Interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a one-dimensional hexagonal quasicrystal with piezoelectric effect

The interaction between a screw dislocation and an elliptical hole with two asymmetrical cracks in a one-dimensional(1 D) hexagonal quasicrystal with piezoelectric effect is considered. A general formula of the generalized stress field, the field intensity factor, and the image force is derived, and the special cases are discussed. Several numerical examples are given to show the effects of the material properties and the dislocation position on the field intensity factors and the image forces.

Some Geometric Properties of the m-Möbius Transformations

Möbius transformations, which are one-to-one mappings of onto have remarkable geometric properties susceptible to be visualized by drawing pictures. Not the same thing can be said about m-Möbius transformations fm mapping onto . Even for the simplest entity, the pre-image by fm of a unique point, there is no way of visualization. Pre-images by fm of figures from C are like ghost figures in Cm. This paper is about handling those ghost figures. We succeeded in doing it and proving theorems about them by using their projections onto the coordinate planes. The most important achievement is the proof in that context of a theorem similar to the symmetry principle for Möbius transformations. It is like saying that the images by m-Möbius transformations of symmetric ghost points with respect to ghost circles are symmetric points with respect to the image circles. Vectors in Cm are well known and vector calculus in Cm is familiar, yet the pre-image by fm of a vector from C is a different entity which materializes by projections into vectors in the coordinate planes. In this paper, we study the interface between those entities and the vectors in Cm. Finally, we have shown that the uniqueness theorem for Möbius transformations and the property of preserving the cross-ratio of four points by those transformations translate into similar theorems for m-Möbius transformations.

Uniform Convergence of Extremal Polynomials When Domains Have Corners and Special Cusps on the Boundary

We study the approximation properties of the extremal polynomials in Ap?norm and C?norm. We prove estimates for the rate of such convergence of the sequence of the extremal polynomials on domains with corners and special cusps.

Analytical solution for steady seepage into a circular deep-buried mountain tunnel with grouted zone in anisotropic strata

Due to the existence of a large number of discontinuous fractures and interfaces in tunnel surrounding rocks,the groundwater inflow into tunnel generally presents significant anisotropy.Therefore,it is of great significance to consider the anisotropic permeability when dealing with water gushing-induced engineering accidents in water-rich mountain tunnels with large burial depth.In this study,based on the complex variable method and the seepage flow theory,a theoretical model of water inflow into a deep-buried circular tunnel in a fully saturated,anisotropic and semi-infinite aquifer is developed.The influence of grouted zone,initial support and secondary lining is fully considered.By comparison to the existing analytical methods and numerical results,the reliability of this proposed analytical solution is well validated.It is indicated from the parametric study that the groundwater inflow into tunnel presents an upward trend with an increasing value of the strata permeability in the vertical direction.Moreover,the water inflow rate and the total water head decrease with the growth of the thickness of grouting circle.It is suggested that reasonable grouting thickness and permeability should be controlled to enhance the grouting effect.This study provides a practical method for estimating the water inflow into a deep-buried,grouted and lined mountain tunnel considering the anisotropic strata permeability.

Optimal thermal design of anisotropic plates with arbitrary cutouts using genetic algorithm

Anisotropic plates in different applications may have geometric defects such as openings and cracks.The presence of the opening disturbs the heat flow,which creates significant thermal stress around the opening.When the heat flux is high enough,these extreme stresses can lead to structural failure.This article aims to obtain the optimal parameters for achieving the minimum value of the normalized stress near the cutout’s boundary in perforated anisotropic plates utilizing the genetic algorithm.Optimization parameters include the curvature of opening’s corners,orientation angle of opening,fibers angle,heat flux angle,and opening’s elongation.The plate is under heat flux,and the opening’s border is thermally insulated.The stress distribution around the opening is calculated using Lekhnitskii’s complex variable method and complex potential functions.The genetic algorithm is then implemented to find the optimal values for design parameters.The results show that by selecting the optimal parameters related to the anisotropic material and the opening’s geometry,the stress intensity factor of the perforated anisotropic plates is remarkably reduced.Furthermore,this optimization algorithm can be extended to find the optimized parameters and achieve the optimal designs in anisotropic and isotropic perforated plates under thermal loadings.

