized. 2. Algebraic interpolation of z from the edges in the new coordinate system. 3. Application of a quasi 2-D grid generation methods for the determination of (x, y. 2

3 4. Rotation of the (x, y, z grid back to the original (X, Y, Z coordinate system. nx and ny are the number of grid nodes along the edges 1 of the region to be discretized. We will designate the two independent variables of the computational domain by (ξ, η and the location of a grid node by x k (j, i, where 1 j ny, 1 i nx and (x 1, x 2, x 3 (x, y, z. 2.1 Definition of the (x, y, z Coordinate System In order to avoid the difficulties of surface grid generation, [1], the block face is first re-oriented. So a coordinate transformation from (X, Y, Z to (x, y, z is carried out to minimize the average z coordinate. The block face is then more or less positioned in a xy-plane. The new coordinate system is obtained as follows: The x axis is obtained from the mean of two straight lines defined by the corner nodes of the face, x = 0.5 ( X1,nx X 1,1 + X ny,nx X ny,1. (1 Similarly, a transverse coordinate direction is determined from z is subsequently computed from t = 0.5 ( Xny,1 X 1,1 + X ny,nx X 1,nx. (2 z = x t, (3 Finally, y is obtained from the cross-product of z and x to obtain a right-handed, rectangular system. 2.2 Algebraic determination of z The shape of the face, i.e. the determination of the z coordinate for the interior grid nodes, is made with simple algebraic interpolation techniques, using the given edge curves. Four possibilities are available: 1. Transfinite interpolation from the four edges. 2. Linear interpolation between the boundaries i = 1 and i = nx. 3. Linear interpolation between the boundaries j = 1 and j = ny. 4. Bi-linear interpolation. 1 Obviously, there are two edges with nx grid nodes and two edges with ny grid nodes. 3

4 First of all the independent variables, (ξ, η, are set from the grid node distributions at the four edges. At the boundaries j = 1 and j = ny for example, ξ is computed from ξ (j, i = i i=2 nx i=2 (x(j, i x(j, i (y(j, i y(j, i (z(j, i z(j, i 1 2 (x(j, i x(j, i (y(j, i y(j, i (z(j, i z(j, i 1 2 (4 with ξ (j, 1 = 0. This means a simple parameter distribution according to the physical distribution of the grid nodes along the edge. Next for all the interpolation techniques, (ξ, η values of the interior nodes are calculated from the values at the four edges. For example, for each value with constant i, ξ (j, i is computed from ξ (j, i = ny j ny 1 ξ (1, i + j 1 ny 1 ξ (ny, i. (5 Finally the z values for the interior nodes can be calculated. Here we only present the transfinite interpolation method which is the most common option: z(j, i = f l 1(ξ z(j, 1 + f l 2(ξ z(j, nx + f l 1(η z(1, i + f l 2(η z(ny, i f l 1(ξ ( f l 1(η z(1, 1 + f l 2(η z(ny, 1 f l 2(ξ ( f l 1(η z(1, nx + f l 2(η z(ny, nx (6 where f1 l and f2 l are one-dimensional linear basis functions in the interval 0 λ 1, given by f1 l = 1 λ, f2 l (7 = λ. Note that these interpolation techniques may also be used to generate an initial approximation of the (x, y grid. The above method to determine z is based on the edge nodes only, but it is also possible to start the grid generation procedure with an available grid. In that case, z may be redefined with any of the interpolation techniques mentioned above or kept equal to the values of the initial grid. The next step is to use more sophisticated methods for determining the other two coordinates x and y. We present four techniques in the following sections. 2.3 Algebraic grid generation The algebraic grid generation technique is based on transfinite interpolation with Hermite or linear basis functions. The basis functions for each direction can be set independently by the user. Although this is the technique that requires the highest level of user 4

