Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 22
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"Line" attraction here is simply attraction to a group of points that form a line,y=f(x). If line attraction is specified then the tangent to the line y=f(x) is computed from theadjacent points on the line. If point attraction is specified, then the "tangent" must be inputfor each point. The unit tangents to the coordinate lines are computed from Eq. (III-3):The presence of branch cuts introduces no complication with this type of attractionsince the distances involved are in terms of the cartesian coordinates, rather than thecurvilinear coordinates. This form of attraction makes the control functions dependent onboth the curvilinear and cartesian coordinates, and thus attraction to space lines and/or pointsinvolves more complicated equations in the transformed region than does attraction to othercoordinate lines and/or points, since for the former, coefficients of the first derivatives arefunctions of the dependent variables.
Attraction to lines and/or points in space has not beenwidely used, and the use of Eq. (31) has not been fully tested.C. Evaluation along a coordinate lineAs has been noted above, if it is desired that the spacing of the coordinate lines in thefield generally follow that of the points on the boundary, the control functions must beevaluated so as to correspond to this boundary point distribution. This can be accomplishedas follows. (The developments in this and the next two sections are generalizations of thatgiven in Ref. [12], and other works cited therein and in Ref. [5].)The projection of Eq. (12) along a coordinate line on whichforming the dot product of this equation with the base vectorthe line.l variesis found by, which is tangent toThus we have(32)Now assume for the moment that the two coordinate lines crossing the coordinate line ofinterest do so orthogonally.
Then on this line we haveandwhich leads to an explicit equation for Pl on the coordinate line of interest:(33)If it is further assumed for the moment that the two coordinate lines crossing thecoordinate line of interest are also orthogonal to each other, i.e., complete orthogonality onthe line of interest,we have on this line gij =sinceijgii and gij =ijgiiAlso, from Eq. (III-38),. But also by Eq. (III-16) g = gllgmmgnn so that gllgll=1. Then Eq. (33) becomes(34)which can also be written, using Eq. (III-3), as(35)By Eq.
(III-7) the derivative of arc length along the coordinate line on whichl variesis(36)Then(37)so that the logarithmic derivative of arc length along this coordinate line is given by(38)which is exactly the i=1 term in the summation in Eq. (34).The unit tangent to a coordinate line on whichm variesis(39)and the derivative of this unit tangent with respect to arc length is a vector that is normal tothis line, the magnitude of which is the curvature, K, of the line. The unit vector in thisnormal direction is the principal normal, , to the line.Thus, using Eq.
(36),(40)Then(41)so that the curvature is(42)The component of Kmm alongthe coordinate line on whichl variesis given by(43)sinceforcan be written as. Then the two terms of the summation in Eq. (34) for which(44)Thus Eq. (34) can be written(45)where (l,m,n) are cyclic, and using Eq. (III-3), we have(46)and(47)with an analogous equation for (Kn n)(1). The arc length in the expressions (45) for thecontrol function Pl along the coordinate line on which l varies can be determined entirelyfrom the grid point distribution on the line using Eq.
(46). The other two terms in Pl,however, involve derivatives off this line and therefore must either be determined byspecifications of the components of the curvature, K , of the crossing lines along the line ofinterest, or by interpolation between values evaluated on coordinate surfaces intersecting theends of this line.If it is assumed that the curvatures of these crossing lines vanish on the coordinate line ofinterest, then the last two terms in Eq. (45) are zero, and the control function becomes simply(48)and then can be evaluated entirely from the specified point distribution on the coordinate lineof interest.The neglect of the curvature terms, however, is ill-advised since the elliptic systemalready has a strong tendency to concentrate lines over a convex boundary, as has beendiscussed earlier in this chapter. Therefore neglect of the curvature terms will result incontrol functions which will produce a stronger concentration than intended over convexboundaries (and weaker over concave).
