Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 38
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(22):Although the distributions produced by the solution arc length forms, Eq. (23) and (24),would have closer spacings near the extrema, the effect is still the same, i.e., to concentratepoints only near gradients, not extrema.Concentration near solution extrema can be achieved by incorporating some effect ofthe second derivative uxx into the weight function. A logical approach is to include thiseffect through consideration of the curvature of the solution curve:If the weight function is taken as(25)then points will be concentrated in regions of high curvature of the solution curve, e.g., nearextrema, with a tendency toward equal spacing in regions of zero curvature, i.e., where thesolution curve is straight (not necessarily flat). This weight function, however, has theserious disadvantage of treating high-gradient regions with little curvature essentially thesame as regions where the curve is flat.
Thus in the curve shown above, nearly all the pointswould be concentrated near the maximum in the curve, with very wide spacing in thehigh-gradient regions on both sides.A combination of the weight functions given by Eq. (24) and (25) provides the desiredtendency toward concentration both in regions of high gradient and near extrema. The effectof the inclusion of the curvature is illustrated below (cf. Ref. [37]) with the functionfollowing):(26)where and are parameters to be specified. Clearly, concentration near high gradients isemphasized by large values of , while concentration near extrema (or other regions of largecurvature) is emphasized by large .Another approach to the inclusion of the second derivative is simply to take the weightfunction as(27)and are non-negative parameters to be specified.whereWith this form, (cf. Ref.
[46] we have by Eq. (15), withand,(28)so that(29)Then with R1 defined as(30)we have(31)Since= 1/N, where I+1 is the number of points on the coordinate line, the maximumpercentage change in the solution over a grid interval,(32)is related to the ratio R1, which measures the relative emphasis put on concentration ofpoints according to the solution gradient by(33)A guide for the choice of to limit the maximum percentage solution change over aninterval to a value r can then be obtained using an equality in Eq. (33) with R1 from Eq.
(30)and neglecting the effect of the term:(34)The smallest possible value of r is 1/N.With the second derivative term included, the value of can be continually updated tokeep the same relative emphasis on concentration according to this term, as measured by theratio R2.(35)The transformation can then be written as(36)where R2 is considered to be constant. In this form, the transformation appears as theweighted average of one based on the solution gradient and one related to the secondderivative.The replacement of Eq. (24) with the form given by Eq. (27), with = 0, still leaves areasonable form for the weight function, but the clear association with the geometricproperties of the solution curve are lost.
In this case the weight function corresponding to Eq.(23) would, after substitution in Eq. (4), leads to the conditionwhich corresponds to an equal distribution of the distance between points on the solutioncurve along a right-angle path formed by x and u from one point to the next. While thisdistance has some indirect relation to arc length on the solution curve (the chord length beingthe hypotenuse of the right triangle formed by this x and u), the direct association witharc length would seem to be preferable.
Following the same reasoning, the use of solutioncurve curvature, rather than simply the second derivative, is also preferable. Therefore, theform given by Eq. (26) is probably more appropriate than that of Eq. (27). A number of othervariations have been used, of course, as is noted in Ref. [45].Since the numerical evaluation of higher derivatives can be subject to considerablecomputational noise, the use of formal truncation error expressions as the weight function isusually not practical, hence the emphasis above on solution gradients and curvature. Someproblems may arise even with solution curvature, i.e., with second derivatives, in roughtransits.
It is common in any case to limit the grid point movement at each time step and/orto smooth the new point distribution.For systems of equations involving more than one physical variable, one approach isto use the most rapidly-varying or dominant physical variable in the definition of the weightfunction. Another is to use some average of the variations of the several variables. It is alsopossible to use entirely different grids for different physical variables, with values transferedamong the grids by interpolation.
Examples of each of these approaches are cited in Ref.[45] and [5].2. Multiple-Dimensional AdaptionA. Adaption along fixed linesIn multiple dimensions, adaption should in general occur in all directions in a mutuallydependent manner. However, when the solution varies predominately in a single direction,one-dimensional adaption of the forms discussed above can be applied with the grid pointsconstrained to move along one family of fixed curvilinear coordinate lines, and applicationsof this approach are noted in Ref.