5 experience and skill in complex geometries, it is also the grid generation method that gives the best control of the interior grid line distances. In domains with smooth boundaries, the algebraic technique may be the most efficient way to generate a structured grid. Some important aspects of the method, especially needed for complex geometries, are: Grid orthogonality at the boundaries. Regions of fixed grid nodes. Guiding grid lines or surfaces that divide the interpolation domain. The possible existence of regions of fixed grid nodes and/or guiding lines implies that the computational domain defined by 1 i nx 1 j ny may have to be subdivided first. This division is performed in such a way that the resulting sub-domains again are structured. This most of the time means that additional edges in the interior of the initial face have to be defined. The grid coordinates of these new edges are obtained with 1-D interpolations using linear or Hermite basis functions according to the input option. So first the sub-domain edges are calculated and next the algebraic grid generation of all sub-domains is performed. Consider a sub-domain bounded by the lines ξ = i o, ξ = i 1, η = j o and η = j 1. For a generic variable x k the present transfinite interpolation is defined in the following way: x k (j, i = f t 1(ξ + x k (j, i o + f t 2(ξ + x k (j, i 1 + f t 3(ξ + xk + (j, i o + f t 4(ξ + xk + (j, i 1 + f1(η t + x k (j o, i + f2(η t + x k (j 1, i + f3(η t + xk + (j o, i + f4(η t + xk + (j 1, i ( f1(ξ t + f1(η t + x k (j o, i o + f2(η t + x k (j 1, i o + f3(η t + xk + (j o, i o + f4(η t + xk + (j 1, i o ( f2(ξ t + f1(η t + x k (j o, i 1 + f2(η t + x k (j 1, i 1 + f3(η t + xk + (j o, i 1 + f4(η t + xk + (j 1, i 1 ( f1(η t + f3(ξ t + xk + (j o, i o + f4(ξ t + xk + (j o, i 1 ( f2(η t + f3(ξ t + xk + (j 1, i o + f4(ξ t + xk + (j 1, i 1 where the local independent variables (ξ +,η + are obtained from (ξ,η with (8 ξ + = ξ (j, i ξ (j, i o ξ (j, i 1 ξ (j, i o and η + = η (j, i η (j o, i η (j 1, i η (j o, i. (9 For the basis functions f t 1 to f t 4 linear or Hermite basis function can be used. If f t n = f l n, the linear basis functions f l 1 and f l 2 are again given by equation (7, while f l 3 and f l 4 are set equal to 0. If f t n = f h n, the Hermite basis functions with an independent variable λ 5

6 varying between 0 and 1 are given by f1 h (λ = 2λ 3 3λ 2 +1 f2 h (λ = 2λ 3 +3λ 2 f3 h (λ = λ 3 2λ 2 +λ f4 h (λ = λ 3 λ 2. (10 The derivatives of the grid coordinates at the boundaries of the sub-domain are computed from the boundary-point distribution combined with the orthogonality condition, x x + y y + z z = 0. (11 The procedure is essentially the same for all four boundaries of each sub-domain, so we will exemplify it for one boundary only. Consider boundary j = j o, where we need to compute the derivatives of the grid coordinates with respect to η + from i = i o to i = i 1. Computation of the derivatives of x and y with respect to η +, i.e. x η + and y η +, includes then the following steps: 1. Calculate the derivatives with respect to ξ +, i.e. x ξ +, y ξ + and z ξ +, using centraldifferences. If i o = 1, first-order forward differences are applied, whereas first-order backward differences are applied when i 1 = nx. 2. Obtain a first estimate of the derivatives with respect to η + using the coordinates at the boundaries j o and j 1. x η + = x(j 1, i x(j o, i, y η + = y(j 1, i y(j o, i and z η + = z(j 1, i z(j o, i (12 and compute the magnitude of the vector (x η +, y η +, z η + r η + = (x η (y η (z η Adjust one of the derivatives with respect to η + from the orthogonality condition. Whether x η + or y η + is adjusted depends on the magnitude of the derivatives with respect to ξ +. x ξ + > y ξ + x η + = y ξ +y η + + z ξ +z η + x ξ + y ξ + x ξ + y η + = x ξ +x η + + z ξ +z η + y ξ + (13 4. Scale the derivatives with respect to η + in order to obtain a vector with magnitude equal to r η +. 6

