Inverse problems in heat conduction

Formulations and principles for obtaining stable solutions of various inverse problems applicable in the physics of heat and in heat engineering are discussed.     

Solving boundary-value problems in heat conduction by the method of successively averaging the unknown function

An approximate analytical method is proposed by which linear boundary-value problems in heat conduction can be solved for an arbitrary distribution of heat sources and for boundary conditions of a general form.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 4, pp. 693–700, April, 1972.     

Inverse boundary-value problems of heat conduction

Possible formulations of the problems of determining heat fluxes and temperatures at the boundary of a solid from known temperatures within the solid are examined. A classification of these formulations is offered. Various methods for solving one-dimensional inverse problems are analyzed.     

Certain problems of transient heat conduction in wedges

Analyzed is the general methodology of solving problems in transient heat conduction in plane finite-length and infinitely long wedges under various boundary conditions with respect to heat transfer at the surfaces. Closed solutions are obtained for special cases.O. N. Ivanova and O. P. Shabatina collaborated in this study.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 24, No. 3, pp. 514–519, March, 1973.     

Discontinuous solutions in problems of nonlinear heat conduction

We propose a method of constructing discontinuous solutions for nonlinear problems in the theory of heat conduction. The method is a modification of the pivot methods for the solution of linear boundary-value problems. As an illustration we present numerical solutions of two problems.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 1, pp. 111–117, July, 1969.     

Reference temperature in two-dimensional problems of heat conduction

The problem of determining the average temperature in calculating the heat transfer between coaxial cylinders is considered. It is shown that the true value of the average temperature depends on the geometry of the heat transfer surfaces and is less than the arithmetic mean value usually employed.     

Some heat conduction problems in an inhomogeneous body

Solutions are obtained for the heat conduction equation in the case when the thermal conductivity is a homogeneous function of the coordinates.     

Additional boundary conditions in unsteady-state heat conduction problems

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).     

BEM approach to inverse heat conduction problems

In the paper the inverse problem of the estimation of the temperature and heat flux on the surface of a heat conducting body is considered. Since the problem belongs to the ill-posed, the method of solving the boundary probelem as well as the method of stabilizing the results of calculations are required. The boundary element method is applied to solve the boundary problem whereas combined ‘future steps’ and the regularization method is applied to obtain stable results. A numerical example is included.     

Variational principles for nonstationary heat conduction problems

Integral variational principles are proposed for nonstationary heat conduction problems. These lead to integration of the wave equation of heat conduction or the Fourier heat conduction equation with the initial and boundary conditions.     

Additional boundary conditions in nonstationary problems of heat conduction

The integral method of thermal balance, with determining the front of temperature disturbance and introducing additional boundary conditions, is used to obtain analytical solutions of heat conduction problems for plate, cylinder, and sphere at the boundary conditions of the third kind. The distribution of isotherms and the velocities of their motion are analyzed.     

Method of integral cross sections in heat conduction problems

We substantiate estimates of the upper and lower bounds of the effective thermal conductivity of piecewise homogeneous bodies. A numerical scheme for calculating the temperature field has been developed and implemented, and a comparison between the results of calculations by different schemes has been carried out.Odessa Polytechnic Institute. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 2, pp. 322–329, March–April, 1995.     

Solution of multi-parameter inverse problems of heat conduction

We develop methods and present results for the solution of inverse problems of heat conduction in connection with the simultaneous determination of thermophysical characteristics and heat exchange parameters.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 60, No. 1, pp. 136–144, January, 1991.     

Approximation procedure for boundary-value problems of heat conduction

An approximation procedure is presented based on the method of moments which significantly reduces the amount of mathematical calculations and formalizes the selection of approximating differential equations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 49, No. 2, pp. 314–321, August, 1985.     

Some problems of heat conduction with phase transitions

An approximation of the problem of heating of a body by a surface heat flux of density 10⁷–10¹⁴ W/m₂ is presented. Expressions are obtained for calculating the movement of the melting boundary and external destruction boundary and temperature conditions on the surface.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 20, No. 3, pp. 497–504, March, 1971.     

Error minimization in nonlinear variational unsteady-state heat conduction problems

The article considers a variational method of solving unsteady-state heat conduction problems.Belarussian State Polytechnical Academy, Minsk, Belarus. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 65, No. 3, pp. 341–349, September, 1993.     

A hybrid flux model for heat conduction problems

A hybrid method of solution for the linear problem of heat conduction in a body is presented. The variational support is a two‐field functional whose arguments are heat flux, which meets a priori inner thermal equilibrium, and temperature on the boundary of the body. The stationary conditions of the functional are the Fourier's law and the prescribed boundary conditions. This variational framework allows to develop a finite element model that exhibits good accuracy, especially in the presence of geometry irregularities in a mesh. Copyright © 2000 John Wiley & Sons, Ltd.     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

Solving heat conduction problems by the straight-lines method

Boundary problems relating to the heat conduction equation are solved by the straight-lines method with spatial quantization.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 4, pp. 709–713, October, 1969.