On conformal mapping of polygonal regions

We develop Kufarev's method for determining unknown parameters in the Schwarz-Christoffel integral in the case of conformal mapping of polygonal regions with boundary normalization.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1484–1494, November, 1993.     

On conformal moduli of polygonal quadrilaterals

The change of conformal moduli of polygonal quadrilaterals under some geometric transformations is studied. We consider the motion of one vertex when the other vertices remain fixed, the rotation of sides, polarization, symmetrization, and averaging transformation of the quadrilaterals. Some open problems are formulated.     

Analysis and design of polygonal resistors by conformal mapping

To compute the electrical resistance ( conformal modulus) of a polygonally shaped resistor cut from a sheet of uniform resistivity, it suffices to find a conformal map of the polygon onto a rectangle. Constructing such a map requires the solution of a Schwarz-Christoffel parameter problem. First we show by examples that this is practical numerically. Then we consider an inverse resistor trimming problem in which the aim is to cut a slit in a given polygon just long enough to increase its resistance to a prescribed value. We show that here the solution can be obtained by solving a generalized parameter problem. The idea of a generalized parameter problem is applicable also in many other Schwarz-Christoffel computations.

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Zusammenfassung Um den elektrischen Widerstand eines polygonalen Resistors aus einem Material homogener Leitfähigkeit zu berechnen, genügt es, eine konforme Abbildung des Polygons auf ein Rechteck zu finden. Die Konstruktion einer solchen Abbildung erfordert die Lösung eines Schwarz-Christof-felschen Parameterproblems. Wir zeigen zunächst anhand von Beispielen, daß dies numerisch durchführbar ist. Dann betrachten wir ein inverses Problem: Die Aufgabe besteht hier darin, einen Schlitz in ein gegebenes Polygon zu schneiden, dessen Länge gerade so gewählt ist, daß der Widerstand auf einen vorgegebenen Wert erhöht wird. Wir zeigen, daß dieses Problem auf ein verallgemeinertes Parameterproblem zurückgeführt werden kann. Die Idee des verallgemeinerten Parameterproblems ist auch auf viele weitere Schwarz-Christoffel-Probleme anwendbar.

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Supported by NSF Mathematical Sciences Postdoctoral Fellowship, and by the U.S. Dept. of Energy under contract DE-AC02-76-ER03077-V. This work was performed at the Courant Institute of Mathematical Sciences, New York University.     

On closed-form integration of some Fuchsian differential equations related to a conformal mapping of circular pentagons with a cut

To solve the problem on a conformal mapping of some circular pentagons with a cut, we suggest to use special methods designed for a class of polygons in polar grids (i.e., bounded by arcs of concentric circles and segments of lines passing through the origin) and based on finding particular solutions of Fuchs type equations in the form of linear combinations of known particular solutions of some simpler equations with three singular points with indeterminate coefficients. The obtained results are first used to solve problems on a conformal mapping of circular quadrangles with a cut which belong to the class of polygons in polar grids, and then, with regard of found solutions, to pentagons of a more complicated structure, which are not polar. In all cases, we present the complete solution of the problem on parameters.     

Numerical conformal mapping of multiply connected domains to regions with circular boundaries

We propose a method to map a multiply connected bounded planar region conformally to a bounded region with circular boundaries. The norm of the derivative of such a conformal map satisfies the Laplace equation with a nonlinear Neumann type boundary condition. We analyze the singular behavior at corners of the boundary and separate the major singular part. The remaining smooth part solves a variational problem which is easy to discretize. We use a finite element method and a gradient descent method to find an approximate solution. The conformal map is then constructed from this norm function. We tested our algorithm on a polygonal region and a curvilinear smooth region.     

On the polynomial approximation of a conformal mapping of a domain with nonzero corner

Let G be a bounded domain with a Jordan boundary that is smooth at all points except a single point at which it forms a nonzero corner. We prove Korevaar’s conjecture on the order of polynomial approximation of a conformal mapping of this domain into a disk. We also obtain a pointwise estimate for the error of approximation.     

On conformal mapping and linear measure

Journal d'Analyse Mathématique -     

Linear integral equations for conformal mapping of bounded multiply connected regions onto a disk with circular slits

In this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region Ω onto a disk with circular slits. The method is based on some uniquely solvable boundary integral equations with classical adjoint and generalized Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.     

On the stressed state of a non-thin transversally isotropic spherical shell with a circular hole

On the basis of a generalized theory constructed using the Fourier-series expansion of the unknowns in Legendre polynomials of the thickness coordinate we give a representation of the general solution of the equilibrium equations of a transversally isotropic spherical shell for an arbitrary approximation. On this basis we study the problem of the stressed state of a shallow spherical shell with a circular cavity on whose boundary surface there are tangential stresses varying nonlinearly over the thickness. Bibliography: 3 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 19–24.     

On the approximate conformal mapping of multiply connected domains

Summary A method is presented for constructing approximations to the standard mappings for multiply connected regions given by Nehari [5]. The case of mapping onto a slit annulus is considered in detail, and computational results are presented for several examples.     

Limiting equilibrium of a plane with a circular hole

The elastoplastic state of a compressible isotropic plane with a circular hole is studied by the method of a small parameter. An unknown boundary separating the domain of limiting equilibrium and the elastic domain is determined. We construct the complex Kolosov-Muskhelishvili functions that describe the elastic state of a plane and compare these with the solutions of Galin's problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 980–981. July, 1993.