Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 8
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Thus a surface grid, having eight"corners", analogous to the four "corners" on the circle in the 2D grid on p. 23, can beconstructed on the surface of a sphere. This serves much better than a latitude-longitude typesystem for joining to adjacent regions. Similarly, the use of the four "corner" system, ratherthan a cylindrical system, in a circular pipe allows T-sections and bifurcations to be treatedeasily by a composite structure, c.f. Ref. [13].Generally, grid configurations with polar axis should not be used in composite gridstructures.E. Overlaid GridsAnother approach to complicated configurations is to overlay coordinate systems ofdifferent types, or those generated for different sub-regions:Here an appropriate grid is generated to fit each individual component of the configuration,such that each grid has several lines of overlap with an adjacent grid.
Interpolation is thenused in the region of overlap when solutions are done on the composite grid, with iterationamong the various grids. This approach has the advantage of simplicity in the gridgeneration, in that the various sub-region grids are only required to overlap, not to fit.However, there would appear to be problems if regions of strong gradients fall on theoverlap regions. Also the interpolation may have to be constructed differently for differentconfigurations, so that a general code may be hard to produce.
Some applications of suchoverlaid grids are noted in Ref. [5].5. Branch CutsAs has been noted in the above discussion of transformed field configurations, it ispossible for discontinuities in coordinate species and/or direction to occur at branch cuts, inthe sense of passage onto another sheet. Continuity can be maintained, however, byconceptually remaining on the same overlapping sheet as the cut is crossed. All derivativesthus do exist at the cut, but careful attention to difference formulations is necessary torepresent derivatives correctly across the cut. Although the correct representation can beaccomplished directly by surrounding the computational region with an extra layer of points,as is discussed in Section 6, it is instructive to consider what is required of a correctrepresentation further here.A.
Point CorrespondencePoints on re-entrant boundaries in the transformed region, i.e., on branch cuts in thephysical region, are not special points in the sense used above. Points on re-entrantboundaries, in fact, differ no more from the other field points than do the points on the 0 and2 lines in a cylindrical coordinate system. Care must be taken, however, to identify theinterior points coinciding with the extensions from such points beyond the field in thetransformed space. This correspondence was noted above in each of the configurationsshown above, being indicated by the dashed connecting lines joining the two members of apair of re-entrant boundaries.
There are essentially four types of pairs of re-entrantboundaries, as illustrated in the following discussion of derivative correspondence. In theseillustrations one exterior point, and its corresponding interior point, are shown for each case.The converse of the correspondence should be evident in each configuration.For the configurations involving a change in the coordinate species at the cut, not onlymust the coordinate directions be taken into account as the cut is crossed, but also thecoordinate species may need to be interpreted differently from that established across the cutin order to remain on the same sheet as the cut is crossed.
For example, points on an -linebelonging to section A in the figure on p. 52, but located outside the right side of this region,are coincident with points on a -line of region B at a corresponding distance (in thetransformed region) below the top of this region.B. Derivative CorrespondenceCare must be taken at branch cuts to represent derivatives correctly in relation to theparticular side of the cut on which the derivative is to be used.
The existence of branch cutsindicates that the transformed region is multi-sheeted, and computations must remain on thesame sheet as the cut is crossed. Remaining on the same sheet means continuing thecoordinate lines across the cut coincident with those of the adjacent region, but keeping thesame interpretation of coordinate line species and directions as the cut is crossed, rather thanadopting those of the adjacent region. As noted above, points outside a region across a cutthe transformed space are coincident with points inside the region across the other memberof the pair of re-entrant boundary segments corresponding to the cut in the transformedspace.
The positive directions of the curvilinear coordinates to be used at these points insidethe region across the other member of the pair in some cases are the same as the defineddirections there, but in other cases are the opposite directions. As noted above, thecoordinate species may change also.For cuts located on opposing sides of the transformed region, the proper form issimply a continuation across the cut. Thus in the configuration on p. 29, with a computationsite on the right side of the transformed region, i.e., on the upper side of the cut in thephysical plane, we have points to the right of the site (below the cut in the physical plane)coinciding with points to the right of the left side of the transformed region (below the cut inthe physical plane) as noted above.
When -derivatives and -derivatives for use outside theright side of the transformed region are represented inside the left side, the positivedirections of and to be used there are to the right and upward, respectively, as isillustrated below. (In this and the following figures of the section, the dotted arrows indicatethe proper directions to be used at the interior points coincident with the required exteriorpoints, i.e., on the same sheet across the cut, while solid arrows indicate the locallyestablished directions for the coordinate lines, i.e., on a different sheet.)With the two sides of the cut both located on the same coordinate line, i.e., on thesame side of the transformed region as in the configuration on p.
