Conformal Mapping Theory Research Articles

Theoretical analysis of -transform decomposition method with applications of nonlinear ordinary differentialequations.

One of the most noteworthy differential equations of the first order is the Riccati differential equation. It is applied in various branches of mathematics, including algebraic geometry, physics, and conformal mapping theory. The -transform Adomian decomposition method is employed in the current study to find exact solutions for different kinds of nonlinear differential equations. We give thorough detailed proofs for new theorems related to the -transform technique. The Adomian decomposition method and the -transform method serve as the foundation for this technique. For certain differential equations, the theoretical analysis of the -transform Adomian decomposition method is examined and is computed using readily computed terms. Our findings are contrasted with exact solutions found in the literature that were produced using alternative techniques. The significant features of the -transform Adomian decomposition method are described in the article. It has been shown that the -transform Adomian decomposition method is very efficient, useful, and adaptable to a broad variety of linear and nonlinear differential equations. Most of the symbolic and numerical calculations were performed using Mathematica.

Achromatic elastic metalens for high-performance confocal piezoelectric energy harvesting

In this study, we analytically, numerically, and experimentally investigated a high-performance confocal piezoelectric energy harvesting system. We achieved a significantly enhanced electrical performance through a Mikaelian lens, which achromatically focuses ambient elastic waves, resulting in the formation of a highly amplified strain energy field in the piezoelectric energy harvester. Previous studies on piezoelectric energy harvesting platforms have limitations, such as the focal position changing with operating frequencies and impedance mismatching owing to inclusions or holes. To address these problems, we utilized the self-focusing ability based on the conformal mapping theory and achromatic ability based on the Kirchhoff–Love thin plate theory to design our Mikaelian lens-based piezoelectric energy harvesting platform. The proposed platform demonstrates a remarkable elastic wave focusing ability at an identical focal position for a broad frequency range. The experimentally visualized wave fields matched well with the numerically calculated full-wave harmonic simulation results. We achieved highly amplified output power up to 1.44 mW within a broad range from 40 to 60 kHz out of the same focal point owing to confined elastic wave energy; the output power extracted at this confocal position was up to 3.76 times higher than that extracted at the lens start position. Our highly performance and broadband achromatic piezoelectric energy harvesting platform lays an attractive foundation for designing potential applications, such as wireless sensing, structural health monitoring, and biomedical devices.

Exact solutions for submerged von Kármán point vortex streets cotravelling with a wave on a linear shear current

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Effects of Curvature on Flexible Bragg Grating in Off-Axis Core: Theory and Experiment

A high curvature flexible Bragg grating, in planar technology, is designed, fabricated and experimentally characterized by employing silica glass. The curvature sensing mechanism for slightly off-axis core is theoretically explained and experimentally demonstrated. The electromagnetic behavior of the proposed sensor is investigated through conformal mapping, modal analysis, and coupled-mode theory. The experimental measurement reveals different sensitivities to positive and negative curvatures. The reflection spectrum of the planar Bragg grating sensor and the relationship between its resonant wavelengths and the curvature is measured for a large curvature range up to 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> = ± 33 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> . The interest toward off-axis core on planar flexible platform for curvature monitoring is due to the lateral stiffness and low Young's modulus of the substrate.

Research on the penetration characteristics of elliptical cross-section projectile into semi-infinite metal targets

To better understand the penetration characteristics of the elliptical cross-section projectile (ECSP) into metal targets, a series penetration experiments of projectiles with different shape cross-section to semi-infinite aluminum target were carried out. The shape ratios of typical ECSPs were 1.25 and 1.56, respectively, and another was circular cross-section projectile (CCSP) which has the same cross-section area in accordance with ECSP. In the penetration experiments, the projectiles were launched with striking velocities ranged from 400 m/s to 1000 m/s, and strain sensors were arranged on the impact surface of the aluminum target to catch the dynamic response during the penetration process. Then, characteristics of cracks on the impact surface of targets, penetration depths and penetration trajectory were measured after the penetration experiments. The results indicated that the dynamic response and damage behavior of targets showed obvious asymmetry under the penetration of ECSPs. Strain data had a decreasing trend from the major axis to the minor axis of the ellipse, and the length and number of cracks of the ECSP were larger than that of the CCSP. Besides, a theoretical model was developed to explain the asymmetric deformation and failure behavior of targets based on the conformal mapping theory and the strain path method. Calculation results showed that the asymmetric deformation of targets was directly caused by the displacement boundary conditions of the ellipse.

