Schwarz-Christoffel Research Articles

String scattering amplitudes and deformed cubic string field theory

We study string scattering amplitudes by using the deformed cubic string field theory which is equivalent to the string field theory in the proper-time gauge. The four-string scattering amplitudes with three tachyons and an arbitrary string state are calculated. The string field theory yields the string scattering amplitudes evaluated on the world sheet of string scattering whereas the conventional method, based on the first quantized theory brings us the string scattering amplitudes defined on the upper half plane. For the highest spin states, generated by the primary operators, both calculations are in perfect agreement. In this case, the string scattering amplitudes are invariant under the conformal transformation, which maps the string world sheet onto the upper half plane. If the external string states are general massive states, generated by non-primary field operators, we need to take into account carefully the conformal transformation between the world sheet and the upper half plane. We show by an explicit calculation that the string scattering amplitudes calculated by using the deformed cubic string field theory transform into those of the first quantized theory on the upper half plane by the conformal transformation, generated by the Schwarz–Christoffel mapping.

Unexpected validity of Schottky's conjecture for two-stage field emitters: A response via Schwarz–Christoffel transformation

The electric field in the vicinity of the top of an emitter with a profile consisting of a triangular protrusion on an infinite line is analytically obtained when this system is under an external uniform electric field. The same problem is also studied when the profile features a two-stage system, consisting of a triangular protrusion centered on the top of a rectangular one on a line. These problems are approached by using a Schwarz–Christoffel conformal mapping, and the validity of Schottky's conjecture (SC) is discussed. The authors provide an analytical proof of SC when the dimensions of the upper-stage structure are much smaller than those of the lower-stage structure, for large enough aspect ratios and considering that the field enhancement factor (FEF) of the rectangular structure is evaluated on the center of the top of the structure, while the FEF of the triangular stage is evaluated near the upper corner of the protrusion. The numerical solution of our exact equations shows that SC may remain valid even when both stages feature dimensions of the same order of magnitude, reinforcing the validity of SC for multistage field emitters.

On the stress concentration around a polygonal cut-out of complex geometry in an infinite orthotropic plate

The best values of fiber volume fractions, fiber arrangements, and cut-out orientations in an orthotropic infinite plate weakened by a polygonal discontinuity of regular and complex geometry are investigated in the present work. Considering the stress concentration factor as a fitness minimization function, the genetic algorithm is employed and, elastic constants and stresses are computed utilizing the Mori-Tanaka theory and the Muskhelishvili’s complex variable method, respectively. The upshot of present work shows a substantial impact of fiber volume fraction, fiber arrangement and, corner radius and orientation of cut-out, on values of stress concentration factor for various in-plane loading conditions. Furthermore, the database of the complex constants used in the Schwarz–Christoffel mapping to develop distinct complex shapes is also reported.

Improved method for field analysis of surface permanent magnet machines using Schwarz–Christoffel transformation

Air gap field solution for an outer rotor permanent magnet (PM) machine is derived based on numerical Schwarz–Christoffel (SC) transformation of a single slot geometry and analytical solution of a point wire field in the rectangular canonical domain. This approach takes into account geometrical distortions of PM edges and field evaluation points located along the air gap centre, which occur due to conformal mapping of the slotted air gap into a slotless domain and impair the accuracy of the existing analytical field solutions using SC transformation and complex relative air gap permeance. The increased accuracy of the improved solution is confirmed by comparing the air gap flux density waveforms produced by the PMs and armature winding, cogging torque, and total torque with the results of time-stepping transient finite element simulation.

On the generation of coastline-following grids for ocean models\u2014trade-off between orthogonality and alignment to coastlines

Regional ocean models usually utilize orthogonal curvilinear grids that are fit to the coastline of the modeled regions. While the orthogonality of the grid is required from the perspective of the numerical algorithms, the alignment to the irregular coastlines improves the characterization of the land-sea distribution and the ocean simulation. In this article, we carry out fractal analysis of two representative coastal regions and discuss the trade-offs between the orthogonality and coastline alignment during the grid generation of these regions. A new grid generation method based on Schwarz-Christoffel conformal mappings is proposed, with automatic coastal boundary retrieval algorithm that generates resolution dependent boundary for grid generation and alleviates the human efforts involved in traditional methods. We show that for the southeastern Pacific region, the coastline is smooth with low fractal dimension and there exists effective trade-off with a coastline boundary that adjusts to the desired grid resolution. On the contrary, there is no effective trade-off for southeast China seas where the coastline is of higher fractal dimension, and a coarser coastline boundary is recommended for better orthogonality with little loss in coastline alignment. Further numerical study of coastal trapped Kelvin waves for the typical regions demonstrate that the new coastline-fitting grids achieve smaller error in numerical dispersion and higher accuracy. Through analysis, we conclude that for grid generation for regional ocean modeling, modelers should bring into consideration of the multi-scale fractal characteristics of the coastline.

