Sharp estimate of electric field from a conductive rod and application

We are concerned with the quantitative study of the electric field perturbation due to the presence of an inhomogeneous conductive rod embedded in a homogenous conductivity. We sharply quantify the dependence of the perturbed electric field on the geometry of the conductive rod. In particular, we accurately characterize the localization of the gradient field (i.e., the electric current) near the boundary of the rod where the curvature is sufficiently large. We develop layer‐potential techniques in deriving the quantitative estimates and the major difficulty comes from the anisotropic geometry of the rod. The result complements and sharpens several existing studies in the literature. It also generates an interesting application in EIT (electrical impedance tomography) in determining the conductive rod by a single measurement, which is also known as the Calderón's inverse inclusion problem in the literature.     

一类各向异性外问题的重叠型区域分解算法

本文以椭圆外调和问题的自然边界归化为基础，提出了求解各向异性常系数椭圆方程的一种重叠型区域分解算法，并分析了算法的收敛性及收敛速度．理论分析及数值实验表明，该方法对于求解各向异性外问题非常有效．     

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.     

The behavior of plasma with an arbitrary degree of degeneracy of electron gas in the conductive layer

We obtain an analytic solution of the boundary problem for the behavior (fluctuations) of an electron plasma with an arbitrary degree of degeneracy of the electron gas in the conductive layer in an external electric field. We use the kinetic Vlasov–Boltzmann equation with the Bhatnagar–Gross–Krook collision integral and the Maxwell equation for the electric field. We use the mirror boundary conditions for the reflections of electrons from the layer boundary. The boundary problem reduces to a one-dimensional problem with a single velocity. For this, we use the method of consecutive approximations, linearization of the equations with respect to the absolute distribution of the Fermi–Dirac electrons, and the conservation law for the number of particles. Separation of variables then helps reduce the problem equations to a characteristic system of equations. In the space of generalized functions, we find the eigensolutions of the initial system, which correspond to the continuous spectrum (Van Kampen mode). Solving the dispersion equation, we then find the eigensolutions corresponding to the adjoint and discrete spectra (Drude and Debye modes). We then construct the general solution of the boundary problem by decomposing it into the eigensolutions. The coefficients of the decomposition are given by the boundary conditions. This allows obtaining the decompositions of the distribution function and the electric field in explicit form.     

Analysis of a quasistatic contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is quasistatic, the material behavior is modeled with an electro-viscoelastic constitutive law and the contact is described with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving two history-dependent hemivariational inequalities in which the unknowns are the velocity and electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on a recent result on history-dependent hemivariational inequalities obtained in Migórski et al. (submitted for publication) [16].     

Reconstruction of inclusions for the inverse boundary value problem with mixed type boundary condition

We consider an inverse boundary value problem for identifying the inclusion inside a known anisotropic conductive medium. We give a reconstruction procedure for identifying the inclusion from the Dirichlet–Neumann map or the Neumann–Dirichlet map associated with the mixed type boundary condition.     

State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity. A general solution to the problem based on the two-temperature generalized thermoelasticity theory (2TT) is introduced. The theory of thermal stresses based on the heat conduction equation with Caputo’s time-fractional derivative of order α is used. Some special cases of coupled thermoelasticity and generalized thermoelasticity with one relaxation time are obtained. The general solution is provided by using Laplace’s transform and state-space techniques. It is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading (thermal shock). Some numerical analyses are carried out using Fourier’s series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically and the effects of two-temperature and fractional-order parameters are discussed.     

Reconstruction of Less Regular Conductivities in the Plane

Abstract

We consider the inverse conductivity problem of how to reconstruct an isotropic electric conductivity distribution in a conductive body from static electric measurements on the boundary of the body. An exact algorithm for the reconstruction of a conductivity in a planer domain from the associated Dirichlet-to-Neumann map is given. We assume that the conductivity has essentially one derivative, and hence we improve earlier reconstruction results. The method relies on a reduction of the conductivity equation to a first order system, to which the ?¯-method of inverse scattering theory can be applied.     

Spannungsfeld der auf den Rand einer elastisch anisotropen Halbebene wirkenden Tangentialkraft

Summary The stress distribution obtained by solving the two-dimensional problem in an anisotropic medium, with boundary conditions of a concentrated tangential load acting on the boundary of a semi-infinite plate, is purely radial. The solution is given in closed form and is combined with the solution for a concentrated normal load to solve the problem of an inclined force acting on the boundary.     

Variational analysis of fully coupled electro‐elastic frictional contact problems

We consider a mathematical model which describes the static frictional contact between a piezoelectric body and a foundation. The material behavior is described with a nonlinear electro‐elastic constitutive law. The novelty of the model consists in the fact that the foundation is assumed to be electrically conductive and both the frictional contact and the conductivity on the contact surface are described with subdifferential boundary conditions which involve a fully coupling between the mechanical and electrical variables. We derive a variational formulation of the problem which is in the form of a system coupling two hemivariational inequalities for the displacement and the electric potential fields, respectively. Then we prove the existence of a weak solution to the model and, under additional assumptions, its uniqueness. The proofs are based on recent results for inclusions of subdifferential type in Sobolev spaces (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Differential transformation approach to thermal conductive problems with discontinuous boundary condition

A two-dimensional differential transformation method is employed to reduce the partial differential equations of the non-continuous thermal conductive boundary value problem to a Taylor series in a polynomial form. The partial differential equations are solved by the two-dimensional T-spectra method of differential transformation and by the use of trial initial polynomial conditions. The investigative parameters include the time-step of the differential transformation and the number of sub-domain segments. The numerical simulation results indicate that the proposed approach using the two-dimensional T-spectra method of differential transformation is applicable to the solution of non-continuous thermal conductive boundary value problems.     

Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account. To cite this article: A. Buffa et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Thermal stress analysis of current-carrying media containing an inclusion with arbitrarily-given shape

The presence of inclusions in metal-based composites subjected to an electric current or a heat flux induces thermal stresses. Inclusion geometry is one of the important parameters in the stress distribution. In this study, the plane problem of an arbitrarily-shaped inclusion embedded in an infinite conductive medium is investigated based on the complex variable method. The shape of the inclusion is defined approximately by a polynomial conformal mapping function. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. The obtained results show that the shape, bluntness and rotation angle of the inclusion have a significant effect on the stress concentration around the inclusion induced by the far-field electric current. In addition, for the considered inclusion-matrix system under given electric loading, a lower amount of the Von Mises stress concentration than that around a circular inclusion could be achieved by appropriate selection of the inclusion shape and orientation.     

一类非线性抛物型分布参数控制系统的研究

对一类有广泛应用的活动边界域上非线性抛物型分布参数系统进行建模。采用将系统转化为固定域系统或转化为集中参数系统的方法,从理论上分析了系统的基本特征,得出控制变量存在两个临界值,大者区分边界活动与否,小者区分边界活动后会自动停止与否。将平面、柱面和球面的一维系统表述成统一形式,通过计算机仿真研究了系统开环控制和闭环反馈控制的动态特性,数值结果与理论结果一致。计算机仿真表明系统是适定的、稳定的,而且是可测的和可控的。     

Iterative solution of the generalized Love's problem in anisotropic media

An integral equation method is used to show the well posedness of the generalized Love's problem for an elastic anisotropic layer superimposed to a homogeneous substrate. The solution is derived using an iterative technique and the dispersion equation is obtained by imposing the pertinent boundary conditions. Inhomogeneous layers with different profiles of the material parameters are considered as examples and numerical results are given. The corresponding generalized problem is discussed for Lamb's modes.     

A dynamic frictional contact problem for piezoelectric materials

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive foundation. The process is dynamic, the material's behavior is modeled with an electro-viscoelastic constitutive law and the contact is described by subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving a second order evolutionary hemivariational inequality for the displacement field coupled with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators.     

On a free boundary problem for a class of anisotropic equations

This paper deals with a certain condenser capacity in an anisotropic environment. More precisely, we are going to investigate a free boundary problem for a class of anisotropic equations on a ring domain N≥2. Our aim is to show that if the problem admits a solution in a suitable weak sense, then the underlying domain Ω is a Wulff‐shaped ring. The proof makes use of a maximum principle for an appropriate P‐function, in the sense of L. E. Payne, a Rellich type identity and some geometric arguments involving the anisotropic mean curvature of the free boundary. Copyright © 2016 John Wiley & Sons, Ltd.     

An Energy-Stable Parametric Finite Element Method for Simulating Solid-State Dewetting Problems in Three Dimensions

We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film\slash vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energy-stable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.     

On the Fredholm property of the electric field equation in the vector diffraction problem for a partially screened solid

We consider a vector problem of diffraction of an electromagnetic wave on a partially screened anisotropic inhomogeneous dielectric body. The boundary conditions and the matching conditions are posed on the boundary of the inhomogeneity domain, and under passage through it, the medium parameters have jump changes. A boundary value problem for the system of Maxwell equations in unbounded space is studied in a semiclassical statement and is reduced to a system of integro-differential equations on the body domain and the screen surfaces. We show that the quadratic form of the problem operator is coercive and the operator itself is Fredholm with zero index.     

The Axisymmetric Deformation of a Thin,or Moderately Thick,Elastic Spherical Cap

A refined shell theory is developed for the elastostatics of a moderately thick spherical cap in axisymmetric deformation. This is a two-term asymptotic theory, valid as the dimensionless shell thickness tends to zero.The theory is more accurate than “thin shell” theory, but is still much more tractable than the full three-dimensional theory. A fundamental difficulty encountered in the formulation of shell (and plate) theories is the determination of correct two-dimensional boundary conditions, applicable to the shell solution, from edge data prescribed for the three-dimensional problem. A major contribution of this article is the derivation of such boundary conditions for our refined theory of the spherical cap. These conditions are more difficult to obtain than those already known for the semi-infinite cylindrical shell, since they depend on the cap angle as well as the dimensionless thickness. For the stress boundary value problem, we find that a Saint-Venant-type principle does not apply in the refined theory, although it does hold in thin shell theory. We also obtain correct boundary conditions for pure displacement and mixed boundary data. In these cases, conventional formulations do not generally provide even the first approximation solution correctly. As an illustration of the refined theory, we obtain two-term asymptotic solutions to two problems, (i) a complete spherical shell subjected to a normally directed equatorial line loading and (ii) an unloaded spherical cap rotating about its axis of symmetry.