Thompson, Warsi, Mastin - Numerical Grid Generation, страница 7

An example of suchan analytical transformation of two-dimensional systems is the construciton of athree-dimensional grid for a curved pipe by rotating and translating (and scaling if thecross-sectional area of the pipe varies) two-dimensional grids generated for the pipecross-section so as to place these transformed two-dimensional grids normal to the pipe axisat successive locations along the axis:Another example is the rotation of a two-dimensional grid about an axis to produce anaxi-symmetric grid:4. Composite GridsAll of the above concepts can be incorporated in a single framework, and the codingcomplexity can be greatly reduced, by considering the physical field to be segmented intosub-regions, bounded by four (six in 3D) generally curved sides, within each of which anindividual coordinate system is generated.

The overall coordinate system, covering the entirephysical field, is then formed by joining the sub-systems at the sub-region boundaries. Thedegree of continuity with which this juncture is made is a design consideration in regard tothe mode of application intended for the resulting grid.This segmentation concept is illustrated in the figure below.The locations of the interfaces between the sub-regions in the physical region are, of course,arbitrary since these interfaces are not actual boundaries. These interfaces might be fixed,i.e., the location completely specified just as in the case of actual boundaries, or might be leftto be located by the grid generation procedure. Also the coordinate lines in adjacentsub-regions might be made to meet at the interface between with complete continuity:---Interfacewith some lesser degree of continuity, e.g., continuous line slope only:---Interfaceor with a discontinuity in slope:or perhaps not to meet at all:Naturally, progressively more special treatment at the interface will be required in numericalapplications as more degrees of line continuity at the interface are lost.

Procedures forgenerating segmented grids with various degrees of interface continuity are discussed later,and conservative interface conditions are given in Ref. [52], [53].Now, with regard to placing these concepts in the framework of segmentation, thesides of an individual subregion (called a "block" hereafter) can be treated as boundaries onwhich the coordinate points, and/or the coordinate line intersection angles, are specified, justas is done for actual boundaries, or a side may be treated as one member of a pair ofre-entrant boundaries, i.e., one side of a branch cut in the physical region across whichcomplete continuity is established. The other member of the pair may be another side (orportion thereof) of the same block or may be all (or part of) a side of an adjacent block in thephysical field. Recall that it is not necessary for a coordinate to remain of the same speciesacross a re-entrant boundary, since the passage is onto a different sheet.

This can introducesome coding complexity, but the treatment is straightforward, and in fact the coding can begreatly simplified by using an extra layer of points surrounding each block as is discussed inSection 6.Some of the general concepts have been embodied in the two-dimensional codediscussed in Ref. [19] and in three recent three-dimensional codes, Ref. [13] Ref.

[14], and[51].A. Simply-connected RegionsThe first L-shaped simply-connected configuration on p. 21 can be interpreted asbeing composed of three blocks, with the sides of adjacent blocks forming pairs of re-entrantboundaries:or two blocks with a portion of a side of one block re-entrant with an entire side of anotherblock:Here, and in the examples to follow, solid lines correspond to physical boundaries, while thedashed lines correspond to the interfaces between the blocks. The dashed arrows indicate thelinkage between the interfaces.

(Obviously, any single block can be broken into any numberof blocks connected by re-entrant boundaries across adjacent sides.) In contrast, theL-shaped configuration on p. 22 corresponds to the use of a single block. Similarly, theconfiguration on p. 24 can be formed with three blocks:while the first configuration for the same boundary on p. 25 is formed with a single block.The slit configuration on p. 25 can be formed of three blocks:or two blocks with only a portion of the adjacent sides of two blocks forming a re-entrantboundary:B. Multiply-connected RegionsThe configuration with a single cut shown on p. 29 corresponds to the use of a singleblock with the left and right sides here being the members of a pair of re-entrant boundaries:The multiply-connected slab configuration on p.

27 can be broken into four blocks:Other decompositions should also be immediately conceivable. The slit configuration on p.28 can be formed with two blocks, again with only portions of adjacent sides serving asre-entrant boundaries:or into four blocks, with entire sides as re-entrant boundaries in all cases:The double-body region on p. 34 opens to a single block as shown there, with portionsof sides as re-entrant boundaries.

A five block configuration would use only entire sides asre-entrant boundaries, however:There is no real advantage, however, to the five-block system here.C. Embedded RegionsThe segmentation concept is most useful in the construction of embedded coordinatesystems. For instance, the system on p. 40 can be considered to be formed of three blocks asfollows:Here portions of adjacent sides of the two larger blocks are re-entrant with each other, whileeach of the remaining portions of these sides is re-entrant with half of one side of the smallerblock. The left and right sides of the smaller block are re-entrant with each other.

Thisconfiguration could also have been constructed with eight blocks:with only entire sides being involved in re-entrant pairs as shown.With embedded systems the coordinate species often changes as the re-entrantboundary is crossed. These systems also show that the blocks need be physically adjacentonly in the physical field, and it is in this sense that "adjacent" is always to be interpreted.The transformed (computational) field should always be viewed as only a bookkeepingstructure.

Various constructions are possible for the configurations on p. 48 and 50, and atwo block structure was actually used on p. 50. A further example follows:D. Three-dimensional RegionsFor general three-dimensional configurations, it is usually very difficult to obtain areasonable grid with the entire physical region transformed to a single rectangular block.

Abetter approach in most cases is to segment the physical region into contiguous sub-regions,each bounded by six curved surfaces, with each sub-region being transformed into arectangular block. An individual grid is generated in each sub-region:These sub-region grids are patched together to form the overall grads, as in thetwo-dimensional cases discussed above. Examples of the use of this segmentation in threedimensions are found, in particular, in Ref. [11] and [12].

Others are noted in the surveygiven by Ref. [9].As noted above, complete continuity can be achieved at the sub-region interfaces bynoting the correspondence of points exterior to one sub-region with points interior toanother. The necessary bookkeeping can be accomplished, and the coding complexity can begreatly reduced, by using an auxiliary layer of points just outside each of the six sides of thecomputational region, analogous to the procedure mentioned above for two dimensions. Acorrespondence is then established in the code between the auxiliary points and theappropriate points just inside other sub-regions.

This approach has recently beenincorporated in an internal region code, Ref. [13], and in two codes for general regions, Ref.[14] and [51]. This is discussed in more detail in Section 6.General three-dimensional regions can be built up using sub-regions as follows: First,point distributions are specified on the edges of a curved surface forming one boundary of asub-region:and a two-dimensional coordinate system is generated on the surface:When this has been done for all surfaces bounding the sub-region, the three-dimensionalsystem within the sub-region is generated using the points on the surface grids as boundaryconditions:In three dimensions it is possible for a line, e.g., a polar axis, in the physical region tomap to an entire side of the computational region as in the illistration below, where the axiscorresponds to the entire left side of the block:The system illustrated here could be one of several identical blocks joining together to forma complete system around the axis.It is illustrated by an exercise that the occurence of a polar axis can be avoided, andthis facilitates the construction of a block structure.