Electric field distribution and current emission in a miniaturized geometrical diode

We study the electric field distribution and current emission in a miniaturized geometrical diode. Using Schwarz-Christoffel transformation, we calculate exactly the electric field inside a finite vacuum cathode-anode (A-K) gap with a single trapezoid protrusion on one of the electrode surfaces. It is found that there is a strong field enhancement on both electrodes near the protrusion, when the ratio of the A-K gap distance to the protrusion height d/h<2. The calculations are spot checked against COMSOL simulations. We calculate the effective field enhancement factor for the field emission current, by integrating the local Fowler-Nordheim current density along the electrode surfaces. We systematically examine the electric field enhancement and the current rectification of the miniaturized geometrical diode for various geometric dimensions and applied electric fields.

Study of Axial-Flux-Type Superconducting Eddy-Current Couplings

A theoretical physical model of axial-flux-type superconducting eddy-current coupling is systematically investigated. A mathematical description of a high-field superconducting bulk magnet used as the magnetic field source is modeled based on the physical superconducting mechanism. Thereafter, a practical two-dimensional (2-D) magnetic equivalent circuit model for the analytical calculation of magnetic field distribution in this coupling design using Schwarz–Christoffel mapping is proposed to predict the air gap magnetic density and the transferred torque, taking nonlinear effects of the magnetic materials into consideration. Finally, a 1/6 section 3-D model of the proposed superconducting eddy-current coupling is designed and analyzed via finite element method. The calculation results by the proposed analytical model are verified with those issued from the finite element method. Also, the efficiency of superconducting eddy current couple is analyzed theoretically.

Analytical Modeling of Manufacturing Imperfections in Double-Rotor Axial Flux PM Machines: Effects on Back EMF

In this paper, a general analytical model is proposed to calculate the back electromotive forces of various manufacturing imperfections in the double-rotor axial flux permanent magnet machine. First, the axial and circumferential flux density in the air gap under healthy condition is developed by Maxwell's equations and Schwarz-Christoffel mapping. Second, the equations of air gap length and radii are introduced to calculate the flux density with manufacturing imperfections. The back EMFs of each case are calculated. For each case, finite-element (FE) method models are developed to validate the analytical model, and the results show that the analytical model predictions match well with the FE results. Finally, the analytical model is verified by experimental results.

Univalent conformal mappings onto polygonal domains with countable set of vertices by generalized Christoffel–Schwarz integral

We propose a formula for the conformalmapping of the upper half-plane onto a polygonal domain, which generalizes the Schwarz–Christoffel equation. It is obtained by terms of partial solution to the Hilbert boundary-value problem with a countable set of singularity points of the coefficients including a turbulence of logarithmic type at the infinity point. We also prove the existence of closed and univalent mappings.

An improved conformal mapping method for magnetic field analysis in surface mounted permanent magnet motors

PurposeThe purpose of this paper is to present an improved conformal mapping (ICM) method that simultaneously considers the influence of relative recoil permeability of PMs, the armature reaction, the stator slotting, and the magnetic saturation on determination of the PM operating point in its different parts.Design/methodology/approachThe ICM method is a time-effective method that considers the magnetic saturation by suitable increments in air-gap length under each tooth and also the width of slot openings. In this paper, the analytical and numerical conformal mappings such as the Schwarz-Christoffel (SC) mapping are used for magnetic field analysis due to the permanent magnets and the armature reaction in one slotted air gap. The field solution in the slotted air gap is obtained through the modulation of field solution in one slotless air-gap using the complex air-gap permeance.FindingsThe ICM method can consider the magnetic saturation in different electric loadings, and also the variation of PM operating points in its different parts.Practical implicationsThe ICM method is applied to one surface mounted permanent magnet (SMPM) motor and is verified by comparing with the corresponding results obtained through finite element method (FEM), and frozen permeability finite element method (FP-FEM).Originality/valueThis paper presents an ICM method with a new technique for saturation effect modeling, which can be used to separate and calculate the on-load components of air-gap field and torque.

A two-dimensional analytical model of vapor intrusion involving vertical heterogeneity.

