Finite element method for the reconstruction of a time-dependent heat source in isotropic thermoelasticity systems of type-III

Abstract

The isotropic thermoelasticity system of type-III, describing both the mechanical and the thermal behaviours of a body occupying a bounded domain with a Lipschitz boundary, is considered. The displacement vector and either the normal heat flux or the temperature are prescribed on the boundary. Both the theoretical and the numerical reconstructions of a time-dependent heat source from the knowledge of an additional weighted integral measurement of the temperature are investigated. It is showed that the appropriate type of measurement depends on the thermal boundary condition available, whilst the existence and uniqueness of a weak solution for exact data is also proved. For each of the two inverse source problems investigated herein, a numerical algorithm is also proposed and the convergence of these numerical schemes for exact data is proved. Four numerical examples with noisy measurements are implemented using the finite element method and thoroughly investigated to validate the convergence and stability of the proposed algorithms.