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doi: 10.3934/eect.2022003
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A shape optimization problem constrained with the Stokes equations to address maximization of vortices

1. 

Graduate School of Natural Science and Technology, Division of Mathematical and Physical Sciences, Kanazawa University, Kanazawa 920-1192, Japan

2. 

Faculty of Mathematics and Physics, Kanazawa University, Kanazawa 920-1192, Japan

*Corresponding author: John Sebastian Simon

Received  August 2021 Revised  November 2021 Early access January 2022

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.

Citation: John Sebastian Simon, Hirofumi Notsu. A shape optimization problem constrained with the Stokes equations to address maximization of vortices. Evolution Equations and Control Theory, doi: 10.3934/eect.2022003
References:
[1]

H. Azegami, Solution of shape optimization problem and its application to product design, In Mathematical Analysis of Continuum Mechanics and Industrial Applications, (eds. H. Itou, M. Kimura, V. Chalupecký, K. Ohtsuka, D. Tagami and A. Takada), Springer Singapore, Singapore, 26 (2017), 83–98. doi: 10.1007/978-981-10-2633-1_6.

[2]

J. Bacani and G. Peichl, On the first-order shape derivative of the Kohn–Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Art. ID 384320, 19 pp. doi: 10.1155/2013/384320.

[3]

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.

[4]

M. S. ChongA. E. Perry and B. J. Cantwell, A general classification of three-dimensional flow fields, Phys. Fluids A, 2 (1990), 765-777.  doi: 10.1063/1.857730.

[5]

C. DapognyP. FreyF. Omnès and Y. Privat, Geometrical shape optimization in fluid mechanics using {F}ree{F}em++, Struct. Multidiscip. Optim., 58 (2018), 2761-2788.  doi: 10.1007/s00158-018-2023-2.

[6]

M. Delfour and J.-P. Zolesio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2$^{nd}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719826.

[7]

M. Desai and K. Ito, Optimal controls of Navier–Stokes equations, SIAM J. Control Optim., 32 (1994), 1428-1446.  doi: 10.1137/S0363012992224972.

[8]

M. F. Eggl and P. J. Schmid, Mixing enhancement in binary fluids using optimised stirring strategies, J. Fluid Mech., 899 (2020), A24, 21 pp. doi: 10.1017/jfm.2020.448.

[9]

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 1998.

[10]

T. L. B. Flinois and T. Colonius, Optimal control of circular cylinder wakes using long control horizons, Physics of Fluids, 27 (2015), 087105.  doi: 10.1063/1.4928896.

[11]

Z. M. GaoY. C. Ma and H. W. Zhuang, Shape optimization for Navier–Stokes flow, Inverse Probl. Sci. Eng., 16 (2008), 583-616.  doi: 10.1080/17415970701743319.

[12]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[13]

K. Goto, K. Nakajima and H. Notsu, Twin vortex computer in fluid flow, New J. Phys., 23 (2021), 063051, 14 pp. doi: 10.1088/1367-2630/ac024d.

[14]

J. HaslingerK. ItoT. KozubekK. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 11 (2009), 317-330.  doi: 10.4171/IFB/213.

[15]

J. HaslingerJ. Málek and J. Stebel, Shape optimization in problems governed by generalised Navier–Stokes equations: Existence analysis, Control Cybernet., 34 (2005), 283-303. 

[16]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[17]

A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, EMS Tracts in Mathematics, 28. European Mathematical Society (EMS), Zürich, 2018. doi: 10.4171/178.

[18]

A. Henrot and Y. Privat, What is the optimal shape of a pipe?, Arch. Ration. Mech. Anal., 196 (2010), 281-302.  doi: 10.1007/s00205-009-0243-8.

[19]

J. C. R. Hunt, A. A. Wray and P. Moin, Eddies, streams, and convergence zones in turbulent flows, In Proc. 1988 Summer Program of Center for Turbulence Research Program, (1998), 193–208.

[20]

K. ItoK. Kunisch and G. H. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539.  doi: 10.1051/cocv:2008002.

[21]

Y. IwataH. AzegamiT. Aoyama and E. Katamine, Numerical solution to shape optimization problems for non-stationary Navier–Stokes problems, JSIAM Lett., 2 (2010), 37-40.  doi: 10.14495/jsiaml.2.37.

[22]

J. Jeong and F. Hussain, On the identification of a vortex, J. Fluid Mech., 285 (1995), 69-94.  doi: 10.1017/S0022112095000462.

[23]

H. Kasumba and K. Kunisch, Vortex control in channel flows using translational invariant cost functionals, Comput. Optim. Appl., 52 (2012), 691-717.  doi: 10.1007/s10589-011-9434-y.

