Introduction to the Combination of Reduced Order Models and Domain Decomposition: State of the Art and Perspectives

TL;DR

The article surveys the state of the art in combining Reduced Order Models with Domain Decomposition to accelerate parametric and multi-physics simulations. It distinguishes intrusive projection-based and non-intrusive data-driven pathways, detailing how local reduced bases are generated and coupled across subdomains. The review covers domain partitioning choices (overlaps, mesh conformity, archetypes), parameterization strategies (sampling, geometry, parameter-space reduction), and varied coupling algorithms (monolithic and iterative, including DG, RDF, RBEM, SCRBEM, PUM, and optimization-based methods). It also highlights data-driven approaches (Schwarz-based, interpolation, LS-Galerkin, Gappy POD, PINNs) and the integration of FFDeformation and active subspaces for parameter-efficiency. Open challenges include interface treatment, stability across heterogeneous regions, and scalable offline training, with perspectives favouring localized training and archetype-based decomposition for large-scale engineering tasks.

Abstract

Reduced Order Models (ROMs) have been regarded as an efficient alternative to conventional high-fidelity Computational Fluid Dynamics (CFD) for accelerating the design and optimization processes in engineering applications. Many industrial geometries feature repeating subdomains or contain sub-regions governed by distinct physical phenomena, making them well-suited to Domain Decomposition (DD) techniques. The integration of ROM and DD is promising to further reduce computational costs by constructing local ROMs and assembling them into global solutions. Due to the complexity and necessity of coupling ROMs, many approaches have been proposed in recent years. This review provides a concise overview of existing methodologies combining ROM and DD. We categorize existing methods into intrusive (projection-based) and non-intrusive (data-driven) frameworks. Various strategies for generating local reduced bases and coupling them across subdomains are illustrated. Particular emphasis is placed on intrusive techniques, including equations, numerical algorithms, and practical implementations. The non-intrusive framework is also discussed, highlighting its general procedures, basic formulations, and underlying principles. Finally, we summarise the state of the literature, identify open challenges, and present perspectives on future implementation from an engineering viewpoint.

Figures (35)

• Figure 1: The procedures and classification for preliminaries of constructing ROMs. Abbreviation: RBF (Radial basis function) and IDW (Inverse Distance Weighting).
• Figure 2: Schematic of domain decomposition: (a) overlapping; (b) non-overlapping. The global domain $\Omega$ is divided into $\Omega_1$ and $\Omega_2$ parts with boundary $\partial \Omega$. For overlapping, two interfaces exist, $\Gamma_1$ and $\Gamma_2$; without it a common interaction $\Gamma_{[12]}$.
• Figure 3: Domain decomposition into $2 \times 2$ subdomains. Left: overlapping partition, redrawn based on diaz2024fast. Right: non-overlapping partition, redrawn based on discacciati2023localized.
• Figure 4: (a) Conforming and (b) non-conforming meshes for adjacent subdomains.
• Figure 5: Examples of domain divisions for problems without repeating geometric parts. (a) Kármán vortex street for flow around a cylinder, redrawn based on xiao2017domain. (b) Pollutant transport in an urban environment, taken with permission from arcucci2020domain, copyright owned by IOS Press.