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.     

The effective conductivity of a randomly inhomogeneous medium

Using a representation of the solution to the diffusion equation in a randomly inhomogeneous medium in the form of a Feynman path integral an explicit expression is obtained for the effective conductivity in a space of arbitrary dimension. A calculation of the path integral only turns out to be possible in the case of a large-scale limit. In particular, it is shown that in the three-dimensional case the expression for the effective conductivity does not admit of an expansion in terms of the conductivity variance. This indicates that the use of standard perturbation theory in the form of an expansion in terms of the conductivity fluctuations is incorrect.     

Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity

Studied in this paper is the Cauchy problem of the two-dimensional magnetohydrodynamics system with inhomogeneous density and electrical conductivity. It is shown that the 2-D incompressible inhomogeneous magnetohydrodynamics system with a constant viscosity is globally well-posed for a generic family of the variations of the initial data and an inhomogeneous electrical conductivity. Moreover, it is established that the system is globally well-posed in the critical spaces if the electrical conductivity is homogeneous.     

Identification of an inclusion in multifrequency electric impedance tomography

The multifrequency electrical impedance tomography is considered to image a conductivity inclusion inside a homogeneous background medium by injecting one current. An original spectral decomposition of the solution of the forward conductivity problem is used to retrieve the Cauchy data corresponding to the extreme case of perfect conductor. Using results based on the unique continuation, we then prove the uniqueness of multifrequency electrical impedance tomography and obtain rigorous stability estimates. Our results in this paper are quite surprising in inverse conductivity problem since in general infinitely many input currents are needed to obtain the uniqueness in the determination of the conductivity.     

Determining the conductivity for a nonautonomous hyperbolic operator in a cylindrical domain

This paper is devoted to the reconstruction of the conductivity coefficient for a nonautonomous hyperbolic operator an infinite cylindrical domain. Applying a local Carleman estimate, we prove the uniqueness and a Hölder stability in the determination of the conductivity using a single measurement data on the lateral boundary. Our numerical examples show good reconstruction of the location and contrast of the conductivity function in 3 dimensions.     

Effect of discontinuity in density and conductivity on the hydromagnetic stability of a two-dimensional jet

The hydromagnetic stability of an incompressible, inviscid, and two-dimensional jet in the presence of a discontinuity in density and conductivity across the jet surfaces is studied against small and large wavelength disturbances. In the equilibrium state the magnetic field is uniform and aligned with the direction of jet velocity. Study of several different cases shows that discontinuity in density or conductivity or in the product of density and conductivity across the jet surfaces has a stabilizing effect on the jet.     

Existence and Regularity of Solutions of a Nonlinear Nonuniformly Elliptic System Arising from a Thermistor Problem

A thermistor is an electric circuit device made of ceramic material whose electric conductivity depends on the temperature. If the only heat source is the electric heating, the temperature and the electric potential satisfy a nonlinear elliptic system which is also degenerate if the electric conductivity is not uniformly bounded from above or away from zero. Under general boundary conditions, we establish existence and Hölder continuity of solutions of such a nonlinear nonuniformly elliptic system. When the elechic conductivity linearly depends on the temperature, we provide a non-uniqueness and non-existence example.     

A superstatistical model for anomalous heat conduction and diffusion

We propose a superstatistical model for anomalous heat conduction and diffusion, which is formulated by the thermal conductivity distribution, overall temperature and heat flux distributions. Our model obeys Fourier's law and the continuity equation at the individual level. The evolution of the thermal conductivity distribution is described by an advection-diffusion equation. We show that the superstatistical model predict anomalous behaviors including the time-dependent effective thermal conductivity and slow long-time asymptotics. The time-dependence of the effective thermal conductivity is determined by the mean square displacement (MSD), which coincides with existing investigations. The superstatistical structure can also be extended into other non-Fourier models including the Cattaneo and fractional-order heat conduction models.     

