Approximation Of Conformal Mappings Research Articles

Approximation of Conformal Mappings via Bieberbach Polynomials inside Regions with Cusps

Bu çalışmada, kompleks düzlemin; sonlu, basit bağlantılı ve aynı zamanda hem iç sıfır açı hem de dış sıfır açıya sahip bölgelerin içerisinde konform dönüşümlere Bieberbach polinomlarıyla yaklaşımı için bazı hesaplamalar elde edimiştir. Ayrıca yaklaşımın hızı bölgenin geometrik özelliğine bağlıdır.

Approximation of conformal mappings and novel applications to shape recognition of planar domains

Our goal is to provide a novel method of representing 2D shapes, where each shape will be assigned a unique fingerprint - a computable approximation to a conformal map of the given shape to a canonical shape in 2D or 3D space (see page 22 for a few examples). In this paper, we make the first significant step in this program where we address the case of simply, and doubly-connected planar domains. We prove uniform convergence of our approximation scheme to the appropriate conformal mapping. Along the way, we affirm a conjecture raised by Ken Stephenson in the 90's which predicts that the Riemann mapping can be approximated by a sequence of electrical networks. In fact, we first treat a more general case. Consider a planar annulus, i.e., a bounded, two-connected, Jordan domain, endowed with a sequence of triangulations exhausting it. We construct a corresponding sequence of maps which converge uniformly on compact subsets of the domain, to a conformal homeomorphism onto the interior of a Euclidean annulus bounded by two concentric circles. The resolution of Stephenson's Conjecture then follows by a limiting argument. Our methods involve harmonic mappings and boundary value problems: discrete and analytic in the plane. The scheme we constructed is programmable.

$$C^\infty $$ C ∞ -Convergence of Conformal Mappings for Conformally Equivalent Triangular Lattices

Two triangle meshes are conformally equivalent if for any pair of incident triangles the absolute values of the corresponding cross-ratios of the four vertices agree. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study discrete conformal maps which are defined on parts of a triangular lattice $T$ with strictly acute angles. That is, $T$ is an infinite triangulation of the plane with congruent strictly acute triangles. A smooth conformal map $f$ can be approximated on a compact subset by such discrete conformal maps $f^\varepsilon$, defined on a part of $\varepsilon T$ for $\varepsilon>0$ small enough, see [U. B\"ucking, Approximation of conformal mappings using conformally equivalent triangular lattices, in "Advances in Discrete Differential Geometry" (A.I. Bobenko ed.), Springer (2016), 133--149]. We improve this result and show that the convergence is in fact in $C^\infty$. Furthermore, we describe how the cross-ratios of the four vertices for pairs of incident triangles are related to the Schwarzian derivative of $f$.

Broadband Microwave Characterization of Nanostructured Thin Film With Giant Dielectric Response

This paper describes the extraction of the dielectric properties of a nanostructured thin film from two-port $S$ -parameter measurements on coplanar waveguide (CPW) lines. The CPW line is on top of a bilayer structure formed by a supporting glass substrate and a dielectric thin film made by dispersing silver nanoparticles inside a polymer host. A dispersion mechanism due to internal inductance of the CPW line when calculating the effective dielectric constant is investigated. The extraction involves conformal-mapping approximation that uses closed-form equations to calculate the dielectric constant and the loss tangent of each layer based on the effective dielectric constant and loss tangent of the entire structure. Additionally, computer-aided-design model with a direct fitting technique are deployed for further investigation of potential multimode propagation for a CPW line with a substrate that has a very large dielectric constant. The results of the two techniques are compared and discussed. The measured dielectric constant ranges from ${\hbox{3}}\times {\hbox{10}}^{4}$ to ${\hbox{4.6}}\times {\hbox{10}}^{3}$ from 1 to 20 GHz with a loss tangent of 0.55 to 1.75 from 1 to 20 GHz.