7 5. Check if the Jacobian of the transformation at the line j o is positive, i.e. x ξ +y η + y ξ +x η + > 0. If the Jacobian is negative, take the symmetric of the derivatives with respect to η +. The fourth step of the procedure is essential to guarantee the smoothness of the interpolation in regions where the two different possibilities included in (13 are applied to consecutive grid lines. The final step of the procedure avoids the problems that may occur when the straight lines defining x η +, y η + and z η + intersect the boundaries of the sub-domain. The computation of the first derivatives at the boundaries of the sub-domains as described above is numerically robust and it ensures grid orthogonality close to the boundary for the transfinite interpolation based on Hermite basis functions. However, as in any algebraic grid generation technique, it does not guarantee that the interior grid lines do not intersect. 2.4 Elliptic grid generation One of the most popular methods of elliptic grid generation is the so-called GRAPE approach, [2]. The main advantage of this approach is the ability to obtain nearlyorthogonal grids at the boundaries of the domain and interior grid line distances that reflect the boundary node distributions. The implementation of the 2-D and 3-D versions has been discussed in many papers, like for example [3]. In the present paper, we will restrict ourselves to a brief description of the consequences of including the z coordinate in the 2-D Grape method. In the current implementation, we assume that the derivatives with respect to ζ obey the following conditions x ζ = 0, y ζ = 0, z ζ = 1. (14 With these assumptions, the coefficients of the system of non-linear elliptical partial differential equations for x and y remain unchanged. These coefficients of the grid generation system are the components of the contravariant metric tensor, g 11, g 22 and g 12, [4] and hence do not depend on z and are equal to the ones of the 2-D grid generation system. However, the z coordinate does affect the calculation of the control functions, which include first and second order partial derivatives of the grid coordinates with respect to ξ and η, [4]. The z coordinate also affects the implementation of one of the boundary conditions. Two types of boundary conditions are available for this method: Dirichlet conditions with x and y fixed. These are independent of z. Neumann-Dirichlet conditions satisfying the orthogonality condition, equation (11 and the edge line definition. The latter is defined by a 1-D cubic spline interpolation 7

8 based on the initial grid node distribution. For this spline definition the influence of z is incorporated. Additional to the incorporation of the z coordinate, the present implementation of the method also allows the definition of regions of fixed grid nodes. At these locations, the grid remains unchanged. This is achieved by a straightforward modification of the coefficients of the linear system of equations that is solved at each iteration. A modification is needed in case a region of fixed grid nodes is attached to an edge of the complete face; the calculation of the non-linear terms of the control-functions is shifted from the boundary line in the direction perpendicular to the boundary, to the last node with fixed grid coordinates. 2.5 Orthogonal grid generation The orthogonal grid generators based on the solution of a system of partial differential equations have been discussed in [5] and [6]. In the present paper, we describe only the consequences of the third coordinate z. With the assumptions (14, the system to generate orthogonal grids in a surface defined by four edge lines should be written as with h ξ = h η = ( hη x h ξ ( hη y h ξ ( hη z h ξ + ( hξ + + x h η ( hξ y h η ( hξ z h η = 0 = 0 = 0. (15 ( x 2 ( + y 2 ( 2 + z ( x 2 ( + y 2 ( 2 (16 + z This would mean that z should also be computed from the system of partial differential equations. However, for the sake of simplicity and due to the limited use of the orthogonal grid generation method in practical applications, we have dropped the z derivatives in the system (15. Hence the influence of the z coordinate is restricted to the incorporation of its derivatives with respect to ξ and η in the definition of h ξ and h η and to the 1-D cubic spline definition that is used for the boundary condition. As a consequence of this simple approach, we do not have a truly orthogonal grid generator for this type of applications. Nevertheless, it is still possible to tune the boundary point distribution with it, which is the main purpose of this grid generation method, [1]. 8