A screw dislocation near one open inhomogeneity and another closed inhomogeneity both permitting constant interior stresses

We prove that the interior stresses within both a non-parabolic open inhomogeneity and another interacting non-elliptical closed inhomogeneity can still remain constant when the matrix is simultaneously under the action of a screw dislocation and uniform remote anti-plane stresses.The constancy of interior stresses is realized through the construction of a conformal mapping function for the doubly connected domain occupied by the surrounding matrix.The mapping function is endowed with the information describing the screw dislocation via the incorporation of two specifically defined logarithmic terms.The constant interior stress fields are observed to be independent of the specific open and closed shapes of the two inhomogeneities and the existence of the screw dislocation.In contrast,the existence of the neighboring screw dislocation significantly affects the open and closed shapes of the two inhomogeneities.

General solutions of plane problem for power function curved cracks

A new exact and universal conformal mapping is proposed.Using Muskhelishvili’s complex potential method,the plane elasticity problem of power function curved cracks is investigated with an arbitrary power of a natural number,and the general solutions of the stress intensity factors(SIFs)for modeⅠand modeⅡat the crack tip are obtained under the remotely uniform tensile loads.The present results can be reduced to the well-known solutions when the power of the function takes different natural numbers. Numerical examples are conducted to reveal the effects of the coefficient,the power,and the projected length along the x-axis of the power function curved crack on the SIFs for modeⅠand modeⅡ.

Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals

This paper provides several solutions to the degenerate scale for the shapes of triangles or quadrilaterals in an exterior boundary value problem (BVP) of the antiplane elasticity or the Laplace equation. The Schwarz-Christoffel mapping is used throughout. It is found that a complex potential with a simple form in the mapping plane satisfies the vanishing displacement condition (or w = 0) along the boundary of the unit circle when the dimension R reaches its critical value 1. This means that the degenerate size in the physical plane is also achieved. The degenerate scales can be evaluated from the particular integrals depending on certain parameters in the mapping function. The numerical results of degenerate sizes for the shapes of triangles or quadrilaterals are provided.

THE VALUE DISTRIBUTION OF GAUSS MAPS OF IMMERSED HARMONIC SURFACES WITH RAMIFICATION

Motivated by the result of Chen-Liu-Ru[1],we investigate the value distribution properties for the generalized Gauss maps of weakly complete harmonic surfaces immersed in R^(n) with ramification,which can be seen as a generalization of the results in the case of the minimal surfaces.In addition,we give an estimate of the Gauss curvature for the K-quasiconfomal harmonic surfaces whose generalized Gauss map is ramified over a set of hyperplanes.

一个复杂边值问题的解

一引言随着微波理论和技术的迅速发展,对新型传输线的研究提出了更高的要求。一系列新型的微波转换过渡部件、微波滤波器和新型定向耦合器的研究,是建立在对新型传输线的理论分析基础之上的。因此,分析菱形外导体—椭园内导体传输线(以下简称外菱—内椭传输线)具有一定的理论意义和实用意义。这种传输线具有极化面固定、功率容量较大等优点。由于其截面复杂,因而无精确解,至今未见专文论述。本文采用“特征点图形逼近法”(具体方法见下文),导出特性阻抗和分布电容的计算公式,并由此推出在特定条件下的另外五种传输线的特性阻抗计算公式。所得结果均用初等函数表示,并附有部分计算结果,以供参考。

Using Refined Theory to Studied Elastic Wave Scattering and Dynamic Stress Concentrations in Plates with Two Cutouts

In this paper, based on complex variables and conformal mapping methods, using the refined dynamic equation of plates, elastic wave scattering and dynamic stress concentrations in plates with two cutouts were studied. Applying the orthogonal function expansion method, the problem to be solved can be reduced into the solution of a set of infinite algebraic equations. According to free boundary conditions, numerical results of dynamic moment concentration factors in thick plates with two circular cutouts analyze that: there will be more complex interaction changes between two-cutout situation than single cutout situation. In the case of low frequency or high frequency and thin plate, the hole-spacing in the absence of coupling interactions was larger or smaller. The numerical results and method can be used to analyze the dynamics and strength of plate-like structures.

