Model reduction for the resolution of multidimensional inverse heat conduction problems

For large linear heat conduction systems, it is proposed here to solve an inverse heat conduction problem (IHCP) that consists in the identification of several time-varying thermal solicitations from simulations of measured temperatures. For this inversion, instead of using a detailed model of large size, this one is first transformed into a reduced model. The latter is built with identified dominant eigenmodes of the system leading to a reduced state representation that links the inputs (unknown solicitations) to the outputs (simulated temperatures). The procedure is sequential and uses future time steps. At first, a numerical 2D IHCP is provided: two time-varying heat flux densities are estimated from various positions of two sensors. A specific study on static and dynamic sensitivities is made. An example of a 3D IHCP is also given. The method is particularly interesting in this last case where, at each time step, the resolution of a system of order 9 (the reduced model) takes the place of a system of order 1331 (the detailed model).