Regularized coupling multiscale method for thermomechanical coupled problems

TL;DR

This work addresses the challenge of efficiently solving strongly coupled thermomechanical equations in heterogeneous media, where the coupling operator can be non-positive definite. The authors develop CGMsFEM, a regularized coupling generalized multiscale finite element method that builds coupling multiscale basis functions from local spectral problems incorporating two relaxation parameters, enabling a reduced global system that still captures coupling effects. They establish convergence through interpolation and prior error analyses, showing energy errors decay with local eigenvalue gaps and are largely independent of the relaxation coefficients. Numerical tests on periodic and random microstructures demonstrate that CGMsFEM achieves higher accuracy with fewer basis functions than uncoupled GMsFEM, confirming robustness and computational efficiency for multiscale thermoelastic problems.

Abstract

The coupling effects in multiphysics processes are often neglected in designing multiscale methods. The coupling may be described by a non-positive definite operator, which in turn brings significant challenges in multiscale simulations. In the paper, we develop a regularized coupling multiscale method based on the generalized multiscale finite element method (GMsFEM) to solve coupled thermomechanical problems, and it is referred to as the coupling generalized multiscale finite element method (CGMsFEM). The method consists of defining the coupling multiscale basis functions through local regularized coupling spectral problems in each coarse-grid block, which can be implemented by a novel design of two relaxation parameters. Compared to the standard GMsFEM, the proposed method can not only accurately capture the multiscale coupling correlation effects of multiphysics problems but also greatly improve computational efficiency with fewer multiscale basis functions. In addition, the convergence analysis is also established, and the optimal error estimates are derived, where the upper bound of errors is independent of the magnitude of the relaxation coefficient. Several numerical examples for periodic, random microstructure, and random material coefficients are presented to validate the theoretical analysis. The numerical results show that the CGMsFEM shows better robustness and efficiency than uncoupled GMsFEM.

Figures (10)

• Figure 1: The fine grid $\mathcal{T}^h$, the coarse grid $\mathcal{T}^H$, the coarse element $K_{i}$ and neighborhood $\omega_{i}$ of the node $x_{i}$.
• Figure 2: Contour plots of the material coefficients in periodic microstructure. (a) Lam$\acute{e}$ coefficients $\mu$, and $\lambda$; (b) Thermal conductivity coefficient $\kappa$, and expansion coefficient $\beta$.
• Figure 3: Contour plots of 8 eigenfunctions of the CGMsFEM and GMsFEM used in Section \ref{['subsec:verification']}. CGMsFEM: (a) $\psi_{\theta}^{cgm}$ (b) $\psi_{u_1}^{cgm}$ and (c) $\psi_{u_2}^{cgm}$; GMsFEM: (d) $\psi_{\theta}^{gm}$ (e) $\psi_{u_1}^{gm}$ and (f) $\psi_{u_2}^{gm}$;
• Figure 4: Contour plots of solutions for periodic microstructure. The reference solutions: (a) $u_1^{ref}$ (b) $u_2^{ref}$ and (c) $\theta^{ref}$; The CGMsFEM solutions: (d) $u_1^{cgm}$ (e) $u_2^{cgm}$ and (f) $\theta^{cgm}$; The GMsFEM solutions: (g) $u_1^{gm}$ (h) $u_2^{gm}$ and (i) $\theta^{gm}$.
• Figure 5: Comparison of relative energy errors of the CGMsFEM and GMsFEM in periodic microstructure. (a) $E_\theta$ (b) $E_u$, and (c) $E_w$;

Theorems & Definitions (15)

• Remark 2.1
• Lemma 3.1
• Proof 3.1
• Corollary 3.1
• Lemma 3.2
• Proof 3.2
• Lemma 3.3
• Theorem 3.1
• Proof 3.3
• Lemma 3.4