[24]

G. MatherI. MezićS. GrivopoulosU. Vaidya and L. Petzold, Optimal control of mixing in Stokes fluid flows, J. Fluid Mech., 580 (2007), 261-281.  doi: 10.1017/S0022112007005332.

[25]

B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, 2$^{nd}$ edition, Numerical Mathematics and Scientific Computations, Oxford University Press, 2010.

[26]

P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.

[27]

M. T. NairM. Hegland and R. S. Anderssen, The trade-off between regularity and stability in Tikhonov regularization, Math. Comp., 66 (1997), 193-206.  doi: 10.1090/S0025-5718-97-00811-9.

[28]

J. Nocedal and S. Wright, Numerical Optimization, vol. 2 of Springer Series in Operations Research and Financial Engineering, 2$^{nd}$ edition, Springer, New York, 2006.

[29]

H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier–Stokes equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 361-380.  doi: 10.1051/m2an/2015047.

[30]

M. Pošta and T. Roubíček, Optimal control of Navier–Stokes equations by Oseen approximation, Comput. Math. Appl., 53 (2007), 569-581.  doi: 10.1016/j.camwa.2006.02.034.

[31]

J. F. T. Rabago and H. Azegami, An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data, J. Engrg. Math., 117 (2019), 1-29.  doi: 10.1007/s10665-019-10005-x.

[32]

J. F. T. Rabago and H. Azegami, A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional, Comput. Optim. Appl., 77 (2020), 251-305.  doi: 10.1007/s10589-020-00199-7.

[33]

S. Schmidt and V. Schulz, Shape derivatives for general objective functions and the incompressible Navier–Stokes equations, Control Cybernet., 39 (2010), 677-713. 

[34]

J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, 1$^{st}$ edition, Springer Series in Computational Mathematics, 16. Springer-Verlag Berlin, 1992. doi: 10.1007/978-3-642-58106-9.

show all references

References:
[1]

H. Azegami, Solution of shape optimization problem and its application to product design, In Mathematical Analysis of Continuum Mechanics and Industrial Applications, (eds. H. Itou, M. Kimura, V. Chalupecký, K. Ohtsuka, D. Tagami and A. Takada), Springer Singapore, Singapore, 26 (2017), 83–98. doi: 10.1007/978-981-10-2633-1_6.

[2]

J. Bacani and G. Peichl, On the first-order shape derivative of the Kohn–Vogelius cost functional of the Bernoulli problem, Abstr. Appl. Anal., 2013 (2013), Art. ID 384320, 19 pp. doi: 10.1155/2013/384320.

[3]

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.

[4]

M. S. ChongA. E. Perry and B. J. Cantwell, A general classification of three-dimensional flow fields, Phys. Fluids A, 2 (1990), 765-777.  doi: 10.1063/1.857730.

[5]

C. DapognyP. FreyF. Omnès and Y. Privat, Geometrical shape optimization in fluid mechanics using {F}ree{F}em++, Struct. Multidiscip. Optim., 58 (2018), 2761-2788.  doi: 10.1007/s00158-018-2023-2.

[6]

M. Delfour and J.-P. Zolesio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, 2$^{nd}$ edition, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719826.

[7]

M. Desai and K. Ito, Optimal controls of Navier–Stokes equations, SIAM J. Control Optim., 32 (1994), 1428-1446.  doi: 10.1137/S0363012992224972.

[8]

M. F. Eggl and P. J. Schmid, Mixing enhancement in binary fluids using optimised stirring strategies, J. Fluid Mech., 899 (2020), A24, 21 pp. doi: 10.1017/jfm.2020.448.

[9]

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 1998.

[10]

T. L. B. Flinois and T. Colonius, Optimal control of circular cylinder wakes using long control horizons, Physics of Fluids, 27 (2015), 087105.  doi: 10.1063/1.4928896.

[11]

Z. M. GaoY. C. Ma and H. W. Zhuang, Shape optimization for Navier–Stokes flow, Inverse Probl. Sci. Eng., 16 (2008), 583-616.  doi: 10.1080/17415970701743319.

[12]

V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, 5. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5.

[13]

K. Goto, K. Nakajima and H. Notsu, Twin vortex computer in fluid flow, New J. Phys., 23 (2021), 063051, 14 pp. doi: 10.1088/1367-2630/ac024d.

[14]

J. HaslingerK. ItoT. KozubekK. Kunisch and G. Peichl, On the shape derivative for problems of Bernoulli type, Interfaces Free Bound., 11 (2009), 317-330.  doi: 10.4171/IFB/213.

