Adjoint-based Control of Three Dimensional Stokes Droplets

2025-05-06 0 0 1.49MB 31 页 10玖币

侵权投诉

Alexandru Fikla,∗, Daniel J. Bodonya

aDepartment of Aerospace Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, United States

Abstract

We develop a continuous adjoint formulation and implementation for controlling the deformation of clean, neutrally

buoyant droplets in Stokes ﬂow through farﬁeld velocity boundary conditions. The focus is on dynamics where surface

tension plays an important role through the Young-Laplace law. To perform the optimization, we require access to

ﬁrst-order gradient information, which we obtain from the linearized sensitivity equations and their corresponding

adjoint by applying shape calculus to the space-time tube formed by the interface evolution. We show that the adjoint

evolution equation can be eﬃciently expressed through a scalar adjoint transverse ﬁeld. The optimal control problem

is discretized by high-order boundary integral methods using Quadrature by Expansion coupled with a spherical

harmonic representation of the droplet surface geometry. We show the accuracy and stability of the scheme on several

tracking-type control problems.

Keywords: Stokes ﬂow, Droplets, Optimal control, Shape optimization.

1. Introduction

In many droplet-based microﬂuidic processes and applications, the precise shape and position of the droplets over

time plays a signiﬁcant role in the performance and eﬃciency of the system. A standard simpliﬁed model for such

problems consists of the two-phase Stokes equations coupled with interfacial forces (surface tension) or additional

surfactants and gravity (several concrete examples can be found in [1]). Optimizing aspects of this type of system

is our focus. However, other types of interface evolution share similar features, e.g. mean curvature ﬂows [2, 3],

ﬂuid-structure interactions or free surface ﬂows, and can be analyzed by similar methods.

In general, the application of the continuous or discrete adjoint method to the ﬁeld of optimal control has been

very successful, with applications to ﬂuid mechanics starting from [4]. However, applications to two-phase ﬂows are

rare due, in part, to the fact that the ﬁeld itself remains an active area of research. Phase-ﬁeld-type models present a

compelling starting point, as the ﬁelds representing the ﬂow quantities are smooth and classic approaches to adjoint

methods can be applied. For example, in [5, 6, 7] the Cahn–Hilliard equations are coupled with the incompressible

Navier–Stokes equations and the corresponding optimal control problem is solved. On the other-hand, sharp interface

∗Corresponding Author

Email address: fikl2@illinois.edu (Alexandru Fikl)

Preprint submitted to Journal of Computational Physics October 24, 2022

arXiv:2210.11916v1 [physics.comp-ph] 21 Oct 2022

models (see [8]) are diﬃcult to handle due to the discontinuities of the state variables and material quantities. Further-

more, incorporating interface jump conditions containing higher-order derivatives of the geometry (mainly curvature)

is problematic, see [9]. Work on two-phase problems has also advanced in recent years, with applications to ﬂuid-

structure interactions [10], two-phase Stefan equations [11], free-surface ﬂows [12, 13], geometric ﬂows [14, 15], etc.

Initial extensions to the incompressible two-phase Navier–Stokes with Volume-of-Fluid formulation are presented

in [16, 17].

In this paper, we extend the work presented initially in [18]. The focus of the previous work was on the optimal

control of a single droplet under axisymmetric assumptions. We relax these assumptions and extend the applica-

tions to fully three-dimensional problems with multiple droplets. While applications to scenarios of practical interest,

where hundreds or thousands of droplets are present in the system, are still prohibitively expensive, we present several

improvements. First, we show that the vector adjoint evolution equation from [18] can be transformed into a scalar

evolution equation for the normal component only. This is a signiﬁcant reduction in cost, especially in three dimen-

sions. Furthermore, we make use of state-of-the art methods for solving boundary integral equations to improve the

general performance of the Stokes solver itself. In this work, the Quadrature by Expansion method [19] is used, for

which accurate Fast Multipole Methods have been developed [20]. This allows constructing a fast solver that scales

approximately linearly in the number of degrees of freedom.

