Nonlinear, Rescaling-Based Inverse Heat Conduction Calibration Method and Optimal Regularization Parameter Strategy

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Abstract
A novel surface heat flux calibration method applicable to nonlinear inverse heat conduction problems and a new strategy for obtaining an optimal regularization parameter independent of the regularization technique are presented herein. For the proposed nonlinear calibration integral equation, quasi-linearization of the nonlinear heat equation is achieved through both time domain and heat flux magnitude rescaling based on the local temperature measurement. The inverse heat conduction problem is then resolved in terms of rescaled variables through a calibration framework. The proposed calibration equation has a form of the Volterra equation of the first kind that relates the rescaled net unknown heat flux to the rescaled net calibration heat flux and their corresponding rescaled in-depth temperature measurements during both calibration and reconstruction tests. Because the functional form of a Volterra integral equation of the first kind is ill-posed, three different regularization methods are considered and compared. The three methods consist of 1) local future time method, 2) singular-value-decomposition-based regularization, and 3) Tikhonov regularization. A new strategy is proposed to determine the corresponding optimal regularization parameters. This strategy uses a Gaussian filter for estimating the variance in a group of predictions. The best regularization parameters are obtained by balancing the weighted bias and variance. Encouraging numerical results are observed independent of applied regularization approaches in presence of a significant noise, verifying the nonlinear calibration method and the merit of the new regularization parameter strategy.

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