Analytic shape recovery of an elastic inclusion from elastic moment tensors

In this paper, we present an analytic non-iterative approach for recovering a planar isotropic elastic inclusion embedded in an unbounded medium from the elastic moment tensors (EMTs), which are coefficients for the multipole expansion of field perturbation caused by the inclusion. EMTs contain information about the inclusion's material and geometric properties and, as is well known, the inclusion can be approximated by a disk from leading-order EMTs. We define the complex contracted EMTs as the linear combinations of EMTs where the expansion coefficients are given from complex-valued background polynomial solutions. By using the layer potential technique for the Lam\'{e} system and the theory of conformal mapping, we derive explicit asymptotic formulas in terms of the complex contracted EMTs for the shape of the inclusion, treating the inclusion as a perturbed disk. These formulas lead us to an analytic non-iterative algorithm for elastic inclusion reconstruction using EMTs. We perform numerical experiments to demonstrate the validity and limitations of our proposed method.