Identification of temperature‐dependent thermophysical properties in a partial differential equation subject to extra final measurement data

We consider an inverse problem for estimating the two coefficient functions c and k in a parabolic type partial differential equation c(u)u_t = ?[k(u)u_x]/?x with the aid of the measurements of u at two different times. The first‐ and second‐order one‐step group preserving schemes are developed. Solving the resultant algebraic equations with a closed‐form, we can estimate the unknown temperature‐dependent thermal conductivity and heat capacity. The new methods possess threefold advantages: they do not require any a priori information on the functional forms of thermal conductivity and heat capacity; no initial guesses are required; and no iterations are required. Numerical examples are examined to show that the new approaches have high accuracy and efficiency, even there are rare measured data. When the measured temperatures are polluted by uniform or normal random noise, the estimated results are also good. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

Inverse problems can be found in many areas of science and engineering and can be applied in different ways. Two examples can be cited: thermal properties estimation or heat flux function estimation in some engineering thermal process. Different techniques for the solution of inverse heat conduction problem (IHCP) can be found in literature. However, any inverse or optimization technique has a basic and common characteristic: the need to solve the direct problem solution several times. This characteristic is the cause of the great computational time consumed. In heat conduction problem, the time consumed is, usually, due to the use of numerical solutions of multidimensional models with refined mesh. In this case, if analytical solutions are available the computational time can be reduced drastically. This study presents the development and application of a 3D-transient analytical solution based on Green’s function. The inverse problem is due to the thermal properties estimation of conductors. The method is based on experimental determination of thermal conductivity and diffusivity using partially heated surface method without heat flux transducer. Originally developed to use numerical solution, this technique can, using analytical solution, estimate thermal properties faster and with better accuracy.     

Asymptotic Localization of Energy in Nondisordered Oscillator Chains

We study two popular one‐dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete nonlinear Schrödinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter ? > 0, is small. We rigorously establish that the thermal conductivity of the chains has a nonperturbative origin with respect to the coupling constant ?, and we provide strong evidence that it decays faster than any power law in ? as ? → 0. The weak coupling regime also translates into a high‐temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature. To our knowledge, it is the first time that a clear connection has been established between KAM‐like phenomena and thermal conductivity. © 2015 Wiley Periodicals, Inc.     

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.     

Residue minimization technique to analyze the efficiency of convective straight fins having temperature-dependent thermal conductivity

In this article, we propose and implement a numerical technique based on residue minimization to solve the nonlinear differential equation, which governs the temperature distribution in straight convective fins having temperature-dependent thermal conductivity. The form of temperature distribution is approximated by a polynomial series, which exactly satisfies the boundary conditions of the problem. The unknown coefficients of the assumed series are optimized using the Nelder–Mead simplex algorithm such that the squared L₂ norm of the residue attains its minimum value within a specified tolerance limit. The near-exact solution thus obtained is further used to calculate the fin efficiency. For the case of constant thermal conductivity, the obtained results are validated with the analytical solutions, while for the case of variable thermal conductivity, the obtained results are corroborated with those previously published in the literature. An excellent agreement in each case consolidates the effectiveness of the proposed numerical technique.     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

Control of thermal stresses and displacements in thermoelastic bodies

By developing the method of the inverse thermoelastic and thermal conductivity problem we study problems of optimal control of thermal stresses and displacements in the case of one- and two-dimensional temperature fields. The solution of the optimization problems reduces to solving integral equations of first or second kind for determining the internal heat sources.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 18–23.     

An inverse problem in estimating simultaneously the effective thermal conductivity and volumetric heat capacity of biological tissue

An inverse problem utilizing the Levenberg–Marquardt method (LMM) is applied in this study to determine simultaneously the unknown spatial-dependent effective thermal conductivity and volumetric heat capacity for a biological tissue based on temperature measurements. The accuracy of this inverse problem is examined by using the simulated exact and inexact temperature measurements in the numerical experiments. A statistical analysis is performed to obtain the 99% confidence bounds for the estimated thermal properties. Results show that good estimation on the spatial-dependent thermal conductivity and volumetric heat capacity can be obtained using the present algorithm for the test cases considered in this study.     

On the Question of Calculation of Temperature Fields in Piecewise-Homogeneous Orthotropic Media

Based on the theory of R-functions, we propose structures of a solution of the boundary-value problem of heat conduction in piecewise-homogeneous orthotropic media, satisfying the conditions of convective heat exchange on the boundary of an orthotropic body and the conditions of ideal thermal contact on the boundary between media. We present the results of a computational experiment confirming the correctness and computational fitness of the proposed structural models.     

Thermal fracture of functionally gradient ceramics

An edge crack in a strip of functionally gradient ceramics (FGC) is studied under thermal loading conditions. Two FGC materials are considered, i.e., one with a spatial variation of shear modulus and the other with a spatial variation of thermal conductivity. Thermal stress intensity factors (TSIF) are numerically calculated based on singular integral equations derived for the dislocation density along the crack faces. It is shown that: (a) for the FGC with a graded shear modulus, the TSIF are reduced for crack lengths longer thanl _c b and remain approximately the same as those of a homogeneous material for shorter crack lengths, wherel _c is about 0.065 andb is the width of the strip; and (b) for the FGC with a thermal conductivity gradient, the TSIF are generally lower compared with those for the bonded two-layer material.     