Case-Specific Focal Sensor Design for Cardiac Electrical Impedance Tomography.

Electrical impedance tomography (EIT) is a non-invasive detection technology that uses the electrical response value at the boundary of an observation field to image the conductivity changes in an area. When EIT is applied to the thoracic cavity of the human body, the conductivity change caused by the heartbeat will be concentrated in a sub-region of the thoracic cavity, that is, the heart region. In order to improve the spatial resolution of the target region, two sensor optimization methods based on conformal mapping theory were proposed in this study. The effectiveness of the proposed method was verified by simulation and phantom experiment. The qualitative analysis and quantitative index evaluation of the reconstructed image showed that the optimized model could achieve higher imaging accuracy of the heart region compared with the standard sensor. The reconstruction results could effectively reflect the periodic diastolic and systolic movements of the heart and had a better ability to recognize the position of the heart in the thoracic cavity.

Acoustic Insulation Mechanism of Membrane-Type Acoustic Metamaterials Loaded with Arbitrarily Shaped Mass Blocks of Variable Surface Density.

Membrane-type acoustic metamaterials (MAMs) have recently received widespread attention due to their good low-frequency sound-transmission-loss (STL) performance. A fast prediction method for the STL of rectangular membranes loaded with masses of arbitrary shapes and surface density values is proposed as a semi-analytical model for calculating the STL of membrane-type acoustic metamaterials. Through conformal mapping theory, the mass blocks of arbitrary shapes were transformed into regular shapes, which simplified the calculation model of acoustic propagation loss prediction, and the prediction results were verified by finite element simulations. The results show that the change in mass surface density was closely related to the size and frequency distribution of STL. The influence of the mass center on the STL and characteristic frequency of the film metamaterial is discussed.

A semi‐analytical solution for displacement‐controlled elliptical cavity expansion in undrained MCC soil

AbstractConventional cavity expansion theory is a one‐dimensional problem and is not applicable for the two‐dimensional cavity expansion issue, which is also required for geotechnical problems. This paper proposes a novel semi‐analytical solution for displacement‐controlled elliptical cavity expansion (DC‐ECE) in undrained soil. The displacement and strain fields for DC‐ECE are obtained from the strain path method (SPM) with the aid of conformal mapping theory. The effective stress can be subsequently obtained by the incremental stress‐strain calculation using a suitable constitutive model such as the MCC model. Then, the computed effective stress results are substituted into the stress equilibrium equation to give the pore pressure solution. The new semi‐analytical solution reveals the non‐axisymmetric characteristics of DC‐ECE and allows the elastic‐plastic (EP) boundary, excess pore pressure, stress, and displacement to be calculated. The proposed method provides a solution to non‐axisymmetric cavity expansion issues and it can be applied to geotechnical engineering problems, including vertical drain installation, flat dilatometer tests and noncylindrical pile penetration in clay.

Stratified Large-Amplitude Steady Periodic Water Waves with Critical Layers

By means of a conformal mapping and bifurcation theory, we prove the existence of large-amplitude steady stratified periodic water waves, with a density function depending linearly on the streamfunction, which may have critical layers and overhanging profiles. We also provide certain conditions for which these waves cannot overturn.

Schwarz–Christoffel accessory parameter for quadrilaterals via isomonodromy

We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz–Christoffel type: in particular, the mapping from the upper half plane to a four-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters—the pre-image of the polygonal vertices—via a special ‘zero curvature limit’ in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.

Thermoelectric field disturbed by two unequal cracks adjacent to a hole in thermoelectric materials

In this paper, the two-dimensional problem of a circular hole with two unequal cracks in infinite thermoelectric materials subjected to a uniform electric current and thermal flux is studied. Meanwhile, the boundary conditions at hole and cracks are assumed to be electrically impermeable and thermally impermeable. By using the complex function method and the conformal mapping theory, the exact solutions of the electric current density and the total energy flux are obtained. Furthermore, according to the theory of fracture mechanics, the intensity factors of electric current density, total energy flux density and stress at the crack tip have been obtained, which are proportional to the applied far-field loads, respectively. On the other hand, these field intensity factors are closely related to the radius of the circular hole and the lengths of cracks. The analytical solution is in good agreement with the regularized simulation results given by COMSOL commercial finite element code. Further numerical experiments reveal the effects of circular hole radius and lengths of cracks on the dimensionless thermoelectric and stress field intensity factors.

