A fast and efficient numerical method for computing the stress concentration between closely located stiff inclusions of general shapes

TL;DR

This work addresses the stress concentration problem in the 2D Lam\'e system for two stiff inclusions that are closely spaced, where the gradient blows up like $\epsilon^{-1/2}$. The authors adopt a singular-term decomposition $\nabla u = \alpha_0 \nabla q + O(1)$ and show that the singular coefficient $\alpha_0$ can be obtained from the touching-domain problem via boundary-integral methods, allowing the residual to be computed on regular meshes. The main contributions are (i) a rigorous strategy to compute $\alpha_0$ for general-shaped inclusions by solving the touching case, (ii) a boundary-element-based numerical scheme that avoids ultra-fine meshing, and (iii) numerical evidence of fast convergence across diverse shapes and background fields. This approach yields accurate stress-concentration factors and solutions with modest discretization, potentially enabling efficient 3D extensions and broad applicability to composite-material reliability analyses.

Abstract

When two stiff inclusions are closely located, the gradient of the solution to the Lamé system, in other words the stress, may become arbitrarily large as the distance between two inclusions tends to zero. To compute the gradient of the solution in the narrow region, extremely fine meshes are required. It is a challenging problem to numerically compute the stress near the narrow region between two inclusions of general shapes as their distance goes to zero. A recent study [15] has shown that the major singularity of the gradient can be extracted in an explicit way for two general shaped inclusions. Thus the complexity of the computation can be greatly reduced by removing the singular term and it suffices to compute the residual term only using regular meshes. The goal of this paper is to numerically compute the stress concentration in a fast and efficient way. In this paper, we compute the value of the stress concentration factor, which is the normalized magnitude of the stress concentration, for general shaped domain as the distance between two inclusions tends to zero. We also compute the solution for two closely located inclusions of general shapes and show the convergence of the solution. Only regular meshes are used in our numerical computation and the results clearly show that the characterization of the singular term method can be efficiently used for computation of the stress concentration between two closely located inclusions of general shapes.

Figures (13)

• Figure 1: General geometry.
• Figure 2: $D_\rho$ for two touching ellipses.
• Figure 3: Left: $\alpha_\rho$ for different values of the grids number $N$. Middle: The relative error: $|(\alpha_\rho-\alpha_*)/\alpha_*|$. Right: The convergent rate: $\log|(\alpha_\rho-\alpha_*)/\alpha_*|$.
• Figure 4: Left: $\alpha_\rho$ for different values of $\rho$. Middle: The relative error: $|(\alpha_\rho-\alpha_*)/\alpha_*|$. Right: The convergent rate: $\log|(\alpha_\rho-\alpha_*)/\alpha_*|$.
• Figure 5: Left: the singular values of $A$ in the decreasing order of $n$ when $\epsilon$ is $0.01$. Right: the condition numbers of $A$ as the distance $\epsilon$ tends to $0$. The dimension of $A$ is $512\times 512$.