Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 39
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Variational ApproachConsidering the grid from a continuous viewpoint, it occurs that something should beminimized by the grid rearrangement, and thus a variational approach is logical. This is thenatural extension of the equidistribution concept discussed above to multiple dimensions.The development in this section is a generalization of that in Ref. [47]. (cf.
Ref. [1] forearlier related work.)A. Variational formulationThe variational formulation for multiple dimensions can be constructed in analogywith the one-dimensional equidistribution discussed in Section 1. Thus in general a weightedintegral measure of the accumulation of some grid property Q, either over the grid points,i.e.,(44)or over the physical field, i.e.,(45)where w is the weight function, will be minimized. The resulting Euler equations then willconstitute the grid generation system. In formulating the variational problem there arebasically three decision points.First, if the integration is taken over then the integral represents a summation overthe grid points, while integration over x represents a summation over cell volumes inphysical space.
With integration over it is thus the accumulation of some property over thegrid points that is minimized, while with integration over x the accumulation over thephysical cell volumes is minimized.The second question concerns the weight function. If the weight function is directlydependent on , then the weight is associated with the grid points, while with weightfunctions dependent directly on x the weight is associated with location in physical space. Asnoted in Section 1 it is this direct dependence of the weight function that figures in thepartial derivativesandin the Euler equations, the fact that a change of variablecould be effected by the transformation x( ) notwithstanding. In most applications theweight function will be based on some solution gradient and hence will be naturally taken asa function of position in physical space, x.Finally, there is the choice of what property is to be accumulated to be minimized.This choice depends, of course, on what is expected from the grid.
Among the gridproperties that might be considered are the following in computational space (integrationover grid points, i.e., d ):(1). square of cell volume:(2). inverse cell volume:(3). sum squares of cell edge lengths (average of squares of diagonal lengths):(4). cell area squared/volume ratio:(5). cell skewness based on edge tangents:(6). cell skewness based on face normals:In two dimensions the two orthogonaly properties, (5) and (6), are equivalent.These six properties correspond in order to the use of the following properties inphysical space, where the integration is over the physical field (dx):(1).
inverse point density:(2). square of point density:(3).(4).(5).(6).Similar representations of other grid properties can also be considered, of course. Theone-dimensional forms of properties (1) and (3) in the computational space reduce to ,while those of properties (2) and (4) become 1/x . Therefore, in analogy with theone-dimensional equidistribution in Section 1, a weight function with properties (1) and (3)that is a function of x should actually be squared in the integral (cf.
Eq. (6)), i.e.,(46a)while w(x) with properties (2) and (4) appears as (cf. Eq. (12))(46b)Similarly, weight functions that are functions ofshould appear as (cf. Eq. (5) and (13))(47a)(47b)The construction for integration in the physical space is analogous, but noting that (1) and(3) correspond to 1/ x, while (2) and (4) correspond to(13) and (12), respectively):, in one dimension (cf. (5), (6),(48a)(48b)The grid for which the weighted accumulation of the property Q is minimized isobtained, by the calculus of variations, as the solution of the Euler variational equations forthe integral I. If the integration is overthese equations are(49)where F is the integrand of the integral I.
With integration over x the variational equationsare(50)These partial differential equations then constitute the generation system for the grid. Notethat the equations resulting from Eq. (50) must be transformed using the relations in ChapterIII so that the curvilinear coordinates become the independent variables. The equations givenby Eq. (49), however, will already be in this form.A grid generation system which involves competitive emphasis on various gridproperties can be constructed by casting the integral to be minimized as a weighted averageof several of the above integrals, each of which represents an accumulation of a differentgrid property.
Since the various grid properties do not all have the same dimensions, it isnecessary to scale the various integrals involved, as is done below for the Brackbill-Saltzmanconstruction.There clearly is no unique construction of the variational formulation for adaptivegrids, and this is an area that is not yet fully developed.
The constructions given later in thischapter are logical and illustrative of the procedure, but should not be considered definitive.B. Euler equationsThe derivation of the Euler equations, hence the grid generation system, isstraightforward but may be algebraically involved. The following developments simplify thederivation somewhat. Consider first the integral over the grid points(51)where g is the covariant metric tensor, with elements gij defined by Eq.
