An elastic inclusion embedded in a homogeneous background induces a perturbation for a given far-field loading. This perturbation admits a multipole expansion with coefficients known by Elastic Moment Tensors (EMTs), which contain information on the material and geometric properties of the inclusion. Iterative optimization approaches to recover the shape of the inclusion involving the EMTs have been reported. In this talk, we focus on the shape derivative of the EMTs for planar inclusion. In particular, we derive asymptotic expressions for the shape deformation of inclusion from a disk, based on the complex formulation for the solution to the plane elastostatic problem.

I will discuss some applications of the spectrum of the Neumann-Poincaré operator.

We consider the spectral structure of the Neumann–Poincaré operators defined on the boundaries of thin domains in two and three dimensions. In two dimensions, we consider rectangle-shaped domains. We prove that as the aspect ratio of the domains tends to $\infty$, or equivalently, as the domains get thinner, the spectra of the Neumann–Poincaré operators are densely distributed in $[−\frac{1}{2}, \frac{1}{2} ]$. In three dimensions, we consider two different kinds of thin domains: thin oblate domains and thin cylinders. We show that in the first case the spectra are distributed densely in the interval $[−\frac{1}{2}, \frac{1}{2} ]$ as as the domains get thinner. In the second case, as a partial result, we show that the spectra are distributed densely in the half interval $[−\frac{1}{2}, \frac{1}{2} ]$ as the domains get thinner.