Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 4
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The transformation can, however, be from any system of coordinates in thephysical space.)B. Boundary value Problem - Transformed RegionThe problem may be simplified for computation, however, by first transforming sothat the physical cartesian coordinates (x,y) become the dependent variables, with thecurvilinear coordinates ( , ) as the independent variables.
Since a constant value of onecurvilinear coordinate, with monotonic variation of the other, has been specified on eachboundary segment, it follows that these boundary segments in the physical field willcorrespond to vertical or horizontal lines In the transformed field. Also, since the range ofvariation of the curvilinear coordinate varying along a boundary segment has been made thesame over opposing segments, it follows that the transformed field will be composed ofrectangular blocks.The boundary value problem in the transformed field then involves generating thevalues of the physical cartesian coordinates, x( , ) and y( , ), in the transformed fieldfrom the specified boundary values of x( , ) and y( , ) on the rectangular boundary of thetransformed field, the boundary being formed of segments of constant or , i.e., vertical orhorizontal lines.
With = constant on a boundary segment, and the increments in taken tobe uniformly unity as discussed above, this boundary value specification is implementednumerically by distributing the points as desired along the boundary segment and thenassigning the values of the cartesian coordinates of each successive point as boundary valuesat the equally spaced boundary points on the bottom (or top) of the transformed field in thefollowing figure.Boundary values are not specified on the left and right sides of the transformed field sincethese boundaries are re-entrant on each other (analogous to the 0 and 2 lines in thecylindrical system), as discussed above, and as indicated by the connecting dotted line on thefigure. Points outside one of these re-entrant boundaries are coincident with points at thesame distance inside the other.
The problem is thus much more simple in the transformedfield, since the boundaries there are all rectangular, and the computation in the transformedfield thus is on a square grid regardless of the shape of the physical boundaries.With values of the cartesian coordinates known in the field as functions of thecurvilinear coordinates, the network of intersecting lines formed by contours (surfaces in3D) on which a curvilinear coordinate is constant, i.e., the curvilinear coordinate system,provides the needed organization of the discretization with conformation to the physicalboundary. It is also possible to specify intersection angles for the coordinate lines at theboundaries as well as the point locations.3.
Transformed Region ConfigurationsAs noted above, the generation of the curvilinear coordinate system is done bydevising a scheme for determination of the field values of the cartesian coordinates fromspecified values of these coordinates (and/or curvilinear coordinate line intersection angles)on portions of the boundary of the transformed region.
Since the boundary of thetransformed region is comprised of horizontal and vertical line segments, portions of whichcorrespond to segments of the physical boundary on which a curvilinear coordinate isspecified to be constant, it should be evident that the configuration of the resultingcoordinate system depends on how the boundary correspondence is made, i.e., how thetransformed region is configured.Some examples of different configurations are given below, from which morecomplex configurations can be inferred.
In these examples only a minimum number ofcoordinate lines are shown in the interest of clarity of presentation tation. In all of theseexamples, boundary values of the physical cartesian coordinates (and/or curvilinearcoordinate line intersection angles) are understood to be specified on all boundaries, bothexternal and internal, of the transformed region except for segments indicated by dottedlines. These latter segments correspond to branch cuts in the physical space, as is explainedin the examples in which they appear. Such re-entrant boundary segments always occur inpairs, the members of which are indicated by the dashed connecting lines on each of theconfigurations shown.
Points outside the field across one segmentof such a pair arecoincident with points inside the field across the other member of the pair. The conceptualdevice of opening the physical field at the cuts is used here to help clarify thecorrespondence between the physical and transformed fields. In many cases an example ofan actual coordinate system is given as well. References to the use of various configurationsmay be found in the surveys given by Ref. [1] and [5], and a number of examples appear inRef. [2].A.
Simply-connected RegionsIt is natural to define the same curvilinear coordinate to be constant on each memberof a pair of generally opposing boundary segments in the physical plane. Thus, asimply-connected region formed by four curves is logically treated by transforming to anempty rectangle:Similarly, an L-shaped region could remain L-shaped in the transformed region:Here, for instance, the cartesian coordinates of the desired points on the physical boundarysegment 4-5 are specified as boundary conditions on the vertical line 4-5, in correspondingorder, which forms a portion of the boundary of the transformed region.The generalization of these ideas to more complicated regions is obvious, thetransformed region being composed of contiguous rectangular blocks.
An example follows:The physical boundary segment on which a single curvilinear coordinate is constantcan have slope discontinuities, however, so that the L-shaped region above could have beenconsidered to be composed of four segments instead of six, so that the transformed regionbecomes a simple rectangle:Here the cartesian coordinates of the desired points on the physical boundary 5-4-3 are thespecified boundary values from left to right across the top of the transformed region.Whether or not the boundary slope discontinuity propagates into the field, so that thecoordinate lines in the field exhibit a slope discontinuity as well, depends on how thecoordinate system in the field is generated, as will be discussed later.It is not necessary that corners on the boundary of the transformed region correspondto boundary slope discontinuities on the physical boundary and a counter-example followsnext:In this case, the segment 1-2 on the physical boundary is a line of constant , while thesegment 1-4 is a line of constant .
Thus at point 1 we have the following coordinate lineconfiguration:The lines through point 1 are as follows:so that the angle between the two coordinate lines is at point 1, and consequently theJacobian of the transformation (the cell area, cf. Chapter III) will vanish at this point. Thecoordinate species thus changes on the physical boundary at point 1. (Differencerepresentations at such special points as this, and others to appear in the following examples,are discussed in Chapter IV.) Since the species of curvilinear coordinate necessarily changesat a corner on the transformed region boundary, the identification of a concave corner on thetransformed region boundary with a point on a smooth physical boundary will always resultin a special point of the type illustrated here.
(A point of slope discontinuity on the physicalboundary also requires special treatment in difference solutions, since no normal can bedefined thereon. This, however, is inherent in the nature of the physical boundary and is notrelated to the construction of the transformed configuration.)Some slightly more complicated examples of the alternatives introduced above nowfollow:Still another alternative in this case would be to collapse the intrusion 2-3-4-5 to a slit in thetransformed region:Here the physical cartesian coordinates are specified and are double-valued on the verticalslit, 2-9-5, in the transformed region. The cartesian coordinates of the desired points on thephysical boundary 2-9 are to be used on the slit in the generation of the grid to the left of theslit in the transformed region, while those on the physical boundary 5-9 are used forgeneration to the right of the slit. Solution values in a numerical solution on such acoordinate system would also be double-valued on the slit, of course.
Thisdouble-valuedness requires extra bookkeeping in the code, since two values of each of thecartesian coordinates and of the physical solution must be available at the same point in thetransformed region so that difference representations to the left of the slit use the slit valuesappropriate to the left side, etc. Difference representations near slits are discussed in ChapterIV.
With the composite grid structure discussed in Section 4, however, this need fordouble-valuedness, and the concomitant coding complexity, with the slit configuration canbe avoided.The point 9 here requires special treatment, since the coordinate line configurationthere is as follows:The coordinate lines through point 9 are as follows:Here the slope of the coordinate line on which varies is discontinuous at point 9, and theline on which varies splits at this point.
Such a special point will always occur at the slitends with the slit configuration.B. Multiply-connected RegionsWith obstacles in the interior of the field, i.e., with interior boundaries, there are stillmore alternative configurations of the transformed region. One possibility is to maintain theconnectivity of the transformed region the same as that of the physical region, as in thefollowing examples showing two variations of this approach using interior slabs and slits,respectively, in the transformed region. The slab configuration is as follows:In coding, points inside the slab in the transformed region are simply skipped in allcomputations.This configuration introduces a special point of the following form at each of thepoints corresponding to the slab corners in the transformed field:The coordinate lines throughpoint 7 are shown below:This type of special point, where the coordinate species changes on a smooth line, occurswhen a convex corner in the transformed field is identified with a point on a smooth contourin the physical field.
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