Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 31
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(5),(86a)and(86b)Now(87)Then the two-directional transfinite interpolation can be constructed by substitution of Eq.(86) and (87) into the projector PP . Here the tensor product form, P P , interpolatesfrom the values of the function, its two first derivatives, and the cross-derivative at the fourcorners of the boundary. The transfinite interpolation form, PP , however, interpolatesfrom the value of the function and its normal derivative on the entire boundary.The triple product corresponding to Eq.
(84) is simply(88)Recall that with the unidirectional form given by Eq. (44), we have in these relationsL=M=N=2 andThe above evaluations of the product projectors serve to illustrate the evaluation ofsuch products for general projectors, i.e., that the effect of the product protectors is simply aninterpolation in one-direction of an interpolant in another direction. This allows themulti-directional transfinite interpolation to be constructed from the Boolean sums of theprotectors given above using any appropriate unidirectional interpolation forms as the basisprojectors.
It should also be noted that the unidirectional interpolation does not have to be ofthe same form in all the directions. Thus Lagrange interpolation could be used in one of thedirections while Hermite is used in the other direction of a two-directional construction. Asnoted above, the blending functions do not have to be polynomials.
In fact, all of theunidirectional interpolation that was discussed earlier in this chapter can be applied in thecontext of multi-directional interpolation based on the protectors. This freedom to combinedifferent types of univariate interpolation gives considerable flexibility to transfiniteinterpolation based on the projector structure, and allows attention to be focused ondeveloping appropriate unidirectional interpolations, the multi-directional format thenfollowing automatically.The protectors allow the transfinite interpolation to be easily set up as a sequence ofunidirectional interpolations, in the manner discussed above.
Thus in the two-directionalcase, Eq. (80) can be written as(89)where I indicates the identity operation. But here the first term is clearly the unidirectionalinterpolation in the -direction, while the parenthesis (I-P ) is the discrepancy on the-lines on which is specified that results from this -interpolation. The second term thenis the unidirectional interpolation of this descrepancy interpolated in the -direction.The two-directional interpolation thus can be implemented by: (1) interpolating inthe -direction, (2) calculating the discrepancy between and this result on the -lines thatare to be used in the -interpolation, (3) interpolating this discrepancy in the -direction,and (4) adding the result of this -interpolation to that of the interpolation.
Symbolicallythese steps can be stated as the following:(90)Obviously, the order of the unidirectional interpolation is immaterial.Similarly, Eq. (83) can be written as(91)The three-directional interpolation thus can be implemented by (1) interpolating in the-direction, (2) calculating the discrepancy between and this result on the -surfaces and-surfaces that are to be used in the interpolation in those directions, (3) interpolating thisdiscrepancy by two-directional interpolation, and (4) adding this result to that of the-interpolation.
These operations can be stated as(92)Exercises1. Show that with N=2 and n constant, the multi-surface interpolation is equivalent to thelinear Lagrange interpolation. Mote that the Lagrange polynomials here satisfy Eq. (25) with. For other choices of 2, other quadratic polynomials result from Eq. (25), so thatthere exists a one-parameter family of cubic forms of the multi-surface interpolation.Similarly, a two-parameter family of quadratic forms exists, etc.2. Show that the quadratic form of the multi-surface interpolation is given by3. Show that the quadratic forms of the multi-surface and Bezier interpolations areequivalent.4.
Show that with N=4 and n the quadradic Lagrange interpolation polynomials given on p.282, the interpolation functions for the multi-surface interpolation are given bythe multi-surface interpolation is equivalent to the cubic Hermite interpolation.5. Show that S( ) is given by Eq. (38) for Lagrange interpolation. (Hint: If all the(2) are the same, the interpolation must reproduce this value, hence the Lagrangeinterpolation polynomials satisfy6.
Show that for quadratic Lagrange interpolation, uniformity requires thatsuch thatNote that this does not completely determine2nin Eq.be selected2.7. Show that uniformity is achieved with cubic Hermite interpolation withorthogonality at the boundaries if the spacings at the boundaries are given bywithwhere is the unit normal to the boundary. This completely specifies the interpolation inthis case.
However, as noted in Exercise 1, 2 is a free parameter.8. Show that for multi-surface interpolation with N=4 andwithuniformity is achieved9. Show also that with orthogonality at the boundary the result of Exercise 8 completelydetermines all of the interpolation parameters, i.e., thatwhere is the unit normal to the boundary. Hint: Use Eq. (7) and (29). For general 2 the1/6 is replaced by 1/2 - 1/[6( 2/I)] and 1/2 - 1/[6(1- 2/I)] in the above expressions involving2and3,respectively.
Some effects of the choice of2are shown in Ref. [32].10. Show that local uniformity on the interval nn+1 for the multi-surfaceinterpolation based on piecewise-linear functions requires thatwhere, with11. Consider a rectangular physical region with equally-spaced points on the bottom and top,but with unequal spacing on the left and right sides (but with the same point distribution onboth of these sides). Show that horizontal interpolation will reflect the unequal spacing of thehorizontal grid lines in the field, but that vertical interpolation will not. Show also that theunequal spacing is reflected with transfinite interpolation.12. Show that transfinite interpolation based on linear blending functions will reflect theunequal boundary point spacing in the field for the rectangular physical region of Exercise11, but will not for a C-grid.
From the consideration of transfinite interpolation as a sequenceof unidirectional interpolations, explain why this is so.13. Show that with cubic Lagrange interpolation the locations of the two intermediatesurfaces, 2 and 3, are related to the slopes at both ends. Note the contrast between thisand the multi-surface interpolation where each of the intermediate surfaces depends on onlythe slope at one end.14.
Give the cubic form of Lagrange interpolation.15. Show that in two dimensions transfinite interpolation is equivalent to a generation systembased on the fourth - order partial differential equation(This is also equivalent to the quadralaterial isoparemetric elements often used to constructfinite element meshes.)IX.
ORTHOGONAL SYSTEMSOrthogonal coordinate systems produce fewer additional terms in transformed partialdifferential equations, and thus reduce the amount of computation required. Also, as hasbeen noted in Chapter V, severe departure from orthogonality will introduce truncation errorin difference expressions. A general discussion of orthogonal systems on planes and curvedsurfaces is given in Ref. [42], and various generation procedures are surveyed in Ref.
[42]and Ref. [1].In numerical solutions, the concept of numerical orthogonality, i.e., that theoff-diagonal metric coefficients vanish when evaluated numerically, is usually moreimportant than strict analytical orthogonality, especially when the equations to be solved onthe system are in the conservative law form.There are basically two types of orthogonal generation systems, those based on theconstruction of an orthogonal system from a non-orthogonal system, and those involvingfield solutions of partial differential equations.
The first approach involves the constructionof orthogonal trajectories on a given non-orthogonal system. Here one set of coordinate linesof the non-orthogonal system is retained, while the other set is replaced by lines emanatingfrom a boundary and constructed by integration across the field so as to cross each line of theretained set orthogonally. Control of the line spacing is exercised through the generation ofthe non-orthogonal system and through the point distribution on the boundary from whichthe trajectories start. The point distributions on only three of the four boundaries can bespecified. Several methods for the construction of orthogonal trajectories are discussed inRef.
[42] and Ref. [1]. If point distributions are to be specified on all boundaries, the fieldapproach must be taken, and it is to this approach that this chapter is primarily directed.1. General FormulationThe characteristic criterion for orthogonal coordinates is the vanishing of theoff-diagonal elements of the metric tensor, i.e., gij = gij = 0 for i j. Thus the Jacobian of thetransformation is simply(1)For brevity, writingit is easy to show from Eq.
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