Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 25

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These equations are nonlinear, and therefore convergence of aniterative procedure requires that the initial guess be within some neighborhood of thesolution. With control functions designed to cause attraction to the boundary, it is possiblefor the coordinate lines to overlap a very sharp convex corner during the course of theiteration, even though a solution with no overlap exists:This problem may be handled by first converging the solution with the coordinatelines artificially locked off the corner. Thus, if newly calculated values of the cartesiancoordinates at a point during the iteration would cause this point to move farther from itspresent location than the distance to the adjacent point on the curvilinear coordinate linerunning to the corner, then these new values are replaced by the average of the coordinatesof the old point and the adjacent point.

After convergence, this lock is removed and finalconvergence to the solution is obtained. Note that this problem does not arise when thecurvilinear coordinate line emanating from the corner is the same as that on the boundary, asin the C-type configuration on p. 30 since then the lines do not wrap around the corner.With very large cell aspect ratio, e.g., for g11>>g22, the generation equation isdominated by the term containing the second derivative along the curvilinear coordinate lineon which the shorter are length lies.

This causes the cartesian coordinates to tend stronglytoward averages of adjacent points on this line during the course of the iteration. Therefore,when strong control functions are used to attract coordinate lines to the boundary in a C-typeconfiguration,the points on the cut are very slow to move from the initial guess during the iteration.Convergence in such a case is very slow, and it is expedient to artificially fix the points onthe cut as if it were a boundary. This will cause the coordinate lines crossing the cut to havediscontinuous slopes at the cut, but since the spacing along these crossing lines is very small,the error thus incurred in difference solutions on the coordinate system is small.B. Control functionsSeveral types of control functions have been discussed in Section 2 which serve tocontrol the coordinate line spacing and orientation in the field. Most of these functions areset before the solution algorithm begins, either directly through input or by calculation fromthe boundary point distributions that have been input.For the attraction to other coordinate lines/points, described in Section 2A, it isnecessary to input the indices of the lines/points, i.e., the i and i of Eq.

(30), to whichattraction is to be made. In the case of attraction to lines, the line is identified by the singleindex which is constant thereon, while a point requires the specification of two indices (in2D, with analogous generalization to 3D). The attraction amplitude and decay factor in Eq.(30) must also be input for each line/point. The control functions are then calculated at each(30) must also be input for each line/point. The control functions are then calculated at eachpoint in the field ( , ) by performing the summations in Eq.

(30), those summations beingover all the attraction lines/points that have been input. As noted in Section 2A, thesesummations must also extend over some lines/points on other sheets across branch cuts insome cases.This type of control function was used in the original TOMCAT code (cf. Ref. [1]),but is not really suitable as a primary means of control function definition because it onlyprovides control--not control to achieve a specified spacing distribution, since theappropriate values of the various parameters involved can only be determined byexperimentation.

This form does, however, still serve as a useful addition to other types ofcontrol, in that it allows particular ad hoc concentrations or adjustments of line spacing andorientation to be made. This can be particularly useful near the special points discussed inChapter II where the grid line configuration departs locally from the usual simple coordinateline intersections.The attraction to lines/points in space, implemented through Eq. (31), requires inputsimilar to that just described, except that here the location of the attraction lines must bedefined in the physical region by inputing a set of points along the line sufficient for itsdefinition in discrete form. For attraction to a point a unit vector must also be input with eachpoint.

Again, attraction amplitude and decay factors must be input.More important is the evaluation of the control functions from the boundary pointdistribution that has been input, as described in Section 2E. With the point distributionspecified on a boundary line, the control functions on this line can be evaluated from Eq.(45)-(47). Here the derviatives in Eq.(46) are best calculated from Eq.

(36) and (37), usingsecond-order, central difference expressions along the line:.) The curvature terms given by Eq. (47), if included, must either be(Recall thatinput at each point on the line, or, as is more likely, must be interpolated from values on theends of the line. In this latter case, the m and n derivatives are off the line and areevaluated from the point distribution on the other coordinate lines intersecting the line ofinterest at its ends, using first-order one-sided difference expressions along these intersectinglines:One-dimensional linear interpolation in l then serves to define the curvature term quantitiesat each point on the line of interest.

Recall that it is the entire curvature term, rather than theindividual vectors involved, that should be interpolated.This evaluation determines the Pl control function on a boundary line on which lvaries. Such an evaluation can be made on each edge of a surface, corresponding to one faceof a block in three dimensions (cf. Section 6 of Chapter II). If it is desired to generate atwo-dimensional grid on this surface, control functions on the surface can be evaluated byinterpolation from the function values on the edges, using linear interpolation between thetwo edges on which i is constant to evaluate P j, and between the two edges on which j isconstant to evaluate Pi (cf. the figure on p.

227). With the control functions thus defined onthe surface, a two-dimensional grid on the surface can now be generated using a surface gridgeneration system described as in Section 3. If the surface is a portion of the physicalboundary, then a parametric definition of the surface will need to be input, so that the systemdefined by Eq.

(107) can be applied. If, however, the surface is simply an interface betweenblocks, then its position is arbitrary and either a plane two-dimensional generation system,such as Eq. (20),can be used, or surface curvature values could be input at each point on thesurface and the surface system Eq. (82) used. The former is the more likely choice.With the grid points on all the block faces defined, either by surface generationsystems or by direct input, two control functions on each face can be evaluated from thesurface point distribution using Eq. (52).

Here the m and n derivatives are along coordinatelines on the surface and thus can be represented by second-order central differences betweenpoints on the surface:(Recall that ( m) m can be expanded tom m for evaluation.) The l-derivatives are offthe surface and must either be specified by input at each point on the surface, or, as is morelikely, must be interpolated from values evaluated along the coordinate lines intersecting thesurface at its edges using first-order, one-sided difference expressions. The interpolationwould here properly be two-dimensional transfinite interpolation discussed in Chapter VIII.This then serves to determine the two control functions P j and Pk, on a surface onwhich i is constant (cf.

the figure on p. 226), so that each control function will be definedon four faces of the block. Transfinite interpolation among these four faces then determinesthis control function in the interior of the block (cf. p. 227).Another possibility is to evaluate the radius of curvature,, of the surface and toreplace the curvature terms in Eq. (45) with(cf. Exercise 9). Here the radius ofcurvature should be interpolated unidirectionally between facing surfaces, and the sametwo-directional transfinite interpolation used for the first term of the control function shouldbe used for the spacing.Still another approach is to solve the three generation system equations for the threecontrol functions at each point using an algebraic grid, but with the off-diagonal metricelements set to zero.

This will produce a grid which will have a greater degree of smoothnessand orthogonality than the algebraic grid and yet has the same general spacing distribution.Here the result of the Computer Exercise 6 in Appendix C must be considered since thealgebraic grid influences the spacing distribution.In generation systems that iteratively adjust the control functions during the course ofthe solution of the difference equations (Section 2F) to achieve a specified spacing and angleof intersection, e.g., orthogonality, at the boundary, this spacing and intersection angle areinput for each boundary point and it is, of course, not necessary to calculate the controlfunctions beforehand.

Several references to discussion of such systems are given in Ref. [5].The GRAPE code is based on this approach, cf. the users manual Ref. [24].C. Surface generation systemsA boundary surface in the physical region will typically be input by giving thecartesian coordinates of points on a series of cross-sections, or other set of space curves:These input points may then be splined to provide a functional definition of these curves.These curves are then parameterized in terms of normalized arc length thereon, i.e., so thatthis normalized parameter varies over the same range on each curve.This normalized arc length then provides one parametric coordinate on the surface. The othercoordinate is defined by connecting points at the same value of the first coordinate on thesuccessive curves, again using a spline fit:This second coordinate is then also expressed in terms of normalized arc length(On a sphere these two parametric surface coordinates could correspond to longitude andlatitude, the latter arising from the cross-sections and the former from the connectingthereof.)There are other techniques of surface definition and parameterization, cf.

especially workson computer-aided design, but the above decription is representative. The end result of thisstage in any case is (u,v), i.e., the cartesian coordinates on the surface in terms of twosurface parametric coordinates.The two parametric coordinates (u,v) used to define the surface can also be adopted asthe curvilinear coordinates defining the surface grid. However it is more likely that thesecoordinates were selected for convenience of input definition of the surface than for thedefinition of an appropriate grid thereon. This is particularly true when two such intersectionsurfaces, e.g., a wing-body, are input, each with its own set of parametric coordinates.Therefore, the surface grid generation system defined by Eq.

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