A finite-difference method for solving inverse boundary-value problems of heat conduction

A finite-difference search method is described for determining the temperature and heat flux on one boundary of the body if the temperature and heat flux on the other boundary are known. The results of numerical experiments, which show that the method has proved to be efficient, are discussed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 3, pp. 502–507, March, 1977.     

A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium

Recently, Hon and Wei proposed a method of fundamental solutions for solving isotropic inverse heat conduction problems (IHCP). It provides an efficient global approximation scheme in both spatial and time domains. In this paper, we try to extend the inherently meshless and integration-free method to solve 2D IHCP in an anisotropic medium. First, we acquire the fundamental solution of the governing equation through variables transformation. Then the truncated singular value decomposition and the L-curve criterion are applied to solve the resulting matrix equation which is highly ill-conditioned. Results for several numerical examples are presented to demonstrate the efficiency of the method proposed. The relationship between the accuracy of the numerical solutions and the value of the parameter T is also investigated.     

Padé extension of Bergman-series solutions to nonlinear boundary-value problems in heat conduction

Summary Bergman-type series solutions involving iterated complementary error integrals are constructed for nonlinear boundary-value problems associated with heat conduction in a region bounded internally by a cylindrical or spherical surface. In particular, a small-time solution is developed when the nonlinear boundary condition is of the Stefan-Boltzmann type. This solution is extended via Padé approximants.     

A simple approach to solve boundary-value problems in gradient elasticity

Summary We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.     

Application of the improved integral method of lines to boundary-value problems in heat conduction

We discuss a generalization of the improved integral method of lines for the solution of the heat equation with boundary conditions of the second and third kinds.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 52, No. 2, pp. 297–300, February, 1987.     

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.     

A new space marching method for solving inverse heat conduction problems

A new space marching method is presented for solving the one-dimensional nonlinear inverse heat conduction problems. The temperature-dependent thermal properties and boundary condition on an accessible part of the boundary of the body are known. Additional temperature measurements in time are taken with a sensor located in an arbitrary position within the solid, and the objective is to determine the surface temperature and heat flux on the remaining part of the unspecified boundary. The temperature distribution throughout the solid, obtained from the inverse analysis, is then used for the computation of thermal stresses in the entire domain, including the boundary surfaces. The proposed method is appropriate for on-line monitoring of thermal stresses in pressure components. The three presented example show that the method is stable and accurate.     

Application of infinite systems to the solution of boundary-value problems of steady thermal conduction in nonuniform media

A method of calculating the temperature field in nonuniform media is described. Examples of the calculation of the temperature distribution for an exponential variation of the thermal conductivity of the medium and also in a multilayer structure are presented.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 3, pp. 524–532, March, 1977.     

A numerical method for the solution of two-dimensional inverse heat conduction problems

A numerical method for the solution of inverse heat conduction problems in two-dimensional rectangular domains is established and its performance is demonstrated by computational results. The present method extends Beck's⁸ method to two spatial dimensions and also utilizes future times in order to stabilize the ill-posedness of the underlying problems. The approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one-dimensional problems which can be decoupled.     

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.     

Resolving power of the iteration method of solving inverse heat-conduction boundary-value problems

A quantitative estimate is obtained for the range of frequencies entering in the boundary condition that is restorable by using a solution of the inverse problem by an iteration method.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 52, No. 6, pp. 986–990, June, 1987.     

Generalization of L. V. Kantorovich's method applied to boundary-value problems of heat conduction

We give a generalization of Kantorovich's method of reduction to an ordinary differential equation [1], as applied to boundary-value problems of heat conduction.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 21, No. 3, pp. 460–466, September, 1971.     

Multi-model method for solving nonlinear transient inverse heat conduction problems

The multi-model inverse method for nonlinear inverse problems is established based on the multi-model control theory. First the model switching variable is chosen and several typical operating balance points in the workspace of the balance variable are selected. Then for each operating balance point the linear local model is established, and the local controller is designed for each linear local model. Finally, according to the instantaneous matching degree between the actual model and the local models, the inversion results of each local controller are weighted and synthesized to obtain the final inversion result. Numerical tests are implemented to solve the one-dimensional nonlinear inverse heat conduction problem by the multi-model inverse method associated with the dynamic matrix control (DMC) and DMC filter, respectively. Numerical results by the multi-model inverse method based on DMC demonstrate that the multi-model inverse method is a highly computationally efficient and accurate algorithm for inverse problems with complicated direct problems. Numerical results by the multi-model inverse method based on DMC filter show that the presented method can extend the applied field of the complicated linear inverse algorithms such as digital filter to the nonlinear inverse problems and it can obtain satisfactory inversion results.     

A two-model iteration algorithm for solving the inverse boundary-value problem of heat conduction

A method is proposed for restoring the heat flux density on the boundary of a body which consists of the sequential solution of the direct problem for an adequate complex model and the invserse problem for a simplified heat transmission model.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 53, No. 6, pp. 1014–1020, December, 1987.     

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.     

Reduction of a class of inverse heat-conduction problems to direct initial/boundary-value problems

The authors present a method of reducing inverse problems of recovery of boundary heat fluxes by means of data of integral or differential temperature measurements on the boundary to direct initial/boundary-value problems. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 73, No. 4, pp. 744–747, July–August, 2000.     

A modified sequential function specification finite element-based method for parabolic inverse heat conduction problems

A method for enhancing the stability of parabolic inverse heat conduction problems (IHCP) is presented. The investigation extends recent work on non-iterative finite element-based IHCP algorithms which, following Becks two-step approach, first derives a discretized standard form equation relating the instantaneous global temperature and surface heat flux vectors, and then formulates a least squares-based linear matrix normal equation in the unknown flux. In the present study, the non-iterative IHCP algorithm is stabilized using a modified form of Becks sequential function specification scheme in which: (i) inverse solution time steps, t, are set larger than the data sample rate, , and (ii) future temperatures are obtained at intervals equal to . These modifications, contrasting with the standard approach in which the computational, experimental, and future time intervals are all set equal, are designed respectively to allow for diffusive time lag (under the typical circumstance where is smaller than, or on the order of the characteristic thermal diffusion time scale), and to improve the temporal resolution and accuracy of the inverse solution. Based on validation tests using three benchmark problems, the principle findings of the study are as follows: (i) under dynamic surface heating conditions, the modified and standard methods provide comparable levels of early-time resolution; however, the modified technique is not subject to over-damped estimation (as characteristic of the standard scheme) and provides improved error suppression rates, (ii) the present method provides superior performance relative to the standard approach when subjected to data truncation and thermal measurement error, and (iii) in the nonlinear test problem considered, both approaches provide comparable levels of performance. Following validation, the technique is applied to a quenching experiment and estimated heat flux histories are compared against available analytical and experimental results.     

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.     

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

Solution of the inverse boundary-value problem of heat conduction in an overdefined formulation

An interaction scheme is considered for the solution of a nonlinear inverse heat-conduction problem with the results of measuring the temperature at an arbitrary number of points within the body taken into account.Notation n number of temperature measurement points - , t time - _p, t_p length of the time interval - x a space coordinate - X_i(t) coordinates of the temperature measurement points - T(x, t) temperature - C(T) bulk specific heat of the material - (T) coefficient of material heat conduction - T(x, 0) initial temperature distribution - q heat flux density - f_i(t) , temperature measurement - ⁱ(z_i, t) , conjugate variable - ⁱ(z_i, t) , temperature variation - , , p parameters of the conjugate gradient method - s number of iteration - l number of points in a discrete representation of the time function - an error estimate Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 45, No. 5, pp. 776–781, November, 1983.