An Inverse Parameter Problem With Generalized Impedance Boundary Condition for Two-Dimensional Linear Viscoelasticity

Abstract

We analyze an inverse boundary value problem in two-dimensional viscoelastic media with a generalized impedance boundary condition on the inclusion via boundary integral equation methods. The model problem is derived from a recent asymptotic analysis of a thin elastic coating as the thickness tends to zero [F. Caubet, D. Kateb, and F. Le Louer, J. Elasticity, 136 (2019), pp. 17-53]. The boundary condition involves a new second order surface symmetric operator with mixed regularity properties on tangential and normal components. The well-posedness of the direct problem is established for a wide range of constant viscoelastic parameters and impedance functions. Extending previous research in the Helmholtz case, the unique identification of the impedance parameters from measured data produced by the scattering of three independent incident plane waves is established. The theoretical results are illustrated by numerical experiments generated by an inverse algorithm that simultaneously recovers the impedance parameters and the density solution to the equivalent boundary integral equation reformulation of the direct problem.