Approximate controllability of linearized shape-dependent operators for flow problems

Abstract

We study the controllability of linearized shape-dependent operators for flow problems. The first operator is a mapping from the shape of the computational domain to the tangential wall velocity of the potential flow problem and the second operator maps to the wall shear stress of the Stokes problem. We derive linearizations of these operators, provide their well-posedness and finally show approximate controllability. The controllability of the linearization shows in what directions the observable can be changed by applying infinitesimal shape deformations.

Residence time optimization of spin pack polymer distributors

“…Figure 3 shows the computed residence times for the initial and the final geometry. As in the wall shear stress based optimization problem considered in [3] the optimized geometry tends to flatten down away from the inflow boundary. Fig.…”

“…Since long residence times and stagnation zones of the fluid have a negative effect on the fiber quality, finding an optimal distributor design plays an important role here. Previous works [2,3] considered cost functionals based on the wall shear stress, which allow for an indirect control of the residence time. By solving an additional advection-diffusion-reaction equation however, the wall shear stress as an objective can be substituted with the residence time directly.…”

Shape design for polymer spin packs: modeling, optimization and validation

“…Remark 1 For θ ∈ Θ 1 it is implied by [11] that Id + θ : R 3 → R 3 is an invertible (1, 1)diffeomorphism and thus Ω θ = (Id + θ )(Ω 0 ) is also of class C 1,1 . Then, a regularity argument similar to [12] would yield u(θ ) ∈ [H 2 (Ω θ )] 3 , thus σ (θ ) ∈ L 2 (Γ w θ ) by the Trace Theorem [13,Thm. 8.7] and the objective (8) is well-defined.…”

“…This general approach to proof the existence of shape derivatives for partial differential equations is shown in [11]. In [3] we have applied this approach to a problem similar to the Surrogate Model considered here.…”

“…Typically, commercial simulation software is used for the simulation, which makes derivative based shape optimization difficult to handle. Therefore, a surrogate model approach Further numerical and theoretical results on the shape optimization of polymer spin packs can be found in [1][2][3][4]. Another interesting application of a similar problem is studied in [5,6].…”