Inverse determination of thermal conductivity using semi-discretization method

In this work a semi-discretization method is presented for the inverse determination of spatially- and temperature-dependent thermal conductivity in a one-dimensional heat conduction domain without internal temperature measurements. The temperature distribution is approximated as a polynomial function of position using boundary data. The derivatives of temperature in the differential heat conduction equation are taken derivative of the approximated temperature function, and the derivative of thermal conductivity is obtained by finite difference technique. The heat conduction equation is then converted into a system of discretized linear equations. The unknown thermal conductivity is estimated by directly solving the linear equations. The numerical procedures do not require prior information of functional form of thermal conductivity. The close agreement between estimated results and exact solutions of the illustrated examples shows the applicability of the proposed method in estimating spatially- and temperature-dependent thermal conductivity in inverse heat conduction problem.     

Single Measurement Detection Of a Discontinuous Conductivity

Let be a bounded region in R$sup:n$esup: 2. with C$sup:2$esup: boundary and conductivity ggr;Suppose that some region D CC may have been replaced with a material which has a differing C$sup:2$esup: conductivity profile. We show that by applying an appropriate current flux on and measuring the resulting potential on an open subset one can "detect" the presence of the region D, that is. the potentials induced on when D is present versus absent must differ. Moreover. if D and its conductivity are known to satisfy certain a priori restrictions, one can assert that the potentials induced on must differ by a fixed aount which does not depend on the domain D or its conductivity.     

Numerical study of dynamic thermal conductivity of nanofluid in the forced convective heat transfer

In this work, forced convective heat transfer of nanofluid in the developing laminar flow (entrance region) in a circular tube is considered. The nanofluid thermal conductivity, as an important parameter, is considered as two parts: static and dynamic part. Simulated results show that the dynamic part of nanofluid thermal conductivity due to the Brownian motion has a minor effect on the heat transfer coefficients, on the other hand, static part of thermal conductivity including nanolayer around nanoparticle has an important role in heat transfer.     

An equivalence in hydromagnetics

In an infinitely extended hydromagnetic system with a unidirectional magnetic field, the following correspondence exists: for every magnetohydrostatic problem of finite conductivity there is a compressible magnetohydrodynamic problem of infinite conductivity.     

New exact multi-coated ellipsoidal inclusion model for anisotropic thermal conductivity of composite materials

The present study deals with a new micromechanical modeling of the thermal conductivity of multi-coated inclusion-reinforced composites. The proposed approach has been developed in the general frame of anisotropic thermal behavior per phase and arbitrary ellipsoidal inclusions. Based on the Green's function technique, a new formulation of the problem of multi-coated inclusion is proposed. This formulation consists in constructing a system of integral equations, each associated to the thermal conductivity of each coating and the reference medium. Thanks to the concept of interior- and exterior-point Eshelby's conduction tensors, the exact solution of the problem of multicoated inclusion is obtained. Analytical expressions of the intensity in each phase and the effective thermal conductivity of the composite, through homogenizations schemes such as Generalized self-consistent and Mori-Tanaka models are provided. Results of the present model are successfully compared with those issued from both analytical models and finite elements methods for composites with doubly coated inclusions. Moreover, the developed micromechanical model has been applied to a three phase composite materials in order to analyze combined effects of the aspect ratio and the volume fraction of the ellipsoidal inclusions, the anisotropy of the thermal conductivity of interphase, the thermal conductivity contrast between local phases on the predicted effective thermal conductivity.     

Comparison between exact thermal boundary condition and harmonic mean conductivity condition at the solid–fluid interface for finite thickness shrouded non-isothermal fin array

A numerical study is made comparing the exact thermal boundary condition and a harmonic mean conductivity condition at the solid–fluid interface for a finite thickness shrouded non-isothermal fin array. Results highlight that there exists a significant deviation of pressure drop across the length of the fin for the exact thermal boundary condition, which is as high as 20% as compared to that obtained using the harmonic mean conductivity condition. The exact thermal boundary condition forecasts relatively more non-isothermal fin as compared to a harmonic mean conductivity condition. The greater the fin spacing the larger the non-isothermal behavior of the fin and it also depends upon Grashof number as well as inlet fluid velocity. The larger the Grashof number the greater is the non-isothermal behavior of fin. The greater the inlet velocity, the larger is the non-isothermal behavior of fin. Bulk fluid temperature is over predicted by as much as 13% by the harmonic mean conductivity condition for larger fin spacing with highest Grashof number coupled with larger velocity. This deviation is only 6% for smaller fin spacing. Overall Nusselt number is over predicted for the harmonic mean conductivity condition as compared to exact thermal boundary condition. This over prediction is limited to about 8%.     