In this work, we present an analytical chlorinated vapor intrusion (CVI) model that can estimate source-to-indoor air concentration attenuation by simulating two-dimensional (2-D) vapor concentration profile in vertically heterogeneous soils overlying a homogenous vapor source. The analytical solution describing the 2-D soil gas transport was obtained by applying a modified Schwarz-Christoffel mapping method. A partial field validation showed that the developed model provides results (especially in terms of indoor emission rates) in line with the measured data from a case involving a building overlying a layered soil. In further testing, it was found that the new analytical model can very closely replicate the results of three-dimensional (3-D) numerical models at steady state in scenarios involving layered soils overlying homogenous groundwater sources. By contrast, by adopting a two-layer approach (capillary fringe and vadose zone) as employed in the EPA implementation of the Johnson and Ettinger model, the spatially and temporally averaged indoor concentrations in the case of groundwater sources can be higher than the ones estimated by the numerical model up to two orders of magnitude. In short, the model proposed in this work can represent an easy-to-use tool that can simulate the subsurface soil gas concentration in layered soils overlying a homogenous vapor source while keeping the simplicity of an analytical approach that requires much less computational effort.

Analytical modeling of axial flux PM machines with eccentricities

In this paper, an analytical quasi three-dimensional method is used to model an axial flux permanent magnet (AFPM) machine with various eccentricities. AFPM machines (AFPMMs) have various advantages but they are sensitive to geometrical imperfections for manufacturing aspect. The main aim of this paper is to propose a general analytical model to analyze the AFPMMs with various types of eccentricities. The radial and tangential magnetic flux densities in the air gap under healthy condition are obtained via combination of Maxwell’s equations and Schwarz-Christoffel (SC) mapping firstly. Next, in order to investigate the eccentricities, equations for air gap length and radii are deduced. The back electromotive force (EMF) is calculated and compared with those from healthy condition and finite element (FE) analysis, respectively. The results show that the analytical predictions agree well with the FE results. Moreover, using this method has a significantly less time consuming than the 3D FM simulation process, which is a great advantage of this method. Finally, the analytical model is verified via experimental results.

Analytic continuation of the Appell function F 1 and integration of the associated system of equations in the logarithmic case

The Appell function F 1 (i.e., a generalized hypergeometric function of two complex variables) and a corresponding system of partial differential equations are considered in the logarithmic case when the parameters of F 1 are related in a special way. Formulas for the analytic continuation of F 1 beyond the unit bicircle are constructed in which F 1 is determined by a double hypergeometric series. For the indicated system of equations, a collection of canonical solutions are presented that are two-dimensional analogues of Kummer solutions well known in the theory of the classical Gauss hypergeometric equation. In the logarithmic case, the canonical solutions are written as generalized hypergeometric series of new form. The continuation formulas are derived using representations of F 1 in the form of Barnes contour integrals. The resulting formulas make it possible to efficiently calculate the Appell function in the entire range of its variables. The results of this work find a number of applications, including the problem of parameters of the Schwarz–Christoffel integral.

Eddy current loss analysis of open-slot fault-tolerant permanent-magnet machines based on conformal mapping method

This paper proposed an analytical method, based on conformal mapping (CM) method, for the accurate evaluation of magnetic field and eddy current (EC) loss in fault-tolerant permanent-magnet (FTPM) machines. The aim of modulation function, applied in CM method, is to change the open-slot structure into fully closed-slot structure, whose air-gap flux density is easy to calculate analytically. Therefore, with the help of Matlab Schwarz-Christoffel (SC) Toolbox, both the magnetic flux density and EC density of FTPM machine are obtained accurately. Finally, time-stepped transient finite-element method (FEM) is used to verify the theoretical analysis, showing that the proposed method is able to predict the magnetic flux density and EC loss precisely.

ПРЕДЕЛЬНО-КВАЗИСТАЦИОНАРНАЯ МОДЕЛЬ ЭЛЕКТРОХИМИЧЕСКОГО ФОРМООБРАЗОВАНИЯ

The paper is focused on modeling of electrochemical machining in quasistationary approximation. The modeling is based on the use of Faraday’s and Ohm’s laws. The dependence of current efficiency on the current density in the form of a step function is used when solving this problem. The model enables us to study the configuration of a machining area, obtained in case of extreme high localization of a solution process. For this function there is a possibility in case of a non-steady process to divide sections of the machining area into zones with different conditions (presence and lack of solution). As the solution in case of such machining happens at a constant value of intensity module, there is a possibility to simulate the change of machining time by embedding of an electrochemical machining electrode into the edge of a workpiece. The evidence from practice shows that the accuracy of approximation of the non-steady process by quasistationary has three and more significant digits. Meanwhile, the computing time decreases by 100 and more times. The problem of shaping while machining by the machining electrode in the shape of a plate with the isolated top surface is solved. The methods of the complex variable theory have been used to solve the problems on the assumption about constancy of electrolytic conduction. The Joukowski and Schwarz-Christoffel transformations are applied to construct conformal mapping. The accurate solution on the problem in quadratures is obtained. The study on non-steady processes using a quasistationary model makes it possible to get information about a shape of the machining area at the beginning, continuation and the end of the machining process. In particular, it’s found out that near the section of the machining start there is a solution like outflow from the board, well-known in hydrodynamics. It allows examining properties of a shape that is formed at the edge of the machining area.

