Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 24
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(Greek coordinate indices are used here to set apart the coordinates generated on asurface from those generated in a three-dimensional region in general).The unit normal, the coefficients band the surface Christoffels ( ) have all beendefined in Eq. (15), (20), and (33) of Appendix A, respectively. The indices , each assumethe two values different from . For each , with ( , , ) taken in cyclic order, we have(70a)(70b)(70c)withassuming the two values, , .A surface grid generation system that is analogous in form to that based onPoisson-type equations in a plane given earlier can be constructed by multiplying Eq.(70a,b,c), respectively, by Galgebra,g, 2Gg,Ggand adding. This given, after some(71)where(72)(73)(74)(75)The quantitiesandare the local principal curvatures of the surface=constant.
It must be noted here the R as defined in (74) is based on the intrinsic valuesof b . That is, the bare solely determined by the data and coordinates as available inthe surface. If, however, it is desired to use Eq. (71) for generating a series of surfaces in athree-dimensional space, as in the following section, from the data of a given surface, then itis desirable to have an extrinsic form for b . To obtain the extrinsic form, we use Eq. (29)of Appendix A, i.e.,(76)Equating the right hand sides of Eq. (69) and (76), taking the dot product withsides, and noting thaton both, we get(77a)where(77b)Thus(78)The operator 2 is called the Beltrami second-order differential operator, and ingeneral is defined as(79)Thus(80a)(80b)The generation system is now formed by taking, in analogy with the system (7),(81a)(81b)where and each assume the two values , in the summation.
Here theare thesymmetric control functions. Thus the equations for the generation of surface grids are (with=x+y+z)(82)where(83)(84)(85)The left-hand side of Eq. (82) here corresponds exactly to that of Eq. (15) for the plane.However, here we have in place of (18) the relations(86a)(86b)(86c)The effect of the surface curvature enters through the inhomogenous term, inparticular through R ( ) which is, in fact, equal to twice the product ofand the meancurvature of the surface. Here, as for the plane, the control functions,, are considered tobe specified. This system corresponds to the following system in the physical space, from(81),(87a)(87b)Thus the Beltrami operator on the general surface replaces the Laplacian operator in theplane. If the surface is a plane, the Beltrami operator reduces to the Laplacian.If only the two control functionsandsystem reduces to the more practical systemare included, the surface grid generation(88)corresponding to the plane system given by Eq.
(20). In the physical plane this system is(89a)(89b)Clearly, we could also replace the system (89) with the simplier system(90a)(90b)in analogy with the sytem (24) in the plane, to obtain the surface grid generation system(91)which is analogous to the plane system (23).Equation (71) is the basic equation for the generation of curvilinear coordinates in agiven surface.
From (74) the function R( ) depends on the principal curvaturesand. The sumis twice the mean curvature of the surface, and its value is invariantto the coordinates introduced in the given surface. If the equation of the surface in the formx3 = f(x1,x2) is available, then from elementary differential geometry(92)whereFor arbitrary surfaces it is always possible to use a nummerical method, e.g., the least squaremethod, to fit anequation in the form x3=f(x1,x2) or F(x1,x2,x3)=0 and to obtain the neededpartial derivatives to find+as a function of x1,x2,x3.
(Surface grids have beenobtained for simply and double connected regions in a surface using the above method.)It may be desired in some applications to generate a new coordinate system based onan already existing coordinate system in a given surface. In the formulation of this problemEq. (71) can have the form of R( ) given in (74), (75) or (78). Let the surface on which thenew grid is to be generated be specified parametrically by(93)(For example, the parameters (u,v) might be latitude and longitude on a spherical surface.) Ifthe specified cartesian coordinates on the surface form a finite set of discrete points, asmooth interpolation scheme is needed to recover the differentiable functions in (93). Toattain the desired smoothness in the parametric representation (93), it is generally preferableto divide the given surface into a suitable number of patches such that each patch isrepresentable by a bicubio spline with suitable blending functions.
Having once establishedthe smooth parametric functions (93), it is now possible to introduce any other desiredcoordinate system, say (,,) on the surface. For example, a surface coordinate systemof the configurationmight be generated on a surface defined by the parametric coordinates (u,v) in alatitude-longitude configuration:Alternatively, a surface may be defined in terms of cross-sections, in which case one of theparametric coordinates (u,v) runs around the section and the other connects the sections:The fundamental equations for the generation of (=constant can be obtained from Eq.