When interpolation from the end points is used todetermine the curvature term, the entire term (K ) should be interpolated, since individualinterpolation of the vectors l and ( m) m can give an inappropriate value for the dotproduct.It should be noted that the assumptions of orthogonality, and perhaps vanishingcurvature, that were made in the course of the development of these expressions for thecontrol functions on a coordinate line are not actually enforced on the resulting coordinatesystem, but merely served to allow some reasonable relations for these control functionscorresponding to a specified point distribution on a coordinate line to be developed.
Thisshould not be considered a source of error since the control functions are arbitrary in thegeneration system (12).D. Evaluation on a coordinate surfaceIn a similar fashion, expressions for the control functions on a coordinate surface onwhich l is constant can be obtained from the projections of Eq. (12) along the twocoordinate lines lying on the surface, i.e., the lines on which m and n vary, (l,m,n) beingcyclic.These projections are given by Eq. (32) with l replaced by m and n, respectively. If it isassumed for the moment that the coordinate line crossing the coordinate surface of interest isorthogonal to the surfacethenyield the equation, so that Pl is removed from both of these two equations toand an analogous equation with m and n interchanged.
Solution of these two equations forPm and Pn then yields(49)with an analogous equation for Pn with m and n interchanged. Since glm = gln = 0, we haveby Eq. (III-38), glm = gln = 0. Therefore only the five terms, ll, mm, nn, mn, nm, arenon-zero in the summation. Also from Eq. (III-38) we havesince hereinterchanging m and n.. An analogous equation for gnn is obtained byThen Eq. (49) can be rewritten as(50)and an analogous equation for Pn with m and n interchanged. But, again using Eq. (III-38),we haveTherefore(51)and the analogous equation with m and n interchanged. This can also be written as(52)and the analogous equation for Pn.All of the terms, except the first, in the above equations can be evaluated completelyfrom the point distribution on the coordinate surface of interest.From Eq.
(47) the first term in (52) can be written(53)where (Kl l)m and (Kl l)n are the components of the curvature K for the coordinate linecrossing the coordinate surface of interest along the two coordinate lines on the surface.These quantities must be either specified on the surface or interpolated from valuesevaluated on its intersections with the other coordinate surfaces. If it is further assumed thatthe curvature of the crossing line vanishes at the surface, then this first term in Eq.
(52)vanishes also.As was noted for the control functions on a line, the curvature terms should not beneglected, however, else the concentration will be stronger than intended over convexboundaries and weaker over concave. Also, it is the entire term K which should beinterpolated, not the individual vectors involved, else the dot product can have inappropriatevalues.E. Evaluation from boundary point distributionUsing the relations developed in the previous two sections for the control functions ona coordinate line and on a coordinate surface, an interpolation procedure can be formulatedfor evaluation of the control functions in the entire field.
If the point distribution is specifiedon all the boundary surfaces of a three-dimensional field, the control functions can beevaluated on these boundaries using the relations in Section D, and then the control functionsin the entire three-dimensional field can be interpolated from these values on the boundingsurfaces using transfinite interpolation (discussed in connection with algebraic gridgeneration in Chapter VIII.)To be definite, consider a general three-dimensional region bounded by six curvedsides:with curvilinear coordinates as shown, which transforms to a rectangular block. From Eq.(52) the two control functions, P j and Pk, can be evaluated from the specified boundary-pointdistribution on the two faces on which i is constant, i.e., the left and right faces in thefigure.
Similar evaluations yield two control functions on each face, with the result that thecontrol function Pk will be known on the four faces on which k varies, i.e., the front, back,left, and right faces in the figure. Thus, in general, interpolation for the control function Pk inthe interior of the region is done from the boundary values on the four faces on whichvaries:k(54), i=1,2,3. In an analogous mannerHere Ii is the maximum value of i, etc., i.e.,all three control functions can be determined in the interior of the region.It may be desirable in some cases to generate a two-dimensional coordinate system ona curved surface, as discussed in Section 3, rather than specifing the point distribution on thesurface.
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