Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System

TL;DR

This paper develops methods for simulating and controlling a complex two-phase flow model governed by a nonsmooth energy potential, introducing new optimality conditions, adaptive algorithms, and model reduction techniques.

Contribution

It introduces two approaches for deriving stationarity conditions in a bilevel control problem and combines POD-based model reduction with adaptive spatial resolution handling.

Findings

Established existence of optimal solutions.

Developed two adaptive solution algorithms with error estimation.

Implemented a POD-MOR approach for efficient low-order modeling.

Abstract

We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn-Hilliard Navier-Stokes system involving a nonsmooth energy potential. We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator. In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.