Christian Kahle

An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system

An adaptive a posteriori error estimator based finite element method for the numerical solution of a coupled Cahn-Hilliard/Navier-Stokes system with a double-obstacle homogenous free (interfacial) energy density is proposed. A semi-implicit Euler scheme for the time-integration is applied which results in a system coupling a quasi-Stokes or Oseen-type problem for the fluid flow to a variational inequality for the concentration and the chemical potential according to the Cahn-Hilliard model [16]. A Moreau-Yosida regularization is employed which relaxes the constraints contained in the variational inequality and, thus, enables semi-smooth Newton solvers with locally superlinear convergence in function space. Moreover, upon discretization this yields a mesh independent method for a fixed relaxation parameter. For the finite dimensional approximation of the concentration and the chemical potential piecewise linear and globally continuous finite elements are used, and for the numerical approximation of the fluid velocity Taylor-Hood finite elements are employed. The paper ends by a report on numerical examples showing the efficiency of the new method.

Numerical Approximation of Phase Field Based Shape and Topology Optimization for Fluids

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided.

A Nonlinear Model Predictive Concept for Control of Two-Phase Flows Governed by the Cahn-Hilliard Navier-Stokes System

We present a nonlinear model predictive framework for closed-loop control of two-phase flows governed by the Cahn-Hilliard Navier-Stokes system. We adapt the concept for instantaneous control from [6,12,16] to construct distributed closed-loop control strategies for two-phase flows. It is well known that distributed instantaneous control is able to stabilize the Burger’s equation [16] and also the Navier-Stokes system [6,12]. In the present work we provide numerical investigations which indicate that distributed instantaneous control also is well suited to stabilize the Cahn-Hilliard Navier-Stokes system.

Model Predictive Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models

Abstract We present a nonlinear model predictive framework for closed-loop control of two-phase flows governed by Cahn-Hilliard Navier-Stokes system with variable density. The control goal consists in achieving a prescribed concentration distribution in the Cahn-Hilliard part through distributed and/or boundary control of the flow part. Special emphasis is taken on quick control responses which are achieved through the inexact solution of the optimal control problems appearing in the model predictive control strategy. The resulting control concept is known as instantaneous control and is applied to feedback control of the Navier-Stokes system in e.g. Choi et al. (1999); Hinze (2005a); Hinze and Volkwein (2002). We provide numerical investigations which indicate that instantaneous wall parallel boundary control of the flow part is well suited to achieve a prescribed concentration distribution in the variable density Cahn-Hilliard Navier-Stokes system.

Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach

We consider shape and topology optimization for uids which are governed by the Navier{Stokes equations. Shapes are modelled with the help of a phase eld approach and the solid body is relaxed to be a porous medium. The phase eld method uses a Ginzburg{Landau functional in order to approximate a perimeter penalization. We focus on surface functionals and carefully introduce a new modelling variant, show existence of minimizers and derive rst order necessary conditions. These conditions are related to classical shape derivatives by identifying the sharp interface limit with the help of formally matched asymptotic expansions. Finally, we present numerical computations based on a Cahn{Hilliard type gradient descent which demonstrate that the method can be used to solve shape optimization problems for uids with the help of the new approach.

Parameter Identification via Optimal Control for a Cahn–Hilliard-Chemotaxis System with a Variable Mobility

We consider the inverse problem of identifying parameters in a variant of the diffuse interface model for tumour growth proposed by Garcke et al. (Math Models Methods Appl Sci 26(6):1095–1148, 2016). The model contains three constant parameters; namely the tumour growth rate, the chemotaxis parameter and the nutrient consumption rate. We study the inverse problem from the viewpoint of PDE-constrained optimal control theory and establish first order optimality conditions. A chief difficulty in the theoretical analysis lies in proving high order continuous dependence of the strong solutions on the parameters, in order to show the solution map is continuously Fréchet differentiable when the model has a variable mobility. Due to technical restrictions, our results hold only in two dimensions for sufficiently smooth domains. Analogous results for polygonal domains are also shown for the case of constant mobilities. Finally, we propose a discrete scheme for the numerical simulation of the tumour model and solve the inverse problem using a trust-region Gauss–Newton approach.

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier–Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the total potential power of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.

Comparative Simulations of Taylor Flow with Surfactants Based on Sharp- and Diffuse-Interface Methods

We present a quantitative comparison of simulations based on diffuse- and sharp-interface models for two-phase flows with soluble surfactants. The test scenario involves a single Taylor bubble in a counter-current flow. The bubble assumes a stationary position as liquid inflow and gravity effects cancel each other out, which makes the scenario amenable to high resolution experimental imaging. We compare the accuracy and efficiency of the different numerical models and four different implementations in total.

Fully Adaptive and Integrated Numerical Methods for the Simulation and Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models

The present work is concerned with the simulation and optimal control of two-phase flows. We provide stable time discretization schemes for the simulation based on both, smooth and non-smooth free energy densities, which we combine with a practical, reliable and efficient adaptive mesh refinement concept for the spatial variables. Furthermore, we consider optimal control problems for two-phase flows and, among other things, derive first order optimality conditions. In the presence of smooth free energies we encounter classical Karush-Kuhn-Tucker (KKT) conditions, while in the case of non-smooth free energies we can prove C(larke)-stationarity. Moreover, we propose a dual weighted residual concept for spatial mesh adaptivity which is based on the newly derived stationarity conditions. We also address future research directions, including closed-loop control concepts and model order reduction techniques for simulation and control of variable density multiphase flows.