The outline of the paper is as follows. We start in Section 2 with a description of the two-phase control problem,

focusing on the constraints and the cost functional. In Section 3, we present the main ideas behind applying shape

calculus to PDEs on moving domains. These concepts are applied to deriving the linearized and adjoint equations

for the control problem. In Section 4, we present a discretization of the state and adjoint systems through a coupled

representation by boundary integral methods (for the Stokes problem) and spherical harmonics (for the geometry).

Finally, Section 5 presents several veriﬁcation tests and applications to three-dimensional multi-droplet systems. We

conclude with some remarks in Section 6.

2. Control of Two-Phase Stokes Flow

In this paper we will consider the evolution of a system of droplets in an otherwise inﬁnite domain. To this end,

we deﬁne an open ﬁnite domain Ω−⊂Rd, where d=3, and its complement Ω+,Rd\¯

Ω−to denote the interior of

the droplets and the surrounding ﬂuid, respectively. The droplet surface Σ,¯

Ω+∩¯

Ω+is assumed to be a ﬁnite set of

disjoint, closed, bounded, and orientable surfaces at each instance of time (see Figure 1).

The interface Σ(t) is the focus in the control of two-phase ﬂows. As such, we deﬁne D ⊂ Rdas a bounded open

set which will contain all admissible interface conﬁgurations Σ(t) for our problem. The general optimization problem

we will be looking at pertains to evolutions equations of the form











X=V(t,X(t),g),t∈[0,T],

X(0) =X0,

(1)

Ω−

Ω+

Figure 1: Example droplet conﬁguration.

where X(t)∈ D is a parametrization of the interface Σ(t) and grepresents a chosen control. For suﬃciently smooth

right-hand sides, we can deﬁne a transformation T(V) : D→Dby X(t)=T(V)(X0) for every ﬂow V. We can

then follow the results from [21] to derive evolution equations for the linearized problem, the adjoint problem and

ultimately an expression for the adjoint-based gradient. This will be discussed in detail in Section 3.

2.1. Quasi-static Two-Phase Stokes Flow

We consider the quasi-static evolution of multiple droplets in a viscous incompressible ﬂow. The droplets experi-

ence no phase change, but are driven by surface tension forces at the ﬂuid-ﬂuid interface. In each phase, the ﬂuid is

described by the velocity ﬁelds u±:Ω±→Rdand the pressures p±:Ω±→R+. Assuming a low-Reynolds number,

the equations governing the ﬂow are the two-phase Stokes equations











∇ · u±=0,x∈Ω±(t),

∇ · σ±[u,p]=0,x∈Ω±(t),

u+→u∞(g),kxk→∞,

(2)

where u∞deﬁnes an intended decay at inﬁnity of the velocity ﬁeld u+. The full velocity ﬁeld uis to be understood as

a superposition of u∞and a ﬂow that decays to zero in the farﬁeld. The Cauchy stress tensor σ±and the rate of strain

tensor ε±are written as

σ±[u,p],−p±I+2µ±ε±[u],

ε±[u],1

2(∇u±+∇uT

±),

where µ±∈R+are the dynamic viscosities of each ﬂuid. We will assume that the surface forces are entirely due

to a constant surface tension. The boundary conditions at the drop surface are the continuity of velocity and the

Young-Laplace law, i.e.











JuK=0,

Jn·σ[u,p]K=γκn,

(3)

where γis a constant surface tension coeﬃcient, κis the total curvature (sum of the principal curvatures) and nis the

exterior normal to Ω−. The jump at the interface is deﬁned as

JfK,f(x+)−f(x−),where f(x±),lim

→0+f(x±n).

The equations are non-dimensionalized by making use of a characteristic velocity magnitude Uand a characteristic

droplet radius R. The resulting non-dimensional parameters are the viscosity ratio λand the Capillary number Ca,

λ,µ−

µ+

and Ca ,µ+U

γ,

The non-dimensional equations will be used going forward. In this form, non-dimensional weighted jump and

average operators are deﬁned as

JfKλ,f(x+)−λf(x−) and hfiλ,1

2(f(x+)+λf(x−)).

