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1.
In electrical impedance tomography, one tries to recover the spatial conductivity distribution inside a body from boundary
measurements of current and voltage. In many important situations, the examined object has known background conductivity but
is contaminated by inhomogeneities. The factorization method of Kirsch provides a tool for locating such inclusions. The computational
attractiveness of the factorization technique relies heavily on efficient computation of Dirichlet boundary values of potentials
created by dipole sources located inside the examined object and corresponding to the homogeneous Neumann boundary condition
and to the known background conductivity. In certain simple situations, these test potentials can be written down explicitly
or given with the help of suitable analytic maps, but, in general, they must be computed numerically. This work introduces
an inexpensive algorithm for approximating the test potentials in the framework of real-life electrode measurements and analyzes
how well this technique can be imbedded in the factorization method. The performance of the resulting fast reconstruction
algorithm is tested in two spatial dimensions.
The work of the second author was supported by the Academy of Finland (project 115013), the Finnish Funding Agency for Technology
and Innovation (project 40084/06), the Finnish Cultural Foundation and the Finnish Foundation for Technology Promotion. 相似文献
2.
Donna J. Cedio-Fengya 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2007,105(4):357-379
In this paper we investigate the electrostatic problem of determining conductivity profiles from the knowledge of boundary
currents and voltages. We obtain an improved estimate for the voltage potential of a two-dimensional conductor having finitely
many circular inclusions and piecewise constant conductivity profile. We derive an asymptotic expansion for the voltage potential
in terms of the reference voltage potential and the location, size, and conductivity of the inhomogeneities. This representation
is used to formulate the nonlinear least squares problem for estimating the location and size of the inhomogeneities. Required
boundary data for the voltage potential are generated numerically by solving a system of integral equations. Computational
experiments are presented to demonstrate the effectiveness of our identification procedure. 相似文献
3.
In this paper we investigate the electrostatic problem of determining conductivity profiles from the knowledge of boundary
currents and voltages. We obtain an improved estimate for the voltage potential of a two-dimensional conductor having finitely
many circular inclusions and piecewise constant conductivity profile. We derive an asymptotic expansion for the voltage potential
in terms of the reference voltage potential and the location, size, and conductivity of the inhomogeneities. This representation
is used to formulate the nonlinear least squares problem for estimating the location and size of the inhomogeneities. Required
boundary data for the voltage potential are generated numerically by solving a system of integral equations. Computational
experiments are presented to demonstrate the effectiveness of our identification procedure. 相似文献
4.
The potential distribution on the scalp produced by current sources in the brain can be measured by an EEG recorder. The relationship between these sources and the scalp potential distribution may be described by a well-known mathematical model where some simplifications are usually introduced. The head is modeled as a multicompartment nested set and the conductivity of the different tissues is approximated by a positive piecewise constant function. This simplified model is used to solve the forward problem (FP), i.e., to calculate the scalp potential for a current source configuration. In this work, we prove that the weak solutions of the FP are continuous with respect to the conductivity values, that is, the difference between the scalp potentials is small if the conductivity values are closed enough. We present numerical examples that illustrates this property. 相似文献
5.
We establish an asymptotic expansion of the steady-state voltage potentials in the presence of a diametrically small conductivity inhomogeneity that is nearly touching the boundary. Our asymptotic formula extends those already derived for a small inhomogeneity far away from the boundary and is expected to lead to very effective algorithms, aimed at determining location and certain properties of the shape of a small inhomogeneity that is nearly touching the boundary based on boundary measurements. Viability of the asymptotic formula is documented by numerical examples. 相似文献
6.
John Sylvester 《纯数学与应用数学通讯》1990,43(2):201-232
We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ? Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ*γ which will produce the same voltage and current measurements on ? Ω. We prove the converse in two dimensions (i.e., if γ1 and γ2 produce the same boundary measurements, then γ1, = Ψ*γ2 for an appropriate Ψ) for conductivities which are near a constant. 相似文献
7.