[45].The fixed family of lines is established by first generating a full multi-dimensionalgrid by any of the grid generation techniques discussed in the earlier chapters, with thecurvilinear coordinate lines of one family therein then being taken as the fixed lines. Thepoints generated for this initial grid, together with some interpolation procedure, e.g., cubicsplines, serve to define the fixed lines along which the points will move during the adaption.The one-dimensional adaption discussed above is then applied with x replaced by arc lengthalong these lines.Examples (cf.
Ref. [46]) of application of the point density form discussed above inthis manner are shown in the following figures. The first figure shows an adaptive grid for acombustion problem, where the adaption is along fixed radial lines. The flame front isclearly visable here because of the strong concentration of points therein:The oscillations evident with the fixed grid are removed by the grid adaption.
An extensionof this problem appears next with a flowing gas. This gives an example of the use of separateadaptive grids for different physical variables of the problem, one for the combustion andone for the fluid mechanics, with values transferred between the two grids by interpolation.Adaption through the spring analogy is illustrated next with adaption along fixed linesbetween the body and outer boundary in a hypersonic flow problem (cf. Gnoffo in Ref.
[45]).Here the concentration of points makes the shock location evident in the grid:Another obvious application of adaption along fixed lines is adaption of boundarypoints along a fixed boundary in two dimensions (cf. Nakamura in Ref. [45]). An example ofsuch adaption along a boundary as a shock forms appears below:B. Uncoupled adaptionOne step beyond this one-dimensional adaption along fixed lines is the application ofsuccessive one-dimensional adaptions separately in each of the curvilinear coordinatedirections. This proceeds in the same manner as for the adaption on the fixed lines, simplyusing the latest grid to re-define the coordinate lines to serve as the "fixed" lines in the nextdirection of adaption, cf.
Ref. [56] and [57]. In the latter a torsion spring analogy is used, aswell as the tension springs discussed above, incorporating resistance to movement awayfrom orthogonality. This is done in effect by adding the term v( )(x-xo)2 to the integral ofEq. (5), where v( ) is a second weight function and xo is the arc length location of theintersection of the normal from the adjacent grid line with the line on which the adaption isoccurring.C. Coupled adaptionThe final grid in the one-dimensional adaption discussed above will, of course, be theresult of the grid point movement along the one family of fixed lines, and therefore thesmoothness of the original grid may not be preserved as the grid adapts.
Some restrictions onthe point movement have generally been necessary in order to prevent excessive griddistortion.In multiple dimensions, in general it is desirable to couple the adaption in the differentdirections in order to maintain sufficient smoothness in the grid. One approach to suchcoupling is to generate the entire grid anew at each stage of the adaption from some basicgrid generation system, be it algebraic or based on partial differential equations. Thestructure of the grid generation system serves to maintain smoothness in the grid as theadaption proceeds.
In this approach, which is analogous to the one-dimensionalequidistribution discussed above, the new point locations are determined directly from thegrid generation system, and then the grid point speeds, , for use in the transformed timederivatives, Eq. (1), are calculated from the change in the point locations by differenceexpressions. Another approach is to determine the grid point speeds directly through someprocess and then to calculate the new point locations by integrating these point speeds.D.
Weight functionsThe one-dimensional weight function, Eq. (23), based on arc length on the solutioncurve can be generalized to higher dimensions as follows: Consider a hyperspace ofdimensionality one greater than that of the physical space, with the solution, u, being theextra coordinate. Let the unit vector in the solution direction be , this being orthogonal tothe physical space.
Then the position vector in this hyperspace is given by(37)where is the position vector in physical space. Now, following Eq. (III-5), the covariantmetric element, denoted Gij, in the hyperspace will be(38)where gij is the metric element in physical space. Now(39)so thatand then(40)It can be shown that(41)(This has been verified for one and two dimensions.)In one dimension this reduces to the expression for arc length on the solution curve,i.e.,In two dimensions Eq. (41) gives an expression for area on the solution surface:(42)Thus the extension of the one-dimensional weight function based on arc length on thesolution curve to two dimensions is that based on area on the solution surface:(43)The extension of this form to three dimensions would also seem logical, but has not beenverified.3.
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