(III-5), and w(x) is aweight function dependent on x. The Euler equations then are given by Eq. (49). As shownin Appendix B, the Euler equations produce the following generation system (withwritten as F’):(52)where(53)Here the gradient of the weight function in the last term is expressed using Eq. (III-42), withi given by Eq. (III-33). It should be noted that if the weight function in the integral (51)had been defined as a function of instead of x, a result different from Eq.(52) would havebeen obtained for the generation system (cf.
Eq. (9) of Appendix B). The two-dimensionalform of Eq. (52) is given as Eq. (10) of Appendix B.With the variational problem formulated in the physical space, and the weightfunctions dependent on x, we have the integral(54)where G is the contravariant metric tensor, i.e., with elements gij from Eq.(III-37). Thenfrom the Euler equations given by Eq. (50), cf. Appendix B, the generation system is (withwritten as F’),(55)withC. Brackbill-Saltzman constructionAs noted in Chapter V there is a need for smoothness in the grid in order to reducecertain terms in the truncation error of a solution done on the grid. The quantitythe extension to multiple dimensions of theused above with the smoothness form in Eq.(12). Therefore to maximize the smoothness of the grid it is logical to minimize the integralof this quantity over the physical field:(57)This amounts to a minimization of the linear point density in the least-squares sense. Theproperty used here is that given as (4) on p.396, which corresponds to the ratio of the squaresof the cell face areas to the cell volume when the accumulation is over the grid points, asgiven by property (4) on p.
395. The corresponding integral over the grid points is(58)Substitution of F from Eq. (57) into Eq. (19) of Appendix B then yields the elliptic gridgeneration system(59)Thus the smoothest grid is that for which the curvilinear coordinates satisfy Laplace’sequation.Emphasis on orthogonality and/or on concentration of grid lines can also beincorporated into the grid generation system by basing the system on the Euler equations foradditional variational principles. Orthogonality can be emphasized by minimizing theintegral Io defined with property (6) on p.
396 as(60)i is normalsince each of these dot products vanishes for an orthogonal grid. (Recall thatto the coordinate surface on which i is constant, cf. Chapter III.) The inclusion of the g3/2,the cube of the Jacobian of the transformation, as a weight function in Io is somewhatarbitrary, and causes orthogonality to be emphasized more strongly in the larger cells. Withthe accumulation over the grid points, this corresponds to the use of the square of the dotproduct of the cell face normals in the variational statement (property (6) on p.395).
Thecorresponding integral over the grid points is(61)Finally, concentration can be emphasized by minimizing the integral Iw defined by(62)where w(x) is a specified weight function. This causes the cells to be small where the weightfunction is large, and uses property (1) on p. 396, i.e., the inverse point density. With theaccumulation over the grid points this corresponds to the use of the square of the cell volume(property (1) on p. 395), and the integral over the grid points is(63)The grid generation system is obtained by minimizing a weighted sum I of these threeintegrals:(64)where N is a characteristic number of points, L is a characteristic length, and W is theaverage weight function over the field:(65)with V being the volume of the field. This sealing in the weighted sum is obtained asfollows: From the above expressions for Is, Io, and Iw we haveTherefore, the three terms in Eq.
(64) should stand in the ratios given. In two dimensions thefactors on Io and Iw both become (N/L)4, since the Jacobian is then proportional to (L/N)3,rather than to (L/N)3. The characteristic length and number of points might logically betaken as the cube roots of the volume and the total number of points in the field,respectively, in three dimensions, the square root being used in two dimensions.Emphasis is varied among the competing features of smoothness, orthogonality, andadaptivity by the choice of the coefficients o and w.
For example, a large o will result ina grid that is nearly orthogonal, at the cost of smoothness and concentration, with ananalogous effect of w. The Euler equations for this variational problem, which will be theweighted sums of those for the individual integrals, form the system of partial differentialequations from which the coordinate system is generated.
These equations will bequasilinear, second-order partial differential equations, with coefficients which are quadraticfunctions of the first derivatives, and are derived in general as described in the preceedingsection and Appendix B.Clearly the integral I3, Eq. (57), is the multi-dimensional generalization of theone-dimensional smoothness integral I3, Eq. (12), without the weight function, and theintegral Iw, in Eq. (63), is the extension of the one-dimensional spring analogy integral I1,Eq.
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