Asymptotic expansions of the voltage potential in terms of the "radius" of a diametrically small(or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities.In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold,σ, are examples of inhomogeneities for which the convergence to the background potential,or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit.The purpose of this paper is to find a "simple" replacement for the background potential, with the following properties:(1) This replacement may be(simply) calculated from the limiting domain Ω\σ, the boundary data on the boundary of Ω, and the right-hand side.(2) This replacement depends on the thickness of the inhomogeneity and the conductivity,a, through its boundary conditions on σ.(3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0. 相似文献
8.
Hoai-Minh Nguyen Michael S. Vogelius 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009
We revisit the asymptotic formulas originally derived in [D.J. Cedio-Fengya, S. Moskow, M.S. Vogelius, Identification of conductivity imperfections of small diameter by boundary measurements. Continuous dependence and computational reconstruction, Inverse Problems 14 (1998) 553–595; A. Friedman, M. Vogelius, Identification of small inhomogeneities of extreme conductivity by boundary measurements: A theorem on continuous dependence, Arch. Ration. Mech. Anal. 105 (1989) 299–326]. These formulas concern the perturbation in the voltage potential caused by the presence of diametrically small conductivity inhomogeneities. We significantly extend the validity of the previously derived formulas, by showing that they are asymptotically correct, uniformly with respect to the conductivity of the inhomogeneities. We also extend the earlier formulas by allowing the conductivities of the inhomogeneities to be completely arbitrary L∞, positive definite, symmetric matrix-valued functions. We briefly discuss the relevance of the uniform asymptotic validity, and the admission of arbitrary anisotropically conducting inhomogeneities, as far as applications of the perturbation formulas to “approximate cloaking” are concerned. 相似文献
9.
Masaru Ikehata Hiromichi Itou Akira Sasamoto 《Mathematical Methods in the Applied Sciences》2016,39(13):3565-3575
A two dimensional version of a reconstruction problem of an unknown weld on the interface between two electric conductive plates is considered. It is assumed that the two plates have a same known isotropic homogeneous conductivity, and the line where the welding area is located is known. Under these assumptions, an explicit extraction formula of the location of the tips of the welding area on the line from a single set of an electric current density and the corresponding voltage potential on the boundary of the material formed by the plates is given. This result may have possibility of application to quality evaluation of spot welding fixation strength of a lamina. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
10.
S. V. Gavrilov A. M. Denisov 《Computational Mathematics and Mathematical Physics》2014,54(11):1690-1699
The two-dimensional electrical impedance tomography problem is considered in the case of a piecewise constant electrical conductivity. The task is to determine the unknown boundary separating the regions with different conductivity values, which are known. Input information is the electric field measured on a portion of the outer boundary of the domain. A numerical method for solving the problem is proposed, and numerical results are presented. 相似文献
11.
We consider the case of a uniform plane conductor containing a thin curve-like inhomogeneity of finite conductivity. In this article we prove that the imperfection can be uniquely determined from the boundary measurements of the first order correction term in the asymptotic expansion of the steady state voltage potential as the thickness goes to zero. 相似文献
12.
V. N. Gol’dberg 《Differential Equations》2012,48(4):524-537
We prove the stability of the mixed problem for a system of telegraph equations under a perturbation of one of the boundary conditions by a sum of a singular perturbation (a small parameter multiplying the highest derivative) and a small regular perturbation. The solution of the problem consists of the current and voltage in a segment of a telegraph line. One of its ends is short-circuited, and a capacitor of small capacity, together with a nonlinear resistance whose volt-ampere characteristic is perturbed by a small term, is connected to the other end. We prove the convergence of the solution of the problem to the unique continuous piecewise continuously differentiable solution of the unperturbed problem bifurcating at some instant of time from its unique classical solution. 相似文献
13.
