Analysis of shape optimization problems for unsteady fluid-structure interaction

Haubner, Johannes; Ulbrich, Michael; Ulbrich, Stefan

doi:10.1088/1361-6420/ab5a11

Cited by 17 publications

(14 citation statements)

References 86 publications

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“…is real analytic with respect to (û0, h0) ∈ U û × U h . Hence, also (16) holds for f * d and we conclude that R * := (0, f * d , g * , g * h ) ∈ F(t0) satisfies the compatibility conditions ( 16), (17) and by construction (û0, h0) ∈ U û × U h → R * ∈ F(t0) is real analytic. Hence, by Theorem 3 the linear problem (29) has a unique solution z * = z * (û0, h0) that is real analytic and by Lemma 6 the first derivative vanishes in 0, i.e., Dz * (0, 0) = 0.…”

Section: Results For the Transformed Problemsupporting

confidence: 51%

“…By using Lp-maximal regularity of a linear system and applying a refined version of a fixed point theorem, we show differentiability of the transformed state with respect to controls in the maximum regularity spaces. A similar technique was recently used in [16] to show differentiability properties for shape optimization of fluid-structure interaction, but the properties and analysis of the fixed point iteration is very different from two-phase flows considered here. In a second step we deduce differentiability results for the control-to-state map in the physcial coordinates.…”

Section: Introductionmentioning

confidence: 99%

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Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

Diehl¹,

Haubner²,

Ulbrich³

et al. 2020

Preprint

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We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on Lp-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting form the Lp-maximal regularity setting. We consider first a formulation, where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state an equivalent Volume-of-Fluid type formulation and use the obtained differentiability results to derive rigorosly the corresponding sensitivity equations of the Volume-of-Fluid type formulation. The results of the paper form an analytical foundation for stating optimality conditions, justifying the application of derivative based optimization methods and for studying the convergence of discrete sensitivity schemes based on Volume-of-Fluid discretizations for optimal control of two-phase Navier-Stokes equations.

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Section: Results For the Transformed Problemsupporting

confidence: 51%

Section: Introductionmentioning

confidence: 99%

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

Diehl¹,

Haubner²,

Ulbrich³

et al. 2020

Preprint

Self Cite

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show abstract

“…These results have been recently extended to more complex systems, see [7]. We also mention [9,10], where the shape of an elastic structure is used to reduce the drag of a fluid flow surrounding this an elastic structure.…”

Section: Setting Of the Problemmentioning

confidence: 88%

Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle

Raymond

Vanninathan

2021

ESAIM: COCV

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We consider a fluid flow in a time dependent domain $\Omega_f(t)=\Omega \setminus \Omega_s(t)\subset {\mathbb R}^3$, surrounding a deformable obstacle $\Omega_s(t)$. We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in $\Omega_f(t)$, $t>0$. We prove that, for any arbitrary exponential decay rate $\omega>0$, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary $\partial \Omega_s(t)$ can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate $\omega>0$.

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“…For a rigorous proof of the required differentiability properties, some progress has been made for stationary FSI-problems in [60]. A rigorous derivation of the corresponding adjoints in the context of shape optimization can be found in [34].…”

Section: Gradient Computationmentioning

confidence: 99%

Optimization with nonstationary, nonlinear monolithic fluid‐structure interaction

Wick

Wollner

2020

Numerical Meth Engineering

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Within this work, we consider optimization settings for nonlinear, nonstationary fluid-structure interaction. The problem is formulated in a monolithic fashion using the arbitrary Lagrangian-Eulerian framework to set-up the fluid-structure forward problem. In the optimization approach, either optimal control or parameter estimation problems are treated. In the latter, the stiffness of the solid is estimated from given reference values. In the numerical solution, the optimization problem is solved with a gradient-based solution algorithm. The nonlinear subproblems of the FSI forward problem are solved with a Newton method including line search. Specifically, we will formally provide the backward-in-time running adjoint state used for gradient computations.Our algorithmic developments are demonstrated with some numerical examples as for instance extensions of the well-known fluid-structure benchmark settings and a flapping membrane test in a channel flow with elastic walls. arXiv:1910.03424v1 [math.NA] 8 Oct 2019 [53,57,60,12]. Here, we notice that the required adjoints are the same as used for adjoint-based error estimation; see for instance [62,51]. Nonlinear (stationary) FSI investigating various partitioned coupling techniques was recently subject in [54]. The by far more challenging situation of nonstationary settings is listed in the following. A nonstationary situation assuming a rigid solid was theoretically studied in [48]. Further theoretical results for a boundary control FSI problem were established in [8].Parameter estimation to detect the stiffness of an arterial wall with a well-posedness analysis and numerical simulations was addressed in [49]. Again in blood flow simulations, a data assimilation problem was formulated in [33], in which however, the arterial walls were not considered. A full FSI problem for data assimilation using a Kalman filter was subject in [6]. In [44], the authors used optimization techniques to formulate the FSI coupling conditions. Adjoints for 1D FSI were derived in [16,47]. Reduced basis methods for FSI-based optimization were developed in [45]. Optimal control of nonstationary FSI applied to benchmark settings was investigated in [3]. A linearized FSI optimization problem was addressed in [23] and detailed results for full-time-dependent FSI optimal control were summarized in [22]. In this respect, we also mention [24] in which the adjoints required for optimization were employed for dual-weighted residual error estimation for time adaptivity. Most recently, a uncertainty quantification framework for fluid-structure interaction with applications in aortic biomechanics was developed in [43].The significance of the current work is on the development of a robust fully monolithic formulation for gradient-based optimization for nonstationary, nonlinear FSI problems. Here, the coupled problem is prescribed in the reference configuration with the help of the ALE approach in a variationalmonolithic way. As previously summarized in our literature review, only very few results exis...

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Analysis of shape optimization problems for unsteady fluid-structure interaction

Cited by 17 publications

References 86 publications

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle

Optimization with nonstationary, nonlinear monolithic fluid‐structure interaction

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