The use of model reduction and function decomposition for identifying bounday conditions of linear thermal system
Résumé
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 Phi(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 Phi 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.