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
References:
[1]

D. ArnoldF. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344. doi: 10.1007/BF02576171. Google Scholar

[2]

D. BucurE. Feireisl and Š. Nečasová, Influence of wall roughness on the slip behavior of viscous fluids, Proc. R. Soc. Edinb., Sect. A, Math., 138 (2008), 957-973. doi: 10.1017/S0308210507000376. Google Scholar

[3]

M. Bulíček and J. Málek On unsteady internal flows of Bingham fluids subject to threshold slip on the impermeable boundary, in Recent Developments of Mathematical Fluid Mechanics (eds. H. Amann et al. ), Birkhäuser, 2016,135-156. Google Scholar

[4]

D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl, 52 (1975), 189-219. doi: 10.1016/0022-247X(75)90091-8. Google Scholar

[5]

G. Farin, Curves and Surfaces for CAGD (Fifth Edition), Morgan Kaufmann, 2002.Google Scholar

[6]

H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak and or slip boundary conditions, Res. Inst. Math. Sci. Kokyuroku, 888 (1994), 199-216. Google Scholar

[7]

G. P. Galdi, Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume Ⅰ: Linearised Steady Problems, vol. 38 of Springer Tracts in Natural Philosophy, Springer, New York, 1994. doi: 10.1007/978-1-4612-5364-8. Google Scholar

[8]

V. Girault and P. A. Raviart, Finite Element Approximation of the Navier–Stokes Equations, vol. 749 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1979. Google Scholar

[9]

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. doi: 10.1007/978-3-662-12613-4. Google Scholar

[10]

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008. doi: 10.1137/1.9780898717761. Google Scholar

[11]

J. Haslinger, A note on contact shape optimization with semicoercive state problems, Appl. Math., 47 (2002), 397-410. doi: 10.1023/A:1021709907750. Google Scholar

[12]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718690. Google Scholar

[13]

J. Haslinger and J. Stebel, Stokes problem with a solution dependent slip bound: Stability of solutions with respect to domains, ZAMM Z. Angew. Math. Mech., 96 (2016), 1049-1060. doi: 10.1002/zamm.201500117. Google Scholar

[14]

J. HaslingerJ. Stebel and T. Sassi, Shape optimization for Stokes problem with threshold slip, Applications of Mathematics, 59 (2014), 631-652. doi: 10.1007/s10492-014-0077-z. Google Scholar

[15]

H. Hervet and L. Léger, Flow with slip at the wall: From simple to complex fluids, C. R. Physique, 4 (2003), 241-249. doi: 10.1016/S1631-0705(03)00047-1. Google Scholar

[16]

L. Holzleitner, Hausdorff convergence of domains and their boundaries in shape optimal design, Control Cybernet., 30 (2001), 23-44. Google Scholar

[17]

C. Le Roux, Steady Stokes flows with threshold slip boundary conditions, Math. Models Methods Appl. Sci., 15 (2005), 1141-1168. doi: 10.1142/S0218202505000686. Google Scholar

[18]

C. Le Roux and A. Tani, Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30 (2007), 595-624. doi: 10.1002/mma.802. Google Scholar

[19]

MATLAB, Release R2014a with Optimization Toolbox 7.0, The MathWorks Inc. , Natick, Massachusetts, 2014.Google Scholar

[20]

C. L. Navier, Mémoire sur les lois du movement des fluids, Mem. Acad. R. Sci. Paris, 6 (1823), 389-416. Google Scholar

[21]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, vol. 28 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1998. doi: 10.1007/978-1-4757-2825-5. Google Scholar

[22]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Series in Computations Physics, Springer Verlag, New York, 1984. doi: 10.1007/978-3-642-87722-3. Google Scholar

[23]

I. J. Rao and K. R. Rajagopal, The effect of the slip boundary condition on the flow of fluids in a channel, Acta Mechanica, 135 (1999), 113-126. doi: 10.1007/BF01305747. Google Scholar

[24]

J. P. Rothstein, Slip on superhydrophobic surfaces, Annual Review of Fluid Mechanics, 42 (2010), 89-109. doi: 10.1146/annurev-fluid-121108-145558. Google Scholar

[25]

N. Saito, On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions, Publ. Res. Inst. Math. Sci., 40 (2004), 345-383. doi: 10.2977/prims/1145475807. Google Scholar

[26]

J. Stebel, On shape stability of incompressible fluids subject to Navier's slip condition, Journal of Mathematical Fluid mechanics, 14 (2012), 575-589. doi: 10.1007/s00021-011-0086-6. Google Scholar

show all references

References:
[1]

D. ArnoldF. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344. doi: 10.1007/BF02576171. Google Scholar

[2]

