Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 9
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Grid points on aboundary segment of the transformed region will be placed in ijk with one index fixed.Now each block has six exterior boundaries, and may also have any number of interiorboundaries (cf. the slab and slit configurations of Section 3), all of which will always beplane segments intersecting at right angles, although the occur ence of interior boundariescan be avoided if desired by breaking the block up into a collection of smaller blocks asdiscussed in Section 4. The boundary segments in the transformed plane may correspond toactual segments of the physical boundary, or may correspond to cuts in the physical region.As discussed in Section 5, these cuts are not physical boundaries, but rather are interfacesacross which the field is re-entrant on itself.
A boundary segment in the transformed regioncorresponding to such a cut then is an interface across which one block is connected withcomplete continuity to another block, or to another side of itself, several examples havingbeen given above in this chapter.Depending on the type of grid generation system used (cf., the later chapters), thecartesian coordinates of the grid points on a physical boundary segment may either bespecified or may be free to move over the boundary in order to satisfy a condition, e.g.,orthogonality, or the angle at which coordinate lines intersect the boundary.To set up the configuration of the transformed region, a correspondence is establishedbetween each (exterior or interior) segment of the boundary of the transformed region andeither a segment of the physical boundary or a segment of a cut in the physical region.
Thisis best illustrated by a series of examples using the configurations of this chapter. The firststep in general is to position points on the physical boundary, or on a cut, which are tocorrespond to corners of the transformed region (exterior or interior). As noted in Section 3,these points do not have to be located at actual corners (slope discontinuities) on the physicalboundary.For example, considering the two-dimensional simply-connected region on p. 23, fourpoints on the physical boundary are selected to correspond to the four corners of the emptyrectangle that forms the transformed region here:Now, considering any one of these four points, one species of curvilinear coordinate will runfrom that point to one of the two neighboring corner points, while the other species will runto the other neighbor:The corresponding species of coordinates will run to connect opposite pairs of corner points:Since the curvilinear coordinates are to be assigned integer values at the grid points, i is tovary from 1 at one corner to a maximum value, Ii, at the next corner, where Ii is the numberof grid points on the boundary segment between these two corners.
Thus, proceedingclockwise from the lower left corner, the cartesian coordinates of the four corner points areplaced in 1,1, 1,J, I,J, and I,1, where I1 = I and I2 = J.The boundary specification is then completed by positioning I-2 points on the lower andupper boundary segments of the physical region as desired, and J-2 points on the left andright segments. The cartesian coordinates of these points on the lower and upper segmentsare placed in i,1 and i,J, respectively, for i from 2 to I-1, and those on the left and rightsegments are placed, respectively, in1,j andI,j forj from 2 to J-1.This process of boundary specification can be most easily understood by viewing therectangular boundary of the transformed region, with I equally-spaced points along twoopposite sides and J equally-spaced points along the other two sides, conceptually, as beingdeformed to fit on the physical boundary. The corners can be located anywhere on thephysical boundary, of course.
Here the point distribution on the sides can be conceptuallystretched and compressed to position points as desired along the physical boundary. Thecartesian coordinates of all the selected point locations on the physical boundary are thenplaced in as described above.This conceptual deformation of the rectangular boundary of the transformed region tofit on the physical boundary serves to quickly illustrate the boundary specification for thedoubly-connected physical field shown on p. 29, which involes a cut. Thus I points arepositioned as desired clockwise around the inner boundary of the physical region from 2 to3, and I points are positioned as desired, also in clockwise progression, around the outerboundary from 1 to 4.
The cartesian coordinates of these points on the inner boundary areplaced in i,1, and those on the outer boundary in i,J, with i from 1 to I. Note that here thefirst and last points must coincide on each boundary, i.e., I,1 = 1,1 and I,J = 1,J.
Theleft and right sides of the transformed region (i=1 and i= I) are re-entrant boundaries,corresponding to the cut, and hence values on these boundaries are not set but will bedetermined by the generation system. The system must provide that the same value appearson both of these sides, i.e., I,j = 1,j for all j from 2 to J-1.The conceptual deformation of the rectangle for a C-type configuration is illustratedon p. 31. Here, with I1 the number of points on the segments 1-2 and 3-4 (which must havethe same number of points), I2 points are positioned as desired around the inner boundary inthe physical region in a clockwise sense from 2 to 3, and the cartesian coordinates of thesepoints are placed in i,1 for i from I1 to I1+I2-1.The first and last of these points must be coincident, i.e.