In order to evolve the interface in time, we use a quasi-static approach in which we ﬁrst compute the velocity

from (2) and then displace the interface using the ODE from (1). For Stokes ﬂow, the motion law is given by

V(u,X),(u·n)n+(I−n⊗n)w,(4)

which is well-deﬁned since JuK=0 allows for a unique interface velocity ﬁeld. At a continuous level, only the

normal component of the velocity is necessary to deﬁne the deformation of the interface. Tangential components from

w: [0,T]×R3→R3can be seen as reparametrizations and represent the same abstract shape. They become important

in the discretization stage, as discussed in Section 4, and should be included in the adjoint to obtain accurate gradients.

2.2. Cost Functionals

Our goal is to ﬁnd controls g(from (1)) that minimize a class of tracking-type cost functionals. The general form

of the cost functionals we will be looking at is

J(g),α1

2ZT

0ZΣ(t)

(u·n−ud)2dSdt+α2

2ZT

0kxc(t)−xd(t)k2dt+α3

2kxc(T)−xd,Tk2,(5)

where udis a target surface normal velocity ﬁeld, while xd(t) and xd,Tare target droplets centroids. We mainly consider

the droplet centroids as a way to describe the position, but any quantity that is invariant to reparametrizations can be

used. This assumption can be relaxed by considering reparametrizations as described in [22]. Then, we consider the

optimization problem











min J(g),

subject to (1) (with (4)), (2) and (3).

(6)

The main control variable we will be focusing on are the farﬁeld boundary conditions u∞(g). For ﬁnite domains,

this is a straightforward choice. However, as the farﬁeld boundary conditions are used to impose a decay in the

velocity ﬁeld, further discussion is required (see Section 3).

In the absence of an evolution equation, such as (1), we can consider a static system. In this case, the interface X

becomes the control and we are left with a standard shape optimization problem. As shown in [18], the static problem

is an important stepping stone in deriving and verifying the required optimality conditions. We will be making use of

the static problem to verify the adjoint equations for the three-dimensional problem presented here as well.

3. Cost Gradients

To understand the optimization problem proposed by (6), we must introduce several important notions from shape

calculus and more general perturbations of moving domains. These ideas are detailed in [21]. First, for an initial

conﬁguration Ω0and interface Σ0, we deﬁne the space-time tubes on R+×Rd, corresponding to a velocity ﬁeld V, by

(see also Figure 2)

Ω(V),[

t∈[0,T]{t} × Ω(t)=[

t∈[0,T]{t} × T(V)(Ω0),

Σ(V),[

t∈[0,T]{t} × Σ(t)=[

t∈[0,T]{t} × T(V)(Σ0).

yΩ−(0)

Ω−(t)

Ω−(T)

Σ(0)

Σ(t)

Σ(T)

Ω−(V)Σ(V)

Figure 2: Schematic of the (two-dimensional) Ω(V) and Σ(V) space-times tubes and transverse slices Ω(t) (blue) and Σ(t) (red) at ﬁxed times t.

The perturbations ˜

Vonly act in the transverse direction (gray slices).

A perturbation in the control gcan then be seen as giving rise to a perturbed space-time tube Ω(V). Perturbations

obtained in this way are called transverse because they occur only in “horizontal” slices of each Ω(t) domain for ﬁxed

t. This allows recovering most of the classic formulae from shape calculus in the context of moving domains. To

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Adjoint-basedControlofThreeDimensionalStokesDropletsAlexandruFikla,,DanielJ.BodonyaaDepartmentofAerospaceEngineering,UniversityofIllinoisUrbana-Champaign,Urbana,IL61801,UnitedStatesAbstractWedevelopacontinuousadjointformulationandimplementationforcontrollingthedeformationofclean,neutrallybuoyantdro...

展开>> 收起<<

Adjoint-based Control of Three Dimensional Stokes Droplets.pdf

共31页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Adjoint-based Control of Three Dimensional Stokes Droplets

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: