Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 30
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(2) can be written in the form(44)wherecan be any function such that(0)=0 and(1)=1. Here we have taken1=1-and12=. The linear polynomial case is obtained here with ( ) = . The function in thisform may contain parameters which can be determined so as to match the slope at theboundary, or to match interior points and slopes.The interpolation function, , in this form is often referred to as a "stretching"function, and the most widely used function has been the exponential:(45)where is a parameter that can be determined to match the slope at a boundary.
Thus, since,from Eq. (44)(46)we can determinefrom the equation(47)with ()1 specified.As noted in Chapter V, the truncation error is strongly affected by the pointdistribution, and studies of distribution functions have been made in that regard. Theexponential, while reasonable, is not the best choice when the variation of spacing is large,and polynomials are not suitable in this case. The better choices are the hyperbolic tangentand the hyperbolic sine. The hyperbolic sine gives a more uniform distribution in theimmediate vicinity of the minimum spacing, and thus has less error in this region, but thehyperbolic tangent has the better overall distribution (cf. Section 3 of Chapter 5).
Thesefunctions are implemented as follows (following Ref. [18]), with the spacing specified ateither or both ends, or a point in the interior, of a point distribution on a curve.Let arc length, s, vary from 0 to 1 asspacing be specified at =0 and =I:varies from 0 to I: s(0)=0, s(I)=1. Then let the(48)The hyperbolic tangent distribution is then constructed as follows.First,(49)(50)Then the following nonlinear equation is solved for:(51)The arc length distribution then is given by(52)where(53)If this is applied to a straight line on whichlocations:varies fromThe points are then located by taking integer values oftoI wehave for the point:Clearly the arc length distribution, s( ), here is the functionWith the spacing hs specified at onlyB is calculated from0of Eq.
(44).=0, the construction proceeds as follows. First(55)and Eq. (51) is solved for. The arc length distribution then is given by(56)With the spacing specified only atreplaced by=I the procedure is the same, except that Eq. (56) is(57)If the spacing s is specified at only an interior point s= , B is again calculated fromEq.
(55), and then is determined as the solution of(58)The value ofat which s =is obtained by solving the nonlinear equation(59)The arc length distribution then is given by(60)This last distribution is based on the hyperbolic sine. From this a distribution baaed onthe hyperbolic sine with the spacing specified at one end can be derived. Here B is evaluatedfrom Eq. (55), and then is determined as the solution of(61)The arc length distribution then is given by(62)if the spacing is specified at=0.
With the specification at=I, the distribution is(63)It is also possible to construct a distribution based on the hyperbolic sine withspecified spacing on each end. Here A and B are again calculated from Eq. (49) and (50), butis determined from(64)The distribution is then given by Eq. (52), but with(65)Finally, a procedure for incorporating the effect of curvature into the distributionfunction is given in Ref. [38], where the arc length distribution is given in the inverse formby(66)where(67)st is the total arc length, K( ) is the curvature, and A( ) is any distribution function (in theinverse form) that would be used without consideration of curvature.2. Multi-Directional InterpolationA.
Transfinite interpolationIn two directions we may write a linear Lagrange interpolation function individually ineach curvilinear direction:(68a)and(68b)This interpolation is now called "transfinite" since it matches the function on the entireboundary defined by =0 and =I in the first equation, or by =0 and =J in the second,i.e., at a nondenumerable number of points, cf.
Ref. [40] and [41]).The tensor product form(69)wherenm =( n, m) matches the function at the four corners:It does not, however, match the function on all the boundary.The sum of Eq. (68a) and (68b),(70)when evaluated on the=0 boundary gives(71)This does not match the function on the =0 boundary because of the second term on theright, which is an interpolation between the ends of this boundary:Similar effects occur on all the other boundaries, and the discrepancy on the=I boundary isThe discrepancies on both of these boundaries can be removed by subtracting froma function formed by interpolating the discrepancies between the two boundaries:( , )(72)But this is simply the tensor product form given by Eq. (69), which matches the function atthe four corners.The function - then matches the function on all four sides of the boundary, so thatwe have the transfinite interpolation form,(73)which matches the function on the entire boundary.
By contrast, the tensor product form(74)matches the function only at the four corners on the boundary. This generalizes to theinterpolation from a set of N+M intersecting curves for which the univariate interpolation isgiven by(75a)and(75b)where now the "blending" functions,cardinality conditionsnandm,are any functions which satisfy the(76)The general form of the transfinite interpolation then is(77)while the tensor-product form is(78)Eq. (77) can be written in the formBut here the first term is the result at each point in the field of the unidirectionalinterpolation in the direction, and the bracket is the difference between the specified valueson the = n lines and the result of the unidirectional interpolation on those lines. Thetwo-directional transfinite interpolation can thus be implemented in two unidirectionalinterpolation steps by first performing the unidirectional interpolation in one direction, say, over the entire field, calling the result 1( , ):(79a)then interopolating the discrepancy on thedirection,here, calling the result= n lines over the entire field in the other2( , ):(79b)and then adding1and2.(79c)The transfinite interpolation form given by Eq.
(77) is the algebraically bestapproximation, while the tensor product from of Eq. (78) is the algebraically worst (cf. Ref.[40]). The difference between these two forms should be fully understood. The transfiniteinterpolation form, Eq. (77), interpolates to the entirety of a set of intersecting arbitrarycurves, while the tensor product form, Eq. (78), interpolates only to the intersections of thesecurves. The interpolation function defined by Eq.
(77) with N=M=2, using the Lagrangeinterpolation polynomials as the blending functions, is termed the transfinite bilinearinterpolant. With N=M=3, this form is the transfinite biquainterpolant. Other immediatecandidates for the blending functions are the Hermite interpolation polynomials and thesplines, since these all can be expressed in the form of Eq. (75). The spline-blended formgives the smoothest grid with continuous second derivatives.B. ProjectorsNow let P ( ) be a one-dimensional interpolation function in thematcheson the N lines,(Note that the subscript-direction which= n (n=1,2,...N),:here does not denote differentiation.) Similarly, let P ( ) matchon the M lines, = m (m=1,2,...M). These interpolations are performed by projectors, Pand P , which are assumed to be idempotent linear n operators.
Protectors are discussedin more detail in Ref. [40]. Some discussion is also given in Ref. [37]. The product projector,P [P ( )], then matches the function P ( ), instead of , on the N lines, = n:Then, since P ( ) matcheson the M lines, = m, it follows that the product projectormwill matchat the NxM points ( n, m):Clearly the same conclusion is reached for the product projector P [P ( )], so that theprotectors P and P commute.= n, and+P ( ) on the M lines = m. It should be clear then that the projector, PThe sum projector, Pmatches( )+P ( ) matches( )+P ( )-P [P ( )] will match+P ( ) on the N lineson the N lines= n, since P [P ( )] matches P( ) on these lines.
Similarly, the projector P ( )+P ( )-P [P ( )] matches onthe M lines = m. Therefore, since P + P = P P , the Boolean sum projector, PP= P +P -P P , will match on the entirety of the N+M lineswhich includes, of course, the entire boundary of the region.In summary, the individual protectors, Pbetween two opposing boundaries:= n and = mand P , interpolate undirectionallyThe product progeotor, P P , interpolates in two directions from the four corners:The Boolean sum projector, PP , interpolates from the entire boundary:In three dimensions, the individual protectors, P , P , and P , interpolateundirectionally between two opposing faces of the six-sided region:(matching on each of the two faces in each case).
The double product projector, P P ,interpolates in two directions from the four edges along which and are constant:(matchingon each of these edges). The Boolean sum projector(80)interpolates in two directions from the four faces on which eitheroris constant:(matchingon all of these faces).The Boolean sum projector(81)interpolates in three directions, matching on the four edges on whichand also on the two faces on which is constant:andare constantThe Boolean sum projector(82)interpolates in three directions, withmatched on all twelve edges:The triple product projector,, interpolatesfrom the eight corners:Finally the Boolean sum projector(83)matcheson the entire boundary.Much cancellation occurs in the algebraic manipulation of the projectors involved indeveloping the above relations, since P P = P , etc.
Thus, for example,This is to be expected since interpolation by P matches the function on all of the two sideson which is constant, while P P matches the function on the four edges on whichand are constant. But these edges are contained on the two sides cited, so that nothing ischanged by adding P P to P in the Boolean sense. The projector formed as the Booleansum of all three of the individual projectors is algebraically maximum, while the tripleproduct projector is algebraically minimal.The importance of the projectors is that the structure given above allowsmulti-directional interpolation to be constructed systematically from unidirectional forms.With one-dimensional interpolation of the form of Eq.
(75) we have(84a)(84b)so that(85)which is just the tensor product form given previously in Eq. (78), so that the two-directionaltransfinite interpolation corresponding to the projector PP is just that given by Eq.(77). As noted above, spline interpolation also falls directly into this form, so that themulti-directional transfinite interpolation based on splines requires only the determination ofthe splines separately in the individual directions.Although Hermite interpolation can be defined in terms of additional points, and thusbe put in this same form also, the use of projectors allows a more direct statement as follows.For the projectors we have, following Eq.
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