Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 6
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Similarly, representation madebelow the left half of the top of the inner region would use points below the slit. The slit isthus a "black hole" into which coordinate lines from the outer system disappear, to reappearas part of the inner system. The slit here, matched with the top of the inner system, is thenclearly a branch cut, and passage through the slit onto the inner system is simply passageonto a different sheet.Note that the embedded system has its own distinctive species and directions for thecoordinate lines, entirely separate from the outer system. Thus for the inner region thedirections are as follows:while for the outer region they are as shown below:Thus at a point on the upper interface, 12-13, between the systems the lines are as follows:while on the lower interface, 9-6 they are as follows:Thus both coordinates reverse direction at the lower interface although the species iscontinuous, while both the species and directions are continuous across the upper interface.This again corresponds to passage onto a different sheet, for the interface between the innerand outer systems, i.e., the segments 12-13 and 9-6, is actually a branch cut.The points 9(12) and 6(13) here require special notice.
For example, at point 9 thecoordinate line configuration is as follows:The lines through point 9 are as shown belowThere are thus several changes in species and direction at this point. This type of specialpoint embodies the form which always occurs with the slit configuration, shown on p. 26,and occurs here because the embedded region inside the contour 9-6-13-12 is essentiallycontained inside a slit defined by the same set of numbers.The above discussion refers to the slit configuration on p.
41. For the configuration onp. 40, the lines in the outer region are still as diagrammed on p. 43, but the lines in the innerregion now are as follows:The coordinate line species and direction given on p. 43 for the upper interface, 12-13, thusapplies here on the entire interface between the two regions.An alternative treatment of the two special points is to place them inside cells asshown below:This results in a six-sided cell surrounding each of these two points which requires specialtreatment as discussed in Chapter IV.Embedded systems can also be constructed in the block configuration:Here the top of the block, 7-8, in the outer system is re-entrant with the correspondingsegment, 7-8, on a portion of the top of the inner system. The left side of the block, 6-7, andthe bottom of the block, 6-5, are similarly re-entrant with single portions of the top of theinner system.
Finally, the right side of the block, 5-12-8, is re-entrant with two portions, 5-12and 12-8, of the top of the inner system. Points outside one of these segments in one systemare thus located at corresponding positions inside the other segment of the re-entrant pair inthe other system. The slab sides, matched with the top of the inner system, are thus branchcuts between the inner and outer systems.Here the coordinate lines proceed as follows for the outer system:while those for the inner system are the same as before, as shown on p. 42. This means thaton the left and right sides of the block, i.e., segments 6-7 and 5-8, the line directions are asfollows:and on the top and bottom, segments 7-8 and 6-5, the directions are as shown below:There are thus changes in coordinate species and/or direction that are different on each sideof the block.The point 8 (and points 7,6 and 5) are special points of the following form:The lines through the point are as shown below:Here the special points occur in the field instead of on the boundary.An example of a C-type system embedded in another C-type system is given next:Here the conceptual opening is as follows: First, considering the system about the upperbody, we have the following configuration:which, with the body collapsed to a slit, opens to the rectangle in the center of thetransformed region.
Next consider the system about the other body:This opens to a rectangle, with the body flattening to a portion of the bottom, which is fittedto the first rectangle along the segment 11-13. Finally, the outermost portion opens asfollows:which opens to a rectangle which is fitted to the first one along the segment 12-14.Again the embedded region inside the contour 14-12-11-13 can be considered to lieinside a slit. This contour, which forms the interface between the inner and outer systems, isactually a branch cut between the two systems, across which there are discontinuities incoordinate species and directon in the same manner as was discussed above for the previousembedded system.
Points below segment 16-12 coincide with points below segment 17-11 inthis case. Points to the left of segment 15-12, above point 15, are coincident with points tothe right of segment 15-11 below point 15. The slit here is formed of the segments 8-15 and9-15. The coincident points 11 and 12 here must be taken as a point boundary in the physicalregion, i.e., fixed at a specified value. Several special points of the types discussed above arepresent here.An alternative arrangement of the transformed region that corresponds to exactly thesame coordinate system in the physical region is as follows:Here points below segment 3-4, to the left of point 4, coincide with points above segment6-5, to the right of point 5.
When calculations are made on or above the segment 12-14 onthe larger block, points below this segment coincide with points below the correspondingsegment on the smaller block. Similarly, when calculations are made on or below thesegment 13-11 on the larger block points above this segment coincide with points below thecorresponding segment on the smaller block. Finally, points below the segment 7-8, to theleft of point 8, on the smaller block are coincident with points above the segment 10-9, to theright of point 9.This configuration displays explicitly the correspondence of the embedded regioninside the contour 14-12-11-13 to a slit.
Conceptually, coordinate lines from the main systemdisappear into the slit and emerge into the embedded system. These coordinate lines thus arecontinued from the main system onto another sheet representing the embedded system. Thisconcept of embedded systems, with continuation onto another sheet through a slit addsconsiderable flexibility to the grid configurations and is of particular importance withmultiple boundaries and in three dimensions.
The composite structure discussed in Section 4removes much of the coding complexity associated with systems of this type.D. Other ConfigurationsAnother arrangement of cuts, where the species of coordinate changes on a continuousline as the cut is crossed, is illustrated below. The transformed region in this case iscomposed of three blocks connected by the cuts.Here points outside one section are coincident with corresponding points inside the adjacentsection.The coordinate line configuration on the interface on the right side of block A here isas follows:This same type of configuration occurs, in different orientations, on each of the interfaces.These interfaces are branch cuts, so that passage onto the adjacent block amounts to passageonto another sheet in the same manner discussed above.As a final configuration for consideration in two dimensions, the following exampleshows a case with fewer lines on one side of a slab than on the other.
This does notnecessitate the use of different increments of the curvilinear coordinates in the numericalexpressions, because, as has been mentioned, these increments always cancel out anyway.E. Three-dimensional RegionsAll the general concepts illustrated in these examples extend directly to threedimensions.
Interior boundaries in the transformed region can become rectangular solids andplates, corresponding to the slabs and slits, respectively, illustrated above for twodimensions. Examples of three-dimensional configurations can be found in the surveys givenby Ref. [8] and [9].It is also possible to use branch cuts, as illustrated above for two dimensions, to bringthe interior boundaries in the physical region entirely to the exterior boundary of thetransformed region:Physical spaceComputational spaceThe correspondence between the physical and transformed fields can, however, becomemuch more complicated in three dimensions, and considerable ingenuity may be required tovisualize this correspondence.
For instance, the simple case of polar coordinates correspondsto a rectangular solid with two opposing sides having the radial coordinate constant thereon,and two re-entrant sides on which the longitudinal coordinate is constant at 0 and 2 ,respectively (corresponding to the cut). The remaining two sides correspond to the north andsouth polar axes, so that an axis opens to cover an entire side.
There is thus a line, i.e., theaxis, in the physical region that corresponds to an entire side in the transformed region.Three-dimensional grids may be constructed in some cases by simply connectingcorresponding points on two-dimensional grids generated on stacks of planes or curvedsurfaces:It should be noted, however, that this procedure provides no inherent smoothness inthe third direction, except in cases where the stack is formed by an analytical transformation,such as rotation, translation or scaling, of the two-dimensional systems.
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