Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 28
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(5), could be equivalentlydefined in terms of the Lagrangian interpolation form, Eq. (2), with 2N points, since both arepolynomial representations. Obviously either approach can be used to control the grid pointspacing in the field.The capability of specifying, as well as, can be used to make the gridorthogonal at the boundary. From Eq.
(III-33) the unit normal to agiven byi-coordinatesurface isUsing Eq. (III-10) this becomesThe condition for orthogonally at the boundary then is thatunit normal to the boundary:ibe in the direction of the(7)is the spacing off the boundary to be specified. Since all thewherequantities with j and k subscripts can be evaluated from the points on the boundary, itremains only to specify the spacing, si, off the boundary and to use Eq. (7) fori on theboundary in the Hermite expressions.C. Other forms of polynomial interpolationAs noted above, Hermite interpolation, which matchesandat N points, can beequivalently constructed as an interpolant which matches at 2N points. Another form ofexpression of the polynomial interpolation uses the direct expression of the polynomial, sothat(8)Here we must havewhere o and I are the boundaries.
This form is not as straightforward as the Lagrangeform for use in grid generation, since in the latter form certain grid point locations can bespecified directly, while in the former the coefficients must be evaluated in terms of thesespecified points.Still another form is that of Bezier, using Bernstein polynomials:(9)withHere we haveThus the coefficients 1 and N-1 specify the slopes at the boundaries. An advantage of theBezier form is that the coefficients define the vertices of an open polygon to which the curveis an approximation.
Thus the general shape of the curve can be inferred by considering thecoefficients to represent points in the field, with the lines from o to 1, and from N-1 toN,defining the slopes at the two ends. The shape of the curve can then be designed by theplacement of the vertices in the field as indicated below. Modifications of the curve can thusbe made by adjusting the positions of these vertices.Still another form can be defined using piecewise polynomials for the interpolationfunctions. Some degree of continuity must be lost in this case, of course.
Continuity of thegrid lines can be achieved using the piecewise-linear polynomials shown below (truncatedversions apply at or near the end points):while slope continuity can be gotten with the following piecewise polynomials:Such piecewise polynomials allow a greater degree of local adjustment to be made, since thepolynomial n which multiplies the interpolation point n vanishes except in the immediatevicinity ofn.By the conditions (3), any interpolation functionnmust vanish at all theinterpolation points except n, but need not vanish between the points.
Adding moreinterpolation points with global polynomials thus means increasing the degree of thepolynomials, since the numbers of zeros must increase, and hence the polynomial becomeshighly oscillitory.D. SplinesThe Lagrange and Hermite interpolation functions given above are completelycontinuous at all points. Complete continuity, however, may be attained at the price ofoscillation. Both of these forms fit a single polynomial from one boundary to the other,matching specified values of the coordinates and perhaps the derivatives thereof (i.e., thepoint spacing). As more interior points are included, or as the first derivatives are included,the order of this global polynomial increases and thus oscillations become more likely.
Analternative approach is to fit a low-order polynomial between each of the specified interiorpoints, with continuity of as many derivatives as is possible enforced at the interior points.The interpolation function is then a piecewise-continuous polynomial.This type of interpolation function is called a spline and is formed as follows for the mostcommon case of the cubic spline.With a cubic polynomial fitted between points i andof the second derivative between these points and thusi+1 wehave a linear variation(10)After two integrations and evaluation of the two constants ofintegration such thatand( i+1)= i+1, we have oni( i)= ii+1(11)Then, after differentiation and setting= i, we have onii+1,(12)Similar evaluation on the adjacent intervali-1i gives(13)Equating these two expressions in order to produce continuity of r’ at the interior points, wehave(14)which is a tridiagonal equation for " at the interior points.
It is necessary to set someconditions on " on the boundaries in order to solve this system, and the "natural" splineuses 1" = I" = 0. This choice minimizes the total curvature, and thus the natural spline isthe smoothest interpolant. This solution defines theof these values fori"i interms of thei,so that substitutioninto Eq. (11) then gives the spline in the general form of Eq.
(2),except that the interpolation functions, n, are, of course, different from the Lagrangeinterpolation polynomials. It should be recalled again that the interior points may or may notbe grid points, the latter being defined by the interpolation formula evaluated at successiveinteger values of after the spline has been constructed over the entire field.E. Tension splinesThe spline tends to give a very smooth point distribution. Stronger localized curvaturearound the specified interior points can be obtained with the tension spline. Here Eq.
(10) onii+1 is replaced by(15)where 2 is a constant to be specified. (The tension spline tends progressively toward alinear function for large values of , and toward a cubic spline for small values.) Integrationand evaluation of constants then yields, on ii+1,(16)The requirement of continuity of first derivatives at the interior points then yields thetridiagonal equation(17)where i =Ref.
[33].F. B-Splinesi+1 -i andi-1 =i-i-1.Some application of tension splines are given inOne further possibility is to use piecewise continuous functions which satisfy thecardinality conditions by vanishing identically outside some interval around n, as discussedin Section C above. This type of function allows the interpolation to be modified locallywithout affecting the interpolation function elsewhere. The B-splines are an example of thisapproach.From Eq. (14) a cubic spline which matches the function at N points, with continuityof second-derivatives, requires N+2 items of data, i.e., the N values of n (n=1,2,...N) andthe values of " at each boundary. Therefore a cubic spline which has = ’ = " = 0 ateach boundary can be defined over five points if is specified at only a single interior point(since N+2=7 data items can be specified here).
If such a spline over five points is joined tothe line = 0 outside these five points, we have a function which is non-zero only over fourintervals and yet which has continuous second derivatives everywhere. Such a function iscalled a B-spline, denoted N4N( ), where the end-points of the non-zero interval are n-4and n. Similarly, quadratic, linear and constant B-splines are non-zero over three, two, andone intervals, respectively, and are denoted Nqn, where q = 3, 2, and 1. The end-points of theinterval of non-zero values for these splines are n-q and n.
The specification of a singlevalue in this interval is usually replaced by the specification of the integral over the intervalso that(18)The practical importance of B-splines is that any spline of order q (the cubic spline isof order 4) can be expressed as a sum of multiples of B-splines. Thus the cubic spline can bewritten as(19)Since the B-splines are non-zero only over four intervals, the modification of one coefficienthere only affects the function over four intervals, thus allowing more localized control of theresulting grid.The B-splines can be calculated from the recurrence relation(20)Thus N4n( ) requires the successive calculation of N1,n-1, N2,n-1, N2,n, N3,n-1, N3,n, andfinally N4,n. The constant B-spline, NN1,n-1, used to start this calculation, is given on the4,ninterval1,n-1n-2For the pointintervals,n-1 by N1,n-1=1 and vanishes elsewhere.nwe have, in view of the vanishing of the B-splines outside four(21)which is a tridiagonal relation (N+1 equations) for the coefficients n, o= o and N= N.Thus, even though the modification of a single coefficient only affects four intervals, themodification of an interpolation point requires a re-determination of all the coefficients andthus affects the function over the entire range.The coefficients, n, in the B-spline representation may be interpreted as the verticesof an open polygon, to which the curve is an approximation, as for the Bezier form discussedabove.
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