Conformal Mapping Technique Research Articles

Stress field of a functionally graded coated inclusion of arbitrary shape

Based on the theory of complex variable functions, the stress field in an infinite matrix containing an arbitrary shape inclusion with a functionally graded coating is analyzed. The elastic properties in the functionally graded coating change continuously and arbitrarily along the normal direction of the inclusion. By using the method of piecewise homogeneous layers and the technique of conformal mapping, the complex potential functions in the matrix, coating and inclusion are derived in the form of Laurent series and Faber series, respectively. The influences of different varying Young’s modulus on the interfacial stresses are discussed by numerical examples for various shape inclusions, including ellipse, triangle, square and rectangle. It is shown that the magnitude and distribution of interfacial stresses for arbitrary shape inclusions can be successfully designed and controlled by adding a functionally graded coating with proper varying elastic properties along the normal direction. The results for some special cases are compared with previous literature and found in good agreement.

A concentrated couple near two non-elliptical inclusions with internal uniform hydrostatic stresses

We employ conformal mapping techniques to study the existence of internal uniform hydrostatic stresses inside two non-elliptical inclusions when the surrounding matrix is simultaneously subjected to a concentrated couple and remote uniform in-plane stresses. The unknown complex coefficients appearing in the corresponding mapping function can be determined analytically for a given pair of loading, one material and three geometric parameters. This allows us to subsequently identify the shapes of the two inclusions. Our analysis further reveals that the shapes of the inclusions depend on the concentrated couple, whereas the corresponding internal uniform hydrostatic stresses do not.

Magnetic Field Analysis of Multi-Flux-Barrier Interior Permanent-Magnet Motors Through Conformal Mapping

An extended analytical method for predicting the open-circuit air-gap field distribution in interior permanent magnet (IPM) machines equipped with multilayer Permanent Magnets (PMs) is proposed in this paper. This presented method is based on the conformal mapping technique. The mapping is used to calculate the relative permeances, which consider the effect of stator slots and rotor saliency on the magnetic field distribution. The open-circuit air-gap field distribution in a multi-flux-barrier IPM can be predicted through the superposition principle. The analytical solution is implemented on: an eight-pole, 50 kW IPM motor used in 2004 Toyota Prius hybrid electric vehicle and four IPMs with two, three, four, and five flux barriers per pole. To verify this extended method, the results are compared with those obtained from the finite-element analysis.

Dynamic analysis of a cylindrical cavity in inhomogeneous elastic half-space subjected to SH waves

Based on complex function methods and a multipolar coordinate system, the scattering induced by a cylindrical cavity in a radially inhomogeneous half-space is investigated. Mass density of the half-space varies depending on the distance from the centre of the cavity while the shear modulus is always constant. The wave velocity is expressed as a function of radius vector and the Helmholtz equation is a partial differential equation with a variable coefficient. By means of a conformal mapping technique, the Helmholtz equation with a variable coefficient is transferred into its normal form. Then, displacement fields and corresponding stress components are deduced. Applying the boundary conditions, dynamic stress concentration factors around the cavity are obtained and analyzed. Typical numerical results are presented to demonstrate the distribution of dynamic stress concentration factors when influencing parameters are assumed.

Harmonic elastic inclusions in the presence of point moment

We employ conformal mapping techniques to design harmonic elastic inclusions when the surrounding matrix is simultaneously subjected to remote uniform stresses and a point moment located at an arbitrary position in the matrix. Our analysis indicates that the uniform and hydrostatic stress field inside the inclusion as well as the constant hoop stress along the entire inclusion–matrix interface (on the matrix side) are independent of the action of the point moment. In contrast, the non-elliptical shape of the harmonic inclusion depends on both the remote uniform stresses and the point moment.