Finite Element Approximation of Conformal Mappings to Unbounded Jordan Domains

In two-dimensional Euclidean space, a simply connected unbounded domain whose boundary is a Jordan curve is called an unbounded Jordan domain. In this article, we discuss a piecewise linear finite element approximation of conformal mappings from the unit disk of the plane to unbounded Jordan domains. Some convergence results and error analysis are presented. Numerical examples are also given.

An approximation of conformal mappings on smooth domains

Let G be a finite domain with z 0 ∈ G and bounded by a Jordan curve L ≔ ∂G. The Bieberbach polynomials π n , n = 1, 2, … , associated with the pair (G, z 0) can be used to approximate the conformal mapping φ0 from G to D(0, r 0) ≔ {w : |w| < r 0} normalized by . In this work we define a new subclass of smooth Jordan curves and give the estimates of in accordance with the parameters defining this subclass.

Approximation of conformal mappings by circle patterns

A circle pattern is a configuration of circles in the plane whose combinatorics is given by a planar graph G such that to each vertex of G corresponds a circle. If two vertices are connected by an edge in G, the corresponding circles intersect with an intersection angle in (0, π). Two sequences of circle patterns are employed to approximate a given conformal map g and its first derivative. For the domain of g we use embedded circle patterns where all circles have the same radius decreasing to 0 and with uniformly bounded intersection angles. The image circle pattern has the same combinatorics and intersection angles and is determined from boundary conditions (radii or angles) according to the values of g′ (|g′| or arg g′). For quasicrystallic circle patterns the convergence result is strengthened to C ∞-convergence on compact subsets.

Application of Conformal Mapping Approximation Techniques: Parallel Conductors of Finite Dimensions

This paper describes a novel approach to the application of conformal mapping to capacitance evaluation. A particular structure composed of an array of identical lines and located below a conductive plate is studied. Results show the application of conformal mapping can reduce computing time when using three-dimensional electrostatic modeling and it can be the basis of algorithms of practical applications, especially in the semiconductor industry.

An Interpolating Polynomial Method for Numerical Conformal Mapping

A simple algorithm is presented for numerical approximation of conformal mappings for simply connected planar regions. The desired mapping is represented by a normalized polynomial of degree N, determined by the boundary values which it is presumed to take at the Nth roots of unity; then its values at the N intermediate 2Nth roots of unity are used to correct the initial guess. Each iteration, which consists essentially of matrix multiplication and boundary projection, costs O(N log N) arithmetic operations. Numerical results are provided to confirm linear convergence of the algorithm.

Approximation of conformal mappings of annular regions

In this paper we examine the convergence rates in an adaptive version of an orthonormalization method for approximating the conformal mapping \(f\) of an annular region \(A\) onto a circular annulus. In particular, we consider the case where \(f\) has an analytic extension in compl(\(\overline{A}\)) and, for this case, we determine optimal ray sequences of approximants that give the best possible geometric rate of uniform convergence. We also estimate the rate of uniform convergence in the case where the annular region \(A\) has piecewise analytic boundary without cusps. In both cases we also give the corresponding rates for the approximations to the conformal module of \(A\).

Circle packings in the approximation of conformal mappings

Circle packings in the approximation of conformal mappings

Transmission-Line Properties of Parallel Wide Strips by a Conformal-Mapping Approximation

A transmission line is made of a symmetrical pair of strip conductors, face-to-face, or a single strip parallel to a ground plane. The strip width is nominally greater than the separation, but may be somewhat less than the separation. The field configuration is evaluated by a conformal mapping procedure which gives a very close approximation in terms of ordinary functions (exponential and hyperbolic) rather than the exact solution in terms of difficult functions (elliptic). Computation procedures are given for synthesis or analysis, the former following more naturally from the derivation. For the transitional case of strip width equal to separation, the mapping approximation is found to leave a relative error of the order of 10/sup -8/, in the wave resistance or shape ratio. Simple, explicit, practical formulas are developed for practical use with slide-rule accuracy. Emphasis is placed on establishing and specifying the residual error of each formula.