Optimal Control of time-discrete two-phase flow driven by a diffuse-interface model

TL;DR

This paper develops a control framework for two-phase flows modeled by diffuse interface equations, deriving optimality conditions and demonstrating numerical solutions with various control strategies.

Contribution

It introduces a novel control framework for diffuse interface two-phase flow models, including derivation of optimality conditions and numerical implementation.

Findings

Successful derivation of necessary optimality conditions for discrete and continuous controls.

Numerical examples demonstrate effectiveness of distributed and boundary controls.

Initial phase field as a control variable offers new control strategies.

Abstract

We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in [Abels, Garcke, Gr\"un, M3AS 22(3):1150013(40), 2012]. On the basis of the stable time discretization proposed in [Garcke, Hinze, Kahle, APPL NUMER MATH, 99:151--171, 2016] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.