A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system

Hintermüller, Michael; Hinze, Michael; Kahle, Christian; Keil, Tobias

doi:10.1007/s11081-018-9393-6

Cited by 24 publications

(16 citation statements)

References 47 publications

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“…Applying the Cauchy-Schwarz inequality to the expressions on the right-hand side, we readily conclude from (4.30) the bound 33) and the claim (4.31) is proved as far as η h N is concerned. But as η h N − η h N L 2 (Q) is by (4.30) bounded, and since A r η 0 N = A r 0 = 0, it also holds true for η h N .…”

Section: Differentiability Of the Control-to-state Mappingmentioning

confidence: 75%

“…The case of Dirichlet and/or Neumann boundary conditions for various types of such systems were the subject of, e.g., the works [13,15,17,23,46,49,50], while the case of dynamic boundary conditions was studied in [10-12, 14, 16, 19, 20, 22, 28]. The optimal control of convective Cahn-Hilliard systems was addressed in [43,47,48], while the papers [26,27,[33][34][35][36][37]40] were concerned with coupled Cahn-Hilliard/Navier-Stokes systems.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

2018

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In the recent paper "Well-posedness and regularity for a generalized fractional Cahn-Hilliard system" by the same authors, general well-posedness results have been established for a a class of evolutionary systems of two equations having the structure of a viscous Cahn-Hilliard system, in which nonlinearities of double-well type occur. The operators appearing in the system equations are fractional versions in the spectral sense of general linear operators A, B having compact resolvents, which are densely defined, unbounded, selfadjoint, and monotone in a Hilbert space of functions defined in a smooth domain. In this work we complement the results given in the quoted paper by studying a distributed control problem for this evolutionary system. The main difficulty in the analysis is to establish a rigorous Fréchet differentiability result for the associated control-to-state mapping. This seems only to be possible if the state stays bounded, which, in turn, makes it necessary to postulate an additional global boundedness assumption. One typical situation, in which this assumption is satisfied, arises when B is the negative Laplacian with zero Dirichlet boundary conditions and the nonlinearity is smooth with polynomial growth of at most order four. Also a case with logarithmic nonlinearity can be handled. Under the global boundedness assumption, we establish existence and first-order necessary optimality conditions for the optimal control problem in terms of a variational inequality and the associated adjoint state system. Colli -Gilardi -Sprekels

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Section: Differentiability Of the Control-to-state Mappingmentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

2018

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“…It is well-known that in phase field applications the variable ϕ ε changes rapidly across the interfacial layers, and an adaptive concept for its spatial resolution is indispensable. Hence, for the mesh generation we use the Dual Weighted Residual (DWR) method [2] where our implementation is guided by [20]. This generates adaptive meshes which well resolve the interfacial regions, and also well reflect the underlying flow physics, compare also [19].…”

Section: Spatial Discretizationmentioning

confidence: 99%

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

et al. 2018

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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 drag 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.

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“…Moreover, we refer to the papers [56,57,49] that address convective Cahn-Hilliard systems, where in the latter contribution (see also its generalization [26] to nonlocal two-dimensional Cahn-Hilliard/Navier-Stokes systems) the fluid velocity was chosen as the control parameter for the first time in such systems. In connection with Cahn-Hilliard/Navier-Stokes systems, we also mention the works [35,36,38,39,42]. In this paper, we study the control by the velocity of convective local Cahn-Hilliard systems with dynamic boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions

Gilardi

Sprekels

2019

Nonlinear Analysis

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In this paper, we study initial-boundary value problems for the Cahn-Hilliard system with convection and nonconvex potential, where dynamic boundary conditions are assumed for both the associated order parameter and the corresponding chemical potential. While recent works addressed the case of viscous Cahn-Hilliard systems, the 'pure' nonviscous case is investigated here. In its first part, the paper deals with the asymptotic behavior of the solutions as time approaches infinity. It is shown that the ω-limit of any trajectory can be characterized in terms of stationary solutions, provided the initial data are sufficiently smooth. The second part of the paper deals with the optimal control of the system by the fluid velocity. Results concerning existence and first-order necessary optimality conditions are proved. Here, we have to restrict ourselves to the case of everywhere defined smooth potentials. In both parts of the paper, we start from corresponding known results for the viscous case, derive sufficiently strong estimates that are uniform with respect to the (positive) Asymptotic limits and control of convective Cahn-Hilliard systems viscosity parameter, and then let the viscosity tend to zero to establish the sought results for the nonviscous case.

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A goal-oriented dual-weighted adaptive finite element approach for the optimal control of a nonsmooth Cahn–Hilliard–Navier–Stokes system

Cited by 24 publications

References 47 publications

Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

Optimal Distributed Control of a Generalized Fractional Cahn–Hilliard System

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

Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions

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