Two-Dimensional Model for Perovskite Nanorod Solar Cells: A Dark Case Study

Nanorod geometry has witnessed a massive focus in the past few years for enhancing the solar cells performance. It improves the cell efficiency because of the carrier transport perpendicular to light absorption directions. Perovskite materials, such as the methylammonium lead iodide also have attracted great attention because of the low cost, simple manufacture process, and good optical properties. These advantages of both nanorod structures and perovskite materials imply that PNSC is a very promising candidate in energy harvesting applications. This increases the demand for accurate and fast models for the analysis and for the optimization of these cells. In this article, we present an analytical model for nanorod solar cells. The model solves the semiconductor equations (Poisson's equation and continuity equations) in two-dimensional cylindrical coordinates. It is based on using the conformal mapping theory as an approximation for potential and electric field estimation. It also uses separation of variables process in solving the two-dimensional continuity equation for both radial and axial directions. The analytical model was applied on a TiO2/ perovskite / spiro-MeOTAD cell to produce the dark current. The results of this model were compared with that of a numerical finite element model and achieved good matching.

Improvements in the Electrical Performance of IC MOSFET Components Using Diamond Layout Style Versus Traditional Rectangular Layout Style Calculated by Conformal Mapping

In the first part of this article, we have proposed an innovative approach to improve the drain current model of the MOSFETs implemented with the diamond layout style (DLS), regarding the longitudinal corner effect (LCE). The proposed model is more accurate than a previous model compared to 3-D Technology Computer-Aided Design (3-D TCAD) simulation results. The new model has an innovative analytical description based on a conformal mapping theory. As a conformal mapping, there has been chosen a Schwarz–Christoffel transformation (SC). The maximal deviation values of the aspect ratio calculated by LCE are in the range from −27% to +38%. In counterpart with the new SC analytical description of DLS, the maximal deviation values are in the range from 0% to −5.5%. The second part of this article describes improvements in the electrical performance of the N-MOSFET components by using DLS counterpart to traditional rectangular layout style (RLS). Both layout style DLS, RLS, respectively, have the same process settings, as well as they are keeping the same gate area ${A}$ , and an aspect ratio width to length ${W}/{L}$ to preserve the same input conditions for their analysis. The maximal drain current increasing for the simulated DLS MOS transistor is over 20% for effective aspect ratio ( ${W}/{L}$ ) $_{\text {eff,SCT,num}}$ equal to 2.0 and angle is set to 60°. The presented model has a very good analytic description with the error level lower than 3%.

Pinning of diffusional patterns by non-uniform curvature

Diffusion-driven patterns appear on curved surfaces in many settings, initiated by unstable modes of an underlying Laplacian operator. On a flat surface or perfect sphere, the patterns are degenerate, reflecting translational/rotational symmetry. Deformations, e.g., by a bulge or indentation, break symmetry and can pin a pattern. We adapt methods of conformal mapping and perturbation theory to examine how curvature inhomogeneities select and pin patterns, and confirm the results numerically. The theory provides an analogy to quantum mechanics in a geometry-dependent potential and yields intuitive implications for cell membranes, tissues, thin films, and noise-induced quasipatterns.

Solutions for Gas-Air Flow Paths and Environmental Protection from Harmful Power-Plant Emissions in the Framework of the Scientific School Created by L. A. Rikhter [On the Centenary of the Birth of L. A. Rikhter

The paper considers the main scientific activities of the doctor of Technical Science, Professor Lev Aleksandrovich Rokhter, such as optimizing the shape of gas-air flow paths of thermal power plants, protecting the environment from harmful emissions, and identifying stack failures, which are currently being developed by his followers. For a systematic study of gas-air flow path components, L.A. Richter applied a technique based on the theory of complex variable and conformal mapping, which makes it possible to find optimal aerodynamic configuration of the gas-air flow path components. By means of the methodology developed by him, gas-air flow path research is currently carried out using Solid Works and Flow Vision application packages. Examples of optimizing the design of flue gas ducts with allowance for aerodynamic forces are given. The professor formulated the reasons and described the mechanism of structural collapse of reinforced concrete and brick stack constructions based on a theory of the occurrence of static pressures in the gas exhaust duct under certain conditions, which contribute to the penetration of aggressive substances of flue gas to the outer structural surface. In the last decade, works that additionally consider the flue gas mass transfer effect on the stack have become widespread. L.A. Richter proved that the problem of estimating the concentrations of impurities in TPP stack emissions is much more complicated under real conditions. This is due to the need to take into account the actual state of the atmosphere and its heterogeneous turbulent structure as well as the obligatory allowance for the rise of the stack plume over the stack mouth. Studies of his followers have shown that the scattering region of harmful substances is significantly reduced when a selfenvelopment phenomenon occurs under certain conditions. The professor’s contribution to the development of the methodology for finding economically optimal rates in various gas-air flow path components, as well as the creation of new structural components of electrostatic precipitators, which are currently widely used in practice, is also shown.

