Adjoint-based optimization of multiphase flows with sharp interfaces

Multiphase phenomena are ubiquitous in any engineering application and significant effort has been put forth into advancing our understanding them. While modeling and numerical simulation of multiphase flows have made significant advances in the last two decades, much less has shown towards their optimal control. Many practical control applications are based on experience and trial-and-error methods due to the large dimensionality of the problem, complexity of the couple systems, and intractable computational effort required. In this work, we develop and apply optimal control methods to standard models for incompressible, immiscible two-phase viscous flows with surface forces. The main focus is on the low-Reynolds number Stokes flow, but extensions to the Navier–Stokes equations are also discussed in detail. We make use of the continuous adjoint method to obtain first-order sensitivity information that can then be used to control the system. At first sight, the two-phase Stokes flow with surface tension is a simple system that has been heavily studied in the literature. However, we will show that linearizing and formulating the adjoint equations involves many subtleties that are not present in more commonly studied single-phased systems. First, the discontinuities of the state variables across the interface make a direct linearization difficult. Then, surface forces introduce high-order derivatives of the geometry, such as the curvature in the Young-Laplace law. This takes a heavy toll on the regularity requirements of both the choice of numerical methods and the theory necessary to show well-posedness of the resulting equations. To deal with the increased regularity requirements, high-order and stable numerical methods are needed. We present a Boundary Integral formulation for Stokes flow and the adjoint equations based on recent state-of-the-art numerical results. To handle the moving geometry, the Boundary Integral formulation is coupled with a spectral representation of the drops based on Fourier modes (in 2D) and Spherical Harmonics (in 3D). These choices form a solid basis for a robust solver that can be used to validate the adjoint-based gradient. They are used to perform surface and boundary control of several static and quasi-static problems. We investigate issues related to shape (interface) optimization in the two-phase Stokes flow with multiple disjoint interfaces (i.e. droplets or bubbles) and show that the control of such systems is feasible. However, there are many remaining open problems before practical applications can become commonplace.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Aerospace Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Fikl, Alexandru

Contributors dc:contributor

• Bodony, Daniel J
• Desjardins, Olivier
• Goza, Andres
• Klöckner, Andreas

• two-phase flows
• Stokes: adjoint
• optimization
• shape calculus
• boundary integral methods
• QBX

Statement dc:rights

• Copyright 2022 Alexandru Fikl

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/115477

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F115477