Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 27
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Then use the volume distribution from this unequally-spaced cylindrical coordinatesystem as V( , ), with corresponding to the points around the circle, ( ), andcorresponding to the radial distribution r( ). An example of grids generated by thisprocedure follows:The specification of the cell volume prevents the coordinate system from overlappingeven above a concave boundary. In this case the line spacing will expand rapidly away fromthe boundary in order to keep the cell volume from vanishing, as in the Following figure.Although this prevents overlap, the rapid expansion that occurs can lead to problems withtruncation error in some eases. This approach is extendable to 3-D with the coordinate linesemanating from the boundary being orthogonal to the other two coordinates, but the lattertwo lines not being orthogonal. There apparently is no system, hyperbolic or elliptic, thatwill give complete orthogonality in 3-D.This hyperbolic grid generation system is faster than the elliptic generation systems byone or two orders of magnitude, the computational time required being equivalent to aboutthat for one iteration in a solution of the elliptic system.
The specification of the cell volumedistribution avoids the grid line overlapping that otherwise can occur with concaveboundaries in a method involving projection away from a boundary. The grid may, however,be somewhat distorted when concave boundaries are involved. The cell volume specificationalso allows control of the gird line spacing, of course, as in the upper part of the secondfigure on p. 275, but again concave boundaries may cause the intended spacing to occur inthe wrong coordinate direction, as in the lower part of this figure, since it is only the volume,and not the spacing in the two separate coordinate directions, that is controlled. As has beennoted, the grid is constructed to be orthgonal.The hyperbolic generation system is not as general as the elliptic systems, however,since the entire boundary of the region cannot be specified.
As noted above, boundary slopediscontinuities are propagated into the field, so that the metric elements will bediscontinuous along coordinate lines emanating from boundary slope discontinuities. Finally,since hyperbolic partial differential equations can have shock-like solutions in somecircumstances, it is possible for very unsuitable grids to result with some specifications ofboundary point and cell volume distributions.
This is in contrast with the elliptic generationsystems which tend to emphasize smoothness because of the nature of elliptic partialdifferential equations.2. Parabolic Grid GenerationParabolic grid generation sytems may be constructed by modifying elliptic generationsystems so that the second derivatives in one coordinate direction do not appear. Thesolution then can be marched away from a boundary in much the same manner as describedabove for the hyperbolic systems. Here, however, some influence of the other boundarytoward which the marching progresses is retained in the equations.In Ref. [30] such a parabolic generation system is formed essentially by firstrepresenting all derivatives in an elliptic generation system with second-order centraldifferences and then replacing all values on the forward line in one coordinate direction, say= j+1, with values specified in some manner in terms of the values on the preceeding linesand specified values on the outer boundary.
This reduces the difference equations to a set of2x2 block tridiagonal equations to be solved on each coordinate line in succession,proceeding away from a specified boundary. Control of the coordinate line spacing can beachieved by certain control functions that are drawn from some analogy with the ellipticsystem. It is possible to use the functional specification of the forward values to cause thegrid to be nearly-orthogonal.The parabolic generation system is also faster than the elliptic generation systems tothe same degree as is the hyperbolic system, since again only a succession of tridiagonalsolutions is required.
The functional specification of the forward values, with an influence ofan outer boundary, introduces a smoothing effect from this second boundary not present inthe hyperbolic system. Orthogonality is not achieved as directly as with the hyperbolicsystem, however. The forms of the forward value specification, and of the control functions,have not yet been well-developed.VIII.
ALGEBRAIC GENERATION SYSTEMSAs noted earlier, the problem of generating a curvilinear coordinate system can beformulated as a problem of generating values of the cartesian coordinates in the interior ofthe rectangular transformed region from specified values on the boundaries.
This, of course,can be done directly by interpolation from the boundaries, and such coordinate generationprocedures are referred to as algebraic generation systems. Thus ( 1, 2, 3) is given as aspecific function of the curvilinear coordinates. This function contains certain coefficientswhich are determined so that the function matches specified values of the cartesiancoordinates, and perhaps derivatives also, on the boundary and perhaps elsewhere.Evaluation of this interpolation function at constant values of the curvilinear coordinatesthen defines the coordinate system. Algebraic grid generation is discussed in Ref. [31] and[8], as well as in the surveys, Ref.
[l], [5] and [37], and in detail in Ref. [32-36].1. Unidirectional InterpolationUnidirectional interpolation means the interpolation is in one curvilinear coordinatedirection only. In this section the cartesian coordinate vector will be shown as a functionof the coordinate involved in the interpolation, as the unidirectional interpolation isfundamentally between points. These points can, however, lie on boundary (and perhapsinterior) curves or surfaces, and in this sense the unidirectional interpolation can beconsidered to be between these curves or surfaces. Therefore the single-variable functionalrelationship ( ) used in this section can be considered to represent dependence on allcoordinates, the interpolation points i being functions of the coordinates along theboundary curves or surfaces.A.
Lagrange interpolationThe simplest type of unidirectional interpolation is Lagrange interpolation, which is,based on polynomials. In the linear form we have, with(1)Here1=(0) and2=(I), so that( ) is defined in terms of the two boundary values,and 2. The grid points are located at the successive integer values of from 0 to I. Onefamily of grid lines will be straight lines connecting corresponding boundary points with thislinear interpolation.1The general form is(2)withn=( n), and the functionssuch thatnbeing polynomials defined on the entire interval(3)In the linear case given above we have, with N=2,From Eq. (2) and (3),so that the interpolation function matchesat the N points=1,2,...,N=I:The specified interior points, n for n=2,3,...N-1, are not necessarily grid points, sincethe grid points are defined by evaluating the interpolation formula at successive integervalues of , but are simply additional parameters that serve to control the distribution.
It ispossible to specify the locations of certain interior grids points, however, by taking the ncorresponding to the specifiednto be the value ofat the grid point of interest.The Lagrange interpolation polynomials, defined to satisfy by Eq. (3), are in general(4)The quadratic forms thus are, with N=3, and2= I/2.for which( ) is defined in terms of the two boundary values,1and3,and one interiorvalue, 2. It should be noted that the purpose of the inclusion of the interior points in gridgeneration is control of the grid point distribution, not to increase the accuracy of theinterpolation as is normally the case. There is, in fact, no question of accuracy of theinterpolation here, since the aim is just to generate a grid from the boundary values of thecoordinates.B.
Hermite interpolationLagrange interpolation matches only function values. It is possible to match bothfunction, , and first-derivative, ’ =, values using Hermite interpolation defined by(5)where the Hermite inerpolation polynomials are defined onconditionsand satisfy theThese polynomials can be obtained from the Lagrange interpolation polynomials by(6a)(6b)where the prime here indicates differentiation of the polynomial with respect to theargument,. With N=2 we haveand the function matches the two boundary values,and’ ,21and2,and the first derivatives,’1at the two boundaries.Extensions of polynomial interpolation to match higher-order derivatives is obviouslypossible, the degree of the polynomial increasing with each additional condition or point tobe matched. The polynomials of high degree exhibit considerable oscillation, however, sosuch procedures are not of great importance to grid generation.
The general form againincludes matches at interior points, which can be used to control the coordinate line spacing,since the first derivative, ’ =, is a measure of the grid point spacing here, withbeing unity between points by construction. As with Lagrange interpolation, these specifiedinterior points may or may not be grid points.It is also possible, of course, to omit points from either of the summations in Eq. (5),so that and its first derivatives are not both matched at all points (deficient Hermiteinterpolation). Thus, with N=2 and the n=1 term omitted from the second summation, thetwo boundary values would be matched, but the first dervative at only the =I boundarywould be matched. Clearly, the Hermite interpolation form, Eq.
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