A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only

TL;DR

This paper introduces a new direct linear method for reconstructing discontinuous elastic parameters from internal displacement data, achieving stable results even with minimal measurements and no boundary information.

Contribution

It presents a novel linear inversion approach for elastic parameters, including discontinuous ones, using internal displacement data without boundary measurements.

Findings

Stable reconstruction of shear modulus with one measurement.

Applicable to both isotropic and anisotropic stiffness tensors.

Proven $L^2$-stability under minimal smoothness assumptions.

Abstract

The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-to-force operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an $L^2$-stability with only one measurement under minimal smoothness assumptions. This stability result is obtained though the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.