Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 35
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Note that theinterpolation scheme may affect the orthogonality of the coordinate system to some degree.2. Schwarz-Christoffel TransformationConformal mappings of circular disks or half-planes onto polygonal regions aredefined by the Schwarz-Christoffel formula.
Suppose the points 1, 2,...., n, lie on the realaxis of the -plane. Then the mapping defined by(19)transforms the upper half plane onto a polygonal region with interior angles of - i = i.However this is not exactly what is needed in most grid generation problems. Presumablyone would be given a polygonal region with vertices z1, z2,...,zn. Thus the parametersA,B, 1, 2,..., n must be determined so that the real axis maps onto the given polygon:There are several numerical techniques for the approximation of the parameters in theSchwarz-Christoffel transformations. Since a conformal mapping of a simply-connectedregion has three degrees of freedom, three of these parameters must be given in order for themapping to be uniquely determined.
In certain infinite regions, the value of B can becalculated from the asymptotic behavior of the mapping function. We can also set zn= n=0,which implies that A=0 from Eq. (19). The remaining parameters to be determined are1, 2,..., n-1.Alternately, as is commonly done in bounded regions, we can choose thevalues 1, 2, 3 which are to map the points z1,z2,z3.
In this case the parameters to bedetermined are A,B, 4, 5,..., n. The basic algorithm for determining the unknown45nparameters consists of computing the distances, using Eq. (19) and a quadratureformula to approximate the integral, and then iterating on the parameters until thesedistances are correct. Once these parameters in the transformation have been computed to thedesired accuracy, the image of any point in the upper half-plane is formed by numericallyevaluating the integral in Eq. (19).Sohwarz-Christoffel transformations are not limited to regions with polygonalboundaries.
They can be used in composition with other conformal mapping methods to mapregions with curved boundaries onto various computational regions. For example, an integralequation method can be used to map a physical region with curved boundary componentsonto the unit disk, which can be easily transformed onto the upper half-plane. Now the upperhalf-plane can be mapped onto the computation region, which may consist of severalrectangular blocks, by Eq.
(19). There are also direct generalizations of theSchwarz-Christoffel transformation for regions with curved boundaries. These are obtainedby considering the limiting case of Eq. (19) as n-> .Recent extensions of the Schwarz-Christoffel transformation to curved contours havemade this procedure a powerful tool for treating complicated internal and otherconfigurations. These improvements also lead to smoother metric coefficients for boundarieswith slope discontinuities than in older methods for the Schwarz-Christoffel transformation.This procedure for the Schwarz-Christoffel transformation may also be more efficient thanother conformal procedures involving an intermediate mapping of a near-circle for mappingcontours and circles in some eases.
Several sources on the recent developments andapplications of the Schwarz-Christoffel transformation are cited in Ref. [1] and [5].3. Construction from Integral EquationsIntegral equations have played a major role in the solution of partial differentialequations. Mathematicians have often resorted to integral equations when attempting toprove the existence and uniqueness of solutions. Numerical analysts turned to the so-calledpanel methods for solving partial differential equations in two and three-dimensionalregions. These mehtods replaced the partial differential equations by a set of integralequations and thereby reduced the dimension of the problem, since panel methods onlyinvolve boundary integrals.
The application of integral equations depends on the availabilityof fundamental solutions of the partial differential equation. Therefore they are especiallyuseful in the solution of Laplace’s equation. Numerous solutions of Laplace’s equation canbe generated by determining the real and imaginary parts of analytic functions. As mostconformal mappings can be reduced to the solution of boundary-value problems forLaplace’s equation, it should come as no surprise that integral equations can be a valuabletool in the construction of conformal mappings.
Only the basic integral equation method ofSymm (cf. Ref. [1]) will be presented here. This method has proven to be robust, yet is easilyderived and involves only the solution of a system of linear equations.Suppose the simply-connected region D, bounded by the contour , is to be=0. If the Dirichlet problemLet z=zo be the point in D which maps to the origin<1.(20)can be solved and the harmonic conjugate h of q can be found, then it can be directly verifiedthat the analytic function(21)maps onto<1. Due to the form of the series expansion for the exponential, it can alsobe shown that this function has a nonvanishing derivative, and hence the conformal mappingof D onto the unit disk is given by Eq.
