Thompson, Warsi, Mastin - Numerical Grid Generation, страница 40

(5), to multiple dimensions, with the spring extension x generalizing to the volume,i.e., (the Jacobianin three dimensions, area in two). This variational approach thus is ageneralization of the one-dimensional equidistribution discussed above to multipledimensions. All of the discussion of weight functions given above in regard toequidistribution therefore has relevance here to the weight function of the integral Iw, Eq.(63). (The role of the constant in the equidistribution weight function, e.g., the 1 in Eq.

(23),etc., which tends to produce a linear transformation, is taken by the smoothness integral Is ofEq. (57), which tends to produce an equally-spaced grid in multiple dimensions.)For the three integrals given by Eq. (58), (61), and (63) we have, respectively, with(i,j,k) cyclic,(66)(67)(68)Here, of course, from Eq. (III-14),In two dimensions, g13 = g23 = 0 and g33 = 1, so that these functionals reduce to(69)(70)(71)(Here an additive constant in Fs has been dropped since only derivatives of F contribute tothe Euler equations.) Then using Eq. (1) of (Appendix B) the two-dimensional generationsystem based on concentration alone is(72)and the generation system based only on orthgonality is(73)The generation system based on smoothness (from Eq.

(66)) is more complicated, but maybe constructed from the relations given in Appendix B. The complete generation system thenis obtained as the linear combination of the concentration system, Eq. (72), the orthgonalitysystem, Eq. (73), and the smoothness system.In Ref. [47] this combination is written in the form(74a)(74b)wherewithHere the coefficients subscripted s, o, and w, arise from the smoothness, orthogonality, andconcentration integrals, respectively. The coefficients ’w and ’o are, taking account of thescaling discussed above in connection with Eq. (64),(75a)(75b)In one dimension, with y=xAlso, for the smoothness integral:= 0, we haveFor the concentration integral:and for the orthogonality integral:ThenNow alsoand, taking the-direction to be the one of interest, we also have yThe generation system in one dimension then reduces to= 0.Thecan be made a part offinally isso that the one-dimensional generation system(76)with, for the scaling,(77)This then is the differential equation that can be applied on a boundary curve,interpreting xas arc length and as the curvilinear coordinate that varies along the particular boundary.In three dimensions the calculation of the required partial derivatives of F in Eq.

(52)for the concentration integral, i.e., Fw given by Eq. (68), may be expedited by noting thatsince,where cij is the signed cofactor of gij. These second derivatives vanish if k=i or l=j, and areequal to gmn otherwise, where (i,k,m) and (j,l,n) are cyclic, the sign being negative whenthe progression from i to k is opposite to that from j to l.D. ApplicationsThe dynamically-adaptive grid is applied by constructing the partial differentialequations which constitute the grid generation system from the Euler equations as discussedabove.

These equations are solved numerically by replacing all derivatives with differenceexpressions (typically second-order, central differences) in the same manner as discussed inChapter IV. As noted in Chapter III, the time derivatives in the equations of the physicalproblem to be solved on the grid are transformed according to Eq. (III-116), with the resultthat, the grid point speeds appear in the difference equations of the physical problem. Thegrid is re-generated at each time step, and these grid point speeds are determined fromdifference representations between time steps. Although the difference equations for the gridand those for the physical solution could be iterated together at each time step, the morecommon procedure is to solve each separately at each time step.Grid points on boundaries may, of course, be held fixed, but it is more appropriate inmost cases to allow the points to move along the boundary to adapt as in the field.

This canbe accomplished either by using Neumann boundary conditions in the grid generationsystems, i.e., making the system orthogonal at the boundary (cf. Chapter VI), or by applyingthe one-dimensional form of the grid generation equations, Eq. (76), in terms of arc length,along the boundary.Some rather spectacular two-dimensional results of the grid adapting to a reflectedshock are shown below for supersonic internal flow over a step.

The formation and multiplereflections of the shock are made evident by the grid adaption into the shock as it develops.Here the magnitude of the pressure gradient was used in the weight function, and bothsmoothing and bounding was applied to the weight function to control grid distortion.E. ExtensionsIn two dimensions, the departure of the grid from conformality can be controlled bybasing F on the Cauchy-Riemann conditions:(79)and some applications are noted in Ref.

[45].Finally, another very useful addition to this composite variational system is a controlon the grid point movement, which can be incorporated by taking F as(80)where ui is the fluid velocity and is the grid speed. With represented by a differenceform, and with the fluid velocity evaluated at the previous time step, this F can be consideredto be a function of xi. Again some applications are noted in Ref.

[45]. The followingexample shows the effectivenss of such control of the grid point movement:4. Other ApproachesSeveral other approaches are discussed in Ref. [45], three of which follow here.A. Attraction-RepulsionAnother approach to adaptive grids is to let the grid points all move as if under themutual influence of forces between all points. Here instead of generating new grid pointlocations through the solution of partial differential equations, the grid points move directlyunder the influence of mutual attraction or repulsion between points. This is accomplishedby assigning to each point an attraction proportional to the difference between the magnitudeof some measure of error (or solution variation) and the average magnitude of this measureover all the points. This causes points with values of this measure that exceed the average toattract other points, and thus to reduce the local spacing, while points with a measure lessthan the average will repel other points and hence increase the spacing.This attraction is attenuated by an inverse power of the point separation distance in thetransformed field.

The collective attraction of all other points is then made to induce avelocity for each grid point. Since each point is influenced by all other points, this iseffectively a type of elliptic generation system. Details of implementation are given in Ref.[48] and other references cited therein.Smoothing through the addition of diffusion -- like terms in the calculation of the gridevolution from the grid speeds has also been used. Reflections in boundaries in thetransformed field are used to provide smooth grid motion near and on the boundaries. Sincethe transformed field is rectangular, this reflection is not complicated by the shape of thephysical boundaries.

A means of including terms that will induce rotational motion into thegrid has been devised to cause the grid lines to align with lines of high gradients such asshocks.This procedure does not exercise any control over either the smoothness ororthogonality of the grid, so that distortion is possible. Collapse of points into each other is,however, impeded because attraction will become repulsion as the points approach eachother, since the measure which drives the motion will drop below the average as the spacingdecreases.

Collapse is further impeded by the fact that the grid velocity decreases with thespacing. It has been found necessary to apply some limits and some damping of the gridspeeds to prevent grid oscillation and distortion. In practice, the computed grid speeds arescaled so that the maximum over the field is a set value, but with the maximum sealing alsolimited. Provision is also made for exponential damping of the grid speeds according to theratio of the maximum Jacobian to a specified value.Since this procedure has all grid points moving to cause some measure to approachuniformity over the field, it can be considered an iterative approach to the equidiatribution ofthis measure over the field.