The Tikhonov regularization method has been used to find the unknown heat flux distribution along the boundary when the temperature measurements are known in the interior of a sample. Mathematically, the inverse problem is ill-posed, though physically correct, and prone to instability. This paper discusses the fundamental issues concerning the selection of optimal regularization parameters for inverse heat transfer calculations. Towards this end, a finite-element-based inverse algorithm is developed. Five different methods, that is, the maximum likelihood (ML), the ordinary cross-validation (OCV), the generalized cross-validation (GCV), the L-curve method, and the discrepancy principle, are evaluated for the purpose of determining optimal regularization parameters. An assessment of these methods is made using 1-D and 2-D inverse steady heat conduction problems where analytical solutions are available. The optimal regularization method is also compared with the Levenberg-Marquardt method for inverse heat transfer calculations. Results show that in general the Tikhonov regularization method is superior over the Levenberg-Marquardt method when the input data errors are noisy. With the appropriately determined regularization parameter, the inverse algorithm is applied to estimate the heat flux of spray cooling of a 3-D microelectronic component with an embedded heating source.
A Theorem for Finding Maximum Temperature in Wet Grinding