An optimisation-based domain-decomposition reduced order model for parameter-dependent non-stationary fluid dynamics problems

TL;DR

This work develops an optimisation-based domain-decomposition approach for parametric, non-stationary Navier–Stokes problems and analyzes its mathematical well-posedness. It compares intrusive POD-Galerkin ROMs with non-intrusive POD-NN ROMs within the DD framework, and provides a gradient-based optimisation strategy to couple subdomains via an interface control. The authors prove a priori estimates, existence of optimal solutions, and convergence to the monolithic solution as the regularisation vanishes, and implement Finite Element discretisation alongside two ROM strategies. Numerical experiments on a backward-facing step and a lid-driven cavity demonstrate substantial computational savings with POD-Galerkin while highlighting POD-NN’s speed advantages and its sensitivity to time discontinuities, offering guidance for hybrid online strategies and future multi-domain extensions.

Abstract

In this work, we address parametric non-stationary fluid dynamics problems within a model order reduction setting based on domain decomposition. Starting from the optimisation-based domain decomposition approach, we derive an optimal control problem, for which we present a convergence analysis in the case of non-stationary incompressible Navier-Stokes equations. We discretize the problem with the finite element method and we compare different model order reduction techniques: POD-Galerkin and a non-intrusive neural network procedure. We show that the classical POD-Galerkin is more robust and accurate also in transient areas, while the neural network can obtain simulations very quickly though being less precise in the presence of discontinuities in time or parameter domain. We test the proposed methodologies on two fluid dynamics benchmarks with physical parameters and time dependency: the non-stationary backward-facing step and lid-driven cavity flow.

Figures (16)

• Figure 1: Domain and boundaries
• Figure 2: Physical domain and domain decomposition for the backward--facing step problem
• Figure 3: Backward--facing step: POD singular eigenvalue decay of the first 50 POD modes (a) and the monolithic solution for a parameter $(\bar{U}, \nu) = (4.5, 0.4)$ at the final time step (b)
• Figure 4: Backward--facing step: high--fidelity solution for the velocities $u_1$ and $u_2$ at 4 different time instances
• Figure 5: Backward--facing step: high--fidelity solution for the pressures $p_1$ and $p_2$ at 4 different time instances

Theorems & Definitions (4)

• Remark : Uniqueness of optimal solutions
• Remark : Weak formulation with "non--symmetric" trilinear form
• Remark : The choice of discrete space $X_h$
• Remark : Sensitivity to the domain splitting