# American Institute of Mathematical Sciences

December  2017, 10(6): 1281-1301. doi: 10.3934/dcdss.2017069

## Shape optimization for Stokes problem with threshold slip boundary conditions

 1 Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,186 75 Prague 8, Czech Republic 2 Faculty of Information Technology, University of Jyvaskyla, P.O. Box 35 (Agora), FIN-40014 Jyvaskyla, Finland 3 Department of Nanotechnology and Informatics, Institute for Nanomaterials, Advanced Technologies and Innovation, Technical University of Liberec, Studentská 1402/2,461 17 Liberec 1, Czech Republic

* Corresponding author

This paper is dedicated to Prof. Tomáš Roubíček in the occasion of his 60th birthday.

Received  July 2016 Revised  October 2016 Published  June 2017

Fund Project: 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

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.

Citation: Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069
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##### References:
Left: reference triangulation $\widehat{\cal T_h}$. Right: Mapped triangulation $\cal T_h$.
Optimized shapes (left) and convergence histories (right) for different values of the penalty/smoothing parameter $\varepsilon$.
Streamlines (left) and pressure contours (right) for $\varepsilon=10^{-5}$.
Tangential velocity and shear stress for $\varepsilon=10^{-5}$
Streamlines (left) and pressure contours (right)
Tangential velocity and shear stress
Optimized Bézier functions $\alpha_m$ for two different values of $\sigma_1$
Contours of the target pressure $p_0$ (left) and computed pressure (right)
Tangential velocity and shear stress on $S(\alpha_{opt})$
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