Use Of Conformal Mapping Research Articles

In Situ Measurement of the Conductance of Regioregular Poly-3′,4′-didodecyl-2,2′:5′,2′′-terthiophene during Potentiodynamic Growth

This work reports a study of regioregular poly-3′,4′-didodecyl-2,2′:5′,2′′-terthiophene (poly-3′4′-DDTT) deposited electrochemically onto a double-band electrode for the in situ measurement of the electrical conductance. The electrodeposition of poly-3′4′-DDTT was conducted in the potentiodynamic mode within the applied potential interval 0 ≤ E appl ≤ 0.9 V vs Ag/Ag+ employing an electrolyte that contained the terthiophenic monomer 3′4′-DDTT (the starting redox species). These electrochemical conditions warrant the oxidation of 3′4′-DDTT (initiation step) and prevent the oxidative degradation of the polymerization product(s). Through the adoption of conformal mapping we could calculate the electrical conductivity of the electrodeposited polymer thanks to the observation of a linear variation of conductance with the consumed charge of polymerization. The use of conformal mapping has allowed also the determination of the volume yield for the poly-3′4′-DDTT under consideration. The electrical conductivity of poly-3′4′-DDTT depended nonlinearly on the scan rate of electrodeposition and varied in the broad range 12 − 34 S cm−1. The variability of poly-3′4′-DDTT conductivity depended on the nature of the electrodeposit which, in turn, depended on the rate of oxidative coupling (determined by the electrical current) and on the rate of precipitation (determined by the conditions of saturation in proximity of the double-band electrode).

Conformally mapped polynomial chaos expansions for Maxwell's source problem with random input data

AbstractGeneralized Polynomial Chaos (gPC) expansions are well established for forward uncertainty propagation in many application areas. Although the associated computational effort may be reduced in comparison to Monte Carlo techniques, for instance, further convergence acceleration may be important to tackle problems with high parametric sensitivities. In this work, we propose the use of conformal maps to construct a transformed gPC basis, in order to enhance the convergence order. The proposed basis still features orthogonality properties and hence, facilitates the computation of many statistical quantities such as sensitivities and moments. The corresponding surrogate models are computed by pseudo‐spectral projection using mapped quadrature rules, which leads to an improved cost accuracy ratio. We apply the methodology to Maxwell's source problem with random input data. In particular, numerical results for a parametric finite element model of an optical grating coupler are given.

Construction of conformal maps based on the locations of singularities for improving the double exponential formula

Abstract The double exponential formula, or DE formula, is a high-precision integration formula using a change of variables called a DE transformation; it has the disadvantage that it is sensitive to singularities of an integrand near the real axis. To overcome this disadvantage, Slevinsky & Olver (2015, On the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. SIAM J. Sci. Comput., 37, A676–A700) attempted to improve the formula by constructing conformal maps based on the locations of singularities. Based on their ideas, we construct a new transformation formula. Our method employs special types of the Schwarz–Christoffel transformation for which we can derive the explicit form. The new transformation formula can be regarded as a generalization of DE transformations. We confirm its effectiveness by numerical experiments.

Analytical solution of steady state heat conduction equations in irregular domains with various BCs by use of Schwarz-Christoffel conformal mapping

Undoubtedly, analytical solutions of conduction heat transfer problems play an important role in formation of both qualitative and quantitative understanding of the heat transfer phenomenon and other phenomena related to it. Analytical methods for solving this class of problems, especially in irregular domains, contain much complexity, making the solution process extremely difficult. This paper demonstrates, in a general manner, the use of conformal mapping, specifically the Schwarz-Christoffel (SC) mapping, to solve the steady-state conduction heat transfer equations (namely Laplace and Poisson) and to obtain the analytical exact solution. The exact temperature distribution can then be obtained for irregular domains (i.e. domains with irregular boundaries) for homogeneous and non-homogeneous conditions. To get to this goal and to clarify the method more, for a specific case, namely a triangular plate with a right angle on the origin and constant-temperature boundary conditions, the steady-state heat conduction equation has been analytically solved using the conformal mapping method. In the analytical solution process, the importance and capabilities of complex variables and their applications, especially the SC mapping, is highlighted. Finally, in order to validate the exact solution, the problem has been solved using the finite-element software COMSOL Multi-physics 5.0 and the results have been compared to the analytical solution. Good agreement between two solutions has confirmed the accuracy of the analytical solution.

