Optimisation-Based Coupling of Finite Element Model and Reduced Order Model for Computational Fluid Dynamics

TL;DR

This work develops an optimisation-based domain-decomposition framework to couple Finite Element and Reduced Order Models for time-dependent Navier–Stokes flow on non-overlapping subdomains, using an interface normal-flux control $g_h$ and gradient-based optimization to decouple subdomain states and locally build ROMs. It formulates both monolithic and discrete DD problems, derives the optimality system with adjoints, and provides a gradient expression $\frac{d\mathcal{J}}{dg} = \left.\xi_{1,h}\right|_{\Gamma_0} - \left.\xi_{2,h}\right|_{\Gamma_0}$, enabling efficient iterative coupling. ROMs are constructed via POD with lifting and supremizer stabilization, and multiple FEM/ROM coupling options are analyzed (FFF, FRF, FRR, RRR) across four hybrid configurations. Numerical tests on a BFS Navier–Stokes problem highlight trade-offs: ROM-enabled couplings reduce interface optimization cost but may require more iterations for accuracy, with RRR showing favorable convergence while FRR can be unstable; overall, monolithic solves remain fastest for this simple test, yet the framework offers a scalable path for large-scale, multi-physics problems where different subdomains demand different discretization fidelities.

Abstract

Using Domain Decomposition (DD) algorithm on non--overlapping domains, we compare couplings of different discretisation models, such as Finite Element (FEM) and Reduced Order (ROM) models for separate subcomponents. In particular, we consider an optimisation-based DD model where the coupling on the interface is performed using a control variable representing the normal flux. We use iterative gradient-based optimisation algorithms to decouple the subdomain state solutions as well as to locally generate ROMs on each subdomain. Then, we consider FEM or ROM discretisation models for each of the DD problem components, namely, the triplet state1-state2-control. On the backward-facing step Navier-Stokes (NS) problem, we investigate the efficacy of the presented couplings in terms of optimisation iterations, optimal functional values and relative errors.

Figures (4)

• Figure 1: Domains and boundaries
• Figure 2: Physical domain and domain decomposition for the backward--facing step problem
• Figure 3: Number of optimisation iterations (a) and the optimal functional values (b) of FFF, FRF, FRR and RRR solutions
• Figure 4: Relative errors of FFF, FRF, FRR, and RRR solutions w.r.t. the monolithic solution