Simple Proofs of Upper and Lower Envelopes of Van Der Pauw’s Equation for Hall-Plates with an Insulated Hole and Four Peripheral Point-Contacts

For plane singly-connected domains with insulating boundary and four point-sized contacts, C0 …C3, van der Pauw derived a famous equation relating the two trans-resistances R01,23, R12,30 with the sheet resistance without any other parameters. If the domain has one hole van der Pauw’s equation becomes an inequality with upper and lower bounds, the envelopes. This was conjectured by Szymański et al. in 2013, and only recently it was proven by Miyoshi et al. with elaborate mathematical tools. The present article gives new proofs closer to physical intuition and partly with simpler mathematics. It relies heavily on conformal transformation and it expresses for the first time the trans-resistances and the lower envelope in terms of Jacobi functions, elliptic integrals, and the modular lambda elliptic function. New simple formulae for the asymptotic limit of a very large hole are also given.

The Scattering of SH Wave in a Nano Hole Embedded the Infinite Inhomogeneous Medium

Scattering of the shear waves by a nano-sized cylindrical hole embedded the inhomogeneous is investigated in this study. The Helmholtz equation with a variable coefficient is transformed the standard Helmholtz equation by the complex function method and the conformal mapping method. By wave function expanding method, the analytical expressions of the displacement field and stress field in the inhomogeneous medium are obtained. Considering the surface effect and using the generalized Young-Laplace equation, we obtain the boundary conditions at nano arbitrary-shaped hole, then the field equations satisfying boundary conditions are attributed to solving a set of infinite algebraic equations. Numerical results show that when the radius of the cylindrical cavity shrinks to nanometers, surface energy becomes a dominant factor that affects the dynamic stress concentration factor (DSCF) around the cylindrical cavity. The influence the density variation of the inhomogeneity on the DSCF is discussed at the same time.

SBOOSP for Massive Devices in 5G WSNs Using Conformable Chaotic Maps

The commercialization of the fifth-generation(5G)wireless network has begun.Massive devices are being integrated into 5G-enabled wireless sensor networks(5GWSNs)to deliver a variety of valuable services to network users.However,there are rising fears that 5GWSNs will expose sensitive user data to new security vulnerabilities.For secure end-to-end communication,key agreement and user authentication have been proposed.However,when billions of massive devices are networked to collect and analyze complex user data,more stringent security approaches are required.Data integrity,nonrepudiation,and authentication necessitate special-purpose subtree-based signature mechanisms that are pretty difficult to create in practice.To address this issue,this work provides an efficient,provably secure,lightweight subtreebased online/offline signature procedure(SBOOSP)and its aggregation(Agg-SBOOSP)for massive devices in 5G WSNs using conformable chaotic maps.The SBOOSP enables multi-time offline storage access while reducing processing time.As a result,the signer can utilize the pre-stored offline information in polynomial time.This feature distinguishes our presented SBOOSP from previous online/offline-signing procedures that only allow for one signature.Furthermore,the new procedure supports a secret key during the pre-registration process,but no secret key is necessary during the offline stage.The suggested SBOOSP is secure in the logic of unforgeability on the chosen message attack in the random oracle.Additionally,SBOOSP and Agg-SBOOSP had the lowest computing costs compared to other contending schemes.Overall,the suggested SBOOSP outperforms several preliminary security schemes in terms of performance and computational overhead.

COMPUTING HARMONIC MAPS AND CONFORMAL MAPS ON POINT CLOUDS

We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space,using point cloud data only.Given a surface,or a point cloud approximation,we simply use the standard cubic lattice to approximate itsϵ-neighborhood.Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices.The conformal map,or the surface uniformization,is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature.We propose algorithms and numerical examples for closed surfaces and topological disks.To the best of the authors’knowledge,our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.