Reconstructing Fine Details of Small Objects by Using Plasmonic Spectroscopic Data. Part II: The Strong Interaction Regime

This paper is concerned with the inverse problem of reconstructing a small object from far field measurements by using the field interaction with a plasmonic particle which can be viewed as a passive sensor. It is a follow-up of the work [H. Ammari et al., Reconstructing fine details of small objects by using plasmonic spectroscopic data, SIAM J. Imag. Sci., to appear], where the intermediate interaction regime was considered. In that regime, it was shown that the presence of the target object induces small shifts to the resonant frequencies of the plasmonic particle. These shifts, which can be determined from the far field data, encodes the contracted generalized polarization tensors of the target object, from which one can perform reconstruction beyond the usual resolution limit. The main argument is based on perturbation theory. However, the same argument is no longer applicable in the strong interaction regime as considered in this paper due to the large shift induced by strong field interaction between the particles. We develop a novel technique based on conformal mapping theory to overcome this difficulty. The key is to design a conformal mapping which transforms the two particle system into a shell-core structure, in which the inner dielectric core corresponds to the target object. We show that a perturbation argument can be used to analyze the shift in the resonant frequencies due to the presence of the inner dielectric core. This shift also encodes information of the contracted polarization tensors of the core, from which one can reconstruct its shape, and hence the target object. Our theoretical findings are supplemented by a variety of numerical results based on an efficient optimal control algorithm. The results of this paper make the mathematical foundation for plasmonic sensing complete.

The effect of core size on the speed of compressible hollow vortex streets

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy &amp; Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

Computational design and additive manufacturing of periodic conformal metasurfaces by synthesizing topology optimization with conformal mapping

In this paper, we present a computational framework for computational design and additive manufacturing of free-form periodic metasurfaces. The proposed scheme rests on the level-set-based topology approach and the conformal mapping theory. A metamaterial with pre-specified performance is created using a level-set-based topology optimization method. The achieved unit cell is further mapped to the 3D quad meshes on a free-form surface by applying the conformal mapping method which can preserve the local shape and angle during the mapping. With embedded geometric information, the proposed level-set-based optimization methods not only can act as a motivator for design synthesis, but also can be seamlessly hooked with additive manufacturing without the need of CAD reconstructions. The current computational framework provides a solution to increasing applications involving innovative metamaterial designs on free-form surfaces for different fields of interests. The performance of the proposed scheme is illustrated through two benchmark examples where a negative-Poisson’s-ratio unit cell pattern, and a stiff and light inner structure are mapped to 3D free-form surfaces and fabricated through additive manufacturing.

Elastoplastic problem for a plate with partially unknown boundary

In this paper there is considered the Elastoplastic problem for infinite plate, that is weakened by two identical square holes. The boundaries of the holes are partially unknown contours. The plate is in a stressed state, a region of plasticity contains only unknown parts of holes contours and does not spread inside of the plate. Applying the theory of functions of a complex variable and the conformal mapping theory the problem is reduced to a boundary value problem of the analytic function theory and the solution of this problem is obtained, the unknown parts of the holes contours are defined.

Improving the Accuracy of Geometric Interpolation for Determining Fundamental Frequency of Parallelogram Plates Vibration

The paper considers a method of geometric interpolation of basic solutions for two-dimensional problems in the theory of elasticity and structural mechanics, particularly as applied to mechanical engineering. The scope of the study is the vibrations of thin elastic parallelogram plates of constant thickness. To determine the fundamental frequency of vibration it is suggested to use an interpolation method with a newly introduced geometric characteristic of plates which is a ratio of inner conformal radius to outer conformal radius. Taken from the theory of conformal mapping, the conformal radii of domains are obtained by mapping the plates onto the interior and exterior of a unit circle. The studies have shown that the use of the suggested ratio ​​gives up to twice more accurate results of the geometric interpolation. The paper presents the basic terms and formulas of the method with comparative analysis of the curve diagrams using a shape factor and a conformal radii ratio.