Entanglement evolution across a conformal interface

For two-dimensional conformal field theories (CFTs) in the ground state, it is known that a conformal interface along the entanglement cut can suppress the entanglement entropy from to , where L is the length of the subsystem A, and is the effective central charge which depends on the transmission property of the conformal interface. In this work, by making use of conformal mappings, we show that a conformal interface has the same effect on entanglement evolution in non-equilibrium cases, including global, local and certain inhomogeneous quantum quenches. I.e. a conformal interface suppresses the time evolution of entanglement entropy by effectively replacing the central charge c with , where is exactly the same as that in the ground state case. We confirm this conclusion by a numerical study on a critical fermion chain. Furthermore, based on the quasi-particle picture, we conjecture that this conclusion holds for an arbitrary quantum quench in CFTs, as long as the initial state can be described by a regularized conformal boundary state.

Enhancing D-bar reconstructions for electrical impedance tomography with conformal maps

We present a few ways of using conformal maps in the reconstruction of two-dimensional conductivities in electrical impedance tomography. First, by utilizing the Riemann mapping theorem, we can transform any simply connected domain of interest to the unit disk where the D-bar method can be implemented most efficiently. In particular, this applies to the open upper half-plane. Second, in the unit disk we may choose a region of interest that is magnified using a suitable Mobius transform. To facilitate the efficient use of conformal maps, we introduce input current patterns that are named conformally transformed truncated Fourier basis; in practice, their use corresponds to positioning the available electrodes close to the region of interest. These ideas are numerically tested using simulated continuum data in bounded domains and simulated point electrode data in the half-plane. The connections to practical electrode measurements are also discussed.

A note on “Quasi-analytical solution of two-dimensional Helmholtz equation”

The recent paper of Van Hirtum in this journal repeats a number of misconceptions about the use of conformal mappings in solving the two-dimensional Helmholtz equation. These are discussed, as is the fact that the numerical approach presented does not lead to accurate results. In general conformal mapping is not useful in solving Helmholtz’s equation. Other, accurate, techniques are briefly reviewed.

Use of Conformal Mapping to Calculate the Mean Level of the Electromagnetic Field Above the Sea Surface

A new method is proposed to calculate the mean level of an electromagnetic field propagating along sea paths, based on a numerical solution of the parabolic equation. A way of taking the influence of waviness of the sea surface on the mean level of the field strength of the radio field is proposed. This method is based on the method of conformal mapping of a curvilinear coordinate system above the uneven sea surface onto a Cartesian coordinate system above the sea surface. A comparison with computer simulations obtained using the Monte Carlo method is carried out.

Solution for the problem of the Stokes flow past a body of arbitrary shape using conformal mapping

An approach to the calculation of flow of an incompressible viscous fluid in the Stokes approximation is developed based on the finite difference method with the use of conformal mapping. The problems of flow around a circular cylinder and ellipse in a periodic circular cell have been solved. The method is particularly well suited for solving problems of flow around bodies of an arbitrary curved shape.

Evolution operators in conformal field theories and conformal mappings: Entanglement Hamiltonian, the sine-square deformation, and others

By making use of conformal mapping, we construct various time-evolution operators in (1+1) dimensional conformal field theories (CFTs), which take the form $\int dx\, f(x) \mathcal{H}(x)$, where $\mathcal{H}(x)$ is the Hamiltonian density of the CFT, and $f(x)$ is an envelope function. Examples of such deformed evolution operators include the entanglement Hamiltonian, and the so-called sine-square deformation of the CFT. Within our construction, the spectrum and the (finite-size) scaling of the level spacing of the deformed evolution operator are known exactly. Based on our construction, we also propose a regularized version of the sine-square deformation, which, in contrast to the original sine-square deformation, has the spectrum of the CFT defined on a spatial circle of finite circumference $L$, and for which the level spacing scales as $1/L^2$, once the circumference of the circle and the regularization parameter are suitably adjusted.

