Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 23
Текст из файла (страница 23)
The two control functions needed on the surface for this purpose can be determinedby interpolation from values evaluated on the four edges of the surface:Eq. (45) allows the control function Pi to be evaluated on the edges on which i varies, i.e.,the top and bottom edges in the figure. This control function on the surface can then beevaluated by interpolation between these two edges:(55)Both of the necessary control functions on the surface can thus be determined from thespecified boundary point distributions on the edges of the surface.F.
Iterative determinationAs noted above, a second-order elliptic generation system allows either the pointlocations on the boundary or the coordinate line slope at the boundary to be specified, butnot both. It is possible, however, to iteratively adjust the control functions in the generationsystem of the Poisson type discussed above until not only a specified line slope but also thespacing of the first coordinate surface off the boundary is achieved, with the point locationson the boundary specified.In three dimensions the specification of the coordinate line slope at the boundaryrequires the specification of two quantities, e.g., the direction cosines of the line with twotangents to the boundary.The specification of the spacing of the first coordinate surface off the boundary requires onemore quantity,and therefore the three control functions in the system (12) are exactly sufficient to allowthese three specified quantities to be achieved, while the one boundary condition allowed bythe second-order system provides for the point locations on the boundary to be specified.The capability for achieving a specified coordinate line slope at the boundary makes itpossible to generate a grid which is orthogonal at the boundary, with a specified pointdistribution on the boundary, and also a specified spacing of the first coordinate surface offthe boundary.
This feature is important in the patching together of segmented grids, withslope continuity, as discussed in Chapter II, for embedded systems.An iterative procedure can be constructed for the determination of the controlfunctions in two dimensions as follows (cf. Ref. [25]): Consider the generation system givenby Eq.
(20). On a boundary segment that is a line of constant we haveandknown from the specified boundary point distributionalso, the spacing off this boundary, is specifiedas is the condition of orthogonality at the boundary, i.e.,,, together with the conditionBut specification ofprovides two equations for the determination of x and y in terms of the already knownvalues of the xand y . Thereforeis known on the boundary.Because of the orthogonality at the boundary, Eq. (20) (Eq.
(23) is used instead in Ref.[25]) reduces to the following equation on the boundary:Dottingandinto this equation, and again using the condition of orthogonality,yields the following two equations for the control functions on the boundary:(56a)(56b)All of the quantities in these equations are known on the boundary except. (On aboundary that is a line of constant , the same equations for the control functions result, butnow withthe unknown quantity.)The iterative solution thus proceeds as follows:(1). Assume values for the control function on the boundary.(2).
Solve Eq. (20) to generate the grid in the field.on -line boundaries, andon -line boundaries, from the(3). Evaluateresult of Step (2), using one-sided difference representations. Then evaluate the controlfunctions on the boundary from Eq. (56).Evaluate the control functions in the field by interpolation from the boundary values.Steps (2) and (3) are then repeated until convergence.This type of iterative solution has been implemented in the GRAPE code of Ref. [24] [26], some results of which are shown below:These grids are orthogonal at the boundary, and the spacing of the first coordinate surface(line in 2D) off the boundary is specified at each boundary point, the locations of which arespecified.An iterative solution procedure for the determination of the three control functions forthe general three-dimensional case can be constructed as follows.
Eq.(52) gives the twocontrol functions, Pm and Pn, for a coordinate surface on which l is constant (1,m,n cyclic)for the case where the coordinate line crossing the surface is normal to the surface. Takingthe projection of the generation equation (12) on the coordinate line along which l varies,we have on this same surface,since glm = gln = 0 on the surface.
Using the relations for the metric components obtained forthis situation in Section D, this equation reduces to(57)Since the coordinate line intersecting the surface is to be normal to the surface, wemay write(58)sinceusing the identity (III-9). Eq. (57) can then be writtenWith the spacing along the coordinate line intersecting the surface specified at thesurface, we haveknown on the surface. Since all the quantities subscripted m orn in Eq. (52) and (59) can be evaluated completely from the specified point distribution onthe surface, we then have all quantities in these equations for the three control functions onthe surface known except for (gll) l and ( l) l.
