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Shape optimization for Stokes problem with threshold slip boundary conditions

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    * Corresponding author 

The first author acknowledges the support of the project 17-01747S of the Czech Science Foundation. The second author was suppported by the Academy of Finland, grant #260076. The third author was supported by the Ministry of Education, Youth and Sports under the projects LM2015084 and LO1201 in the framework of the targeted support of the Large Infrastructures and of National Programme for Sustainability Ⅰ

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  • This paper deals with shape optimization of systems governed by the Stokes flow with threshold slip boundary conditions. The stability of solutions to the state problem with respect to a class of domains is studied. For computational purposes the slip term and impermeability condition are handled by a regularization. To get a finite dimensional optimization problem, the optimized part of the boundary is described by Bézier polynomials. Numerical examples illustrate the computational efficiency.

    Mathematics Subject Classification: Primary: 49J20, 35J86, 65K15; Secondary: 65N30, 90C30.


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  • Figure 1.  Shape of admissible domains

    Figure 2.  Left: reference triangulation $\widehat{\cal T_h}$. Right: Mapped triangulation $\cal T_h$.

    Figure 3.  Optimized shapes (left) and convergence histories (right) for different values of the penalty/smoothing parameter $\varepsilon$.

    Figure 4.  Streamlines (left) and pressure contours (right) for $\varepsilon=10^{-5}$.

    Figure 5.  Tangential velocity and shear stress for $\varepsilon=10^{-5}$

    Figure 6.  Streamlines (left) and pressure contours (right)

    Figure 7.  Tangential velocity and shear stress

    Figure 8.  Optimized Bézier functions $\alpha_m$ for two different values of $\sigma_1$

    Figure 9.  Contours of the target pressure $p_0$ (left) and computed pressure (right)

    Figure 10.  Tangential velocity and shear stress on $S(\alpha_{opt})$

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