Replacing voids and localized parameter changes with fictitious forcing terms in boundary-value problems

We present a general observation on how to replace changes of material properties in limited regions within a domain with fictitious forcing terms in initial- and boundary-value problems associated with wave propagation and diffusion. Then, by considering a paradigmatic heat conduction problem on a domain with a cavity, we prove that the presence of the void can be replaced by a fictitious heat source with support contained within the cavity. We illustrate this fact in a situation where the source term can be analytically recovered from the values of the temperature and heat flux at the boundary of the cavity. Our result provides a strategy to map the nonlinear geometric inverse problem of void identification into a more manageable one, that involves the identification of forcing terms given the knowledge of external boundary data. To set the stage for a systematic study of the inverse problem, we present algebraic reconstructions, based on a finite-element discretization of the domain, that give an approximation of the fictitious source from different sets of temperature measurements. We show how the accuracy of the reconstruction is reflected on the void identification.