Explicit boundary heat flux reconstruction employing temperature measurements regularized via truncated eigenfunction expansions

摘要

The present work is aimed at developing and validating an explicit inverse problem solution for boundary heat flux reconstruction in thermally thin plates, supposing a large amount of temperature measurements is available, both in space and time, such as typically obtained by modern infrared thermography systems with fine spatial resolution and high frequency. In order to handle the magnification of the measurement errors when applying the derived explicit inversion formula, a regularized representation is proposed for the measured temperatures, consisting of truncated eigenfunction expansions in which only the most important modes, based on the discrepancy principle, are employed. Two applications are considered, the first one consists of a spatially uniform but time varying heat flux with fast transitions, and the second one consists of a spatially varying heat flux concentrated in two small spots. Good results are obtained with very little computational effort, indicating the feasibility of the proposed approach. This work adds to the available standard inverse problems tools, either as an alternative approach or as a companion in obtaining a starting approximate solution for iterative methods.