Thompson, Warsi, Mastin - Numerical Grid Generation (523190), страница 34
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[1]. Some specific recommendations of techniques and tools are given in Ref.[7]. Conformal systems have advantage the of introducing the fewest additional terms intransformed partial differential equations. Considerable understanding of the theory offunctions of a complex variable may be necessary for effective applications, though.Although the complex variable techniques by which conformal transformations areusually generated are inherently two-dimensional, certain more general cases can be treatedby rotating or stacking two-dimensional systems:Systems can also be generated on curved surfaces, as has been done by cartographers,for stacking.
Examples of the use of conformal mapping in the construction ofthree-dimensional configurations are noted in Ref. [5].A curvilinear coordinate system generated by a conformal mapping is very rigid in thesense that little control can be exerted over the distribution of the grid points. Conformalmappings also do not exist in three dimensions (except for trivial cases).
Furthermore, thecoordinate system tends to be more difficult to construct than when using algebraic orelliptic systems. In spite of these facts, conformal mappings continue to play a significantrole in grid generation. A number of recent developments and applications of conformaltransformations are noted in Ref. [1], [5], and [7].The desirability of a coordinate system generated by a conformal transformation lies inthe form of the transformed equations.
For example, consider the diffusion equation(1)Nowandsatisfy the Cauchy-Riemann equations(2)or equivalently(3)It follows that in this case g12=g21=0 and g11=g22=curvilinear coordinates, using Eq. (III-46), as. Equation (1) can be written in(4)where the Laplacian is defined in terms of the curvilinear coordinates. Therefore it isobserved that the diffusion equation remains essentially unchanged. The only effect of thetransformation is a change in the diffusion coefficient. Neumann boundary conditions arealso unchanged in conformal coordinates. The boundary conditionwhereis normal to a=constant coordinate line, is expressed in curvilinear coordinates as1.
Construction by Finite-DifferencesThe literature abounds with methods for constructing conformal mappings. As can beseen in the review article, Ref. [1], these methods may include the construction ofSchwarz-Christoffel transformations, the solution of integral equations, or expansions interms of power series or Fourier series. Since this chapter is not intended to be acomprehensive treatment of conformal mapping, only the simple, yet frequently used, finitedifference method based on elliptic systems is discussed here.Consider the problem of conformally mapping the interior of the contour onto theinterior of a rectangle.
The Riemann Mapping Theorem states that such a mapping exists,and it also implies that the mapping is uniquely determined by specifying three realparameters. Suppose we wish to indicate four specific points on which are to map to thevertices of the rectangle. If the rectangle is fixed, then the problem is over-determined andno conformal mapping exists. Therefore, the mapping must determine one of the dimensionsof the rectangular region which we will now denote as the setRather than allow a rectangle with variable width, one can equivalently introduce theparameter M in (3) so that(5)whereThe mapping is no longer conformal, but the conformal mapping can be easily obtained bysimply multiplying the coordinate by M.
On the unit square the functions x and y nowsatisfy(6)Two boundary conditions are needed in order to determine a unique solution for thiselliptic system. One condition is derived from the equation of the boundary curve I whichmight be(7)The other condition comes from applying the orthgonality equation, g12=0. This conditionalso follows on eliminating the parameter M in Eq. (5). The implementation of the boundaryconditions is done in the following order. First a boundary value for x or y is computed fromthe orthogonality constraint.
If the boundary point lies along =0, then we may use(8)A forward difference is used to approximate the derivatives and central differences for the-derivatives. The same equations are used along = 1 with backward differences for the-derivatives. Once an x or y value has been computed from Eq. (8), the other coordinatevalue is given by writing (7) in the form(9)Although either equation in (8) could be used, it is advisable to choose either the first orsecond equation, depending on whether x or y has the largest absolute value. Thisavoids not only the possibility of division by zero but also problems with the solvability ofthe implicit equation (7).
The same techniques are used along an =constant coordinate line.In this case the orthogonality constraint can be written as(10)Now the parameter M must also be determined. It follows from Eq. (5) that(10)An iterative algorithm is used to construct the mapping, and any algorithm which canbe used for the elliptic systems in Chapter VI can also be used here. At each iteration a newset of boundary values for x and y are computed using Eq. (8)-(10). There are two options incomputing a value for M. Either a different value at each point can be computed from Eq.(11), or a constant value can be computed from a relation such as(12)where 01.
Eq. (12) is derived from the equation in Eq. (5) by integrating along anconstant coordinate line. This same technique can also be used to derive an alternate formulafor finding the boundary values at x and y. Along = 0, for example, we haveThe constant M, called the conformal module of the region by complex analysts, has asimple geometric interpretation. From Eq. (11) it is noted that M is simply the aspect ratio ofthe grid cells.
There exist highly accurate numerical methods for computing both M and theboundary values for x and y. If these values are computed first, then the system (6) can besolved by a direct elliptic solver.The only control over the distribution of grid points with a nonformal mapping is bychanging the points which map to the vertices of the rectangular region.
However, most ofthe advantageous features are retained when the conformal mapping is combined withone-dimensional stretching transformations. Thus we will consider a new set ofcomputational variables, and , with and serving as intermediate variables defined bythe one-dimensional equations(13)If x and y are solutions of Eq. (6), then in terms of the new computational variables,(14)(15)In Eq. (15), the covariant metric tensor components are defined relative to the transformationfrom the physical x,y variables to the computational , variables.The application of this transformation to the diffusion equation (1) results in thefollowing transformed equation:(16)Note that the coefficients of the and derivatives in Eq. (16) are functions of and ,respectively.
Therefore, only one-dimensional arrays are needed to store these coefficients. Itcan be further noted that the steady-state equation (At=0) is a separable elliptic equationwhich can be solved using a direct elliptic solver.In the above development the stretching functions given by Eq. (13) are used tocontrol the grid point distribution. Clearly the derivatives of these functions must benonvanishing, and these derivatives may as well be taken to be positive so that theorientation of the physical boundary is preserved.
The function f( ) is a contraction mappingif f’( ) < 1 and an expansion mapping if f’( ) > 1. Therefore, relative to a conformalmapping, the =constant coordinate lines will be closer together where f’( ) < 1 and fartherapart when f’( ) > 1. The same control over the =constant coordinate lines is exerted bythe function h( ). Several one-dimensional functions are discussed in Chapter VIII.On any particular boundary segment, say =0, it is in theory possible to match anydesired distribution of grid points provided the correct stretching function f( ) can bedetermined.
However, there is no known way of generating this stretching function so thatEq. (14), together with the boundary conditions given by Eq. (7) and g12=0, will have asolution with the prescribed boundary values along =0. The solution to this problem lies inthe implicit determination of the stretching function in the solution of the elliptic system.Suppose that h( ) = , so that Eq. (15) can be written as(17)This equation allows Eq. (14) to be written as(18)This quasilinear system can be solved with the orthogonality condition on all boundarycomponents except = 0 where we will now impose the Dirichlet conditionwheredefines the desired distribution of grid points.The ability to specify grid points along a boundary component extends the usefulnessof conformal mappings.
For example, one can assign coordinates around an airfoil and alongthe branch cut in a C-type coordinate system so that the coordinate lines pass smoothlythrough the cut. In many segmented systems the grid points can be chosen so that coordinatelines pass smoothly from one sub-region into the next. One disadvantage of this method isthe reported slow convergence in the iterative solution of (18) for certain problems. Analternate method of achieving the same result would be to generate a conformal mappingfrom Eq. (6) and then use interpolation to redistribute the grid lines.
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