Direct reconstruction of general elastic inclusions

TL;DR

This paper extends the monotonicity method for inverse linear elasticity problems to reconstruct inclusions with both positive and negative contrasts, including extreme cases like infinitely stiff or perfectly elastic materials.

Contribution

It proves the general outer approach of the monotonicity method that handles both positive and negative inclusions with finite or extreme contrast.

Findings

The method can reconstruct inclusions with infinite contrast.

It handles both positive and negative deviations from background parameters.

The approach is applicable to a wider class of elastic inclusions.

Abstract

The inverse problem of linear elasticity is to determine the Lam\'e parameters, which characterize the mechanical properties of a domain, from pairs of pressure activations and the resulting displacements on its boundary. This work considers the specific problem of reconstructing inclusions that manifest themselves as deviations from the background Lam\'e parameters. The monotonicity method is a direct reconstruction method that has previously been considered for domains only containing positive (or negative) inclusions with finite contrast. That is, all inclusions have previously been assumed to correspond to a finite increase (or decrease) in both Lam\'e parameters compared to their background values. We prove the general outer approach of the monotonicity method that simultaneously allows positive and negative inclusions, of both finite and extreme contrast; the latter refers to either infinitely stiff or perfectly elastic materials.