An asymptotic formula for the voltage potential in the case of a near-surface conductivity inclusion

We rigorously derive an asymptotic expansion of the steady-state voltage potentials in the presence of a conductivity inclusion of small volume that is close to a planar surface. This new formula is motivated by the practically important inverse problem of imaging a conductivity inclusion near a planar interface.     

Linear elliptic equations in composite media with anisotropic fibres

Linear elliptic equations in composite media with anisotropic fibres are concerned. The media consist of a periodic set of anisotropic fibres with low conductivity, included in a connected matrix with high conductivity. Inside the anisotropic fibres, the conductivity in the longitudinal direction is relatively high compared with that in the transverse directions. The coefficients of the elliptic equations depend on the conductivity. This work is to derive the Hölder and the gradient $L^p$ estimates (uniformly in the period size of the set of anisotropic fibres as well as in the conductivity ratio of the fibres in the transverse directions to the connected matrix) for the solutions of the elliptic equations. Furthermore, it is shown that, inside the fibres, the solutions have higher regularity along the fibres than in the transverse directions.     

Modeling the Spatio-Temporal Electrical Activity of Neuron Sources

An evolutionary dipole model is constructed describing the spatio-temporal behavior of the electric potential on the surface of the head (EEG data). An approach is proposed to the solution of the direct three-dimensional EEG problem (finding the induced field). This approach finds the solution as a semi-analytical representation of an approximate solution in spherical functions with indeterminate coefficients. The coefficients are then determined by least squares. The method works with arbitrary (nonspherical) boundary surfaces, unbounded regions, finite conductivity outside the head, and complex spatial dependence of electrical conductivity. A nonhomogeneous conductivity model is considered with conductivity varying sharply across layers. An accurate numerical solution can be obtained if the conductivity of the layers differs by a factor of 80, which ensures sufficient accuracy in estimating dipole localization. Optimal dipole placement is reconstructed by a genetic algorithm, which also determines the best combination and the best number of dipoles. The method works also when several brain zones are active simultaneously. During the iterative fitting of the dipole parameters to minimize the error functional, the evolution of the genetic algorithm is directly linked with the temporal variation of the EEG signal.__________Translated from Prikladnaya Matematika i Informatika, No. 17, pp. 55–71, 2004.     

Orthotropic conductivity reconstruction with virtual‐resistive network and Faraday's law

We obtain the existence and the uniqueness at the same time in the reconstruction of orthotropic conductivity in two‐space dimensions by using two sets of internal current densities and boundary conductivity. The curl‐free equation of Faraday's law is taken instead of the elliptic equation in a divergence form that is typically used in electrical impedance tomography. A reconstruction method based on layered bricks‐type virtual‐resistive network is developed to reconstruct orthotropic conductivity with up to 40% multiplicative noise. Copyright © 2015 John Wiley & Sons, Ltd.     

Global existence of strong solutions to compressible Navier–Stokes system with degenerate heat conductivity in unbounded domains

In one-dimensional unbounded domains, we prove the global existence of strong solutions to the compressible Navier–Stokes system for a viscous and heat conducting ideal polytropic gas, when the viscosity is a constant and the heat conductivity is proportional to a positive power of the temperature. Note that the conditions imposed on the initial data are the same as those of the constant heat conductivity case (Kazhikhov, A. V. Siberian Math. J. 23 [1982], 44-49) and can be arbitrarily large. Therefore, our result generalizes Kazhikhov's result for the constant heat conductivity case to the degenerate and nonlinear one.     

Global existence of solutions for the 1-D radiative and reactive viscous gas dynamics

In this paper, we prove the existence of a global solution to an initial-boundary value problem for 1-D flows of the viscous heat-conducting radiative and reactive gases. The key point here is that the growth exponent of heat conductivity is allowed to be any nonnegative constant; in particular, constant heat conductivity is allowed.