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Direct reconstruction of general elastic inclusions

Sarah Eberle-Blick, Henrik Garde, Nuutti Hyvönen

TL;DR

This work tackles the inverse boundary value problem for linear elasticity with inclusions that perturb the Lamé parameters, including extreme cases of perfectly elastic and infinitely stiff materials. It develops an outer monotonicity framework that jointly detects positive and negative inclusions, extends to extreme contrasts, and proves convergence of the forward problem via truncated parameter limits, all anchored by the Neumann-to-Dirichlet map $\Lambda(\lambda,\mu)$. Central contributions include localization results with extreme inclusions, new monotonicity inequalities, and virtual measurement operators that enable a direct, parameter-agnostic reconstruction of inclusion domains without a priori contrast bounds. The paper also provides a linearized outer reconstruction for non-extreme inclusions, offering a fast, provably correct method for practical imaging while accommodating both inner and outer strategies and ensuring robustness to extreme material behavior. Overall, the results yield a direct, globally convergent inclusion-detection methodology for elastic media that integrates extreme-material models into a rigorous monotonicity-based imaging framework.

Abstract

The inverse problem of linear elasticity is to determine the Lamé 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 Lamé 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 Lamé 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.

Direct reconstruction of general elastic inclusions

TL;DR

This work tackles the inverse boundary value problem for linear elasticity with inclusions that perturb the Lamé parameters, including extreme cases of perfectly elastic and infinitely stiff materials. It develops an outer monotonicity framework that jointly detects positive and negative inclusions, extends to extreme contrasts, and proves convergence of the forward problem via truncated parameter limits, all anchored by the Neumann-to-Dirichlet map . Central contributions include localization results with extreme inclusions, new monotonicity inequalities, and virtual measurement operators that enable a direct, parameter-agnostic reconstruction of inclusion domains without a priori contrast bounds. The paper also provides a linearized outer reconstruction for non-extreme inclusions, offering a fast, provably correct method for practical imaging while accommodating both inner and outer strategies and ensuring robustness to extreme material behavior. Overall, the results yield a direct, globally convergent inclusion-detection methodology for elastic media that integrates extreme-material models into a rigorous monotonicity-based imaging framework.

Abstract

The inverse problem of linear elasticity is to determine the Lamé 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 Lamé 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 Lamé 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.

Paper Structure

This paper contains 16 sections, 15 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Remarks on notation
  3. Forward problem with extreme inclusions
  4. Variational formulation
  5. Extreme measurements as a limit
  6. Reconstruction of extreme inclusions
  7. Linearized outer reconstruction of non-extreme inclusions
  8. Orthogonal projections
  9. Proof of Theorem \ref{['thm:convergence']}
  10. Extreme monotonicity inequalities
  11. Virtual measurement operators
  12. Localization results
  13. Proof of Theorem \ref{['thm:monoext']}
  14. Proof of Theorem \ref{['thm:monopos']}
  15. Proof of Theorem \ref{['thm:mononeg']}
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $C=C_0\cup C_\infty$ satisfy Assumption assump, let $\lambda$ and $\mu$ be as in eq:extremeLame, and define the $\epsilon$-truncated versions of $\lambda$ and $\mu$, with $\epsilon>0$, by Then As a direct consequence,

Theorems & Definitions (36)

  • Definition 2.2
  • Remark 2.3
  • Theorem 3.1
  • proof
  • Definition 4.1
  • Remark 4.2
  • Definition 4.3
  • Theorem 4.5
  • proof
  • Theorem 4.6
  • ...and 26 more