I1,1 = I1 + I2-1,1. Now thetop, and the left and right sides, of the rectangle are deformed here to fit on the outerboundary of the physical region. (In the illustration given, the two top corners are placed onthe two corners that occur in the physical boundary, a selection that is logical but notmandatory.) The cartesian coordinates of the J points(positioned as desired on the segment4-5 of the physical boundary) are placed in I,j, proceeding upward on the physicalboundary from 4 to 5 for j= 1 to J, and those on the segment 1-6 are placed in 1,j, butproceeding downward on the physical boundary from 1 to 6 for j1 to J.
Finally, the cartesiancoordinates of the I selected points on the physical boundary segment 6-5 are placed in i,J,proceeding clockwise from 6 to 5 for i=1 to I. Since the same number of points must occuron the top and bottom of the rectangle, we must have I=2(I1-1)+I2. Here the portions of thelower side of the rectangle, i.e., i from 2 to I1-1, and from I1+I2 to I-1 with j=1, arere-entrant boundaries corresponding to the cut, and hence no values are to be specified onthese segments.
The generation system must make the correspondence i,1 = I-i+1,1 fori=2 to I1-1 on these segments.The conceptual deformation of the boundary of the transformed regions also serves forthe slab configuration on p. 27, where the interior rectangle deforms to fit the interiorphysical boundary, while the outer rectangle deforms to fit the outer physical boundary. Onthe inner boundary, the cartesian coordinates of J2-J1+1 selected points on the segment 5-8of the physical boundary are placed in I1,j for j from J1 to J2, proceeding upward on thephysical boundary from 5 to 8, where J1 and J2 are the j-indices of the lower and uppersides, respectively, of the interior rectangle and I1 is the i-index of the left side of thisrectangle.
Similarly, J2-J1+1 points are positioned as desired on the segment 6-7 of thephysical boundary and are placed in I2,j, where I2 is the i-index of the right side of theinner rectangle. Also I2-I1+1 points on the segments 5-6 and 8-7 of the physical boundaryare placed in i,J1 and i,J2, respectively, for i from I1 to I2, proceeding to the right on eachsegment. The outer boundary is treated as has been described for an empty rectangle. Herethere will be J1-1 coordinate lines running from left to right below the inner boundary, andJ-J2 lines running above the inner boundary.
Similarly, there will be I1-1 lines runningupward to the left of the interior boundary and I-I2 lines to the right. Thus the specificationsof the desired number of coordinate lines running on each side of the inner boundary servesto determine the indicies I1, I2, J1, and J2. Note that the points inside the slab, i.e., I1 < i <I2 and J1 < j < J2 are simply excluded from the calculation.The slit configuration, illustrated on p. 28, can also be treated via the conceptualdeformation, but now with a portion of a line inside the rectangle opening to fit the interiorboundary of the physical region. This requires that provision be made in coding for twovalues of the cartesian coordinates to be stored on the slit. If the i-indices of the slit ends, 5and 6, are I1 and I2, respectively, then the cartesian coordinates of I2-I1+1 points positionedas desired on the lower portion of the physical interior boundary, again proceeding from 5 to6, are placed in a one-dimensional array, while the coordinates of the same number of pointsselected on the upper portion of the physical interior boundary, again proceeding from 5 to 6,are placed in another one-dimensional array.
The first and last points in one of these arraysmust, of course, coincide with those in the other. Then the generation system must readvalues into i,J1 for i from I1 to I2 (J1 being the j-index of the slit) from the former array foruse below the slit, or values from the latter array for use above. (As has been noted, the useof a composite structure eliminates the need for these two auxiliary arrays.) Note that theindex values I1 and I2 are determined by the number of lines desired to run upward to theleft and right of the interior boundary, respectively, i.e., I1-1 lines on the left and I-I2 on theright. Similarly, there will be J1-1 lines below the interior boundary, and J-J1 above.Configurations, such as those illustrated on pp.
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