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

TL;DR

The paper addresses forward and inverse linear elasticity in composite materials on complex domains with multiple singularities by marrying a domain decomposition into star-shaped subdomains with the direct-line method, yielding fast convergence and sharp handling of singularities. It develops a TV-regularized inverse framework where the forward elasticity problem is solved via the proposed direct-line solver and the Lamé coefficients are reconstructed with Adam optimization, enabling joint recovery of material properties and interfaces. Theoretical error estimates establish first-order convergence in the energy norm and second-order convergence in $L^2$, while numerical experiments across three forward and inverse problems demonstrate rapid eigenvalue convergence, accurate interface localization, and robustness to noise. The approach offers a scalable, accurate tool for forward simulations and inverse reconstructions in heterogeneous elastic domains, with potential impact on non-destructive evaluation and materials design.

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 lamé coefficient of multiple singular points in general regions. Our method can simultaneously deduce heterogeneous $μ$ and $λ$ 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.

Figures (16)

• Figure 1: Composite materials in a general domain with multiple singular points and material interfaces
• Figure 2: Decomposition of domain $\Omega$ into two subdomains, each containing one singular point
• Figure 3: (a) Original domain with four singular points; (b) Partition into four subdomains
• Figure 4: Domain $\Omega$ in Example \ref{['ex1']}
• Figure 5: $\frac{\partial u}{\partial r}(r, \phi_0)$ with $\phi_0 = \frac{3\pi}{4}$ in Example \ref{['ex1']}

Theorems & Definitions (10)

• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4
• Example 4.5
• Example 4.6
• Example 4.7