These two quantities are not independent,and using Eq. (58), we have(60)Recall also that ( l) =ll.Therefore, with the control functions in the field determined from the values on theboundary by interpolation, as discussed in the preceeding section, Eq. (52) and (59) can beapplied to determine the new boundary values of the control functions in terms of the newvalues of ( l) l in an iterative solution. Upon convergence, the coordinate system then willhave the coordinate lines intersecting the boundary normally at fixed locations and with thespecified spacing on these lines off the boundary.A similar iterative determination of the two control functions for use in generating acoordinate system on a surface can be set up using only Eq.
(52), and the analogous equationfor Pn, with the first term either omitted, amounting to the assumption of vanishing curvatureof the crossing line at the surface or with this term considered as specified on the surface,either directly or by interpolation from the edges of the surface (the edges assumed to be oncoordinate lines) to provide two equations for the two control functions, Pm and Pn, on theseedges.
Here the two dimensional surface coordinate system is to be orthogonal on thebounding edges of the surface with the spacing off the edges, and the point distributionthereon, specified on these edges.Since the coordinate system is to be orthogonal on the edges, gmn = 0 there so that thelast term in the bracket in Eq. (52) vanishes. Eq. (52) then reduces to the followingexpression(61)and the analogous equation for Pn is(62)These equations can also be written(63)and(64)If the point distribution is specified along an edge on whichand ( mmm varies,then gmm,can all be calculated on this edge. The specification of the spacing from thism,edge to the first coordinate line off the edge determines gnn on this edge. Also, because ofthe orthogonality on the edge, we have(65)whereis the unit normal to the surface.
Note thatwill vary along the edge if thesurface is curved. Since the surface normal, , will be known, all quantities in Eq. (63) and(64) are known except (gnn) n and ( n n. These two quantities are not independent and,in fact,(66)On edges along whichnvaries, Eq. (65) and (66) are replaced by(67)and(68)and it is (gmm)mand ( m)mthat are not known.The iterative solution then proceeds as described above, with the new controlfunctions being determined from Eq. (63) and (64), together with Eq.
(65) and (66) on edgesalong which m varies, or with Eq. (67) and (68) on edges along which n varies.3. Surface Grid Generation SystemsThe grid generation systems discussed in the preceeding sections of this chapter havebeen for the generation of curvilinear coordinate systems in general three-dimensionalregions. Two-dimensional forms of these systems serve to generate curvilinear coordinatesystems in general two-dimensional regions in a plane.
It is also of interest, however, togenerate two-dimensional curvilinear coordinate systems on general curved surfaces.Here the surface is specified, and the problem is to generate a two-dimensional grid onthat surface, the third curvilinear coordinate being constant on the surface. Theconfigurations of the transformed region will be the same as described in Chapter II fortwo-dimensional systems in general, i.e., composed of contiguous rectangular blocks in aplane, with point locations and/or coordinate line slopes specified on the boundaries. Theseboundaries now correspond to bounding curves on the curved surface of the physical region.The problem is thus essentially the same as that discussed above for two-dimensional planeregions, except that the curvature of the surface must now enter the partial differentialequations which comprise the grid generation system.As for general regions, algebraic generation systems based on interpolation can beconstructed, and such systems are discussed in Chapter VIII.
The problem can also beconsidered as an elliptic boundary-value problem on the surface with the same generalfeatures discussed above being exhibited by the elliptic generation system.A. Surface grid generationAn elliptic generation system for surface grids can be devised from the formulae ofGauss and Beltrami, of. Ref. [27]. Some related, but less general, developments are noted inRef. [9] and [5]. The starting point is the set formed by the formulae of Gauss for a surface,which for a surface,= constant, ( = 1,2, or 3) are given by Eq. (34) of Appendix A:(69)where the variation of the indicies , and is over the two coordinate indices differentfrom .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
