Numerical Conformal Mapping with Rational Functions

In this study, the Green function of the (interior) Dirichlet problem for the Laplace (also Poisson) differential equation in a rectangular domain is expressed in terms of elliptic functions and the solution of the problem is based on the Green function and therefore on the elliptic functions. The method of solution for the Dirichlet problem by the Green function is presented; the Green function and transformation required for the solution of the Dirichlet problem in the rectangular region is found and the problem is solved in the rectangular region. An example for the problem in the rectangular region is given in order to present an application of the solution of Dirichlet problem. The equation is solved first by the known method of separation of variables and then in terms of elliptic functions; the results of both methods are compared. The results are found to be consistent but the advantage of this method is that the solution is obtained in terms of elementary functions.

The traditional view in numerical conformal mapping is that once the boundary correspondence function has been found, the map and its inverse can be evaluated by contour integrals. We propose that it is much simpler, and 10–1000 times faster, to represent the maps by rational functions computed by the AAA algorithm. To justify this claim, first we prove a theorem establishing root-exponential convergence of rational approximations near corners in a conformal map, generalizing a result of D. J. Newman in 1964. This leads to the new algorithm for approximating conformal maps of polygons. Then we turn to smooth domains and prove a sequence of four theorems establishing that in any conformal map of the unit circle onto a region with a long and slender part, there must be a singularity or loss of univalence exponentially close to the boundary, and polynomial approximations cannot be accurate unless of exponentially high degree. This motivates the application of the new algorithm to smooth domains, where it is again found to be highly effective.

Numerical conformal mapping method based Voronoi cell finite element model for analyzing microstructures with irregular heterogeneities

River reconstruction using a conformal mapping method

We present a unified boundary integral method for approximating the conformal mappings from any bounded or unbounded multiply connected region $G$ onto the five classical canonical slit domains. The method is based on a uniquely solvable boundary integral equation with the generalized Neumann kernel. Using the proposed method, the approximate mapping functions onto the five canonical slit domains can be computed in a unified way by solving linear systems with a common coefficient matrix. The method can be also used for calculating the conformal mappings of simply and doubly connected regions. The performance of the method is illustrated by several examples for regions with smooth boundaries and with piecewise smooth boundaries.

Generalised Wagner model of water impact by numerical conformal mapping

The 2D capacitance of a thin (zero thickness) strip in the center of rectangular cross section envelope can be evaluated using the formulas for 2D "corner" geometric capacitances or permeances. If the width of the strip is larger than the height of the envelope the known corner capacitance for infinitely wide strip can be used, In the paper, using conformal mapping with special functions and numerical conformal mapping, more exact approximate formulas than existing. are proposed for the case of narrow strip, with the width smaller than envelope height.

This paper presents a fast boundary integral equation method with for computing conformal mappings of multiply connected regions. We consider the canonical region consists of the entire complex plane bounded by a finite straight slit on the line Im ω = 0 and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method.

This paper presents analytical and numerical conformal mapping (CM) to analyze magnetic fields originating from permanent magnets and armature winding currents in a slotted air-gap of a surface mounted radial permanent-magnet synchronous motor (SPM), taking into account the effect of arbitrarily curved motor surfaces. We also studied the slotless configuration of the external rotor permanent-magnet motor for the purpose of calculating the complex relative air-gap permeance. The model is able to predict the air gap field and torque with high accuracy, which cannot be achieved using the previously available analytical methods in both cases: internal and external rotor permanent-magnet motor. We obtained one of the conformal mappings used in this study, the numerical solution of the Schwarz-Christoffel (SC) integral, by using the Matlab SC Toolbox. We used this field solution to calculate cogging torque and electromagnetic torque by integrating the Maxwell stress tensor inside the air gap. We verified the accuracy of the developed method by comparing its results with those obtained from the developed numerical finite-element (FE) model.

A simplified nonlinear transient analysis method for gas bearings was recently published by the authors (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” J. Tribol., 134(1), 011704). The method uses the fact that linearized dynamic characteristics of gas bearings, namely the impedances, can be approximated by rational transfer functions. The method gave good results if the rational transfer function approach approximated the linearized dynamic characteristics well. Indeed, each of the four complex impedances Zαβ,α,β={x,y} had one or two poles depending on the order of the rational function that were used. These poles appear as supplementary eigenvalues of the extended matrix of the homogeneous system of first order differential equations describing the model of the rotor. They govern the stability of the dynamic model in the same way as the original eigenvalues do and therefore they impose non-negligible constraints on the rational function approximation of the impedances of gas bearings. The present improvement of the method overrides this problem. The basic idea is to impose the same set of poles for Zxx, Zxy, Zyx, and Zyy. By imposing this constraint, the poles are stable and the introduction of artificial instability or erratic eigenvalues is avoided. Campbell and stability diagrams naturally taking into account the variation of the dynamic coefficients with the excitation frequency can now be easily plotted. For example, the method is used for analyzing the stability of rigid and flexible rotors supported by two identical gas bearings modeled with second order rational transfer functions. The method can be applied to any bearing or seal whose impedance is approximated by rational transfer functions.

Numerical Conformal Mapping onto the Parallel Slit Domains by the Charge Simulation Method

The determination of the poles of the mapping function and their use in numerical conformal mapping

The first investigations of nonlinear approximation problems were made by P.L. Chebyshev in the last century, and the entire theory of uniform approxima tion is strongly connected with his name. By making use of his ideas, the theories of best uniform approximation by rational functions and by polynomials were developed over the years in an almost unified framework. The difference between linear and rational approximation and its implications first became apparent in the 1960's. At roughly the same time other approaches to nonlinear approximation were also developed. The use of new tools, such as nonlinear functional analysis and topological methods, showed that linearization is not sufficient for a complete treatment of nonlinear families. In particular, the application of global analysis and the consideration of flows on the family of approximating functions intro duced ideas which were previously unknown in approximation theory. These were and still are important in many branches of analysis. On the other hand, methods developed for nonlinear approximation prob lems can often be successfully applied to problems which belong to or arise from linear approximation. An important example is the solution of moment problems via rational approximation. Best quadrature formulae or the search for best linear spaces often leads to the consideration of spline functions with free nodes. The most famous problem of this kind, namely best interpolation by poly nomials, is treated in the appendix of this book.

The existence of infinitely many solutions for an autonomous elliptic Dirichlet problem involving the p-Laplacian is investigated. The approach is based on variational methods.

The strongly elliptic system with constant m × m matrix-valued coefficients for a vector-valued functions u = (u 1, … , u m ) in the half-space as well as in a domain with smooth boundary ∂ Ω and compact closure is considered. A representation for the sharp constant in the inequality is obtained, where |·| is the length of a vector in the m-dimensional Euclidean space, , and ‖·‖ p is the L p -norm of the modulus of an m-component vector-valued function, 1 ≤p ≤∞. It is shown that where is a point at ∂ Ω nearest to x ∈ Ω, u is the solution of Dirichlet problem in Ω for the strongly elliptic system with boundary data from , and is the sharp constant in the aforementioned inequality for u in the tangent space to ∂ Ω at . As examples, Lamé and Stokes systems are considered. For instance, in the case of the Stokes system, the explicit formula is derived, where 1 < p < ∞.