Data-driven Optimization for the Evolve-Filter-Relax regularization of convection-dominated flows

TL;DR

This work tackles stabilization of convection-dominated, under-resolved flows by making the Evolve-Filter-Relax (EFR) parameters time-dependent and data-driven. It introduces three Opt-EFR strategies to optimize the filter radius $\delta$, the relaxation parameter $\chi$, or both, using a DNS reference to define objective functions that are either local or global (including velocity, gradient, and pressure terms). The results on a cylinder wake at $Re=1000$ show that Opt-EFR methods, especially those with global objectives that include $\nabla\mathbf{u}$ (and sometimes $p$), deliver substantial accuracy improvements (up to ~99% over standard EFR) with comparable computational cost, while highlighting the crucial role of gradient information in the loss. The findings establish data-driven adaptive regularization as a robust approach for under-resolved turbulent flows and point to future directions in ML-based training and reduced-order models for real-time parameter adaptation across a broader range of flows.

Abstract

Numerical stabilization techniques are often employed in under-resolved simulations of convection-dominated flows to improve accuracy and mitigate spurious oscillations. Specifically, the evolve--filter--relax (EFR) algorithm is a framework which consists in evolving the solution, applying a filtering step to remove high-frequency noise, and relaxing through a convex combination of filtered and original solutions. The stability and accuracy of the EFR solution strongly depend on two parameters, the filter radius $δ$ and the relaxation parameter $χ$. Standard choices for these parameters are usually fixed in time, and related to the full order model setting, i.e., the grid size for $δ$ and the time step for $χ$. The key novelties with respect to the standard EFR approach are: (i) time-dependent parameters $δ(t)$ and $χ(t)$, and (ii) data-driven adaptive optimization of the parameters in time, considering a fully-resolved simulation as reference. In particular, we propose three different classes of optimized-EFR (Opt-EFR) strategies, aiming to optimize one or both parameters. The new Opt-EFR strategies are tested in the under-resolved simulation of a turbulent flow past a cylinder at $Re=1000$. The Opt-EFR proved to be more accurate than standard approaches by up to 99$\%$, while maintaining a similar computational time. In particular, the key new finding of our analysis is that such accuracy can be obtained only if the optimized objective function includes: (i) a global metric (as the kinetic energy), and (ii) spatial gradients' information.

Figures (35)

• Figure 1: Flowchart of the Opt-EFR and standard EFR approaches.
• Figure 2: Meshes taken into account to run the simulations.
• Figure 3: The domain considered in the test cases.
• Figure 4: From top to bottom: velocity (left plots) and pressure (right plots) fields at final $t=4$ for DNS on the fine mesh (the reference solution), and noEFR method on the coarse mesh (i.e., the solution to improve with regularization). The results of standard EFR and EF approaches on the coarse mesh are also displayed.
• Figure 5: Optimal value of $\chi(t)$ in the $\chi$-Opt-EFR algorithms (A), and count of observations of the parameter values (B).