Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations

Drossos Gintides, Leonidas Mindrinos

TL;DR

The paper develops an inverse-elasticity framework that combines Eshelby’s equivalent inclusion method for static inhomogeneities with a Lippmann–Schwinger-type integral formulation for elastodynamics to recover spatially varying elastic parameters from limited far-field data. The static problem is recast as a Fredholm integral equation of the second kind with a contraction operator, enabling stable reconstruction from a few measurements, while the dynamic problem yields a nonlinear forward map whose Fréchet derivative is used in a Newton-type, Tikhonov-regularized inversion. The authors derive explicit representations for the far-field patterns and the LS operator, formulate the linearized inverse problem, and validate the approach with 2D numerical experiments that demonstrate feasibility and serve as a foundation for extending to 3D anisotropic settings. Overall, the work provides a coherent, data-efficient methodology for inverse elasticity that can inform seismic, materials imaging, and nondestructive testing applications, and it outlines clear directions for advancing to full 3D anisotropic reconstructions.

Abstract

We present two applications of the integro-differential volume equation for the eigenstrain, building on Eshelby's inclusion method [15,16], in the contexts of both static and dynamic linear elasticity. The primary objective is to address the inverse problem of recovering the elastic moduli of the inhomogeneity using a limited number of incident fields. In the static case, we adopt an efficient reformulation of Eshelby's equation proposed by Bonnet [7]. By employing a first-order approximation in addition with a limited number of incident loadings and measurements, we numerically determine the material coefficients of the inclusion. In elastodynamics, we focus on the inverse scattering problem, utilizing the Lippmann-Schwinger integral equation to reconstruct the elastic properties of the inclusion through a Newton-type iterative scheme. We construct the Frechet derivative and we formulate the linearized far-field equation. Additionally, the corresponding plane strain problems are analyzed in both static and dynamic elasticity.

On the inverse elastic problem for isotropic media using Eshelby and Lippmann-Schwinger integral formulations

TL;DR

The paper develops an inverse-elasticity framework that combines Eshelby’s equivalent inclusion method for static inhomogeneities with a Lippmann–Schwinger-type integral formulation for elastodynamics to recover spatially varying elastic parameters from limited far-field data. The static problem is recast as a Fredholm integral equation of the second kind with a contraction operator, enabling stable reconstruction from a few measurements, while the dynamic problem yields a nonlinear forward map whose Fréchet derivative is used in a Newton-type, Tikhonov-regularized inversion. The authors derive explicit representations for the far-field patterns and the LS operator, formulate the linearized inverse problem, and validate the approach with 2D numerical experiments that demonstrate feasibility and serve as a foundation for extending to 3D anisotropic settings. Overall, the work provides a coherent, data-efficient methodology for inverse elasticity that can inform seismic, materials imaging, and nondestructive testing applications, and it outlines clear directions for advancing to full 3D anisotropic reconstructions.

Abstract

We present two applications of the integro-differential volume equation for the eigenstrain, building on Eshelby's inclusion method [15,16], in the contexts of both static and dynamic linear elasticity. The primary objective is to address the inverse problem of recovering the elastic moduli of the inhomogeneity using a limited number of incident fields. In the static case, we adopt an efficient reformulation of Eshelby's equation proposed by Bonnet [7]. By employing a first-order approximation in addition with a limited number of incident loadings and measurements, we numerically determine the material coefficients of the inclusion. In elastodynamics, we focus on the inverse scattering problem, utilizing the Lippmann-Schwinger integral equation to reconstruct the elastic properties of the inclusion through a Newton-type iterative scheme. We construct the Frechet derivative and we formulate the linearized far-field equation. Additionally, the corresponding plane strain problems are analyzed in both static and dynamic elasticity.

Paper Structure

This paper contains 10 sections, 2 theorems, 80 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The static elastic inhomogeneity problem
  3. Three-dimensional setup
  4. Plane strain problem
  5. The elastodynamic inhomogeneity problem
  6. Three-dimensional problem
  7. The inverse problem
  8. Plane strain problem
  9. Numerical examples
  10. Conclusions

Key Result

Theorem 1

The volume integral equation eq_sol for $\bm x \in \Omega$ is written equivalently as for the modified unknown density function $\bm h^\ast := \bm{\mathcal{B}}^{-1} : \Delta \bm{\mathcal{C}} : \bm\epsilon (\bm u^-),$ where and $\nu$ is the Poisson's ration and $\bm{\mathcal{I}}$ is the fourth-order identity tensor. The operator $\bm{\mathcal{Q}} : L^2_c (B;\mathbb{R}^{3\times 3}_{sym}) \rightarr

Figures (3)

  • Figure 1: Left: The interior (red) and the boundary (blue) grid points for $n=32$ and $N=21$. Right: The Gaussian basis functions for $K=5.$
  • Figure 2: The exact Lamé parameter $\lambda^\ast$ of the first example (top left), the initial guess $\lambda^\ast_0 = 0.5$ (top right) and the reconstructed values for exact (bottom left) and data with $3\%$ noise (bottom right).
  • Figure 3: The exact Lamé parameter $\lambda^\ast$ of the second example (top left), the initial guess $\lambda^\ast_0 = 0.5$ (top right) and the reconstructed values for exact (bottom left) and data with $3\%$ noise (bottom right).

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2