Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 29
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The slopes at the ends are defined by the directions 1- o and N- N-1. The curvepasses close to the mid-point of each side, with the exception of the first and last sides. Thecurve also passes through the points ( k-1 + 4 k + k+1)/6 for k = 2,3,...,N-2. These pointsare one-third of the way along the straight line joiningk-1 andkto the mid-point of the line joiningk+1.Since the B-splines are non-zero only on four intervals, the alteration of onevertex only affects the curve in its immediate vicinity.
An application of B-splines in gridgeneration is given in Ref. [39].G. Multi-surface interpolationThe multi-surface method, discussed in Ref. [32]-[36], is also a unidirectionalinterpolation procedure. This procedure is constructed from an interpolation of a specifiedvector field, followed by vector normalizations at each interpolation point in order to cause adesired telescopic collapse so that the boundaries are matched. The specified vector field isdefined from piecewise-linear curves determined by the boundaries and successiveintermediate control surfaces.
Normals to such surfaces are special cases. Polynomialinterpolants for the vector field yield all of the classical polynomial cases along with arational method for avoiding disasters such as can occur with direct Hermite interpolationwith excessively large or discontinuous derivatives. Here the immediate surfaces are notcoordinate surfaces, but are used only to define the vector field.
These vectors are taken to betangents to the coordinate lines intersecting the surfaces, so that integration of this vectorfield produces the position vector field for the grid points.A collection of subroutines which automatically perform the necessary parts of gridconstruction using this multi-surface procedure has been written and is described in Ref.[34]. Some of the automation features of this collection are applicable to other gridconstruction procedures as well.
These subroutines can rotate and move curves, prospect onecurve from another, normalize and parameterize curves, cluster points on a curve, andperform other such utilitarian functions to aid in the setup of an overall configuration.In the multi-surface interpolation we have(22a)where(22b)and where the n are specified points, with 1= (0) and N= (I), on the boundarysurfaces. (Recall the discussion at the beginning of this section, i.e., that the points can beconsidered to lie on curves or surfaces and thus the interpolation, while being fundamentallybetween points, can be considered to be between the surfaces on which those points lie.)Here the telescopic collapse for the series for =I matches the boundary at I.
Theintermediate points here,2,3....,N-1,are not grid points, but serve only to define theslopes, as given by Eq. (24) below. Eq. (22) is a polynomial if the functionspolynomials, but such is not required.nareSince, by differentiation of Eq. (22),(23)we have, for 0= 1< 2<....< n-1=I,(24)if the functionsnsatisfy the cardinality conditions(25)The polynomials that satisfy these conditions are simply the Lagrange polynomials given byEq. (4), here stated as(26)Using Eq.
(24), Eq. (23) can be written as(27)This form thus is based on an interpolation of the first-derivatives, instead of, theinterpolation expression for coming from an integration of the interpolation function for. Note, however, that this amounts to the specification of the slopeat particularvalues of the curvilinear coordinate n, and not at a specified position in space as is done inthe Hermite form. It is clear from Eq. (24) that the intermediate points,serve to define the slopesnfor n=2,3,....,N-2,( n):Because of the integration involved, the degree of the interpolation polynomial will be onegreater than that of the functions n.Also with Eq.
(24), the interpolation for, Eq. (22), can be written(28)and thus is equivalent to a form of deficient Hermite interpolation. In implementation,however, it is the points n that are specified, as in Eq. (22). Again it should be recalled thatthe pointnisnot the grid point atAs has been noted,1andn,nexecept for the boundaries1=0andN=I.are determined by the boundaries:(29a)For N 4, 2 andthrough Eq. (24):N-1 aredetermined by the intended values ofat the boundaries(29b)For N=3 only one of the above equations can be used, i.e.,can be specified at eitherFor N=3 only one of the above equations can be used, i.e.,can be specified at eitherboundary but not at both. The use of the intermediate surfaces, instead of direct specificationof the derivativesas in classical Hermite interpolation, provides a geometricinterpretation that serves to help avoid the overlapping of grid lines that can occur if toolarge a value is given for.Following Ref.
[35], consider now for thediagramed below:nthe piecewise-linear functionswith the normalizationwheren = n+1 for n=2,3,....,N-2,nso that eachnintegrates to unity. These functions are given by,(30)With these functions we have(31)Note that Gn(I) = 1 here for all n and that Gn( 1) = 0,These interpolation functions have the formNow on the intervaln+1 wenhaveGm( )=1 for m=1,2,....,n-1 and Gm( )=0 for m=n+2,n+3,....,N-1.
Therefore, on this intervalwhich, because of the telescopic collapse of the summation, reduces to, fornn+1,(32)Then(33a)and(33b)Also, from Eq. (32), fornn+1(34)so that(35a)and(35b)We thus have on this intervalThus on the intervalnn+1,( ) is affected only byn,n+1,andn+2.Conversely n affects only the grid point locations on the interval n-2n+1.Therefore local adjustments in the grid point locations can be made without affecting all ofthe points.With the grid points located at unity increments ofpoints, we have, from Eq.
(33), the grid points given by, so that I is the total number ofThe local control provided by these piecewise linear interpolants can be used torestrict undesirable mesh forms or to embed desirable ones within a global system withcontinuous first derivatives (cf. Ref.
[34]-[35]). The second derivatives are, however,discontinuous. As examples, the propagation of boundary slope discontinuities can bearbitrarily restricted and general rectilinear Cartesian systems can be embedded to simplifyproblems over a large part of their domain.In a further development (cf. Ref. [36]) the procedure is extended to use piecewisequadratic local interpolants, thus achieving continuity of second derivatives, withdiscontinuous third derivatives.
The conceptual extension to higher order piecewisepolynomial local interpolants, with consequent higher degree of continuity, is also discussed.Note that because of the integration of n, the level of derivative continuity is always onegreater than that of the piecewise polynomials.H. UniformityIt may be desirable for purposes of control of the grid point distribution to have auniform distribution of the relative pro)ection of ( )- (0) along the straight lineconnecting the boundary points, i.e., (I)- (0). This property has been called "uniformity"by Eiseman, cf.
Ref. [32] - [36], and can be realized as follows: The unit vector along thisstraight line isso that the relative pro)ection of( )- (0) along this line is given by(36)where(37)Uniformity is then achieved by choosing the interpolation parameters such that S( ) islinear. This does not completely determine all the interpolation parameters, however, so thatsome remain to be specified as desired. Uniformity is trivially assumed for linearinterpolation of course.For Lagrange interpolation we have, from Eq. (2),(38)so that uniformity is achieved by selecting the n, for n=2,3,....,N-1, to cause all of the termsin S that are quadratic or higher to vanish.
For Hermite interpolation, (5),(39)For the multi-surface interpolation defined by Eq. (22) we have(40)For S( ) to be linear we must haveor(41)But, using Eq. (25), we then have(42)as the uniformity condition on the n’s (cf. Ref. [35]). Both the n (for n=2,3,....,N-1) andthe n (for n=2,3,....,N-2) are free to be chosen in order to satisfy the uniformity conditions(42). Thus a one-parameter family of cubic forms (N=4) results, a two-parametric family ofquartic forms, etc. Substitution of Eq. (42) back into (41) yields a restriction on the choice ofthe functions n since these must satisfy the relation(43)Uniformity is particularly useful when the distribution function, such as thosediscussed in the next section, is used to redistribute the points on the grid lines set up by theinterpolation (cf. Ref.
[34]-[36]). Thus the interpolation is first applied with I=1 and with theuniformity conditions enforced. The final grid points then are placed according to thedistribution function on the grid lines set up by the interpolation variable in place of the arclength, s, in the distribution function s( ).I. Functions other than polynomialsThe interpolation functions in the general forms given by Eq. (2), (5), and (22) do nothave to be polynomials, and, in fact, if the variation in spacing over the field is large, otherfunctions are better suited for grid generation. With N=2, Eq.
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