A two-step regularized inverse solution for 2-D heat source reconstruction

An inverse problem is considered here that consists in the reconstruction of heat source maps from the measurement of 2D temperature fields. The endorsed application concerns the thermomechanical characterization of materials submitted to mechanical loads. An infrared camera is used to record the temperature evolutions resulting from reversible or more usually irreversible heat dissipation. An algorithm is presented and checked theoretically on simulated data. It relies on a formulation in terms of a constrained optimization problem. One constraint imposes the observance of the heat governing equation. The second constraint is a penalty term of Tikhonov type. A discretization of the equations resulting from the minimization conditions yield a linear matrix form. The main problem concerns the conditioning of the sparse matrix so obtained, which is seriously damaged if one tries to get a very fine spatial resolution for the source. The strategy used for optimal regularization is obtained in two steps. First, prior information is obtained from experiments in a light inversion procedure. Second, this information is used for efficient regularization.This algorithm has the advantage of being easy to implement and independent from hypothesis on boundary conditions. The L-curve criterion is shown to produce good results and a robust processing for noisy data.