Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 20
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This evaluation of the control functions from theboundary point distribution is discussed more fully in Section 2 of this chapter.D. General Poisson-type systemsIf a curvilinear coordinate system,i (i=1,2,3),which satisfies the Laplace systemis transformed to another coordinate system, (i = 1,2,3), then the new curvilinearcoordinates, i satisfy the inhomogeneous elliptic system (cf. Ref.
[19])(4)where(5)with thedefined by the transformation fromi toi:(6)(It may be noted that if the subsequent transformation is one-dimensional, i.e., ifthen only the three functionswith i=1,2,3, are nonzero.)These results show that a grid with lines concentrated by applying a subsequenttransformation (often called a "stretching" transformation) to a grid generated as the solutionof the Laplace system could have been generated directly as the solution of the Poissonsystem (4) with appropriate "control functions",derived from the subsequentconcentrating transformation according to Eq.
(6). Therefore, it is appropriate to adopt thisPoisson system (4) as the generation system, but with the control functions specified directlyrather than through a subsequent transformation.Thus an appropriate generation system can be defined by Eqs. (4) and (5):(7)with the control functions,considered to be specified. The basis of the generation system(7) is that it produces a coordinate system that corresponds to the subsequent application of astretching transformation to a coordinate system generated for maximum smoothness.
FromEq. (6), the three control functions(i = 1,2,3) correspond to one-dimensional stretchingin each coordinate direction and thus are the most important of the control functions. Inapplications, in fact, the other control functions have been taken to be zero, i.e.,so that the generation system becomes(8)It may be noted that, using Eq.
(III-37), Eq. (7) can be written as(9)Actual computation is to be done in the rectangular transformed field, as discussed inChapter II, where the curvilinear coordinates, i, are the independent variables, with thecartesian coordinates, xi, as dependent variables. The transformation of Eq. (9) is obtainedusing Eq. (III-71). Thus we have(10)But2=0 and then using Eq. (7), we have(11)This then is the quasi-linear elliptic partial differential equation which is to be solved togenerate the coordinate system.
(In computation, the Jacobian squared, g, can be omittedfrom the evaluation of the metric coefficients, gij in this equation since it would cancelanyway, cf. Eq. III-38.) As noted above, the more common form in actual use has been thatwith only three control functions, Eq. (8), which in the transformed region is(12)Most of the following discussion therefore will center on the use of this last equation as thegeneration system. This form becomes particularly simple in one dimension, since then wehave(13)which can be integrated to give, withThe one-dimensional control function corresponding to a distribution x( ) thus is given by(14)In two dimensions, Eq.
(11) reduces to the following form, using the two-dimensionalrelations given in Section 8 of Chapter III (with 1 = and 2 = )(15)where(16)(17a)(17b)with=x+y and(18a)(18b)(18c)This corresponds to the following system in the physical space, from (7),(19a)(19b)where g = (x y- x y )2.The two-dimensional form of the simpler generation system (12) with only two controlfunctions is(20)for which the system in the physical space is, from (8),(21a)(21b)This generation system has been widely used, and a number of applications are noted in Ref.[1] and [5]. Several examples appear in Ref. [2].Substitution of (3) in (10) gives the transformation of the original Poisson system (3)as(22)This generation system has also been widely used, cf.
Ref. [1] and [2], and thetwo-dimensional form is(23)corresponding in the physical space to(24a)(24b)This system has also been widely used (cf. Ref. [1] and [5]), and its use predates thatof Eq. (21). In general, however, the form of (12), corresponding to the system (8), isprobably preferable over that of (22), which corresponds to (3), because of the simple formto which the former reduces in one dimension, and because the control functions in (8) areorders of magnitude smaller than those in (3) for similar effects.E. Other systemsOther elliptic systems of the general form (4) have been considered, such as withPi=gPi where the Pi are the specified control functions, and with,where D is the control function.
The latter form puts Eq. (4) in the form of a diffusionequation with the control function in the role of a variable diffusivity:(25)This system also corresponds to the Euler equations for maximization of the smoothness, butnow with the coefficient, D, serving as a weight function, i.e., multiplying the integrand inEq. (2), so that the smoothness is emphasized where D is large. Both of these systems haveactually been implemented only in two dimensions, although the formulations are general.Specific references to these and other related systems are given in Ref. [1] and [5].Another elliptic system for the generation of an orthogonal grid has been constructedby combining the orthogonality conditions,) with a specified distribution ofthe Jacobian over the field,.
(This system is discussed further in Chapter IX.)Some two-dimensional applications appear in Ref. [2], as noted in Ref. [5].The second-order systems allow the specification of either the point distribution on theboundary (Dirichlet problem):or the coordinate line slope at the boundary (Neumann problem):but not both. Thus it is not possible with such systems to generate grids which are orthogonalat the boundary with specified point distribution thereon. (This assumes that the controlfunctions are specified.
It is possible to adjust the control functions to achieve orthogonalityat the boundary as is discussed in Section 2.)2,A fourth-order elliptic system can be formulated by replacing the Laplacian operator,with the biharmonic operator, 4. The analogous form to (4) then is(26)which can be implemented as a system of two second-order equations:(27a)(27b)From (III-71) and (22) above, the transformed system is(28a)(28b)This generation system, being of higher order, allows more boundary conditions, so that thecoordinate line intersection angles, as well as the point locations, can be specified on theboundary. It is therefore possible with this system to generate a coordinate system which isorthogonal at the boundary with the point distribution on the boundary specified, and forwhich the first coordinate surface off the boundary is at a specified distance from theboundary:This allows segmented grids to be patched together with slope continuity as discussed inChapter II.In the above discussions, generation systems have been formulated based on lineardifferential operators in the physical space, e.g., the Laplacian with respect to the cartesiancoordinates, resulting in quasi-linear equations in the transformed space where thecomputation is actually performed.
It is also possible to formulate the generation systemusing linear differential operators in the transformed space, e.g., the Laplacian with respectto the curvilinear coordinates:(29)The use of some such generation systems is noted in Ref. [1], and such a biharmonic systemis noted in Ref. [5]. Although this certainly produces simpler equations to be solved, sincethe computation is done in the transformed space, such systems transform to quasi-linearequations in the physical space, and hence the extremum principles are lost in the physicalspace. This means that there is a possiblity of coordinate lines overlapping in generalconfigurations.
Therefore it is generally best to formulate the generation system using linearoperators in the physical space.As noted above, other variations of elliptic systems of the type discussed here arenoted in Ref. [1] and [5]. Elliptic generation systems may also be produced from the Eulerequations resulting from the application of variational principles to produce adpative grids,as is discussed in Chapter XI. Still another system, based on the successive generation ofcurved surfaces in the three-dimensional region, is given in Section 3B of this chapter.Finally, quasiconformal mapping (Ref. [22] and [23]) is another example of an ellipticgeneration system.2. Control FunctionsFor the elliptic generation system given by Eq.
(12), the control functions that willproduce a specified line distribution for a rectangular region, and for an annular region, aregiven as Eq. (14) and in Exercise 8, respectively. These functions could be used in otherregions, of course, with the same general effect. In such extended use, the former would bemore appropriate for simply-connected regions, while the latter would be appropriate formultiply-connected regions. Use of the rectangular function in a multiply-connected regionproduces a stronger concentration than was intended because of the concentration overconvex boundaries that is inherent in Poisson-type generation systems (cf.
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