The Use of Model Reduction and Function Decomposition for Identifying Boundary Conditions of A Linear Thermal System

Abstract

This numerical study deals with the identification of space and time varying inputs applied to a linear diffusive thermal system. Such an Inverse Heat Conduction Problem (IHCP) is ill-posed, its resolution is difficult for a large amount of unknowns and requires large memory size and computing time for multidimensional cases. Consequently, we propose a procedure to reduce both the number of unknowns and the model order. A 2D example is presented, with a heat flux density φ(y,t) to be identified from simulated transient temperature measurements. Starting from a Classical Detailed Model (CDM), two steps are performed. Firstly, a decomposition of the spatial distribution of φ on a functions basis leads to a small number of unknowns. Secondly, a Reduced Model (RM) is built using the Modal Identification Method. When RM is used to solve the inverse problem instead of CDM, computing time is drastically reduced (up to a factor 1000) whilst preserving accuracy. A procedure to determine the number of unknown coefficients is proposed. The inversion algorithm is sequential and requires no iterations. Future time steps with a function specification are used as a regularisation procedure. Tikhonov's regularisation is needed with CDM but not with RM.