Generalised technique for calculation of plane director profiles in bounded nematic liquid crystals

ABSTRACTWe propose an improved conformal mapping technique for analytical calculation of two-dimensional profiles of the nematic liquid crystal (NLC) director in a cell with a simply connected cross-sectional region. We consider the case of the strong anchoring and the piecewise constant director pretilt on the piecewise smooth curve which bounds the region. Obtained expressions for the director profile explicitly depend on the conformal mapping which maps the region onto the upper half plane of a complex plane. An advantage of our method in comparison with the standard conformal mapping technique is that it does not require the knowledge of the inverse mapping and the calculation of the integral in the Poisson formula. Proposed technique allows to take into account topological defects in the bulk of the NLC on the symmetry axis of the region. As an example of how the method can be used, we find an analytical expression for the director profile in a horizontal cylindrical groove partly filled with the NLC. We consider the case where a disclination line parallel to the axis of the groove occurs in the bulk of the NLC. The equilibrium position of the disclination line is found.

Three-Dimensional Printed Fluid-Filled Electrostatic Rotating Machine Designed with Conformal Mapping Methods

Recently, fluid-filled electrostatic machines demonstrated specific and volumetric torque densities that hold promise to be competitive with electromagnetic machines in niche applications, e.g., air-cooled, low-speed, and direct-drive machines. These demonstrations of variable capacitance (or elastance, which is the dual of reluctance) machines were nonoptimized from an electrostatics perspective as their geometry was heavily constrained by manufacturability issues. This paper proposes a semi-analytical design method that combines conformal mapping techniques with finite element analysis, leading to more optimal electrostatic machine geometries. Parametric sweeps of key relative dimensions establish best practices/guidelines for design. A fractional horsepower proof of concept machine was designed using the new approach and was built using stereolithographic three-dimensional printing to circumvent manufacturing constraints. The machine is mostly plastic, plated with conductor, and is, therefore, lightweight. This manufacturing approach suggests that a machine can be injected molded or cast in a single step. Measurements of the prototype demonstrate a torque density of 0.31 Nm/L and specific torque density of 0.22 Nm/kg, comparable with similar size NEMA frame fractional horsepower induction motors. Also the Nm/kV2 of the prototype machine is two orders of magnitude greater than prior nonliquid-filled work.

Influence of nanoscale deformation twins near a slant edge crack tip on crack blunting in nanocrystalline metals

A theoretical model is developed that discusses the effect of nanoscale deformation twins on dislocation emission from the tip of slant edge crack in nanocrystalline materials. By combining complex variable method of Muskhelishvili and conformal mapping technique, the explicit solutions of complex potentials are obtained analytically. The critical stress intensity factors (SIFs) for the emission of first lattice edge dislocation from a slant edge crack tip are calculated. The effects of vital parameters such as the crack length, the inclined angle of the crack, the relative thickness of the nanotwin on critical SIFs for dislocation emission are evaluated in detail. The results show that the emission of lattice dislocation from the slant edge crack tip is significantly influenced by nanoscale deformation twins. The smaller the inclined angle is, the more difficult it is for dislocation emission from the slant edge crack tip. Particularly, the dislocation near the tip of slant edge crack is prone to emit when the inclined angle of the crack is about 45°. As a special case, when the inclined angle of slant edge crack is 0°, the present results will reduce to those of the problem of nanoscale deformation twins interacting with mode I straight crack.

Charge-Based Compact Model for DC Current in Organic TFT Including Non-Linear Injection Effects with a Close Link to Electrical Device Parameter