9 2.6 Hyperbolic grid generation The hyperbolic grid generation method follows the description given in [7], but in some cases we have introduced our own options. The basic idea of this type of methods is to grow layers of grid nodes from an available line or surface to obtain a nearly-orthogonal grid. In its most general form, only one of the boundaries of the domain is known. The shape of the other boundary in the marching direction is a result of the grid generation process and the 2 lines at the outer ends of the growing grid are defined by the boundary conditions applied to the system of hyperbolic equations. In the present implementation, we have only considered Dirichlet boundary conditions in the hyperbolic generation system, i.e. we have assumed that the grid coordinates at the computational boundaries parallel to the marching direction are known. The hyperbolic system of equations that determines x and y may be written in the following way: x x + y y x y y x = z z (17 = h ξ h η This system may be solved starting from a boundary ξ = constant or η = constant. The solution procedure is similar in both directions so we will describe it only for the case where the line j = 1 is the starting line. This means that the marching is performed in the η direction. The generating system of equations is not linear, so one has to be careful with the linearization and discretization procedures. A simple Newton linearization is applied to the system of equations (17, but as suggested in [7], the unknown derivatives with respect to ξ are approximated by x i (1 + θ x i (j θ x i (j 1, (18 where θ is a smoothing parameter, j 1 is the η value at the last available ξ line and j is the index of the new line to be generated. Using central-difference approximations for the derivatives with respect to ξ and first-order two-point forward differences for the derivatives with respect to η, we obtain a coupled tri-diagonal system of 2 (nx 2 linear equations that may be written as A ξ [ x(i y(i ] + 0.5(1 + θa η [ x(i + 1 x(i 1 y(i + 1 y(i 1 ] = z z, (19 h ξ h η where x = x(j x(j 1, y = y(j y(j 1 9

10 and A ξ = ( x ( y o o ( y ( x o o ( x, A η = ( y o o ( y o ( x o. (20 The subscript o indicates that these derivatives are supposed to be calculated with the information available at the line j 1. As we have mentioned above, the values of x and y at i = 1 and i = nx are supposed to be known. The linear system of equations (19 has a straightforward solution. However, the smoothing introduced by θ is not sufficient to make a successful hyperbolic grid generator, specially in regions where the line j 1 is concave. Therefore we have introduced three additional types of smoothing: 1. Elliptic smoothing of the cell areas, h ξ h η, which is applied only in concave regions. 2. Corrector steps with the derivatives in the marching direction computed from the last values available of the two layers of grid nodes. 3. Explicit elliptic smoothing of the grid coordinates. In summary, the generation of a layer of grid nodes at η = j from a known layer at η = j 1 consists of the following steps: 1. Compute h η from the boundaries i = 1 and i = nx and the grid node distribution at j Calculate the derivatives of x and y with respect to ξ at j 1 and the corresponding value of h ξ. 3. Obtain the derivatives of the grid coordinates with respect to η using the orthogonality condition, h ξ and h η. 4. Determine the angles at each grid node of the j 1 grid line. The purpose of this determination is to identify the concave and convex regions. 5. Calculate the right-hand side of ( Apply smoothing to the cell areas if required. 7. Solve the coupled tri-diagonal system (19 with fixed grid coordinates at i = 1 and i = nx. 8. Execute the required number of corrector sweeps with the derivatives of the grid coordinates with respect to η computed from the last available values at j and the known coordinates at j 1. This includes the update of the matrix A η and a new solution of the system (19. 10

