Uniqueness in the inverse boundary value problem for piecewise homogeneous anisotropic elasticity

TL;DR

This paper proves the uniqueness of determining anisotropic elastic tensors in a piecewise homogeneous medium from boundary measurements, even when the subdomains are unknown but have curved boundaries, advancing inverse elasticity problem understanding.

Contribution

It establishes global uniqueness results for inverse boundary value problems in anisotropic elasticity with unknown subdomains having curved boundaries.

Findings

Global uniqueness for known subdomains with curved interfaces.

Extension of uniqueness to unknown subdomains that are subanalytic with curved boundaries.

Applicable to localized boundary measurements for a single subdomain.

Abstract

Consider a three dimensional piecewise homogeneous anisotropic elastic medium $\Omega$ which is a bounded domain consisting of a finite number of bounded subdomains $D_\alpha$, with each $D_\alpha$ a homogeneous elastic medium. One typical example is a finite element model with elements with curvilinear interfaces for an ansiotropic elastic medium. Assuming the $D_\alpha$ are known and Lipschitz, we are concerned with the uniqueness in the inverse boundary value problem of identifying the anisotropic elasticity tensor on $\Omega$ from a localized Dirichlet to Neumann map given on a part of the boundary $\partial D_{\alpha_0}\cap\partial\Omega$ of $\partial\Omega$ for a single $\alpha_0$, where $\partial D_{\alpha_0}$ denotes the boundary of $ D_{\alpha_0}$. If we can connect each $D_\alpha$ to $D_{\alpha_0}$ by a chain of $\{D_{\alpha_i}\}_{i=1}^n$ such that interfaces between adjacent regions contain a curved portion, we obtain global uniqueness for this inverse boundary value problem. If the $D_\alpha$ are not known but are subanalytic subsets of $\mathbb{R}^3$ with curved boundaries, then we also obtain global uniqueness.