In this paper a closed-form current model for organic TFTs is presented which leads to a charge-based continuous current model with threshold voltage and slope as model parameters, while still keeping a single expression valid from below to above threshold operation. Most compact models for organic field-effect transistors (OFETs) are based on a two-piece description: a set of equations for above threshold operation is combined with a current model valid in subthreshold region by means of an analytical smoothing function [1]. Thereby the threshold voltage is introduced as fitting parameter without any relation to physical parameters. In [2] a charge-based current model with a continuous equation valid in all regions of operation has been presented. While solving Poisson’s equation the model preserves a close link to physical parameters like shallow and deep trap densities and avoids the introduction of a threshold voltage and subthreshold slope as model parameters. However, from a circuit designer’s perspective a model providing a close link to electrical device parameters is desirable because parameters as threshold voltage and slope can easily be extracted from I/V curves and are performance measures of a device. In this paper an improvement of the model from [2] is presented which leads to a charge-based C∞-continuous current model with threshold voltage and slope as model parameters, while still keeping a single expression valid from below to above threshold operation. To include the effect of hopping transport the empirical power-law mobility model [3] and a contact resistance [4] have been incorporated in the effective mobility. Channel length modulation effects have been included according to [5]. Furthermore, from physical parameters given by trap densities the threshold voltage and subthreshold slope of the device can be calculated. In the case of co-planar device structures the output characteristics in the linear operation regime show a superlinear behaviour due to non-linear injection effects. If the barrier height between source electrode and the channel material is not negligible the channel current is limited by the source injection current. The Schottky barrier height is modulated by the gate and drain potential. A strong electric field causes thinning of the barrier, allowing more carriers to enter channel. An analytical equation for the electric field at the injection barrier has been derived by applying the conformal mapping technique. The bias-dependent lowering of the Schottky barrier, and hence the maximum injection current can be calculated. An equivalent resistance of the Schottky barrier is calculated from the voltage drop Vds and the maximum possible injection current. To arrive at a closed-form current equation for the intrinsic device together with the injecting Schottky barrier this equivalent resistance is incorporated in the current equation by a first order approximation which is commonly used for the consideration of parasitic resistances. In this case the equivalent resistance is bias dependent and reduced with increasing Vds due to Schottky barrier lowering. The model has been validated vs. measurements on devices with small molecule organic semiconductors (Fig. 1, 2). The results show good agreement, even for the 1st and 2nd derivatives of the transfer characteristics. Acknowledgements: This project was funded by the German Federal Ministry of Education and Research contract No.03FH013PX2 (FhProfUnt). This project was also funded by the Horizon 2020 Programme of the European Union under contract RISE 2014 645760 ("DOMINO"). We acknowledge CEA-Liten (Grenoble) and Max Planck Institute for Solid State Research (Stuttgart) for providing measurement data. We thank AdMOS GmbH for support and Keysight Technologies for license donation of IC-CAP.

Two-dimensional problem of thermoelectric materials with an elliptic hole or a rigid inclusion

The two-dimensional problems of an elliptic hole or a rigid inclusion embedded in a thermoelectric material subjected to uniform electric current density and energy flux at infinity are studied based on the complex variable method of Muskhelishvili and conformal mapping technique. The closed-form solutions of electric potential, temperature and stress components are presented according to electrical insulated and thermal exact boundary conditions on the rim of the hole or inclusion. Numerical results are carried out to illustrate the influence of the value of major to minor axis ratio of the elliptic geometry and heat conductivity of inhomogeneity on thermoelectric and stress fields. It is found that energy flux at surfaces of the hole or rigid inclusion does not vanish due to the Joule heat and Seebeck effect when the electric field is applied. In addition, stress induced by applied electric field has a non-linear relationship with the electric current density. The heat conductivity of the air inside the elliptic hole reduces the concentration factors of energy flux and stress. However, the concentration factors of energy flux and stress at the bonding interface increase with the increasing values of heat conductivity of the flat rigid inclusion.

Effect of cooperative grain boundary sliding and migration on dislocation emission from a branched crack tip in deformed nanocrystalline solids

A theoretical model is established to describe the effect of cooperative grain boundary (GB) sliding and migration on dislocation emission from the tip of branched crack in deformed nanocrystalline solids. The explicit solutions of complex potentials are obtained by means of complex variable method and conformal mapping technique. The critical stress intensity factors (SIFs) for the first lattice dislocation emission from the tip of branched crack are calculated. The effects of the lengths of branched crack and main crack, and the angle between their planes on the critical SIFs for dislocation emission are evaluated in detail. The results indicate that the emission of lattice dislocations from the tip of branched crack is strongly influenced by cooperative GB sliding and migration. When main crack approaches the branched crack, dislocation emission from the tip of branched crack will be suppressed. The main crack tends to propagate while shorter branched crack is prone to be blunted by emitting lattice dislocations from its tip. As a special case, when the planes of main crack and the branched crack are flattened out into one, the present results are in good agreement with previously known results.