(21). We now turn to the problem of solving theboundary value problem in Eq. (20). Suppose there exists a solution of the form(22)for z on . Regardless of the value of the function ( ), the function q(z) is harmonic on D.In order that q(z) satisfy the boundary condition, it is clear that we need to choose ( ) suchthat, for z on ,(23)This is then the integral equation for determining the unknown function ( ). The harmonicis arg(z). Thus the function of h(z) can be expressed asconjugate of(24)Note that the function h(z) is only unique up to an addition constant.
The addition of aconstant to h(z) results in a rotation of the conformal mapping defined in Eq. (21).The practicality of this method depends on the efficient solution of the integralequation in Eq. (23). In order to solve this equation numerically, divide into n intervals,j,j=1,2,...,n and assume ( ) has a constant value j, on j. Let z j be a fixed point ofNow Eq. (23) can be approximated by the linear system of equationsj.(25)There are two alternatives in computing the coefficients in this system. If the j are assumedto be straight lines, then the integrals can be calculated analytically. Otherwise, each integralmust be computed numerically.
Once these coefficients have been computed, the system canbe solved to yield a step function which approximates the function ( ). The values of jare now used to estimate the functions q(z) and h(z):(26)Again the above integrals would, in general, be computed numerically. These values of q(z)and h(z) would be substituted in Eq. (21) to yield the image in the unit disk of any givenpoint z in the region D.This integral equation method is a very efficient and accurate method. However, it hasone deficiency in regard to grid generation and the numerical solution of partial differentialequations. The transformation which is constructed maps the physical region D onto thecanonical region, which in this case is the unit disk. The unit disk could be the computationalregion, or it could be mapped onto a rectangular region by an auxiliary transformation. Inany case, what is needed is the mapping from the unit disk onto the physical region.Therefore an interpolation scheme would be needed to approximate the inverse of thecomputed mapping.It is sometimes more efficient to generate the final grid by solving the Laplace systemnumerically with Dirichlet boundary conditions from the conformal transformations,especially if a fast Poisson solver can be applied.4.
Elementary Complex TransformationsAn extensive list of complex mappings is compiled in Ref. [44]. However, thesemappings are only for regions with special boundary curves. If a strictly conformaltransformation is not necessary, then these mappings may be used to create what are callednearly conformal mappings. For example, suppose an airfoil shape can be modeled as theimage of a circle under the Joukowski transformation(27)Under the inverse transformation, a given airfoil will map to a curve which is nearly circular.The region about the nearly circular curve can be mapped onto the region about a circularregion by a simple algebraic transformation.
One scheme for accomplishing this finalmapping would be to divide each complex number on a given ray from the center by themodulus of the complex number on the curve. The composite mapping in this case would bea nearly-conformal mapping of the exterior of the airfoil onto the exterior of a circle. Theinverse mapping, which could be explicitly defined, would define a nearly-orthogonal 0-typegrid about the airfoil.Analytic functions are not only of value in mapping regions about airfoils, but are alsohelpful in the more general problem of generating grids in the neighborhood of boundarypoints with slope discontinuities.
With most algebraic methods of grid generation, theseslope discontinuities will propagate into the physical region resulting in non-smooth gridlines and the associated increase in truncation error in the numerical solution of partialdifferential equations. The general idea can be conveyed with the following example.Suppose we have a region where the boundary has an interior angle of at the point zo.Under the mapping(28)the corner is eliminated. While this simple mapping may be useful in transforming theinterior of a contour, the mapping of the exterior region would not be one-to-one.
Theelimination of corners for regions surrounding a contour can be effected by applying theKarman-Trefftz mapping defined by(29)whereis the conjugate of zo. The exponent a depends on the exterior angle and the regionshould be translated, if necessary, so thatis an interior point of the contour.This transformation may be applied sucoessively to eliminate any number of corners on theboundary of the physical region.Elementary complex functions can therefore serve to precondition a region. Cornerswhich are to map to sides of a computational rectangle can be eliminated. Conversely,right-angle corners can be formed at points of the physical reigon which are to map tovertices of the computational region thereby eliminating problems of extremenonorthogonality.The trend in treating more complicated regions is to break the mapping up into asequence of more simple mappings. Contours, such as airfoils, are generally mapped tonear-circles by one or more simple transformations, and then the near-circle is mapped to acircle by a series transformation, e.g., the Theodorsen procedure.