A Fast Analytic Model of Axial Flux Permanent Magnet Machines with Static/Dynamic Axis Eccentricity

This paper presents a general analytical model to calculate the characteristics of axial-flux permanent-magnet machines with axis eccentricities. The radial and tangential magnetic flux densities in the air gap under normal conditions were first obtained using a combination of Maxwell’s equations and Schwarz-Christoffel (SC) transformation. Next, equations for the radii were deduced to investigate the static/dynamic eccentricities. The back electromotive forces (EMFs) were calculated and compared with those obtained from finite element (FE) analysis. The analytical predictions show good agreement with the FE results. Detection approaches were obtained by comparing with normal conditions, and the analytical model was verified experimentally.

Analytical Modeling of Static Eccentricities in Axial Flux Permanent-Magnet Machines with Concentrated Windings

The aim of this paper is to calculate the static eccentricity (SE) of a double rotor axial flux permanent magnet (AFPM) machine by using a general analytical model. The flux density in the air gap under healthy conditions is calculated firstly, where the axial and circumferential magnetic flux densities are obtained using a coupled solution of Maxwell’s equations and Schwarz-Christoffel (SC) mapping. The magnetic flux densities under SE conditions are calculated afterwards using a novel bilinear mapping. Some important electromagnetic parameters, e.g., back electromotive force (EMF), cogging torque and electromagnetic (EM) torque, are calculated for both SE and healthy conditions, and compared with the finite element (FE) model. As for the double rotor AFPM, SE does not contribute much effect on the back EMF and EM torque, while the cogging torque is increased. At each calculated section, FE models were built to validate the analytical model. The results show that the analytical predictions agree well with the FE results. Finally, the results of analytical model are verified via experimental results.

A unified solution approach for a large variety of antiplane shear and torsion notch problems: Theory and examples

In this work, a unified solution approach is proposed for the analytical evaluation of the stress fields close to notches under antiplane shear and torsion loadings, which allows a large variety of notch problems to be tackled. The method is based on the complex potential approach for antiplane elasticity combined with the use of proper conformal mappings. In particular, it is shown that a well defined analytical link does exist between the complex potential to be used to determine stresses and the first derivative of the conformal mapping used to mathematically describe the notch profile. This makes some methodologies such as Schwarz-Christoffel transformation, which allows describing any polygonal domain automatically, very attractive for the direct solution of notch problems. A bulk of solutions are provided to support this finding, from cracks and pointed notches, to radiused notches. In addition, the accuracy of each proposed solution is discussed in detail taking advantage of a bulk of results from FE analyses.

Electrostatic repulsive out-of-plane actuator using conductive substrate.

A pseudo-three-layer electrostatic repulsive out-of-plane actuator is proposed. It combines the advantages of two-layer and three-layer repulsive actuators, i.e., fabrication requirements and fill factor. A theoretical model for the proposed actuator is developed and solved through the numerical calculation of Schwarz-Christoffel mapping. Theoretical and simulated results show that the pseudo-three-layer actuator offers higher performance than the two-layer and three-layer actuators with regard to the two most important characteristics of actuators, namely, driving force and theoretical stroke. Given that the pseudo-three-layer actuator structure is compatible with both the parallel-plate actuators and these two types of repulsive actuators, a 19-element two-layer repulsive actuated deformable mirror is operated in pseudo-three-layer electrical connection mode. Theoretical and experimental results demonstrate that the pseudo-three-layer mode produces a larger displacement of 0–4.5 μm for a dc driving voltage of 0–100 V, when compared with that in two-layer mode.

On the coupling of analytical and FEM solution in stress analysis around the polygonal hole shape in a finite two-dimensional domain

Stress analysis of finite plates with arbitrary hole shapes is performed using a mixed analytical-FEM approach. The analytical solution for the infinite plate is corrected using the FEM correction for the finite plate. The analytical method uses a modified Muskhelishvili complex variable method for an arbitrary polygonal hole shape utilizing the Schwarz–Christoffel mapping approach. Later, the FEM correction is calculated for the finite two-dimensional domain with appropriate boundary conditions. Coupled solutions accurately solve the problem of stress concentration at the hole corners. It is impossible to solve stress concentrations accurately using only the finite element method (FEM) approach because the solution is mesh dependent. The proposed method rapidly solves for the stress concentrations with arbitrary accuracy irrespective of hole shape and domain geometry.