11 9. If required, apply extra smoothing with the Laplace equation. 10. Check if the Jacobian of the new layer of grid nodes is positive. The first item is straightforward to implement. h η is computed at the two boundaries, i = 1 and i = NX, from the distances between the grid nodes at lines j and j 1. The values of h η at the interior grid nodes, 1 < i < nx, are obtained by linear interpolation using weights computed from the grid node distribution at line j 1 as in the ξ definition given by equation (4. The derivatives with respect to ξ at the line j 1 are approximated by central-differences schemes and h ξ is computed from these derivatives. The derivatives with respect to η are then obtained from ( x o = h η h ξ ( y o and ( y o = h η h ξ ( x o. (21 The local angle at each node of line j 1, φ(i, is estimated with first-order backward and forward differences applied to the ξ derivatives. In this determination, we have to ensure that we are computing the angle in the marching direction and not the interior angle that is given by 2π φ. In the ideal situation φ(i = π. To identify concave and convex regions we use the ratio φ(i/π, which is greater than one for convex regions and smaller than one for concave regions. The calculation of the right-hand side of the system (19 is straightforward. However, before we solve the system, it is possible to apply smoothing to the cell areas. The next step is the simple solution of a coupled tri-diagonal system of linear equations, (19. An extra type of smoothing may be introduced using a predictor-corrector approach for the solution of the system (19. The previous solution is the predictor step, where the matrix A η is computed with the grid coordinates available at j 1. A number of corrector sweeps may be performed computing the derivatives with respect to η included in A η from the last grid coordinates available at j and the grid coordinates at j 1 using first-order differences. The number of corrector sweeps is an input parameter. After the solution of (19 at the line j, extra smoothing may be applied using the Laplace equation. The number of smoothing sweeps and the amount of smoothing are controlled by input parameters and by φ. Although several smoothing alternatives are available, there is no guarantee that the newly generated grid line has positive Jacobian at all the grid nodes. Therefore, the final step of the procedure is to check if there are grid nodes with negative Jacobian. The procedure described above is applied to the required number of grid lines, which is selected from an input variable. In the remaining part of the domain, the grid coordinates are generated with simple one-dimensional interpolations in the marching direction using ξ or η. Both procedures, hyperbolic grid generation and algebraic fill-in, again allow the existence of regions with fixed grid coordinates. 11

12 3 EXAMPLES OF APPLICATION This section presents two examples of application of the grid generation package which are typical of viscous flow calculations around ship hulls: a the external boundary of an H-O grid and b a cross-section of a ship hull at a block boundary between different parts of a volume grid. In both examples the z coordinate is obtained by transfinite interpolation. 3.1 External boundary of an H-O grid In viscous flows calculations around ship hulls with H-O grids, the external domain boundary can be of any shape, provided that suitable boundary conditions can be imposed. Hence the external boundary can be defined by the face edges only. An example of this type of grids is presented in figure 1 for a grid defined by two straight lines and two elliptical arcs. Four grids are plotted in figure 1: two algebraic grids obtained with Hermite and linear basis functions; a grape grid and a grid produced with the orthogonal grid generator. Table 1 presents the mean, (θ m, and maximum, (θ M, deviations from orthogonality in the interior grid nodes and at the four boundaries. The grid coordinate derivatives required to compute the deviation from orthogonality are approximated by second-order central schemes at the interior grid nodes and first-order one-sided schemes at the boundaries. Method Interior I = 1 I = NX J = 1 J = NY (θ m (θ M (θ m (θ M (θ m (θ M (θ m (θ M (θ m (θ M Hermite Linear Grape Orthogonal Table 1: Deviations from orthogonality of the grids on a typical external boundary of viscous ship flow calculation in a H-O grid. Although this is a rather simple geometry there are several interesting details in the grids plotted in figure (1. The algebraic grid with Hermite basis functions presents the smallest deviations from orthogonality at the boundaries. However, at the interior grid nodes, the linear basis functions lead to smaller values of (θ m and (θ M than the Hermite functions. The largest deviations from orthogonality are found in the Grape grid. Nevertheless, (θ m and (θ M are reasonably small and the interior grid line distances vary smoothly. 12

13 Hermite Linear Grape Orthogonal Figure 1: Grids on a typical external boundary of an H-O grid for ship viscous flow calculation. 3.2 Cross-section of a ship hull Next we show an example of a grid problem often encountered in grid generation for dredgers. This kind of ship hulls is quite challenging for the generation of structured grids. Nevertheless, single block or simple multi-block (2 blocks H-O grids are feasible. But even if the final grid is a single-block grid, it is common practice to generate the volume grid in several blocks to improve the grid quality. In that case block faces are moderately curved cross-sections of the ship hull where the present grid generation techniques can be applied efficiently. Typically, these cross-sections have an edge along the ship surface, an edge on the external boundary, an edge on the symmetry plane of the ship and an edge on the still water plane 2. The selected example is a difficult cross section, not only because a headbox is fitted to the main hull which leads to several sharp corners in the cross-sectional shape of the vessel, but also because the block edge along the ship surface is extended along the longitudinal symmetry plane of the ship, implying a grid singularity. Furthermore, the flow problem to be solved requires a high grid resolution close to the ship surface for a proper application of the no-slip condition. Figure 2 presents the grid node distribution nodes are used in circumferential and hull normal direction respectively. For grid generation a combination of the hyperbolic and grape method is applied. The grid is generated in two steps: the first step generates the eleven grid lines closest to the ship surface with the hyperbolic method. For the 2 The grid presented in this paper is used for calculations neglecting the gravity waves. 13

