Functional Identification of the Nonlinear Thermal-Conductivity Coefficient by Gradient Methods. II. Numerical Modeling

Abstract

Algorithms for the gradient method of solution of the inverse problem on determination of the nonlinear thermal-conductivity coefficients are given. Results of numerical experiments are discussed.Introduction. In [1], we consider the problem of functional identification of the nonlinear thermal-conductivity coefficient λ(T). Behind the approach proposed is the gradient method of numerical solution of inverse heat-conduction problems [2][3][4]. We note that, in the traditional approach to finding λ(T), one uses a finite-dimensional approximation of a coefficient by the system of basis functions [4], whereas in [1], we propose a method of solution of inverse heat-conduction problems without preliminary approximation of the functions sought; this method uses new representations of the operator conjugate to the internal-superposition operator, which makes it possible to obtain formulas convenient for numerical calculations of the values of the conjugate operator.In the present work, we propose algorithms of functional identification of the coefficient λ(T), describe computational experiments, and discuss calculation results.Computational Formulas. We give the formulas from [1], which are necessary for numerical realization of the algorithms of solution of inverse heat-conduction problems. The system of equations for finding λ(T) has the form

Estimation of the apparent thermal diffusivity coefficient using an inverse technique

“…Different techniques such as: the conjugate gradient method [8][9][10][11] and the Levenberg-Marquardt method [8,12] were performed in the literature and were used to estimate the thermophysical properties [13][14][15][16][17]. New methodologies are being used to solve inverse problems, particularly stochastic methods, which usually supply potential solutions, but the computational time required by stochastic methods generally exceeds that of deterministic optimization methods [18][19].…”

“…An iterative procedure, based on minimizing a sum of squares function with the Levenberg-Marquardt method was used to solve the inverse problem. Borukhov et al [17] considered the problem of functional identification of the non-linear thermal-conductivity coefficient. For numerical solution of inverse heat-conduction problems, Borukhov et al [17] used a gradient method.…”

Global optimization of thermal conductivity using stochastic algorithms

“…Unlike the conventional techniques, the resolution of the IHTP permits the determination of more than one thermo-physical property and the understanding of complex materials. Different optimization techniques, such as conjugate gradient and Levenberg-Marquardt methods were employed to estimate the thermo-physical properties in recent literature [1][2][3][4][5][6][7]. Modern optimization methodologies are being used to solve inverse problems, particularly stochastic methods, which usually supply potential solutions, but the computational time required by stochastic methods generally exceeds that of deterministic optimization methods [8,9].…”

“…An iterative procedure, based on minimizing a sum of squares function with the Levenberg-Marquardt method was used to solve an inverse problem. Borukhov et al [7] considered the problem of functional identification of the non-linear thermal-conductivity coefficient. For numerical solution of inverse heat-conduction problem, Borukhov et al [7] used an optimization method based on gradient information.…”

“…Borukhov et al [7] considered the problem of functional identification of the non-linear thermal-conductivity coefficient. For numerical solution of inverse heat-conduction problem, Borukhov et al [7] used an optimization method based on gradient information. Mendonc¸a et al [16] estimated the thermal conductivity and volumetric thermal capacity of apples through of the methodology based in the inverse problem of heat conduction solution.…”