A Moving Pseudo-boundary Mfs for Void Detection in Two-dimensional Thermoelasticity

We investigate the numerical reconstruction of smooth star-shaped voids contained in a two-dimensional isotropic linear thermoelastic material from the knowledge a single non-destructive measurement of the temperature field and both the displacement and traction vectors (over-specified boundary data) on the outer boundary. The primary fields, namely the temperature and the displacement vector, which satisfy the thermo-mechanical equilibrium equations of linear thermoelasticity, are approximated using the method of fundamental solutions (MFS) in conjunction with a two-dimensional particular solution proposed in [37]. The fictitious source points are located both outside the (known) exterior boundary of the body and inside the (unknown) void. The inverse geometric problem reduces to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void. The interior source points are anchored to and move with the void during the iterative reconstruction procedure. The stability of the numerical method is investigated by inverting measurements contaminated with noise.