Estimation of the parameters of a Gaussian heat source by the Levenberg–Marquardt method: Application to the electron beam welding

In this paper, we estimate a heat source in a longitudinal section during an electron beam welding. The aim of this work is the parameter identification of the Gaussian source term representative of the dissipated heat flux in the liquid zone from measured temperatures in the solid zone. In a previous work [J. Guo, P. Le Masson, E. Artioukhine, T. Loulou, P. Rogeon, M. Carin, M. Dumons, J. Costa, Estimation of a source term in a two-dimensional heat transfer problem: Application to an electron beam welding, in: 4th Int. Conf. Inverse Problems, Russia, 2003], we have analyzed the feasibility of the estimation for a source term S(x,z,t) in a transversal section. This work has an application in the electron beam welding of steels of thickness about 8 cm. The direct thermo-metallurgical problem is presented in a two-dimensional longitudinal section (x,y) for a quasi-steady state. This non-linear problem is considered in the thesis of J. Guo [J. Guo, Estimation de la distribution énergétique induite par un faisceau d'électrons dans un matériau métallique – Application au cas du soudage d'un acier, Thèse de l'université de Bretagne Sud, 2005]. Here, we solve only a linear case. The sample is divided in the axial direction z in few sections. At each section, a source term is defined with a part of the beam and creates a vaporized zone and a fused zone. The goal of this work is the rebuilding of the complete source term with the estimations at each section. In this paper, the feasibility of the parameter estimation by Levenberg–Marquardt method is analyzed.