Amel Ben Abda Fahmi Ben Hassen Juliette Leblond Moncef Mahjoub 《Mathematical and Computer Modelling》2009,49(11-12):2213-2223
We are concerned with an inverse problem related to sources detection from boundary data in a 2D medium with piecewise constant conductivity. It stands as a 2D version of the inverse problem of electroencephalography, where pointwise sources model epilepsy foci, with the so-called multi-layer spherical model of the head (scalp, skull, brain). When overdetermined electrical measurements (potential and current flux) are available on the scalp, one wants to recover the current sources (conductivity defaults) located in the brain (inner boundary). This recovery issue reduces to a number of inverse problems, where the sources identification process makes use of best rational approximation in the disk, whereas the preliminary cortical mapping step (Cauchy type issue) relies on best constrained harmonic or analytic approximation in an annulus (bounded extremal problems). 相似文献
14.
Studies on ZnO ceramic varistors by deep heat treatment at 650–900 C are reported. The current creep time curve exhibits a
peak during the continuous action of a dc biasing voltage; the forwardV-l characteristic is improved rather than degraded after the action of the biasing voltage. We assume that the zinc interstitial
cations Zni are out diffused rapidly and the concentration of Zni in the depletion layer is decreased rapidly during deep heat treatment; the oxygen anions O’o could be accumulated at the grain interface if the out diffusion quantity of Zni is not enough to react with the O’o; the current creep phenomenon above results from the migration of the interface O’o by the biasing voltage. We suggest an improved grain boundary defect model for the ZnO varistors by deep heat treatment,
and examine the model using the experimental data of lifetime positron-annihilation spectroscopy.
Project supported by the National Natural Science Foundation of China. 相似文献
15.
In this article the constant and the continuous linear boundary elements methods (BEMs) are given to obtain the numerical solution of the coupled equations in velocity and induced magnetic field for the steady magneto-hydrodynamic (MHD) flow through a pipe of rectangular and circular sections having arbitrary conducting walls. In recent decades, the MHD problem has been solved using some variations of BEM for some special boundary conditions at moderate Hartmann numbers up to 300. In this paper we develop this technique for a general boundary condition (arbitrary wall conductivity) at Hartmann numbers up to 105 by applying some new ideas. Numerical examples show the behavior of velocity and induced magnetic field across the sections. Results are also compared with the exact values and the results of some other numerical methods. 相似文献
16.
D. R. Westbrook 《Numerical Methods for Partial Differential Equations》1989,5(3):259-273
The thermistor is an electrical device that can be used as a current surge protector. The basis of its operation is the temperature-dependent electrical conductivity that drops by 4 to 5 orders of magnitude over a temperature range of 100–200°C. In the present work the coupled heat and electrical current problems are examined with the conductivity modeled by a step function. A numerical scheme is suggested for the nonlinear coupled problem. Convergence of this scheme for some boundary conditions is demonstrated and some numerical results are given. 相似文献
17.
Houcine Meftahi 《Mathematical Methods in the Applied Sciences》2017,40(7):2505-2521
In this paper, we consider the conductivity problem with piecewise‐constant conductivity and Robin‐type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non‐monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius‐type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first‐order and second‐order shape derivative of the proposed cost functional, which is performed rigorously by means of shape‐Lagrangian formulation without using the shape sensitivity of the states variables. © 2016 The Author. Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd. 相似文献
18.
Min‐Gi Lee Min‐Su Ko Yong‐Jung Kim 《Mathematical Methods in the Applied Sciences》2016,39(5):1183-1196
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. 相似文献
19.
We show that a near constant conductivity of a two-dimensional body can be uniquely determined by steady state direct current measurements at the boundary. Mathematically, we show that the coefficient γ in the operator ?.γ? is uniquely determined by its Dirichlet integrals. 相似文献
20.
We consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current‐voltage observation. We develop a real time algorithm for determining the location of the anomaly. This new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage. Combined with the size estimation result, this algorithm gives a good initial guess for Newton‐type schemes. We give the rigorous proof for the location search algorithm. Both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast, stable and efficient. © 2001 John Wiley & Sons, Inc. 相似文献