[15]

J. HaslingerJ. Málek and J. Stebel, Shape optimization in problems governed by generalised Navier–Stokes equations: Existence analysis, Control Cybernet., 34 (2005), 283-303. 

[16]

F. Hecht, New development in FreeFem++, J. Numer. Math., 20 (2012), 251-265.  doi: 10.1515/jnum-2012-0013.

[17]

A. Henrot and M. Pierre, Shape Variation and Optimization: A Geometrical Analysis, EMS Tracts in Mathematics, 28. European Mathematical Society (EMS), Zürich, 2018. doi: 10.4171/178.

[18]

A. Henrot and Y. Privat, What is the optimal shape of a pipe?, Arch. Ration. Mech. Anal., 196 (2010), 281-302.  doi: 10.1007/s00205-009-0243-8.

[19]

J. C. R. Hunt, A. A. Wray and P. Moin, Eddies, streams, and convergence zones in turbulent flows, In Proc. 1988 Summer Program of Center for Turbulence Research Program, (1998), 193–208.

[20]

K. ItoK. Kunisch and G. H. Peichl, Variational approach to shape derivatives, ESAIM Control Optim. Calc. Var., 14 (2008), 517-539.  doi: 10.1051/cocv:2008002.

[21]

Y. IwataH. AzegamiT. Aoyama and E. Katamine, Numerical solution to shape optimization problems for non-stationary Navier–Stokes problems, JSIAM Lett., 2 (2010), 37-40.  doi: 10.14495/jsiaml.2.37.

[22]

J. Jeong and F. Hussain, On the identification of a vortex, J. Fluid Mech., 285 (1995), 69-94.  doi: 10.1017/S0022112095000462.

[23]

H. Kasumba and K. Kunisch, Vortex control in channel flows using translational invariant cost functionals, Comput. Optim. Appl., 52 (2012), 691-717.  doi: 10.1007/s10589-011-9434-y.

[24]

G. MatherI. MezićS. GrivopoulosU. Vaidya and L. Petzold, Optimal control of mixing in Stokes fluid flows, J. Fluid Mech., 580 (2007), 261-281.  doi: 10.1017/S0022112007005332.

[25]

B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, 2$^{nd}$ edition, Numerical Mathematics and Scientific Computations, Oxford University Press, 2010.

[26]

P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, 2003. doi: 10.1093/acprof:oso/9780198508885.001.0001.

[27]

M. T. NairM. Hegland and R. S. Anderssen, The trade-off between regularity and stability in Tikhonov regularization, Math. Comp., 66 (1997), 193-206.  doi: 10.1090/S0025-5718-97-00811-9.

[28]

J. Nocedal and S. Wright, Numerical Optimization, vol. 2 of Springer Series in Operations Research and Financial Engineering, 2$^{nd}$ edition, Springer, New York, 2006.

[29]

H. Notsu and M. Tabata, Error estimates of a stabilized Lagrange-Galerkin scheme for the Navier–Stokes equations, ESAIM Math. Model. Numer. Anal., 50 (2016), 361-380.  doi: 10.1051/m2an/2015047.

[30]

M. Pošta and T. Roubíček, Optimal control of Navier–Stokes equations by Oseen approximation, Comput. Math. Appl., 53 (2007), 569-581.  doi: 10.1016/j.camwa.2006.02.034.

[31]

J. F. T. Rabago and H. Azegami, An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data, J. Engrg. Math., 117 (2019), 1-29.  doi: 10.1007/s10665-019-10005-x.

[32]

J. F. T. Rabago and H. Azegami, A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional, Comput. Optim. Appl., 77 (2020), 251-305.  doi: 10.1007/s10589-020-00199-7.

[33]

S. Schmidt and V. Schulz, Shape derivatives for general objective functions and the incompressible Navier–Stokes equations, Control Cybernet., 39 (2010), 677-713. 

[34]

J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, 1$^{st}$ edition, Springer Series in Computational Mathematics, 16. Springer-Verlag Berlin, 1992. doi: 10.1007/978-3-642-58106-9.

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
Parameter Value Parameter Value
$ \alpha $ 6.0 $ \ell_0 $ 20
$ b_0 $ $ 1\times10^{-4} $ $ \tau $ 1.05
$ \overline{b} $ 10
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
Parameter Value Parameter Value
$ \alpha $ 1.3 $ \ell_0 $ .5
$ b_0 $ $ 1\times10^{-2} $ $ \tau $ 1.05
$ \overline{b} $ 10
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
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
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