Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 36
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It is necessary forconvergence that the near circle be sufficiently near to being a circle. A series for thedifferential form is generally superior to the usual Theodorsen form for general bodies. Thisseries appears in terms of arc length and surface angle, rather than the polar coordinates ofthe Theodorsen form which can lead to infinite derivatives and multiple values. The orderingof the points can break down in the Theodorsen form for closely spaced points also. Thedifferential form is applicable, however, as long as there are no corners, even for twistedcontours.
In this and other series transformations, the differential form is usually moretolerant of odd shapes.Multiple-body configurations can be treated by a sequence of transformations whichmap each body to a circle in succession, while maintaining previously established circles.Another procedure, invovles iteratively mapping each body to a circle with no specialconsideration of the others. This process generally requires only a few iterations to converge.Some recent applications are noted in Ref. [1] and [5].XI. ADAPTIVE GRIDSIn an adaptive grid, the physics of the problem at hand must ultimately direct the gridpoints to distribute themselves so that a functional relationship on these points can representthe physical solution with sufficient accuracy.
The idea is to have the grid points move as thephysical solution develops, concentrating in regions of large variation in the solution as theyemerge. The mathematics controls the points by sensing the gradients in the evolvingphysical solution, evaluating the accuracy of the discrete representation of the solution,communicating the needs of the physics to the points, and finally by providing mutualcommunication among the points as they respond to the physics. The basic techniquesinvolved then are as follows:(1) a means of distributing points over the field in an orderly fashion, so that neighborsmay be easily identified and data can be stored and handled efficiently.(2) a means of communication between points so that a smooth distribution ismaintained as points shift their position.(3) a means of representing continuous functions by discrete values on a collection ofpoints with sufficient accuracy, and a means for evaluation of the error in this representation.(4) a means for communicating the need for a redistribution of points in the light of theerror evaluation, and a means of controlling this redistribution.Several considerations are involved here, some of which are conflicting.
The pointsmust concentrate, and yet no region can be allowed to become devoid of points. Thedistribution also must retain a sufficient degree of smoothness, and the grid must not becometoo skewed, else the truncation error will be increased as noted in Chapter V. This meansthat points must not move independently, but rather each point must somehow be coupled atleast to its neighbors. Also, the grid points must not move too far or too fast, else oscillationsmay occur.
Finally the solution error, or other driving measure, must be sensed, and theremust be a mechanism for translating this into motion of the grid. The need for a mutualinfluence among the points calls to mind either some elliptic system, thinking continuously,of some sort of attraction (repulsion) between points, thinking discretely.
Both approacheshave been taken with some success, and both are discussed below. It should be noted that theuse of an adaptive grid may not necessarily increase the computer time, even though morecomputations are necessary, since convergence properties of the solution may be improved,and certainly fewer points will be required.With the time derivatives at fixed values of the physical coordinates transformed totime derivatives taken at fixed values of the curvilinear coordinates, no interpolation isrequired when the adaptive grid moves.
Thus, as given by Eq. (III-116),(1)where xi andi arethe cartesian and curvilinear coordinates, respectively. The computationthus can be done on a fixed grid in the transformed space, without need of interpolation,even though the grid points are in motion in physical space. The influence of the motion ofthe grid points is registered through the grid speeds, (xi)t, appearing in the transformed timederivative. This is the appropriate approach when the grid evolves with the solution at eachtime step. Some methods, however, change the grid only at selected time steps, and hereinterpolation must be used to transfer the values from the old grid to the new since the gridmovement is not continuous.In the following discussion, the problem of grid adaption will be formulated as avariational problem, the ideas being developed first in one dimension and then extended tomultiple dimensions.1.
One-Dimensional AdaptionA. EquidistributionA number of studies of numerical solutions of boundary-value problems in ordinarydifferential equations have shown that the error can be reduced by distributing the grid pointsso that some positive weight function, w(x), is equally distributed over the field, i.e.,(2)or, in discrete form,(3)where xi is the grid interval, i.e., xi = xi+1 - xi. (The subscript here indicates position onthe line in this one-dimensional case.) With this condition, the grid interval will, of course,be small where the weight function is large, and vice versa.
Thus if the weight function issome measure of the error, or the solution variation, the grid points will be closely spaced inregions of large error, or solution variation, and widely spaced where the solution is smooth.(It may be more appropriate in some cases to replace the equal sign in Eq. (2) and (3) with"less than or equal", and thus to "sub-equidistribute" the weight function.)This approach has also been applied to redistribute the grid points (or to add points) ateach time step, or at certain intervals, in numerical solutions of initial/boundary-valueproblems in one-dimensional partial differential equations.
A number of references to the useof equidistribution are cited in Ref. [45]. It can be shown that the point distribution isasymptotically optimal if some error measure is distributed evenly, and that this optimumerror is rather stable under perturbations of the point distribution. Thus it is not necessary tolocate the grid points with excessive accuracy.B. Equidistribution by transformationThe nonuniform point distribution can be considered to be a transformation, x( ),from a uniform grid in -space, with the coordinate serving to identify the grid points.The grid points are conveniently defined by successive integer values of , making=1by construction and the maximum value of , i.e., N, equal to the total number points on theline.
Then x = x= x so that x represents the variation in x between grid points.Hence the equidistribution statement, Eq. (3), can be represented as(4)With the weight function w taken as a function ofminimization of the integralthis is just the Euler equation for the(5)(From the calculus of variations, the function x( ) for which the integral F(x,x )dis anextremum is given by the solution of the differential equation.This equation is called the Euler’s variational equation.) The integral (5) can be taken torepresent the energy of a system of springs, with spring constants w( ), spanning each gridinterval, considering all the points to have been expanded from a common point so that xis the extension of the spring at .
The grid point distribution resulting from theequidistribution thus represents the equilibrium state of such a spring system, i.e., the state ofminimum energy. Since x represents the distance between grid points, this variationalproblem can also be interpreted as the minimization of the cumulative spacing between thegrid points in the least-squares sense, subject to the weight function w( ).If the weight function is taken to be a function of x, instead ofwhich Eq. (4) is the Euler equation is, then the integral for(6)The variational problem in this case is the least-squares minimization over the grid of thecummulative grid point spacing weighted by the weight function.Integration over , as in both these cases, constitutes a summation over the gridpoints, with x representing the spacing between grid points.
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