Hybrid coupling with operator inference and the overlapping Schwarz alternating method

TL;DR

This paper tackles the bottlenecks of multiscale solid-mechanics simulations by proposing a hybrid domain-decomposition framework that couples subdomain-local non-intrusive OpInf ROMs with subdomain-local FOMs via overlapping Schwarz alternating method (O-SAM). The approach extends SAM to OpInf, introduces boundary POD bases and a systematic regularization-parameter selection, and supports heterogeneous time integrators across subdomains. Across four challenging 3D solid-mechanics problems, the method achieves substantial online speedups (up to $106\times$) while preserving accuracy, including when mixing linear, quadratic, and cubic OpInf models with FOMs. The work provides a minimally intrusive, plug-and-play workflow for integrating data-driven ROMs into production-like simulations, offering a scalable path for efficient multiscale engineering analyses and potential extensions to a broad class of PDEs.

Abstract

This paper presents a novel hybrid approach for coupling subdomain-local non-intrusive Operator Inference (OpInf) reduced order models (ROMs) with each other and with subdomain-local high-fidelity full order models (FOMs) with using the overlapping Schwarz alternating method (O-SAM). The proposed methodology addresses significant challenges in multiscale modeling and simulation, particularly the long runtime and complex mesh generation requirements associated with traditional high-fidelity simulations. By leveraging the flexibility of O-SAM, we enable the seamless integration of disparate models, meshes, and time integration schemes, enhancing computational efficiency while maintaining high accuracy. Our approach is demonstrated through a series of numerical experiments on complex three-dimensional (3D) solid dynamics problems, showcasing speedups of up to 106x compared to conventional FOM-FOM couplings. This work paves the way for more efficient simulation workflows in engineering applications, with potential extensions to a wide range of partial differential equations.

Figures (25)

• Figure 1: Illustration showing an domain decomposition of a 2D domain $\Omega$ into two overlapping subdomains, $\Omega_1$ and $\Omega_2$, for the application of O-SAM.
• Figure 2: Field transfer in the O-SAM algorithm within one controller time-step $I_k$ using the spatial projectors $~\Pi_1$ and $~\Pi_2$, and temporal interpolants $~\Xi_1^j$ and $~\Xi_2^m$. (a) $\Omega_1$ serves as the source subdomain, while $\Omega_2$ functions as the destination domain; (b) the roles of the domains are reversed. The super-scripts $j+l$ and $m+l$ correspond to times $t_{k-1} + l\Delta t_1$ and $t_{k-1} + m\Delta t_2$, respectively, and the integers $n_i$ denote the number of time-steps in $\Omega_i$, as described in \ref{['eq:schwarz_discrete']}. Thus, $~u_i^{n_i(k-1)} \approx ~u_i(t_{k-1}, ~X)$ and $~u_i^{n_ik} \approx ~u_i(t_{k}, ~X)$ for $i=1,2$.
• Figure 3: 1D linear elastic wave propagation problem: singular value decay (a) and projection errors calculated in $\Omega_2$ for the Symmetric Gaussian (b) and Rounded Square (c) problem variant, as functions of the POD basis size.
• Figure 4: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, reproductive regime: displacement (top row) and velocity (bottom row) relative errors with respect to the exact analytical solution for various FOM-OpInf (a) and OpInf-OpInf (b) O-SAM couplings. Dashed horizontal lines show relative errors for FOM-FOM O-SAM couplings with the colors designated in the legend.
• Figure 5: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, reproductive regime: mean number of Schwarz iterations required to reach convergence for various FOM-OpInf (a) and OpInf-OpInf (b) O-SAM couplings. Dashed horizontal lines show the number of Schwarz iterations needed to reach convergence for FOM-FOM O-SAM couplings with the colors designated in the legend.

Theorems & Definitions (8)

• Remark 1
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• Remark 8