Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 37
Текст из файла (страница 37)
In the first case above, i.e.,Eq. (5), the weight function w( ), being a function of is associated with the grid pointsthemselves, not with their locations. In the second case, Eq. (6), however, the weightfunction w(x) is associated with the locations of the grid points, rather than directly with thepoints. Since there is a relation x( ) representing the locations of the grid points, any weightfunction can obviously be transformed from one argument to the other. However, in derivingthe Euler equations for a variational problem it is only the direct dependence that isconsidered in the partial derivativesor, i.e., whether the weight function isdetermined by the identity of the grid point or by the location of the grid point, w(x),although implicit differentiation is used in the total derivatives.andThe constant in Eq.
(4) can be evaluated by normalizing x to the interval (O,L). Ifnormalized to (1,N) we have from Eq. (4),isand hence(7)so that(8)Sincex=1/x, the transformation is then determined by(9)Thus(10)so that Eq. (2) is realized by taking equal increments in , i.e., varying by equalincrements between grid points as was stated initially. From Eq. (8) the grid point spacing isgiven by(11)An alternative viewpoint results from integrating over x, instead of over , i.e.,summing over the grid intervals rather than over the grid points.
Since identifies the gridpoints, x represents the change in ,i.e., the number of grid points per unit distance, andhence is the grid point density. Eq. (4) is now the Euler equation for minimization of theintegral(12)Here the integral in question is F( x,.)dx, so that the Euler equation is given bySince x can be considered to represent the point density, this variational problemrepresents a minimization over the field of the density of grid points in the least-squaressense, subject to the weight function, and thus produces the smoothest point distributionattainable. Here the weight function w(x) is associated with the grid point locations, notdirectly with the points. If the weight function is associated with the points themselves,rather than the locations, then w = w( ) and the integral for which Eq.
(4) is the Eulerequation is(13)This variational problem is the least-squares minimization over the field of the cumulativepoint density weighted by the weight function.The constant in Eq. (4) is evaluated in this form by writing,so that with the normalization as defined above,(14)The transformation then is given by(15)Thus(16)so that again Eq. (2) is realized by taking equal increments ingiven by.
The point spacing is now(17)The grid and solution may be determined separately, perhaps even in an iterativefashion. However, the transformation allows the grid and solution to be dynamically coupledso that both evolve together. With the spring analogy approach, Eq. (8) supplies thefollowing differential equation for the grid:(18)which supplies an additional differential equation to be solved simultaneously with thedifferential equation system of the physical problem at hand, with the grid point location x asthe independent variable.
Similarly, with the smoothness approach, the differential equationfor the grid is(19)Eq. (18) and (19) really differ only by the way the constant is evaluated, i.e., whetherby integration over or over x. This is a real difference in implementation, though, sinceintegration over is dependent on the grid, but integration over x is not. Thus, with thespring analogy approach, the weight function is associated with the grid points, i.e., with ,and the grid adjusts to achieve a uniform value of w x. The uniform value reached,however, is dependent on the grid since the right-hand side of Eq. (18) is dependent on thepoint distribution. In contrast, in the smoothness approach, where the weight function isassociated with the the grid points, i.e., with x, the grid adjusts to achieve a specifieduniform value of w x, since the right-hand side of Eq.
(19) is an integral in physical space,independent of the grid. In the first approach, the points move to change the spacing xbetween points, while in the second the points move to change the point density x. (Notethat Eq. (4) can also be written as x/w=constant.) Either approach is viable, unless it isintended that the uniform value of w x be fixed beforehand, as would be the case if theweight function is taken to be representative of truncation error and a certain bound is to beimposed on this error. The smoothness approach, i.e., integration over x, has been the mostwidely used because it is natural in most physical problems to associate the weight functionwith some physical property which varies in space.Implementation of the two forms proceeds as follows: The form based on the gridpoint density is implemented using Eq.