Applications of quadrilateral and quadrilateral-prism mesh generation in overland and subsurface simulations

This paper developed an automatic mesh generator used for simulating real-world problems of hydrological cycles, fluid flow, salinity and thermal transport, sediment and water quality transport, etc. The 2-D overland area was discretized with quadrilaterals based on conformal mapping technique. A position-percent interpolation equation was established to determine nodal vertical-coordinates and match boundaries. For 3-D simulations of subsurface zones, an additional mesh generation method was proposed to create quadrilaterals for filling the empty regions along river reaches and around junctions with storage, ponds or lakes. In order to deal with special storage region with an odd number of boundary nodes, an additional triangle was inserted. Five topological optimization templates were proposed to improve the quality of degenerated elements. This paper proposed the basic algorithms to generate quadrilateral-prism meshes through stretching the quadrilateral mesh along vertical directions based on material domains. Aiming at four types of fractures, corresponding extraction methods were presented to preserve geometric features. In order to accommodate the need for overland and subsurface simulations, 1-D/2-D/3-D correspondence was established between rivers, junctions, ponds, lakes, control structures and finite element meshes. Finally, several realistic hydrologic and geologic examples for meshing multi-domain areas were provided to demonstrate the accuracy and effectiveness of the mesh generator developed in this paper.

Simply and doubly periodic arrangements of the equi-stress holes in a perforated elastic plane: The single-layer potential approach

The layer potentials of two-dimensional linear elastostatics are applied as a novel building block for the semi-analytical design of non-standard arrangements of the equi-stress holes in an infinite plate under a given bulk-type loading. Thematically, this paper begins where our previous work left off with the main attention being focused on periodic structures of low rotational symmetry which are hard to tackle by the more customary conformal mapping technique. The analytical derivations are backed up by numerical simulations, in which a genetic algorithm is utilized to identify the optimal interface shapes for different geometries and a wide range of the governing parameters.

Successive Conformal Mapping Technique to Extract Inner Fringe Capacitance of Underlap DG-FinFET and Its Variations With Geometrical Parameters

We propose a new analytical model based on successive conformal mapping to compute the bias dependent inner fringe capacitance in nonplanar multigate MOSFET structure with doping modulated source/drain (S/D) and gate underlap for sub 20-nm node. The conventional analytical model of capacitance and resistance for planar MOSFET cannot be applied to the nonplanar multigate MOSFET. This model considers 3-D devices fabricated on bulk oxide, gate-S/D extension with nonuniform doping gradient and spacers. The percentage variation of inner fringe capacitance with respect to underlap length is studied for various fin width and oxide thickness.

Screw dislocation in a thin film with surface effects

We use conformal mapping techniques to derive a semi-analytical solution to the problem of a (straight) screw dislocation embedded in a thin solid film. The surfaces of the film are assumed to incorporate surface effects which result in deformation-dependent tractions imposed on the surfaces of the film. A number of examples are used to illustrate the stress distribution in the film and the image force acting on the dislocation. We show that in the absence of surface effects significant errors are induced in the determination of both the stress field in the film and the mobility of the dislocation, in particular as the dislocation approaches each of the film's surfaces or as the film becomes thinner. In fact, we demonstrate that the incorporation of surface effects with either positive or negative (surface) shear modulus can either relieve or intensify the stress concentration on the surfaces of the film. We find also that the dislocation tends to move towards a surface which possesses a smaller surface shear modulus and a smaller distance to the dislocation. Moreover, we show that when the thickness of the film exceeds ten times the (smaller) distance between the film's surfaces and the dislocation, the film can be treated essentially as a half-space without inducing large errors in the determination of the stress field in the film and the image force imposed on the dislocation.