Surface-tension-driven Stokes flow: A numerical method based on conformal geometry

A novel numerical scheme is presented for solving the problem of two dimensional Stokes flows with free boundaries whose evolution is driven by surface tension. The formulation is based on a complex variable formulation of Stokes flow and use of conformal mapping to track the free boundaries. The method is motivated by applications to modelling the fabrication process for microstructured optical fibres (MOFs), also known as holey fibres, and is therefore tailored for the computation of multiple interacting free boundaries. We give evidence of the efficacy of the method and discuss its performance.

Optimization of Conformal Mapping Functions used in Developing Closed-Form Solutions for Underground Structures with Conventional cross Sections

Elastic solutions applicable to single underground openings usually suffer from geometry related simplification. Most tunnel shapes possess two axes of symmetry while a wide range of geometries used in tunneling practice involve only one symmetry axis. D-shape or horse-shoe shape tunnels and others with arched roof and floor are examples of the later category (one symmetry axis). In the present paper, with the use of conformal mapping, two methods were developed to determine the appropriate mapping functions on which an analytical elastic solution for a tunnel with one vertical axis of symmetry is based. These conformal mapping functions turn complicated geometries into a unit circle for the sake of simplification. These two approaches were introduced into a computer program used for an arbitrary tunnel cross section. Results showed that the second approach has more accuracy and is able to produce any shape, since it uses a nonlinear structure in its constitutive equations. Besides, the values for different coefficients have been presented for a variety of tunnel geometry curvature, as well as acceptable variation for the coefficients to represent tunnels with conventional shapes.

On The Use of Conformal Maps for the Acceleration of Convergence of the Trapezoidal Rule and Sinc Numerical Methods

We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the $\sinh$ map, and allows the treatment of a finite number of singularities in the complex plane. In the case where locations are unknown, the so-called Sinc-Padé approximants are used to provide approximate results. This adaptive method is shown to have almost the same convergence properties. We use the conformal maps to generate high-accuracy solutions to several challenging integrals, nonlinear waves, and multidimensional integrals.

Conformal reduction of boundary problems for harmonic functions in a plane domain with strong singularities on the boundary

We consider the Dirichlet, Neumann and Zaremba problems for harmonic functions in a bounded plane domain with nonsmooth boundary. Purpose: We wish to construct explicit formulas for solutions of these problems when the boundary curve belongs to one of the following three classes: sectorial curves, logarithmic spirals and spirals of power type. Methods: To study the problem, we apply the familiar Vekua-Muskhelishvili method which consists in the use of conformal mapping of the unit disk onto the domain to pull back the problem to a boundary problem for harmonic functions in the disk. This in turn later reduces to a Toeplitz operator equation on the unit circle with symbol-bearing discontinuities of the second kind. Results: We develop a constructive invertibility theory for Toeplitz operators and thus derive solvability conditions as well as explicit formulas for solutions. Conclusions: Our results raise Fredholm theory for boundary value problems in domains with singularities which are not necessarily rectifiable.

Conformal Mappings in Relativistic Astrophysics

We describe the use of conformal mappings as a mathematical mechanism to obtain exact solutions of the Einstein field equations in general relativity. The behaviour of the spacetime geometry quantities is given under a conformal transformation, and the Einstein field equations are exhibited for a perfect fluid distribution matter configuration. The field equations are simplified and then exact static and nonstatic solutions are found. We investigate the solutions as candidates to represent realistic distributions of matter. In particular, we consider the positive definiteness of the energy density and pressure and the causality criterion, as well as the existence of a vanishing pressure hypersurface to mark the boundary of the astrophysical fluid.

