Conformal Mapping Technique Research Articles

A circular Eshelby inclusion with linear eigenstrains interacting with a coated non-parabolic inhomogeneity with internal uniform anti-plane stresses

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

Interior eigensolver for sparse Hermitian definite matrices based on Zolotarev’s functions

This paper proposes an efficient method for computing selected generalized eigenpairs of a sparse Hermitian definite matrix pencil $(A,B)$. Based on Zolotarev's best rational function approximations of the signum function and conformal mapping techniques, we construct the best rational function approximation of a rectangular function supported on an arbitrary interval via function compositions with partial fraction representations. This new best rational function approximation can be applied to construct spectrum filters of $(A,B)$ with a smaller number of poles than a direct construction without function compositions. Combining fast direct solvers and the shift-invariant generalized minimal residual method, a hybrid fast algorithm is proposed to apply spectral filters efficiently. Compared to the state-of-the-art algorithm FEAST, the proposed rational function approximation is more efficient when sparse matrix factorizations are required to solve multi-shift linear systems in the eigensolver, since the smaller number of matrix factorizations is needed in our method. The efficiency and stability of the proposed method are demonstrated by numerical examples from computational chemistry.

A hole of irregular shape interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

We use conformal mapping techniques together with analytic continuation to show that a non-parabolic open elastic inhomogeneity continues to admit a state of uniform internal stress when a hole with closed curvilinear traction-free boundary is placed in its vicinity and the surrounding matrix is subjected to uniform remote anti-plane stresses. The internal uniform stress field inside the inhomogeneity is found to be independent of the existence of the nearby hole and the specific non-parabolic shape of the inhomogeneity. In contrast, the non-parabolic shape of the inhomogeneity is influenced solely by the existence of the nearby hole.

Phoretic self-propulsion of Janus disks in the fast-reaction limit

Using a combination of singular perturbation analysis and conformal mapping techniques, the self-propulsion velocity is found in the limit of fast reaction for a Janus particle with half of the boundary active and the other half inert.

Semi-analytical approach for calculation of intercore crosstalk in multicore fibers

In this paper, a semi-analytical method to calculate intercore crosstalk (ICXT) in bent and twisted multicore fibers (MCFs) is proposed. The modified refractive index of bent MCF is first computed using geometrically exact beam theory (GEBT) and conformal mapping technique. The computed refractive index profile is used to obtain mode field profiles of bent MCF by solving the wave equation using vectorial finite element method in COMSOL. To take twisting of MCF into account, outer cores of the MCF are approximated to be helically bent around the central core. Using the mode field profiles, mode coupling coefficient between the MCF cores is calculated for different bend radii which is then used to calculate ICXT by using closed form analytical expressions reported in the literature. The ICXT values obtained using our proposed approach are in very good agreement with the experimental values reported in the literature. The effect of change in core pitch and change in relative refractive index contrast of MCF on the mode coupling coefficient between the MCF cores is analysed. The influence of twisting rate of MCF on the ICXT values for different bend radii is studied and it is shown that, ICXT is constant with respect to twisting rate of MCF for bending radii greater than the critical bending radius of the MCF. The proposed approach is also extended to calculate ICXT between different modes of trench assisted few-mode MCF (FM-MCF). The ICXT values obtained for the trench assisted FM-MCF follow the expected trend.

Equivalent DG Dimensions Concept for Compact Modeling of Short-Channel and Thin Body GAA MOSFETs Including Quantum Confinement

In this work, short-channel effects (SCEs) in cylindrical gate-all-around (GAA) MOSFETs with intrinsic or lightly doped channels are analytically described by using the conformal mapping technique for 2-D double-gate (DG) FETs. An equivalent capacitor model leads to an equivalent channel length concept, which allows to correctly determine the SCEs relevant center and surface potentials ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi _{{C}} $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi _{{S}} $ </tex-math></inline-formula> ) at the potential barrier. Furthermore, we make use of the rotational symmetry of GAA FETs and modify a compact drain current model of a DG device to use it for GAA transistors. Also, a mathematical correlation regarding quantum confinement for thin body transistors is derived, which shows that the mostly unwanted quantum effects occur in GAA structures already for thicker channels compared to DG transistors. Both transistor types experience a comparable influence with regard to quantization if the channel thickness of GAA FETs is 53% more than that of DG FETs. The dc behavior of our adapted model is verified with 3-D TCAD simulation data and applied to experimental data of ultrascaled silicon on insulator (SOI) omega-gate nanowire N-MOSFETs.

