Heterogeneous optimized Schwarz Methods for heat conduction in composites with thermal contact resistance

TL;DR

This paper introduces an optimized Schwarz method for heat conduction in composites with thermal contact resistance, demonstrating improved convergence properties and providing theoretical and numerical validation for complex heterogeneous problems.

Contribution

The paper develops a scaled Robin transmission condition with an optimized free parameter, enhancing convergence speed and robustness for heterogeneous heat conduction problems with TCR.

Findings

Larger TCR leads to faster convergence of the method.

Mesh-independent convergence is achieved asymptotically.

Higher heterogeneity contrast improves convergence speed.

Abstract

Heat transfer in composites is critical in engineering, where imperfect layer contact causes thermal contact resistance (TCR), leading to interfacial temperature discontinuity. We propose solving this numerically using the optimized Schwarz method (OSM), which decouples the heterogeneous problem into homogeneous subproblems. This avoids ill-conditioned systems from monolithic solving due to high contrast and interface jumps. Both energy estimate and Fourier analysis are used to prove the convergence of this algorithm when the standard Robin condition is applied to transmit information between subdomains. To achieve fast convergence, instead of the standard Robin, the scaled Robin transmission condition is proposed, and the involved free parameter is rigorously optimized. The results reveal several new findings due to the presence of TCR: first, the larger the TCR, the faster the OSM converges; second, mesh-independent convergence is achieved in the asymptotic sense, in contrast to the mesh-dependent results without TCR; and last, the heterogeneity contrast benefits the convergence, with a larger contrast leading to faster convergence. Interestingly, different from the case without TCR, the thermal conductivity also benefits the convergence, similar to the effect of heterogeneity. Numerical experiments confirm the theoretical findings and demonstrate the method's potential for nonlinear problems on irregular domains.