(15). With the solution u( ) known on the currentgrid points at a given time step, the weight function is evaluated at each point and then theintegral in the denominator of Eq. (15) is evaluated by numerical quadrature, i.e., bysumming the product w x over the grid points using coefficients in the summationappropriate to whatever type of numerical quadrature is intended. The integral in thenumerator is similarly evaluated out to values of the upper limit x that produce thesuccessive integral values of which define the grid points. Thus we have xi defined by(20)These values of xi then are the new grid point locations, and the solution proceeds to the nexttime step.The spring analogy form, however, requires iteration.
Here we have, from Eq. (8), thepoint locations xi defined by(21)With the solution known at a given time step, the weight function is evaluated at each gridpoint, and the integral in the denominator is evaluated numerically as before. Then theintegral in the numerator is evaluated with the upper limit set at the successive integralvalues of as indicated, and this defines a changed point distribution, xi. The complicationhere is that the integral in the denominator, i.e., the constant in Eq. (4), depends on the pointdistribution, amounting to a sum of 1/w over the points since=1 by constructionregardless of the distribution.(By contrast, the corresponding integral in Eq.
(20), i.e., the constant in Eq. (4), does notdepend on the point distribution, being simply an integral of a function in physical space.)Therefore, this integral must be re-evaluated using the changed point distribution.The integral in the numerator is then also re-evaluated for each point, thus changing the pointdistribution again. This process must be continued until convergence before the final newpoint distribution is obtained. The solution then proceeds to the next time step.
The necessityfor iteration with the spring analogy form clearly makes this form more difficult toimplement than the grid point density form. Since no particular advantages of the formerhave been noted, preference naturally falls to the latter.A number of examples of both the point density form and the spring analogy form, aswell as other applications of the use of one-dimensional equidistribution are cited in thesurvey of adaptive grids given as Ref. [45].C. Weight functionsAs noted above, the effect of the weight function w is to reduce the point spacing xwhere w is large, and therefore the weight function should be set as some measure of thesolution error, or as some measure of the solution variation.
The simplest choice is just thesolution gradient, i.e.,(22)In this case, Eq. (4) becomeswhich then reduces toWith the solution gradient as the weight function the point distribution adjusts so that thesame change in the solution occurs over each grid interval, as illustrated below:This choice for the weight function has the disadvantage of making the spacing infinitelylarge where the solution is flat, however.A closely-related choice, also based on the solution gradient, is the form(23)An increment of arc length, ds, on the solution curve u(x) is given byso that this form of the weight function may be writtenand then Eq.
(4) becomeswhich reduces toThus, with the weight function defined by Eq. (23), the grid point distribution is such that thesame increment in arc length on the solution curve occurs over each grid interval. For thecurve shown above this gives the following point distribution:Unlike the previous choice, this weight function gives uniform spacing when thesolution is flat. The concentration of points in the high-gradient region, however, is not asgreat. This concentration can be increased, while still maintaining uniform spacing where thesolution is flat, by altering the weight function to(24)where is a parameter to be specified. Considering u to be plotted against x/ , we have foran increment of arc length on this solution curveso that this weight function is equivalent toand Eq.
(4) becomeswhich reduces toThus we have equal increments of arc length on the solution curve with u plottedagainst x/ in this case. Now division of the abscissa by a for a flat curve would simplyreduce the spacing by the same factor. However, since the slope steepens as the curve iscompressed to the left by this change of scale, the effect on the spacing where the curve isnot flat will be a greater reduction in spacing.In fact, since the 1 in the weight function given by Eq. (24) tends to produce equal spacing,while the 2ux2 tends to produce concentration in the high-gradient regions, with infinitespacing in flat regions, this weight function involves a weighted average between thetendency toward equal spacing and that toward concentration entirely in the high-gradientregions. The larger the value of , the stronger will be the concentration in the high-gradientregions and the wider the spacing in the flat regions.Now a disadvantage of all the above forms of the weight function is that regions nearsolution extrema, i.e., where ux=0 locally, are treated similar to flat regions, as is illustratedbelow for the form given by Eq.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