Numerical modeling of two microwave sensors for biomedical applications

AbstractThe purpose of this invited paper is to give readers a critical and comprehensive overview on how to extract dielectric properties of a bioliquid within a broad frequency range. Two sensors are used in the paper to characterize saline solutions by measuring the broadband complex permittivity. The two sensors are based on transmission line and interdigital electrodes designed for low‐ and high‐ frequency measurements, respectively, on the basis of the coplanar waveguide structure due to its convenience of fabrication and integration with microfluidic structures for liquid measurements. Different from traditional work where the finite element simulation method is used, the characterization theories of the two sensors are built based on a numerical modeling procedure, which can dramatically increase the device design efficiency, taking just a few seconds. Differently from the finite element method, the proposed numerical analysis utilizes a conformal mapping technique for both sensors. The characterization theories of the two sensors are validated by measuring de‐ionized water. The platform is finally used to measure 0.1 and 0.5 mol/L saline solutions within a broadband frequency range going from 10 up to 50 GHz, with the repeatability error within 5%.

Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple.

Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes.

Analytical Modeling of Participation Reduction in Superconducting Coplanar Resonator and Qubit Designs Through Substrate Trenching

A strategy aimed at decreasing dielectric loss in coplanar waveguides (CPW) and qubits involves the creation of trenches in the underlying substrate within the gaps of the overlying metallization. Participation of contamination layers residing on surfaces and interfaces in these designs can be reduced due to the change in the effective dielectric properties between the groundplane and conductor metallization. Although finite element method approaches have been previously applied to quantify this decrease, an analytical method is presented that can uniquely address geometries possessing small to intermediate substrate trench depths. Conformal mapping techniques produce transformed CPW and qubit geometries without substrate trenching but a non-uniform contamination layer thickness. By parametrizing this variation, one can calculate surface participation through the use of a two-dimensional, analytical approximation that properly captures singularities in the electric field intensity near the metallization corners and edges. Examples demonstrate two regimes with respect to substrate trench depth that capture an initial increase in substrate-to-air surface participation due to the trench sidewalls and an overall decrease in surface participation due to the reduction in the effective dielectric constant, and are compared to experimental measurements to extract loss tangents on this surface.

A circular inhomogeneity interacting with an open inhomogeneity designed to admit internal uniform hydrostatic stresses

We use analytic continuation and conformal mapping techniques to prove the existence of an internal uniform hydrostatic stress field inside a two-dimensional non-parabolic open inhomogeneity interacting with a nearby circular elastic inhomogeneity. The circular inhomogeneity and the surrounding matrix have distinct Poisson's ratios but identical shear moduli. The single complex constant appearing in the conformal mapping function is determined analytically. A simple condition is found that ensures that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. We also establish the existence of an internal uniform hydrostatic stress field inside a non-parabolic open inhomogeneity near an arbitrary number of circular elastic inhomogeneities. The circular inhomogeneities and the matrix have distinct Poisson's ratios but a common shear modulus. All of the complex constants in the conformal mapping function are uniquely determined by solving a set of non-linear equations via iterations.

Fracture analysis for a cusp-type crack problem

This paper provides an exact solution for a cusp-type crack problem in plane elasticity. The solution of the studied problem depends on the conformal mapping technique and complex potentials. The stress intensity factors at the cusp-type crack tip can be evaluated accordingly. Some particular characters for the SIF (stress intensity factor) solution have been found. Particularly, if a compression loading is applied along the direction of the cusp-type crack, the relevant SIF will take a positive value. This phenomenon is useful in the rupture of the rock with the cusp-type crack.

Edge corrections for parallel-plate capacitors

As a result of field fringing, the capacitance of a parallel-plate capacitor differs from that predicted by the textbook formula. Using singular perturbations and conformal mapping techniques, we calculate the leading-order correction to the capacitance in the limit of large aspect ratio. We additionally obtain a comparable approximation for the electrostatic attraction between the plates.

Interaction of a screw dislocation with a parabolic cavity and a semi-infinite crack

We derive an analytic solution to the problem of a screw dislocation interacting with a parabolic cavity and a semi-infinite sharp crack using conformal mapping techniques and the method of images. Closed-form expressions for the image force acting on the screw dislocation, the mode III stress intensity factor at the crack tip and the generalized mode III stress intensity factor for the parabolic cavity are obtained. The correctness of the solution is validated by comparison with existing solutions in the literature.

An edge dislocation interacting with two semi-infinite cracks

We use conformal mapping techniques and analytic continuation to derive an analytic solution to the problem of an edge dislocation interacting with two semi-infinite vertical cracks. Closed-form expressions for the rigid-body rotation at infinity of the left half-plane, the image force acting on the edge dislocation due to its interaction with the two cracks and mode I and mode II stress intensity factors at the two crack tips induced by the edge dislocation are obtained.

Parametrization of CPW for Photonics Application with the Use of Closed Form Formulas

In this paper parametrized analysis of CPW (Coplanar Waveguide) applied in photonics is presented. The analysis is based on analytical formulas, which were delivered and modified from conformal mapping technique, and also after review of available literature from analytical models for the conventional CPW. Adjustment of the already reported CPW formulas was performed to properly model thin films CPW structures in PIC (Photonic Integrated Circuit). Thus, a closed expressions were proposed to simulate basic parameters, like: per unit length (p.u.l.) R (resistance), L (inductance), C (capacitance), G (conductance) and propagation quantity like characteristic impedance Zo versus frequency. The CPW line dedicated for the PIC application, is based on InP (Indium Phosphide) substrate and with air or BCB (Benzocyclobutene) surroundings of the metallization on the top. The modified formulas are curve-fitting approximations, with new fitting coefficients. Essential modifications are for p.u.l. C and p.u.l. L. Usefulness and effectiveness of the modified method was verified through results comparison with simulations in full-wave Electronic Design Automation software, with the circuit simulator QUCS and with real measurements, conducted with the use of PNA (Performance Network Analyzer). The paper novelty are new coefficients for analytical formulas, implementation of modified formulas in the octave code and model parametrization for geometrical parameters.