D. BucurE. Feireisl and Š. Nečasová, Influence of wall roughness on the slip behavior of viscous fluids, Proc. R. Soc. Edinb., Sect. A, Math., 138 (2008), 957-973. doi: 10.1017/S0308210507000376. Google Scholar

[3]

M. Bulíček and J. Málek On unsteady internal flows of Bingham fluids subject to threshold slip on the impermeable boundary, in Recent Developments of Mathematical Fluid Mechanics (eds. H. Amann et al. ), Birkhäuser, 2016,135-156. Google Scholar

[4]

D. Chenais, On the existence of a solution in a domain identification problem, J. Math. Anal. Appl, 52 (1975), 189-219. doi: 10.1016/0022-247X(75)90091-8. Google Scholar

[5]

G. Farin, Curves and Surfaces for CAGD (Fifth Edition), Morgan Kaufmann, 2002.Google Scholar

[6]

H. Fujita, A mathematical analysis of motions of viscous incompressible fluid under leak and or slip boundary conditions, Res. Inst. Math. Sci. Kokyuroku, 888 (1994), 199-216. Google Scholar

[7]

G. P. Galdi, Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume Ⅰ: Linearised Steady Problems, vol. 38 of Springer Tracts in Natural Philosophy, Springer, New York, 1994. doi: 10.1007/978-1-4612-5364-8. Google Scholar

[8]

V. Girault and P. A. Raviart, Finite Element Approximation of the Navier–Stokes Equations, vol. 749 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, 1979. Google Scholar

[9]

R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. doi: 10.1007/978-3-662-12613-4. Google Scholar

[10]

A. Griewank and A. Walther, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008. doi: 10.1137/1.9780898717761. Google Scholar

[11]

J. Haslinger, A note on contact shape optimization with semicoercive state problems, Appl. Math., 47 (2002), 397-410. doi: 10.1023/A:1021709907750. Google Scholar

[12]

J. Haslinger and R. A. E. Mäkinen, Introduction to Shape Optimization: Theory, Approximation, and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718690. Google Scholar

[13]

J. Haslinger and J. Stebel, Stokes problem with a solution dependent slip bound: Stability of solutions with respect to domains, ZAMM Z. Angew. Math. Mech., 96 (2016), 1049-1060. doi: 10.1002/zamm.201500117. Google Scholar

[14]

J. HaslingerJ. Stebel and T. Sassi, Shape optimization for Stokes problem with threshold slip, Applications of Mathematics, 59 (2014), 631-652. doi: 10.1007/s10492-014-0077-z. Google Scholar

[15]

H. Hervet and L. Léger, Flow with slip at the wall: From simple to complex fluids, C. R. Physique, 4 (2003), 241-249. doi: 10.1016/S1631-0705(03)00047-1. Google Scholar

[16]

L. Holzleitner, Hausdorff convergence of domains and their boundaries in shape optimal design, Control Cybernet., 30 (2001), 23-44. Google Scholar

[17]

C. Le Roux, Steady Stokes flows with threshold slip boundary conditions, Math. Models Methods Appl. Sci., 15 (2005), 1141-1168. doi: 10.1142/S0218202505000686. Google Scholar

[18]

C. Le Roux and A. Tani, Steady solutions of the Navier-Stokes equations with threshold slip boundary conditions, Math. Methods Appl. Sci., 30 (2007), 595-624. doi: 10.1002/mma.802. Google Scholar

[19]

MATLAB, Release R2014a with Optimization Toolbox 7.0, The MathWorks Inc. , Natick, Massachusetts, 2014.Google Scholar

[21]

J. Outrata, M. Kočvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, vol. 28 of Nonconvex Optimization and its Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 1998. doi: 10.1007/978-1-4757-2825-5. Google Scholar

[22]

O. Pironneau, Optimal Shape Design for Elliptic Systems, Springer Series in Computations Physics, Springer Verlag, New York, 1984. doi: 10.1007/978-3-642-87722-3. Google Scholar

[23]

I. J. Rao and K. R. Rajagopal, The effect of the slip boundary condition on the flow of fluids in a channel, Acta Mechanica, 135 (1999), 113-126. doi: 10.1007/BF01305747. Google Scholar

[24]

J. P. Rothstein, Slip on superhydrophobic surfaces, Annual Review of Fluid Mechanics, 42 (2010), 89-109. doi: 10.1146/annurev-fluid-121108-145558. Google Scholar

[25]

N. Saito, On the Stokes equation with the leak and slip boundary conditions of friction type: regularity of solutions, Publ. Res. Inst. Math. Sci., 40 (2004), 345-383. doi: 10.2977/prims/1145475807. Google Scholar

[26]

J. Stebel, On shape stability of incompressible fluids subject to Navier's slip condition, Journal of Mathematical Fluid mechanics, 14 (2012), 575-589. doi: 10.1007/s00021-011-0086-6. Google Scholar

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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