Transmission Conditions for the Non-Overlapping Schwarz Coupling of Full Order and Operator Inference Models

TL;DR

These findings highlight NO-SAM's potential for enabling flexible, non-intrusive, and efficient multi-model coupling across independently meshed subdomains, while emphasizing the need for careful interface condition design in higher-dimensional and predictive settings.

Abstract

This work investigates transmission conditions for the domain decomposition-based coupling of subdomain-local models using the non-overlapping Schwarz alternating method (NO-SAM). Building on prior efforts involving overlapping SAM (O-SAM), we formulate and assess two NO-SAM variants, based on alternating Dirichlet-Neumann and Robin-Robin transmission conditions. For the subdomain-local models, we consider a mix of full order models (FOMs) and non-intrusive reduced order models (ROMs) constructed via an emerging model reduction technique known as operator inference (OpInf). Of particular novelty is the first application of NO-SAM to couple non-intrusive OpInf ROMs with each other, and with FOMs. Numerical studies on a one-dimensional linear elastic wave propagation benchmark problem demonstrate that transmission condition choice and parameter tuning significantly impact convergence rate, accuracy, and stability. Robin-Robin coupling often yields faster convergence than alternating Dirichlet-Neumann, though improper parameter selection can induce spurious oscillations at subdomain interfaces. For FOM-OpInf and OpInf-OpInf couplings, sufficient modal content in the ROM basis improves accuracy and mitigates instability, in some cases outperforming the coupled FOM-FOM reference solutions in both accuracy and efficiency. These findings highlight NO-SAM's potential for enabling flexible, non-intrusive, and efficient multi-model coupling across independently meshed subdomains, while emphasizing the need for careful interface condition design in higher-dimensional and predictive settings.

Figures (8)

• Figure 3.1: Illustration showing an example non-overlapping domain decomposition.
• Figure 3.2: FOM-FOM coupling: Pareto plot illustrating the relationship between the error in the converged Schwarz solution relative to the corresponding monolithic solution and the mean number of Schwarz iterations per time step required for convergence, for varying $\alpha_{ij}$ and $\beta_{ij}$ parameters. The unrelaxed alternating Dirichlet-Neumann result is also plotted as a reference in gray.
• Figure 3.3: FOM-FOM coupling: plots of the NO-SAM displacement, velocity, and acceleration solutions obtained with alternating Dirichlet–Neumann transmission conditions, compared to the corresponding monolithic solution (dashed line). The blue subdomain $\Omega_1$ employs a Dirichlet transmission condition, and the red subdomain $\Omega_2$ employs a Neumann transmission condition.
• Figure 3.4: FOM-FOM coupling: plots of the NO-SAM displacement, velocity, and acceleration solutions obtained with Robin–Robin transmission conditions (lowest error parameter set, $\left[ \overline{\alpha}_{12}, \overline{\alpha}_{21}, \beta_{12}, \beta_{21} \right] = \left[ 10^{-3}, 10^{-3}, 1, 1\right]$) compared to the corresponding monolithic solution (dashed line). The solutions computed for subdomains $\Omega_1$ and $\Omega_2$ are shown in blue and red, respectively.
• Figure 3.5: FOM-FOM coupling: plots of the NO-SAM displacement, velocity, and acceleration solutions obtained with Robin–Robin transmission conditions (highest error parameter set, $\left[ \overline{\alpha}_{12}, \overline{\alpha}_{21}, \beta_{12}, \beta_{21} \right] = \left[ 10^{-1}, 1, 1, 3 \right]$) compared to the corresponding monolithic solution (dashed line). The solutions computed for subdomains $\Omega_1$ and $\Omega_2$ are shown in blue and red, respectively.