Integral Transforms and Bayesian Inference in the Identification of Variable Thermal Conductivity in Two-Phase Dispersed Systems

Abstract

This work illustrates the use of Bayesian inference in the estimation of spatially variable thermal conductivity for one-dimensional heat conduction in heterogeneous media, such as particle-filled composites and other two-phase dispersed systems, by employing a Markov chain Monte Carlo (MCMC) method, through the implementation of the Metropolis-Hastings algorithm. The direct problem solution is obtained analytically via integral transforms, and the related eigenvalue problem is solved by the generalized integral transform technique (GITT), offering a fast, precise, and robust solution for the transient temperature field, which are desirable features for the implementation of the inverse analysis. Instead of seeking the function estimation in the form of a sequence of local values for the thermal conductivity, an alternative approach is proposed here, which is based on the eigenfunction expansion of the thermal conductivity itself. Then, the unknown parameters become the corresponding series coefficients. Simulated temperatures obtained via integral transforms are used in the inverse analysis. From the prescription of the concentration distribution of the dispersed phase, available correlations for the thermal conductivity are employed to produce the simulated results with high precision in the direct problem solution, while eigenfunction expansions with reduced number of terms are employed in the inverse analysis itself, in order to avoid the so-called inverse crime. Both Gaussian and noninformative uniform distributions were used as priors for comparison purposes. In addition, alternative correlations for the thermal conductivity that yield different predictions are also employed as Gaussian priors for the algorithm in order to test the inverse analysis robustness.

The authors would like to acknowledge the financial support provided by CNPq, CAPES, and FAPERJ, Brazilian agencies for the fostering of science. The authors are also deeply grateful to the reviewers for the very careful revisions and helpful suggestions.

Notes

^a Case 1, uniform prior; case 2, normal prior from Lewis and Nielsen with 40% standard deviation; case 3, normal prior from Lewis and Nielsen with 80% standard deviation; case 4, normal prior from Maxwell with 40% standard deviation; case 5, normal prior from Maxwell with 80% standard deviation.