Filtering Strategies – Anna Ivagnes

Motivation

The starting point is the incompressible Navier–Stokes system,

\begin{equation} \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} - \nu\,\Delta\mathbf{u} + \nabla p = \mathbf{f}, \qquad \nabla\cdot\mathbf{u} = 0, \label{eq:ns} \end{equation}

discretized in time and space, e.g. by finite elements with a semi-implicit scheme.

When convection dominates, simulating the incompressible Navier-Stokes equations is characterized by turbulence, a chaotic, three-dimensional, and multi-scale phenomenon. In 3D, it manifests via the energy cascade: the energy is trasferred from the large to the smallest scales, and then dissipated. In 2D, there is a direct enstrophy cascade and an inverse energy cascade.

To capture all the scales of turbulence, we need a very refined grid, which can be very expensive and prohibitive. Let's focus on the 2D flow past a cylinder case at Re=1000. We consider two grids, a more refined grid (where we perform the DNS), and a coarser one.

Performing (more affordable) simulations on coarser grids may produce spurious oscillations, as we cut off the smallest scales. The numerical instabilities pollute the solution and eventually lead to a blow up.

noEF-coarse grid — noEF simulation (coarse grid).

A spatial filter is a smoothing operator that alleviates the oscillations, and reduces the energy at the large wavenumbers (small scales).

The EFR method

The classical evolve–filter–relax (EFR) approach implements the idea of filtering with a fixed, hand-tuned filter. Within EFR, each time step is split into three stages: an evolve step \({\mathbf{w}}^{n+1}\) from the current state (obtained via discretized Navier-Stokes), a filter step

\begin{equation} \bar{\mathbf{w}}^{n+1} = F_\delta\big({\mathbf{w}}^{n+1}\big), \label{eq:filter} \end{equation}

and a relax step

\begin{equation} \mathbf{u}^{n+1} = (1-\chi)\,\bar{\mathbf{w}}^{n+1} + \chi\,{\mathbf{w}}^{n+1}, \label{eq:relax} \end{equation}

where \(\delta\) is the filter width and \(\chi\in[0,1]\) the relaxation parameter. We will focus here on the differential filter.

\begin{equation} \bar{\mathbf{w}}^{n+1} -2\delta^2 \Delta \bar{\mathbf{w}}^{n+1}= {\mathbf{w}}^{n+1}, \label{eq:filter-diff} \end{equation}

There are standard values that are used in literature for \( \delta\) and \(\chi\): the filter radius \(\delta\) is typically related to the Kolmogorov length scale \(\eta=L \,Re^{-3/4}\) or to the mesh size, while the relax parameter \(\chi\) is typically related to the time step size \(\mathcal{O}(\Delta t)\). However, the classical values lead to poor results, either insufficient filtering in EFR or overdiffusive velocity in EF.

Standard EF (\(\delta=\eta\)) simulation (coarse grid).

This motivates this line of research on filtering strategies.

Can we optimize \(\delta\) and \(\chi\) such that our EFR solution is as close as possible to a reference solution? We can perform a data-driven optimization at each time step (or every \(k\) time step), obtaining time-dependent optimal values \(\delta(t)\) and \(\chi(t)\). However, we always need reference data, so we cannot extrapolate in time or to unseen settings.

adaptive

Reinforcement learning can be leveraged to learn the optimal \(\delta(t)\) value as the simulation evolves. We introduce a fully data-free reinforcement-learning based evolve-filter approach (DD-RL-EF), where the reference data are not used at all, not even in training stage.

data-driven filter

We introduced a data-driven and energy-conserving EFR approach. The filter becomes:

\begin{equation} \bar{\mathbf{w}}^{n+1} = \mathbf{F}\mathbf{w}^{n+1}, \label{eq:filter-dd} \end{equation}

where \(\mathbf{F}\) is computed once offline just solving a simple least-squares problem. Such filter is now always dissipative. Hence, to guarantee stability, in the relax step \(\chi\) is tuned online such that the EFR solution dissipates energy and enstrophy.

Related publications

A new data-driven energy-stable evolve-filter-relax model for turbulent flow simulation

doi

arXiv

Data-driven optimization for the evolve-filter-relax regularization of convection-dominated flows

doi

arXiv

Reinforcement learning-based filters for convection-dominated flows: reference-free and reference-guided training

arXiv