Modeling of Surface Potential and Fringe Capacitance of Selective Buried Oxide Junctionless Transistor

The Selective Buried Oxide-Charge Plasma based Junctionless Transistor (SELBOX-CPJLT) is a new upcoming device used mainly for reducing self-heating effects and second order effects. In this paper, 2D mathematical modeling of Surface Potential for SELBOX-CPJLT using Poisson’s equation has been proposed. We have modelled the fringe capacitance for SELBOX-CPJLT with the help of conformal mapping. SELBOX structure is a novel idea to counter the self-heating effect which is present in the SOI technologies. Implementation of the charge plasma on SELBOX based JLT MOSFETs is called as SELBOX-CPJLT. The electrostatic performance and the lattice temperature of SELBOX-CPJLT is also compared with the SELBOX-JLT in this paper. The simulation results show that the SELBOX-CPJLT device has low Drain Induce Barrier Lowering (DIBL) effect, Subthreshold Slope (SS) with high Ion/Ioff ratio. It also provides good thermal stability as compared to SELBOX-JLT. The Fringe capacitance of SELBOX-CPJLT MOSFET has also been tabulated for different device parameters using conformal mapping technique. The proposed device is simulated and validated by using 2D-Silvaco (ATLAS) simulation.

Uniformity of stresses inside a non-elliptical inhomogeneity near an irregularly shaped hole in plane elasticity

We analyze the uniformity of stresses inside a non-elliptical elastic inhomogeneity interacting with a nearby irregularly shaped hole when the surrounding matrix is subjected to uniform remote in-plane stresses. Using Muskhelishvili's complex variable formulation and the technique of conformal mapping for a doubly connected domain, we demonstrate that Eshelby's uniformity property is indeed valid when the inhomogeneity and the matrix have the same shear modulus but distinct Poisson's ratios. In this case, the shapes of the inhomogeneity and the hole are completely determined by two material, one loading and three geometric parameters.

Results for Initial Trials Comparing Conformal with Non-Conformal Hypocenter Scanning Algorithms

This present paper sets out to perform a limited comparison between that method of P2P (point to point) scanning for Hypocenters, which uses the structure, imparted by a conformal mapping, to accelerate its ray tracing activity, and a method of scanning which incorporates the Earth spheroidal geometry directly within its ray tracing technique and which does not currently use a conformal mapping technique. The “conformal” method uses a mean-radius spherical Earth or can take a mean radius over the region covered by Epicenter and the sensor-field.In this comparison an attempt is made to compare and contrast a set of some possible functions (“indicator functions”) which serve to define the position of a Hypocenter within their response curve which is formed during the scan. Depending on the function, the expected indication may be either a minimum or a maximum, occurring within the range of the response.

Anti-plane analysis of a reinforced nano-elliptical cavity or nano-crack in a magnetoelectroelastic matrix with surface effect

Taking the surface effect into account, anti-plane analysis of a reinforced (coated) nano-elliptical cavity or nano-crack embedded in an infinite magnetoelectroelastic (MEE) matrix is considered under remotely mechanical, electrical and magnetic loadings. Analytical treatment of the problem is presented based on the technique of conformal mapping and the method of complex variable function. Exact solutions of stress, electric displacement and magnetic induction in the reinforced layer and matrix are derived explicitly in closed form. Numerical examples are provided to show the influence of material properties of surface layer and reinforced layer, aspect ratio of elliptical hole and coatings, thickness of coatings and crack length on the field concentration around the nano-hole and field intensity factors near the nano-crack tip. The result indicated that the material properties of surface layer and reinforced layer have a great effect on the fields concentration around the nano-hole or nano-crack, which is helpful for optimum design of MEE nanocomposites in engineering practice.

Effective elastic stiffness of 2D materials containing nanovoids of arbitrary shape

This paper aims to determine the in plane effective elastic properties of two-dimensional (2D) materials containing nanovoids of arbitrary shape. To achieve this objective, the complex variable and the associated conformal mapping techniques are used to solve the heterogeneity problem of a single nanovoid with arbitrary shape embedded in an infinite matrix. In this particular problem, to capture the edge and size effects of nanovoids, line elasticity model is used for the void boundary. The results of the heterogeneity problem are then used to determine the elastic properties of 2D nanoporous materials by applying the dilute and Mori–Tanaka schemes. Applications to the case of aluminum and the study the shape and size effects are also presented.