Deformation problem for glued elastic bodies and an alternative iteration method

TL;DR

The paper addresses a stationary deformation problem for two elastic bodies glued along a Lipschitz interface $\Gamma$, modeled as linear elasticity with an adhesive term $\zeta [u]$. It develops a variational framework with spaces $X$, $V$, and a symmetric bilinear form $a_0$ and energy $E(u)=\frac{1}{2}a_0(u,u)-l_0(u)$, proving the unique existence of a weak solution under $\zeta\ge 0$ and $\|\zeta\|_{L^{\infty}(\Gamma)}>0$. The authors introduce a Gauss–Seidel type alternating iteration that monotonically decreases the total energy and converges to the discrete monolithic FE solution, with a proven rate $O(r^m)$ where $r=1/(1+\alpha_1\beta_1)$. Numerical discretization is developed with a matrix form for the monolithic problem and a discrete alternating method; three-dimensional experiments demonstrate that the alternating scheme can be faster and more memory-efficient for large-scale problems while maintaining accuracy.

Abstract

We study a mathematical model for deformation of glued elastic bodies in 2D or 3D, which is a linear elasticity system with adhesive force on the glued surface. We reveal a variational structure of the model and prove the unique existence of a weak solution based on it. Furthermore, we also consider an alternating iteration method and show that it is nothing but an alternating minimizing method of the total energy. The convergence of a monolithic formulation and the alternating iteration method are numerically studied with the finite element method.

Figures (4)

• Figure 1: $\partial_D\Omega\subset\partial\Omega_1\cup\partial\Omega_2$
• Figure 2: $\partial_D\Omega\subset\partial\Omega_1$
• Figure 3: Relative errors of P1 finite element solution computed by monolithic formulation
• Figure 4: Convergence history of Gauss-Seidel type iteration

Theorems & Definitions (15)

• Definition 2.1: Weak solution
• Theorem 3.1: Coercivity of $a_0$
• Theorem 3.2: Unique existence
• proof
• Proposition 3.3
• Theorem 4.1
• proof
• Remark 5.1
• Lemma 5.2
• proof