A modified sequential function specification finite element-based method for parabolic inverse heat conduction problems

Abstract

A method for enhancing the stability of parabolic inverse heat conduction problems (IHCP) is presented. The investigation extends recent work on non-iterative finite element-based IHCP algorithms which, following Beck’s two-step approach, first derives a discretized standard form equation relating the instantaneous global temperature and surface heat flux vectors, and then formulates a least squares-based linear matrix normal equation in the unknown flux. In the present study, the non-iterative IHCP algorithm is stabilized using a modified form of Beck’s sequential function specification scheme in which: (i) inverse solution time steps, Δt, are set larger than the data sample rate, Δτ, and (ii) future temperatures are obtained at intervals equal to Δτ. These modifications, contrasting with the standard approach in which the computational, experimental, and future time intervals are all set equal, are designed respectively to allow for diffusive time lag (under the typical circumstance where Δτ is smaller than, or on the order of the characteristic thermal diffusion time scale), and to improve the temporal resolution and accuracy of the inverse solution. Based on validation tests using three benchmark problems, the principle findings of the study are as follows: (i) under dynamic surface heating conditions, the modified and standard methods provide comparable levels of early-time resolution; however, the modified technique is not subject to over-damped estimation (as characteristic of the standard scheme) and provides improved error suppression rates, (ii) the present method provides superior performance relative to the standard approach when subjected to data truncation and thermal measurement error, and (iii) in the nonlinear test problem considered, both approaches provide comparable levels of performance. Following validation, the technique is applied to a quenching experiment and estimated heat flux histories are compared against available analytical and experimental results.