Surface Motion of a Half‐Space with a Semicylindrical Canyon underP,SV, and Rayleigh Waves

A mathematically rigorous closed‐form solution is presented for the surface motion of a half‐space under P , SV , and Rayleigh waves scattered by a semicylindrical canyon using the complex variable function method. The scattered wave potentials are obtained in terms of the image method. With the aid of the conformal mapping technique, the half‐surface in the original is mapped onto a semicircumference in the transformed plane. The boundary conditions along the half‐surface are expressed in the transformed domain. The boundary conditions along the canyon surface are formulated in the original domain. The boundary value problem, once integrated along the corresponding angular weights, leads to a series of linear algebraic equations with respect to the unknown coefficients, which can be solved straightforwardly. The convergence of the solution results is tested by truncating the series number. The accuracy of the proposed solution is examined by comparing the present results with those using numerical methods. A parametric study is performed to evaluate the effects of the geometry of the canyon and the material properties on the dynamic response of the domain.

A mode-III crack under adhesion studied by non-uniform linear spring models

A model-III crack under adhesion is studied by non-uniform linear spring models. Employing complex variable method and conformal mapping technique, a closed-form solution is obtained, and an explicit relation between the stress intensity factor and spring stiffness is established. Results show that adhesion reduces the stress intensity factor, and the reduction effect increases with increasing spring stiffness and crack length. As a result of adhesion, the stress intensity factor given by the present model first increases quickly with increasing crack length when the crack length is relatively small, and decreases slowly with increasing crack length after reaching its maximum at a critical crack length.

The fracture behavior of two asymmetrical limited permeable cracks emanating from an elliptical hole in one-dimensional hexagonal quasicrystals with piezoelectric effect

By modeling that the surfaces of the cracks and hole are in limited permeable boundary conditions, the anti-plane problem of an elliptical hole with two asymmetrical cracks in one-dimensional (1D) hexagonal quasicrystals (QCs) with piezoelectric effect is investigated by adopting the technique of conformal mapping and the Stroh-type formulism. The analytic solutions of the field intensity factors and energy release rate are presented. With the variation of the hole-size and the crack length, the proposed model can be reduced to some classic crack models. In the absence of the electric load or the phason field, the present results match with the classical results of 1D hexagonal QCs or piezoelectric materials provided in the open literature. Numerical examples are then conducted to reveal the effects of the hole-size, the crack length, the coupling coefficient and applied mechanical/electric loads on the field intensity factors and the energy release rate.

Stress field in a porous material containing periodic arbitrarily-shaped holes with surface tension

In the mechanical analysis of a structure/composite with periodic holes/inhomogeneities based on analytic techniques, the holes/inhomogeneities are usually assumed to be circular. In this paper, we develop an efficient method (based on complex variable techniques) to calculate the surface tension-induced stress field in a porous material containing a periodic array of unidirectional holes of arbitrary shape. In this method, we use conformal mapping and Faber series techniques to address a finite representative unit cell (RUC) consisting of a single arbitrarily-shaped hole with a constant surface tension imposed on the hole’s boundary and periodic deformations imposed on the edge of the RUC. Several numerical examples are presented to verify the accuracy of our method and to study the influence of the shape and volume fraction of the periodic holes on the stress concentration in the structure. We show that the maximum hoop stress around periodic holes of some shapes (such as triangle, pentagon or hexagon) may appear exactly at the point(s) of maximum curvature when the hole volume fraction exceeds a certain value. Moreover, when the hole volume fraction falls below about 7%, it is found that the surface tension-induced stress concentration around periodic holes can be treated approximately as that around a single hole with the same hole shape and size in an infinite plane.

Analysis and design of a broadband coplanar Wilkinson power divider at X‐band

ABSTRACTThe analysis and design methods of a broadband coplanar Wilkinson power divider are proposed in this paper. The novelty is that cascaded two sections of coupled asymmetrical coplanar strips (ACPSs) are utilized to obtain broadband performance over X‐band and two resistors are used for isolation. The design equations are derived using even‐ and odd‐mode analysis. The specific structure parameters for broadband performance over X‐band are obtained through electronic design simulation tool and quasi‐static conformal mapping technique. The power divider is fabricated on GaAs substrate and is compatible with GaAs MMIC process. The measured results of the power divider show close agreement with the simulation. The measured return loss S11 is better than −15 dB across the entire X‐band, 8–12 GHz. Good dividing and high isolation properties are also revealed through measurements. © 2016 Wiley Periodicals, Inc. Microwave Opt Technol Lett 59:307–312, 2017