(75) in the form,) on the surface(94)Here we have taken(95a)(95b)Now using the chain rule of differentiation, we can writeterms of u,for example,v,uu,,etc., inetc. Thus,(96)(97)(98)with the quantities as defined below. Substituting these derivatives in (94), and also in theexpressions forand, we get(99)where(100)(101)(102)(103)To isolate the differential equations for u and v as dependent variables from (99), wetake the dot product of Eq.
(99) with u, and then with v, and use the conditionsWritingthe required equations are(104a)(104b)where(105a)(105b)(105c)(106a)(106b)Note that the metric quantities with an overbar relate to the surface definition in terms of theparametric coordinates and therefore can be calculated directly from the surfacespecification, Eq. (92).Clearly we could redefine the control functions so that (104) is replaced by thefollowing system, which is analogous in form to the plane system (20):(107a)(107b)B.
Three-dimensional gridsAs mentioned earlier, the system of Eq. (71), or Eq. (82), is also capable of generatingthree-dimensional grids. This capability in the set of equations is incorporated through( ) as defined in (78).The strategy of the method is to generate a series of surfaces on each of which twocurvilinear coordinates vary while the third remains fixed. The variation along the thirdcoordinate is specified as a surface derivative condition, which in turn depends on the givenboundary data.A study of Eq.
(82) - (84) shows immediately that for the solution of Eqs. (82) weneed to specify the values of andon certain curves=constant. To fix ideas, letus consider the problem of coordinate generation between two given surfacesandas shown below:The coordinates on these surfaces areand. To start solving these equations weneed the values of andon the surfacesand.
These values ofare the input conditions for the solution of Eqs. (82), and are either prescribed analytically ornumerically. On the other hand, the values ofat B and are available easily, basednumerically. On the other hand, the values ofat B and are available easily, basedon the values of , simply by numerical differentiation. The values of rin the field foreach surface to be generated are then obtained by interpolation between the available valuesof ()B and () . A simple formulae which has been used with success is(108)where4. ImplementationThe setup of the transformed region configuration is done as described in Chapter II.This includes the placing of the cartesian coordinates of the selected points on the boundaryof the physical region into ijk for each block and the setting of the interfacecorrespondence between points on the surrounding layer for each block and points inside thesame, or another, block via input to an image-point array as described in Section 6 ofChapter II.A.
Difference equationsImplementation of an elliptic generation system then is accomplished by devising analgorithm for the numerical solution of the partial differential equations comprising thegeneration system. Recall that the use of the surrounding layer for each block, as describedin Section 6 of Chapter II, allows the same difference representations that are used in theinterior to be used on the interfaces.
The usual approach is to replace all derivatives in thepartial differential equations by second-order central difference expressions, as given inChapter IV, and then to solve the resulting system of algebraic difference equations byiteration. As noted above, most generation systems of interest are quasilinear, so that thedifference equations are nonlinear.A number of different algorithms have been used for the soltuion of these equations,including point and line SOR, ADI, and multi-grid iteration (cf.
Ref. [1] and [5]). For generalconfigurations, point SOR is certainly the most convenient to code and has been found to berapid and dependable, using over-relaxation, for a wide variety of configurations. Theoptimum acceleration parameters and the convergence rate decrease as the control functionsincrease in magnitude. Some consideration has been given to the calculation of a field oflocally-optimum acceleration parameters (cf. Ref.
[1]), but the predicted values generallytend to be too high, and the desired increases in convergence rate were not obtained.Since the system is nonlinear, convergence depends on the initial guess in iterativesolutions. The algebraic grid generation procedures discussed in Chapter VIII can serve togenerate this initial guess, and transfinite interpolation generally produces a more reliableinitial guess than does unidirectional interpolation because of the reduced skewness in theformer. In fact with strong line concentration, convergence may not be possible from aninitial guess constructed from unidirectional interpolation, while rapid convergence occursfrom an initial guess formed with transfinite interpolation. With the slab and slitconfigurations, the interpolation must be unidirectional between the closest facing boundarysegments as illustrated below:In a block structure, however, the slab/slit configuration can be avoided so that transfiniteinterpolation can be used.Since the coordinate lines tend to concentrate near a convex boundary, very sharpconvex corners may cause problems with the convergence of iterative solutions of thegeneration equations.
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