14 remaining 40 grid lines a simple linear interpolation is applied. The final grid is obtained in a second step using the Grape method and keeping the grid lines generated with the hyperbolic method fixed. The four plots included in the figure present a view of the complete grid and three detailed views of the most difficult regions. Figure 2: Grid at a cross-section of a ship hull. At the interior grid nodes the mean and maximum deviation from orthogonality are: (θ m = 8.1 o and (θ M = 50.5 o. These are very reasonable values for such a complicated geometry. The most important feature of the grid is the remarkably good approximation of orthogonality at the ship surface, which is fundamental for accurate viscous flow cal- 14

15 culations, (θ m = 0.6 o and (θ M = 11.3 o. At the external boundary the deviations from orthogonality are also small, (θ m = 0.5 o and (θ M = 1.4 o. Naturally, at the symmetry plane of the ship the deviations from orthogonality have to be much larger due to the grid singularity, (θ M = 80.9 o. Nevertheless, bearing in mind that these values are computed with one-sided differences, the mean value is still acceptable, (θ m = 23.7 o. At the still water plane, (θ m = 10.0 o and (θ M = 37.4 o, as a consequence of the non-orthogonality at the corner between the ship surface and still water plane. 4 FINAL REMARKS In this paper we have presented a grid generation package for making structured grids on surfaces defined by four edge lines, which are typical of block connections in external flow applications. The method uses an algebraic technique to obtain one of the grid point coordinates and a quasi two-dimensional grid generation procedure to obtain the other two coordinates. The algebraic interpolation of the third coordinate is performed in a coordinate system that minimizes its value. The influence of the known coordinate is incorporated in the four methods of two-dimensional grid generation available: elliptic grid generation based on the Grape approach; orthogonal grid generation based on a system of partial differential equations; hyperbolic grid generation; and algebraic grid generation based on transfinite interpolation with Hermite and/or linear basis functions. The available methods allow the definition of regions of fixed grid nodes and so it is possible to combine different techniques in the generation of a grid. This capability is important for the generation of structured grids for viscous flow calculations in complex geometries. The two examples presented in this paper illustrate the capabilities of the grid generation package described. In particular, it was demonstrated that nearly-orthogonal grids can be produced close to the ship surface in a complex geometry including a grid singularity, corners and knuckles. REFERENCES [1] Eça L., Hoekstra M., Windt J. - Practical Grid Generation Tools with Applications to Ship Hydrodynamics - 8 th International Conference in Grid Generation in Computational Field Simulations, Hawaii, June [2] Sorenson R.L. - Grid Generation by Elliptical Partial Differential Equations for a Tri-Element Augmentor-Wing Airfoil - Numerical Grid Generation, Ed. Thompson J.F., North-Holland [3] Thompson J.F. - A General Three-Dimensional Elliptic Grid Generation System on a Composite Block Structure - Computer Methods in Applied Mechanics and Engineering, Vol. 64, March 1987, pp

16 [4] Eça L. - Practical Tools for 2-D, 3-D and Surface Grid Generation. - IST Report D72-10, Instituto Superior Técnico, Lisbon, November [5] Eça L. - 2-D Orthogonal Grid Generation with Boundary Point Distribution Control - Journal of Computational Physics 125, pp , [6] Eça L. - Orthogonal Generation Systems - Chapter 7, Handbook of Grid Generation, CRC Press, [7] Chan W.M. - Hyperbolic Methods for Surface and Field Grid Generation - Chapter 5, Handbook of Grid Generation, CRC Press,