30, however, the situationis not as simple as the above. In this case, when the computation site is on the left branch ofthe cut in the transformed region (on the lower branch in the physical region), the pointsbelow this boundary in the transformed region coincide with points located above the rightbranch of the cut (above the cut in the physical region) at mirror-image positions, as hasbeen noted earlier. The -derivatives for use at such points below the left branch thus mustbe represented at these corresponding points above the right branch.
The positive directionof for purposes of this calculation of derivatives above the right branch, for use below theleft branch, must be taken as downward, not upward. There is a similar reversal in theinterpretation of the positive direction of . This is in accordance with the discussion on p.31. These interpretations are illustrated below:In the configuration on p. 40, where two sides of a cut face each other across a void,there is really no problem of interpretation, since the directions in the configuration aretreated simply as if the void did not exist. This correspondence is as shown below:In all cases the interpretation of the positive directions of the curvilinear coordinatesmust be such as to preserve the direction in the physical region, i.e., on the same sheet, as thecut is crossed. In the cases where the coordinate species change at the cut, the situation iseven more complicated.
Thus on the left side, segment 6-7, of the slab interface between theinner and outer systems in the embedded configuration on p. 46, where the species changesacross the cut, the correspondence is as follows:Thus, when a -derivative is needed outside the outer sytem, for use inside the left slabinterface, the positive -direction at the corresponding points inside the inner system mustbe taken to coincide with the negative -direction of the inner system. Similarly, an-derivative would be represented taking the positive -direction to coincide with thepositive -direction of the inner system. In an analogous fashion, a -derivative neededoutside the inner system, for use inside the segment 6-7, would be represented at thecorresponding point inside the outer system, i.e., to the left of the left slab side, but with thepositive -direction taken to be the positive -direction of the outer system.
An -derivativewould be represented similarly, taking the positive -direction to be the negative -directionof the outer system.A -derivative to the left of the right side of the slab in the outer system would berepresented below segment 12-5 or 8-12, as the case may be, but with the positive-direction taken to be the positive -direction of the inner system. Similarly, an-derivative would be represented taking the positive -direction to be the negativedirection of the inner system. For a -derivative above the bottom of the slab in the outersystem, the correspondence is to below the segment 5-6 inside the inner system, with thepositive -direction taken to be the negative -direction of the inner system. The-derivative is represented taking the positive -direction to be the negative -direction ofthe inner sytem.
Finally, for derivatives below the top of the slab in the outer system, thecorrespondence is to below the segment 7-8 inside the inner system, with both the speciesand direction of the coordinates unchanged.The proper interpretation of coordinate species and direction across branch cuts for allthe other configurations discussed above can be inferred directly from these examples. Aconceptual joining of the two members of a pair of re-entrant boundaries in accordance withthe dashed line notation used on the configurations given in this chapter will always showexactly how to interpret both the coordinate species and directions in order to remain on thesame sheet and thus to maintain continuity in derivative representation across the cut.Examples of the proper difference representation are given in the following section.
Thecomplexities of this correspondence can be completely avoided, however, by usingsurrounding layers around each block in a segmented structure as discussed in the nextsection.6. ImplementationAs discussed above, the transformed region is always comprised of contiguousrectangular blocks by construction. This occurs because of the essential fact that one of thecurvilinear coordinates is defined as constant on each segment of the physical boundary.Consequently, each segment of the physical boundary corresponds to a plane segment of theboundary of the transformed region that is parallel to a coordinate plane there.
The completeboundary of the transformed region then is composed of plane segments, all intersecting atright angles. Although the transformed region may not be a simple six-sided rectangularsolid, it can be broken up into a contiguous collection of such solids, here called blocks.i cancel from all differenceNow it is noted in Chapter III that the incrementsexpressions, and that the actual values of the curvilinear coordinates i are immaterial. Thecoordinates in the transformed region can thus be considered simple counters identifying thepoints on the grid. This being the case, and the transformed region being comprised of acollection of rectangular blocks, it is convenient to identify the grid points with integervalues of the curvilinear coordinates in each block, and thus to place the cartesiancoordinates of a grid point in ijk, where the subscripts (i,j,k) here indicate position ( i, 2,3)in the transformed region. (In coding, a fourth index may be added to identify theblock.) In each block, the curvilinear coordinates are then taken to vary as i = 1,2,...,Ii overthe grid points, where Ii is the number of points in the i-direction.
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