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A shape optimization problem constrained with the Stokes equations to address maximization of vortices

  • *Corresponding author: John Sebastian Simon

    *Corresponding author: John Sebastian Simon 
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  • We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the $ L^2 $-norm of the curl and the det-grad measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.

    Mathematics Subject Classification: Primary: 49Q10, 49J20, 49K20; Secondary: 35Q93.

    Citation:

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  • Figure 1.  Set up of the domain

    Figure 2.  Initial geometry of the domain with the refined mesh

    Figure 3.  From curlaL-problem, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 4.  From detgradaL-problem, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 5.  From curldF-problem, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 6.  (left) Comparison of final shapes between aL-algorithm and dF-algorithm for the problem with the parameters $ \gamma_1 = 1 $, $ \gamma_2 = 0 $ and $ \alpha = 5 $, (upper right) Comparison of objective value trends between aL-algorithm and dF-algorithm, (lower right) Comparison of volume trends between aL-algorithm and dF-algorithm

    Figure 7.  From detgraddF-problem, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 8.  (left) Comparison of final shapes between aL-algorithm and dF-algorithm with the parameters $ \gamma_1 = 0 $, $ \gamma_2 = 1 $, and $ \alpha = 1 $, (upper right) Comparison of objective value trends between aL-algorithm and dF-algorithm, (lower right) Comparison of volume trends between aL-algorithm and dF-algorithm

    Figure 9.  (left) Comparison of final shapes between aL-algorithm and dF-algorithm with the parameters $ \gamma_1 = 0 $, $ \gamma_2 = 1 $, and $ \alpha = 1.2 $, (upper right) Comparison of objective value trends between aL-algorithm and dF-algorithm, (lower right) Comparison of volume trends between aL-algorithm and dF-algorithm

    Figure 10.  From configuration 1 of mixeddF-problem, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 11.  (left) Plots of the solutions of mixeddF-problem (using configuration 1), of curldF-problem and of detgraddF-problem generated boundaries, (upper right) Comparison of the objective function value of the curldf-problem and the curl part of the mixeddF-problem, (lower right) Comparison of the objective function value of the curldF-problem and the detgrad part of the mixeddf-problem

    Figure 12.  (left) Plots of the solutions of mixeddF-problem (using configuration 2), of curldF-problem and of detgraddF-problem generated boundaries, (upper right) Comparison of the objective function value of the curldF-problem and the detgrad part of the mixeddF-problem, (lower right) Comparison of the objective function value of the curldF-problem and the detgrad part of the mixeddF-problem

    Figure 13.  (left) Plots of the solutions of mixeddF-problem (using configuration 4), of curldF-problem and of detgraddF-problem generated boundaries, (upper right) Comparison of the objective function value of the curldF-problem and the detgrad part of the mixeddF-problem, (lower right) Comparison of the objective function value of the curldF-problem and the detgrad part of the mixeddF-problem

    Figure 14.  (left) Plots of the solutions of mixeddF-problem (configurations 1 and 10), of curldF-problem, and of detgraddF-problem, (upper right) Hausdorff distance between mixeddF-solution and curldF-solution on each configuration, (lower right) Hausdorff distance between mixeddF-solution and detgraddF-solution on each configuration

    Figure 15.  The figure shows the configurations for the two-obstacle experiment: (top) Set-up of the domain where the obstacles are placed parallel to the flow; (bottom) Set-up of the domain where the obstacles are placed perpendicular to the flow

    Figure 16.  From the first configuration of the two-obstacle experiment, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 17.  From the second configuration of the two-obstacle experiment, the figure features the following: (left) Evolution of the free-boundary $ \Gamma_{\rm f} $, (upper right) Normalized trend of the objective functional, (lower right) Normalized trend of the volume

    Figure 18.  The figure shows the comparison of the flows using the initial domain (lower part), and the final shape from curldf-problem (upper part)

    Figure 19.  The figure shows the comparison of the flows using the initial domain (lower part), and the final shape from detgraddf-problem (upper part)

    Figure 20.  The figure shows the comparison of the flows using the the final shape from curldf-problem (lower part), and the final shape from detgraddf-problem (upper part)

    Table 1.  Parameter values for curlaL-problem

    Parameter Value Parameter Value
    $ \alpha $ 6.0 $ \ell_0 $ 20
    $ b_0 $ $ 1\times10^{-4} $ $ \tau $ 1.05
    $ \overline{b} $ 10
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    Table 2.  Parameter values for detgradaL-problem

    Parameter Value Parameter Value
    $ \alpha $ 1.3 $ \ell_0 $ .5
    $ b_0 $ $ 1\times10^{-2} $ $ \tau $ 1.05
    $ \overline{b} $ 10
     | Show Table
    DownLoad: CSV

    Table 3.  Parameter Values for mixeddF-problem

    Configuration $ \alpha $ $ \gamma_1 $ $ \gamma_2 $
    1 6.0 1.0 1.0
    2 7.0 1.0 2.0
    3 8.0 1.0 3.0
    4 9.0 1.0 4.0
    5 10.0 1.0 5.0
    6 11.0 1.0 6.0
    7 12.0 1.0 7.0
    8 13.0 1.0 8.0
    9 14.0 1.0 9.0
    10 15.0 1.0 10.0
     | Show Table
    DownLoad: CSV
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