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

With the increased interest in complex problems, such as multiphysics and multiscale models, as well as real-time computations, there is a strong need for domain-decomposition (DD) segregated solvers and reduced-order models (ROMs). Segregated models decouple the subcomponents of the problems at hand and use already existing state-of-the-art numerical codes in each component. In this manuscript, starting with a DD algorithm on non-overlapping domains, we aim at the comparison of couplings of different discretisation models, such as Finite Element (FEM) and ROM for separate subcomponents. In particular, we consider an optimisation-based DD model on two non-overlapping subdomains where the coupling on the common interface is performed by introducing a control variable representing a normal flux. Gradient-based optimisation algorithms are used to construct an iterative procedure to fully 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. We perform numerical tests on the backward-facing step Navier-Stokes problem to investigate the efficacy of the presented couplings in terms of optimisation iterations and relative errors.