Seepage flow around twin circular tunnels in anisotropic ground revealed by an analytical solution

A new analytical solution is proposed for steady seepage flow around twin circular tunnels in fully saturated anisotropic ground.The solution is an exact one that fully satisfies all the boundary conditions and precisely considers the different permeabilities along two direc-tions and the interactions between twin tunnels.The solution provides a fast approach for the estimation of the seepage field and a useful tool for design optimization.The solution is successfully addressed using problem equivalence and the Schwartz alternating method com-bined with a mapping function.Using a coordinate transformation of the governing equation,the anisotropic problem of circular tunnels is first equivalent to that of isotropic elliptical tunnels,and the length of the ellipse along the anisotropic axis depends on the anisotropic permeability ratio.The Schwartz alternating method is then employed to address the solution of equivalent elliptical twin tunnel prob-lems,where a mapping function,with which an elliptical tunnel in the half-plane can be mapped into an annulus in the image plane,is introduced to solve the single tunnel problems in each iterative step.The iterative procedure is quite simple and efficient in its calculations and achieves good convergence,and the analytical solution agrees very well with the numerical results to reflect its high precision in the entire ground.Finally,parametric studies are performed to investigate the influences of the anisotropic permeability and tunnel spacing on the seepage field.This is the first study to provide the exact analytical solution of the seepage field of twin tunnel problems in aniso-tropic ground,and the procedure can be extended to multiple tunnel problems.

Dynamic characteristics of large permanent magnet direct‐drive generators considering electromagnetic–structural coupling effects

Dynamic characteristics of large permanent magnet direct‐drive generators(PMDGs)considering electromagnetic–structural coupling effects are analyzed in this study.Using the conformal mapping method,the scalar magnetic potential of the air gap magnetic field considering the slot effect is calculated.On the basis of the discrete current element and magnetic equivalent circuit model,the local magnetic saturation effect of the stator and rotor is quantitatively simulated and the air gap magnetic field intensity distribution is obtained via numerical simulation.A series of uniformly distributed equivalent electromagnetic springs are introduced to develop an electromagnetic–structural coupling finite element PMDG model.The proposed air gap field analysis method is verified by the finite element analysis results.On the basis of the test platform for the Goldwind 1.5MW PMDG,both modal and dynamic response tests for the stator/rotor coupling system are conducted,and the results are compared with the natural frequencies,mode shapes,and vibration responses obtained using the numerical model.The effects of the air gap length and rotor speed on the natural frequencies of the coupling system are analyzed.The proposed model has the potential to accurately evaluate the PMDG vibration energy,avoiding resonance points,and maintaining stable operations of the unit.

Anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids

By constructing a new numerical conformal mapping and using the Stroh-type formulism, an anti-plane problem of four edge cracks emanating from a square hole in piezoelectric solids is investigated. The explicit expressions of the complex potential function, field intensity factors, energy release rates and mechanical strain energy release rate near the crack tip are obtained under the assumptions that the surfaces of the cracks and hole are electrically permeable and electrically impermeable. Numerical examples are presented to show the influences of the geometrical parameters of defects and applied mechanical/electrical loads on the energy release rate and mechanical strain energy release rate under two electrical boundary conditions.

On Local Singularities in Ideal Potential Flows with Free Surface

Despite important advances in the mathematical analysis of the Euler equations for water waves,especially over the last two decades,it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems.For ideal free-surface flow with zero surface tension and gravity,the authors review existing works that describe"splash singularities",singular hyperbolic solutions related to jet formation and"flip-through",and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure.The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation.Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.

Square Maxwell’s fish-eye lens for near-field broadband achromatic super-resolution imaging

Broadband super-resolution imaging is important in the optical field.To achieve super-resolution imaging,various lenses from a superlens to a solid immersion lens have been designed and fabricated in recent years.However,the imaging is unsatisfactory due to low work efficiency and narrow band.In this work,we propose a solid immersion square Maxwell's fish-eye lens,which realizes broadband(7-16 GHz)achromatic super-resolution imaging with full width at half-maximum around 0.2λ based on transformation optics at microwave frequencies.In addition,a super-resolution information transmission channel is also designed to realize long-distance multi-source super-resolution information transmission based on the super-resolution lens.With the development of 3D printing technology,the solid immersion Maxwell's fish-eye lens is expected to be fabricated in the high-frequency band.