The direct-line method for forward and inverse linear elasticity problems of composite materials in general domains with multiple singularities

TL;DR

This paper introduces a combined domain decomposition and direct-line method to efficiently solve forward and inverse linear elasticity problems in complex composite materials with multiple singularities, achieving high accuracy and convergence.

Contribution

The work presents a novel integrated approach that handles multiple singularities in general regions, providing optimal error estimates and effective inverse problem reconstruction.

Findings

Rapid convergence of semi-discrete eigenvalues to exact eigenvalues

Effective handling of multiple singular points in general regions

Accurate reconstruction of heterogeneous Lamé coefficients

Abstract

In this work, a combined strategy of domain decomposition and the direct-line method is implemented to solve the forward and inverse linear elasticity problems of composite materials in general domains with multiple singularities. Domain decomposition technology treats the general domain as the union of some star-shaped subdomains, which can be handled using the direct-line method. The direct-line method demonstrates rapid convergence of the semi-discrete eigenvalues towards the exact eigenvalues of the elliptic operator, thereby naturally capturing the singularities. We also establish optimal error estimates for the proposed method. Especially, our method can handle multiple singular point problems in general regions, which are difficult to deal with by most methods. On the other hand, the inverse elasticity problem is constructed as a energy functional minimization problem with total variational regularization, we use the aforementioned method as a forward solver to reconstruct the lam\'{e} coefficient of multiple singular points in general regions. Our method can simultaneously deduce heterogeneous $\mu$ and $\lambda$ between different materials. Through numerical experiments on three forward and inverse problems, we systematically verified the accuracy and reliability of this method to solve forward and inverse elastic problems in general domains with multiple singularities.