Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 32
Текст из файла (страница 32)
(III-74) that(2)The general differential equations satisfied in the transformed region are, from Eq. (VI-10),(3)Substituting Eq. (2) in (3) for the Laplacians, these grid generation equations take thefollowing simpler form for an orthogonal system:(4)whereis the cartesian coordinate vector.On the other hand, starting from Eq. (2), by writingand using the chain rule of differentiation, we get the generation equations in the physicalregion as(5)Another fundamental set of equations for orthogonal coordinates are known as Lame’sequations, stated as(6a)(6b)where (i,j,k) are cyclic.
Equations (6) express essentially the condition that the curvilinearcoordinates are to be introduced in an Euclidean space. (cf. Ref. [27]). In three dimensions,Eq. (6) represents six equations, although thereare only three distinct metric coefficients,h1,h2,h3.In summary the equations (2), (4), (5) and (6), together with the vanishing of theoff-diagonal metric elements, are the fundamental equations which any orthogonalcoordinate system must satisfy.2.
Two-Dimensional Orthogonal CoordinatesThe fundamental equations for two-dimensional orthogonal coordinates are collectedbelow as a particular case of the equations (2) - (6):I. Transformed plane: g12=0 and(7a)(7b)(7c)II. Physical plane: g12=0 and(8a)(8b)(8c)Also, the Laplacians (2) take the simple forms(9a)(9b)Considering Eq. (7a) and (8a), either of which provide the orthogonality condition, itis a straightforward matter to conclude that there exists a positive function F such that(10)and the Eq. (7a) is identically satisfied. In the same manner, from Eq. (8a),(11)It is obvious that the positive function F is related to the grid aspect ratio:(12)The choice of the sign in Eq. (10) and (11) follows from the right-handedness of the system1, 2.Introducing (12) into Eq.
(7b), while using Eq. (9), we get(13)which forms the basic generation system for plane orthogonal coordinates. Though thegenerating equations (7b) and (13) are completely equivalent, nevertheless, the apparentdifference in their structures must be taken into consideration to decide about the type ofboundary conditions for their solution.With Eq.
(7b) as the generating system then the two options are: (i) Specify F=h2/h1 asa known function of 1, 2. This case covers the cases F= and=constant. For any constant , Eq. (9) reduce to the Laplace equationsand Eq. (7b) becomeswhere2 1=0,2 2=0,(14)For =1, the coordinates 1, 2 are isothermic, i.e., h2=h1, and so are conformal. Cases inwhich1 have also been considered, and specific references are given in Ref. [1]. It isalso of interest to state that starting from a conformal system ( 1, 2), yet another system( 1, 2) can be established by transforming the Laplace equations 2 1=0, 2 2=0, suchthat1 and is a product of a function of 1 and a function of 2. (cf. Ref. [1]).
(ii)The other option is to calculate F iteratively. In this case the field values of F are updated byiteratively changing its values at the boundaries under the orthogonality condition g12=0.With Eq. (13) as the generating system, the two Laplaciansbe specified.
Following the nonorthogonal case, let2 1and2 2have to(15a)(15b)where P1,....,Q2 are arbitrary specified functions ofrewrite these equations as1, 2.Using Eqs. (9) and (12) one can(16a)(16b)Thus if P1,....,Q2 are specified, the above equations provide a way to determine F.
(Using theconditionone can establish a fourth order algebraic equation in F.) It is therefore concluded that theuse of Eq. (13) with P1,....,Q2 specified is equivalent to using Eq. (7b) in which F hasexplicitly been specified.The above noted considerations are important in deciding about the type of boundarydata needed for the solution of either Eq. (7b) or Eq. (13). The solution of Eq. (7b) withspecified F, or the solution of Eq.
(13) with specified P1,....,Q2, does not allow an arbitrarypoint distribution on the domain boundaries. The reason for this as follows: For example, ona boundary segmentnormal derivative2==constant if x1( 1,) is prescribed, then from Eq. (10) thebecomes available. If in addition to x1( 1,one also specifiesx2( 1, which amounts to specifying the complete boundary point distribution, then theproblem becomes overdetermined. Thus for the cases under consideration, specification ofthe complete boundary point distribution is not possible.