A finite-element approach to solving a thermal conductivity problem for thick plates

A method is proposed for the numerical solution of a problem in the thermal conductivity a plate heated by a moving heat source, taking into account the dependence of the thermal properties of the material on temperature. A hypothesis of N. N. Rykalin concerning the state of the thermal field is adopted. The method is based on a spline variant of the finite-element method; this is based on the synthesis of parametrization and finite elements with splines. A FORTRAN algorithm is given for performing the method on an ES series computer.Translated from Matematichekie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 58–62, 1989.     

On Inverse Problems for Semiconductor Equations

This paper is devoted to the investigation of inverse problems related to stationary drift-diffusion equations modeling semiconductor devices. In this context we analyze several identification problems corresponding to different types of measurements, where the parameter to be reconstructed is an inhomogeneity in the PDE model (doping profile). For a particular type of measurement (related to the voltage-current map) we consider special cases of drift-diffusion equations, where the inverse problems reduces to a classical inverse conductivity problem. A numerical experiment is presented for one of these special situations (linearized unipolar case).Lecture held by P.A. Markowich in the Seminario Matematico e Fisico on April 5, 2004Received: June, 2004     

The inverse Abel transform for an exceptional bounded symmetric domain

By using the first shift operator of thebc ₂- root system, the elementary spherical functions on the exceptional bounded symmetric domainE ₆/spin(10) ×T are obtained, and the inverse Abel transform for this space is derived. Finally an expression of the heat kernel of this domain is given. Project partially supported by the National Natural Science Foundation of China.     

Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.     

The Calderón Problem with Partial Data for Less Smooth Conductivities

ABSTRACT

In this article we consider the inverse conductivity problem with partial data. We prove that in dimensions n ≥ 3 knowledge of the Dirichlet-to-Neumann map measured on particular subsets of the boundary determines uniquely a conductivity with essentially 3/2 derivatives.     

Uniqueness and stable determination in the inverse Robin transmission problem with one electrostatic measurement

In this paper, we consider the inverse Robin transmission problem with one electrostatic measurement. We prove a uniqueness result for the simultaneous determination of the Robin parameter p, the conductivity k, and the subdomain D, when D is a ball. When D and k are fixed, we prove a uniqueness result and a directional Lipschitz stability estimate for the Robin parameter p. When p and k are fixed, we give an upper bound to the subdomain D. For the reconstruction purposes of the Robin parameter p, we set the inverse problem under an optimization form for a Kohn–Vogelius cost functional. We prove the existence and the stability of the optimization problem. Finally, we show some numerical experiments that agree with the theoretical considerations. Copyright © 2013 John Wiley & Sons, Ltd.     

Approximation of the conductivity coefficient in the heat equation

In this paper, we study the conductivity coefficient determination in the heat equation from observation of the lateral Dirichlet-to-Neumann map. We define a bilinear form function Q_γ associated to the boundary condition and the Dirichlet-to-Neumann map, and prove that the linearized problem d?Q_γ is injective. Based on the idea of complex geometrical optics solutions, we give two approximations to the conductivity coefficient by using the Fourier truncation method and the mollification method. Under the a priori assumption of the conductivity, we estimate the errors between the conductivity coefficient and its approximations by setting a suitable bound of the frequency.     

Quality rating of a metal matrix-diamondcomposite from its thermal conductivity and electric resistance

The efficiency of hot-pressed diamond-containing composite materials (DCM) for various tool applications is greatly affected by microdefects, namely, the residual porosity of the metal matrix, damaged diamond grains, and imperfect diamond-matrix interfaces. An instrumental evaluation of these microdefects, predetermining the quality of a tool equipped with DCM, is rather difficult due to the small size, the nonstandard shape, and the strong heterogeneity of specimens. Proposed here is an alternative, nondestructive technique of DCM quality rating, which includes the measurement of electric resistance and thermal conductivity of diamond-containing composites and processing the obtained data by the methods of composite mechanics. It exploits the fact that diamond, being a dielectric, possesses an extremely high thermal conductivity, which allows estimating the residual porosity of a sintered metal matrix from the ratio of specific electric resistances, one being measured and another predicted by a theory. These data, in turn, are utilized to predict the thermal conductivity ofDCMwith an imperfect matrix. Matching with experiments, after solving the inverse problem gives the thermal resistance of diamond-matrix interface, which, within the frame work of the given model, simulates the damage of both the diamond grains and their bonds with the matrix. Thus, the numerical rating of quality is given in terms of two dimensionless parameters. The first one, 0 < K < 1, reflects the quality of the sintered metal matrix, whereas the second one, 0 < R <1, is an aggregate measure of the integrity of diamond grains and the perfection degree of composite interfaces. The quite satisfactory agreement observed between the theory and experiment confirms the efficiency of the technique and the reliability of the data obtained. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 361–374, May–June, 2006.     

Stability Estimates for the Inverse Conductivity Problem for Less Regular Conductivities

A log-type stability estimate for the inverse conductivity problem in space dimension n ≥ 3, if the conductivity has C ^3/2+ε regularity is proven.     

Determination of the coefficient of thermal conductivity and heat capacity per unit volume in a multilayer medium

We consider the inverse problem of simultaneously determining two time-dependent thermophysical characteristics—the coefficient of thermal conductivity and the heat capacity per unit volume—for a body having the shape of a layer situated between two other layers with known thermophysical characteristics. The necessary measurements are carried out on their outside boundaries. The problem is reduced to a system of nonlinear equations for which the existence of a solution is established by using Schauder's fixed-point theorem. We find conditions that guarantee that the solution of the inverse problem is unique. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 153–159.