Designing acoustic transformation devices using fluid homogenization of an elastic substructure

The design of devices using finite embedded coordinate transformations presents an unique approach to control acoustic waves. Though combining the use of conformal mappings may provide a pathway to more realizable material properties, many device geometries still require combinations of density and sound speed which are unavailable in isotropic materials. Here, we present a design strategy based on a multiple scattering homogenization method to approximate the unique values required within such a device. We apply the method, using full-wave simulations, to the design of an aqueous cylindrical-to-plane wave lens, which can be constructed from simple materials.

Generalized Poisson Integral Formula applied to potential flow solutions for free and confined jets with secondary flow

A generalized form of the Poisson Integral Formula is presented and applied to obtain a new class of two-dimensional, incompressible, potential flowfield solutions for free and confined jets. Two basic configurations are considered that demonstrate the procedure, a jet issuing from a straight wall or flowing from a semi-infinite, constant area duct. In either case the jet can be free or confined, with or without uniform external flow. A discretized form of the Poisson Integral Formula is also presented so that arbitrary boundary value distributions (i.e., boundary conditions) can be handled, including those with functional and/or slope discontinuities. The procedure makes use of conformal mapping and complex variable theory so that jet flowfield determination is reduced to solution of a boundary value problem on a simple domain (i.e., upper half-plane). The entire velocity field is described analytically; stream function and velocity potential contour maps are readily constructed. These solutions can serve as the zero-order inviscid outer solution (replacing the usual assumption of uniform or quiescent outer flow) in a matched asymptotic expansion process whereby viscous effects are introduced near the jet axis by means of a boundary-layer-type analysis. A number of example solutions are presented, most without any form of singularity in the flowfield as a result of having the freedom to arbitrarily select boundary conditions. Solutions are also presented having separation and recirculation regions adjacent to sharp jet exit corners. Several additional examples of the generalized Poisson Integral Formula applied to other types of problems are also described, including periodic boundary distributions and distributions having discontinuities.

Generalization of the van der Pauw relationship derived from electrostatics

In an earlier paper, this author, along with two others Weiss et al. (2008) [1], demonstrated that the original van der Pauw relationship could be derived from three-dimensional electrostatics, as opposed to van der Pauw’s use of conformal mapping. The earlier derivation was done for a conducting material of rectangular cross section with contacts placed at the corners. Presented here is a generalization of the previous work involving a square sample and a square array of electrodes that are not confined to the corners, since this measurement configuration could be a more convenient one. As in the previous work, the effects of non-zero sample thickness and contact size have been investigated. Buehler and Thurber derived a similar relationship using an infinite series of current images on a large and thin conducting sheet to satisfy the conditions at the boundary of the sample. The results presented here agree with theirs numerically, but analytic agreement could not be shown using any of the perused mathematical literature. By simply equating the two solutions, it appears that, as a byproduct of this work, a new mathematical relationship has been uncovered. Finally, the application of this methodology to the Hall Effect is discussed.

Eigenvalue bounds, spectral partitioning, and metrical deformations via flows

We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n -vertex graph of genus g and maximum degree D satisfies λ 2 ( G )= O (( g +1) 3 D / n ). This recovers the O ( D / n ) bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of O (( g +1)poly( D )/ n ), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that λ 2 ( G ) = O ( D h 6 log h / n ) whenever G is K h -minor free. This shows, in particular, that spectral partitioning can be used to recover O (√ n )-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on λ 2 for graphs that exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find separators of sublinear size in a general class of geometric graphs. Moreover, while the standard “sweep” algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the well-known nonplanar separator theorem of Alon, Seymour, and Thomas that states that every excluded-minor family of graphs has O (√ n )-node balanced separators.

A Group-Theoretical Method for Natanzon Potentials in Position-Dependent Mass Background by Means of Conformal Mappings

A new manner for deriving the exact potentials is presented. By making use of conformal mappings, the general expression of the effective potentials deduced under the \(\mathfrak{su}(1,1)\) algebra can be brought back to the general Natanzon hypergeometric potentials.