That is, Eq. (7b) with F specified,or Eq. (13) with specified P1,,....,Q2, cannot be solved when the complete boundary pointdistribution is prescribed. The appropriate boundary conditions for such problems arediscussed in the context of conformal coordinates in Section A.The specification of the complete boundary point distribution is possible in the casewhen Eq. (7b) is solved without specifying F. An iterative approach can be used to updatethe values of F based on the changed values at the boudnaries.
(cf. Section B).A. Conformal systemsConsidering first conformal systems, i.e., with h2=h1 and F=1, the basic equationsfrom (9a,b) are(17a)(17b)Let the domain in which the conformal coordinates are to be generated be bounded bya piecewise-smooth curve on which s is the arc length and n the outward normal. TheCauchy-Riemann equations (17b) on the boundary take the form(18)Referring to the figure below, let the curves1=constant,and the curvesreadily find that on1and32and41andbe those on whichthe condition2 be those portions on which2=constant.
From Eq. (18) we, and on3and4the condition, are to be imposed, where the subscript n indicates the normal derivative.Therefore, for the generation of conformal coordinates, the properly posed boundary valueproblems areon1and2:on3and4:on1and2:on3and4:1=,1=, respectively(19)2=,2=, respectively(20)In the transformed plane the governing equations for conformal coordinates areobtained from (13):(21a)(21b)1Takingand2as monotonically increasing parameters having the ranges,12,, the given equations of the curvesrespectively, can be expressed .in parametric form as1,2,3,4,(22)The specification of the boundary data in the form of (21) should at best be regardedas a statement of the problem, rather than as a procedure, since the exact boundarypoint-distribution in this form is not possible a’priori.
To develop the procedure itself weregard the specification in (22) as an initial guess. However, this type of specificationproduces an overdetermined situation. For example, if on2)are specified, then from the first equation in (21b),boundary. Thus both1both x1(,2)and x2(,can be calculated on thisbecome specified, which makes the problem overdetermined. Following this logic, we canisolate the proper arbitrarily specifed boundary values for Eq.
(21) as follows: specifying x1(,2)on1, x1(,2)on2, x2(1,) on3,and x2( 1,) on4.Thus, forthe x1-equation the normal derivative conditions on 3 and 4 are provided by the secondequation in (21b) through the specified x2 values. Similarly, for the x2-equation the normalderivative conditions onspecified x1-values.1and2are provided by the second equation in (21b) through theIn any numerical procedure, the values of x1 are determined by integration through theformula(23)and these values in turn give the new values of x2 through the exact functional relationsbetween x1 and x2 for these curves.
Similarly, the values of x2 are calculated by the formula(24)and then the new values of x1 are determined by the functional relations between x1 and x2for these curves. Further discussion of conformal systems is given in Chapter X.B. Other systemsFor general orthogonal systems, the basic equations for x1 and x2 remain Eq. (13). Asnoted earlier, the other constraint besides orthogonality (g12=O) is now to specify thefunction F defined in Eq.(12), which is the ratio of the scale factors, i.e., the grid aspect ratio.One approach is to specify the function F explicitly, in which case, as with the conformalcoordinates, it is not possible to specify an arbitrary point distribution on the boundaries.
Theset of equations in (7a) must be used to find the proper x1 and x2 values by integration on theappropriate boundaries. Another alternative is to specify an arbitrary point distribution on theboundaries, and leave the function F to be determined iteratively in the course of the solutionfor the grid. This is done in a manner similar to that used in the GRAPE code, discussed inChapter VI, with new boundary values of the function F being calculated from the presentiterate for the coordinates. The function F in the field is then determined from theseboundary values by either transfinite interpolation or as the solution of Laplace’s equations,the former being found preferable in the cases considered.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
