Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 5
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Both coordinate lines experience slope discontinuities at this point.The slit configuration is as shown below:(An obvious varition would be to have the slit vertical.) In this slit configuration, the point 5and 6 are special points of the form shown on p. 26 characteristic of the slit configuration,and will require special treatment in difference solutions.The transformed region could, however, be made simply-connected by introducing abranch cut in the physical region as illustrated below:Conceptually this can be viewed as an opening of the field at the out and then a deformationinto a rectangle:Here the coincident coordinate lines 1-2 and 4-3 form a branch cut, which becomesre-entrant boundaries on the left and right sides of the transformed region.
All derivatives arecontinuous across this cut, and points at a horizontal distance outside the right-side boundaryin the transformed region are the same as corresponding points at the same horizontaldistance on the same horizontal line inside the left-side boundary, and vice versa.
(In alldiscussions of point correspondence across cuts, "distance" means distance in thetransformed region). In coding, the use of a layer of points outside each member of a pair ofre-entrant boundaries in the transformed region holding values corresponding to theappropriate points inside the other boundary of the pair avoids the need for conditionalchoices in difference representations, as discussed in Section 6 of this chapter.Boundary values are not specified on the cut.
(This cut is, of course, analogous to thecoincident 0 and 2 lines in the cylindrical coordinate system discussed above.) At the cutwe have the following coordinate line configuration, as may be seen from the conceptionaldeformation to a rectangle:so that the coordinate species and directions are both continuous across the cut.This type of configuration is often called an O-type. Another possible configuration isas shown below, often called a C-type:Opening the field at the cut we have, conceptually,with 1-2-3-4 to flatten to the bottom of the rectangle.
Here the two members of the pair ofsegments forming the branch cut are both on the same side of the transformed region, andconsequently points located at a vertical distance below the segment 1-2, at a horizontaldistance to the left of point 2, coincide with points at the same vertical distance above thesegment 4-3, at the same horizontal distance to the right of point 3. The point 2(3) is aspecial point of the type shown on p. 26 for slit configurations.The coordinate line configuration at the cut in this configuration is as follows:where it is indicated that varies to the right on the upper side of the cut, but to the left onthe lower side.
The direction of variation of also reverses at the cut, so that although thespecies and slope of both lines are continuous across the cut, the direction of variationreverses there.It is possible to pass onto a different sheet across a branch cut, and discontinuities incoordinate line species and/or direction occur only when passage is made onto a differentsheet. It is also possible, however, to remain on the same (overlapping) sheet as the cut iscrossed, in which case the species and direction are continuous, and this must be theinterpretation when derivatives are evaluated across the cut, as is discussed in Section 5 tofollow.
These concepts are illustrated in the following figure, corresponding to the C-typeconfiguration given on p. 30:In the present discussion of configurations, the behavior of the coordinate lines across the cutwill always be described in regard to the passage onto a different sheet, since this is in factthe case in codes. It is to be understood that complete continuity can always be maintainedby conceptually remaining on the same sheet as the cut is crossed. Much of this complexitycan, however, be avoided with the use of an extra layer of points surrounding thetransformed region as will be discussed in Section 6.Although in principle any region can be transformed into an empty rectangular blockthrough the use of branch cuts, the resulting grid point distribution may not necessarily bereasonable in all of the region. Furthermore, an unreasonable amount of effort may berequired to properly segment the boundary surfaces and to devise an appropriate pointdistribution thereon for such a transformation.
Some configurations are better treated with acomputational field that has slits or rectangular slabs in it.Regions of higher connectivity than those shown above are treated in a similarmanner. The level of connectivity may be maintained as in the following illustration:Here one slit is made horizontal and one vertical just for generality of illustration. Bothcould, of course, be of the same orientation.
Slabs, rather than slits, could also have beenused. The example has three bodies.With the transformed region made simply-connected we have, using two branch cuts,a configuration related to the 0-type shown above for one internal boundary:The conceptual opening here is as follows:with segment 2-3-4-5-6-7 opening to the bottom. Here the pairs of segments (1-2,8-7) and(3-4,6-5) are the branch cuts, which form re-entrant boundaries in the transformed region asshown.
In this case, points outside the right side of the transformed region coincide withpoints inside the left side, and vice versa. This cut is of the form described on p. 30, whereboth the coordinate species and direction are continuous across the cut. Points below thebottom segment 3-4, to the left of point 4, coincide with points above the bottom segment6-5 to the right of point 5. This cut is of the form discussed on p. 31, for which thecoordinate species is continuous across the cut but the direction changes there.
There are anumber of other possibilities for placement of the two cuts on the boundary of thetransformed region, of course, some examples of which follow.It is not necessary to reduce the connectivity of the region completely; rather, a slit or slabcan be used for some of the interior boundaries, while others are placed on the exteriorboundary of the transformed region.One other possibility in two dimensions is the use of a preliminary analyticaltransformation of infinity to a point inside some interior boundary, with the coodinatesresulting therefrom replacing the cartesian coordinates in the physical region. The gridgeneration then operates from these transformed coordinates rather than from the cartesiancoordinates. This typically gives a fine grid near the bodies, but may give excessively largespacing away from the body.Thus, for example, if points on the two physical boundaries shown beloware transformed according to the complex transformationz’ = 1/zwhere z = x+iy and z’ = x’+iy’, infinity in the x,y system will transform to the origin in thex’, y’ system, as shown below.Then with the grid generated numerically from the x’, y’ system the following configurationresults:References to the use of this approach are made in the survey of Ref.
[1]. Somewhat relatedto this are various two-dimensional configurations which arise directly from conformalmapping, cf. Ref. [6] and the survey of Ives on this subject, Ref. [7]. (Conformal mapping isdiscussed in Chapter X.)C. Embedded RegionsIn more complicated configurations, one type of coordinate system can be embeddedin another. A simple example of this is shown below, where an 0-type system surrounding aninternal boundary is embedded in a system of a more rectangular form, using what amountsto a slit configuration.The conceptual opening of this system is best understood in stages: First consideringonly the embedded 0-type system surrounding the interior boundary, we have the regioninside the contour 12-13-6-9 opening as follows:This then opens to the rectangular central portion of the transformed region shown above,with the inner boundary contour 8-7-8 collapsing to a slit.
The rest of the physical regionthen opens as shown below:These two regions then deform to rectangles and are fitted to the top and bottom of therectangle corresponding to the inner system along the contours 12-13 and 9-6 as shown.Here points at a vertical distance below the segment 11-12 are coincident with pointsat the same vertical distance below the segment 10-9 on the same vertical line, and viceversa, with similar correspondence for the pair of segments 13-14 and 6-5. Points at ahorizontal distance to the left of the segment B-12, at a vertical distance above point 8,coincide with points at the same horizontal distance to the right of the segment 8-9, at thesame vertical distance below point 8.
Similar correspondence holds for the pair 7-13 and 7-6.Boundary values are specified on the slit 8-7.The composite system shown on p. 40 can also be represented as a slit configuration inthe transformed region:with the inner system represented asand the lower side of the slit considered re-entrant with the left half of the top boundary ofthe rectangle corresponding to the inner system, the upper side of the slit being re-entrantwith the right half of this top boundary of the inner region. Now the conceptual opening is asfollows for the inner region:Difference representations made above the slit thus would use points below the right half ofthe top of the inner region in the transformed region, etc.
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