# American Institute of Mathematical Sciences

March 2018, 8(1): 89-115. doi: 10.3934/mcrf.2018004

## Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

 1 Faculty of Mathematics and Computer Science, Banacha 22, 90-238 Lodz, University of Lodz, Poland 2 Institut Elie Cartan de Nancy, UMR 7502, 54506 Vandoeuvre lès Nancy Cedex, France 3 Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-338 Warsaw, Poland

Received  April 2017 Revised  September 2017 Published  January 2018

In the paper the shape optimization problem for the static, compressible Navier-Stokes equations is analyzed. The drag minimizing of an obstacle immersed in the gas stream is considered. The continuous gradient of the drag is obtained by application of the sensitivity formulas derived in the works of one of the co-authors. The numerical approximation scheme uses mixed Finite Volume - Finite Element formulation. The novelty of our numerical method is a particular choice of the regularizing term for the non-homogeneous Stokes boundary value problem, which may be tuned to obtain the best accuracy. The convergence of the discrete solutions for the model under considerations is proved. The non-linearity of the model is treated by means of the fixed point procedure. The numerical example of an optimal shape is given.

Citation: Anna Kaźmierczak, Jan Sokolowski, Antoni Zochowski. Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes. Mathematical Control & Related Fields, 2018, 8 (1) : 89-115. doi: 10.3934/mcrf.2018004
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] H. Beirão da Veiga, Stationary motions and the incompressible limit for compressible viscous fluids, Houston J. Math., 13 (1987), 527-544. [3] H. Beirão da Veiga, An $L^p$-theory for the $n$-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Comm. Math. Phys., 109 (1987), 229-248. [4] H. Beirão da Veiga, Existence results in Sobolev spaces for a transport equation, Ricerche Mat., 36 (1987), 173-184. [5] V. Berinde, Iterative Approximation of Fixed Points Lecture Notes in Mathematics 1912, Springer Verlag, 2007. [6] C. Brandenburg, F. Lindemann, M. Ulbrich and S. Ulbrich, Advanced numerical methods for PDE constrained optimization with application to optimal design in Navier Stokes flow, Constrained Optimization and Optimal Control for partial Differential Equations, in Series: Internat. Ser. Numer. Math. , Birkhäuser/Springer Basel AG, Basel, 160 (2012), 257-275. [7] E. Casas, An optimal control problem governed by the evolution Navier-Stokes equations. In S. S. Sritharan, editor, Optimal Control of Viscous Flows, SIAM, Philadelphia, (1998), 79-95. [8] R. Eymard, T. Gallouet, R. Herbin and J. Latche, A convergent finite element-finite volume scheme for the compressible Stokes problem; part Ⅱ -The isentropic case, Mathematics of Computation, 79 (2010), 649-675. [9] E. Feireisl, T. Karper and M. Pokorný, Mathematical Theory of Compressible Viscous Fluids, Birkhäuser, Basel, 2016. [10] T. Gallouet, R. Herbin and J. Latche, A convergent finite element-finite volume scheme for the compressible Stokes problem; part â… -The isothermal case, Mathematics of Computation, 78 (2009), 1333-1352. [11] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001. [12] T. Kondoh, T. Matsumori and A. Kawamoto, Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration, Struct. Multidiscip. Optim., 45 (2012), 693-701. [13] B. Mohammadi and O. Pironneau, Shape optimization in fluid mechanics, Ann. Rev. Fluid Mech., 36 (2004), 255-279. [14] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, 2$^{nd}$ edition, Oxford Univ. Press, Oxford, 2010. [15] S. Novo, Compressible Navier-Stokes model with inflow-outflow boundary conditions, J. Math. Fluid Mech., 7 (2005), 485-514. [16] A. Novotný, About steady transport equation. Ⅰ. $L^p$-approach in domains with smooth boundaries, Comment. Math. Univ. Carolin., 37 (1996), 43-89. [17] A. Novotný, About steady transport equation. Ⅱ. Schauder estimates in domains with smooth boundaries, Portugal. Math., 54 (1997), 317-333. [18] A. Novotný and M. Padula, Existence and uniqueness of stationary solutions for viscous compressible heat conductive fluid with large potential and small non-potential external forces, Sibirsk. Mat. Zh. , 34 (1993), 120-146 (in Russian). [19] A. Novotný and M. Padula, $L^p$-approach to steady flows of viscous compressible fluids in exterior domains, Arch. Ration. Mech. Anal., 126 (1994), 243-297. [20] A. Novotný and M. Padula, Physically reasonable solutions to steady compressible Navier-Stokes equations in $3D$-exterior domains $(v_{âˆž}\ne 0)$, Math. Ann., 308 (1997), 439-489. [21] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Univ. Press, Oxford, 2004. [22] M. Padula, Existence and uniqueness for viscous steady compressible motions, Arch. Ration. Mech. Anal., 97 (1987), 89-102. [23] P. I. Plotnikov, E. V. Ruban and J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier-Stokes equations: well-posedness and sensitivity analysis, SIAM J. Math. Anal., 40 (2008), 1152-1200. [24] P. I. Plotnikov, E. V. Ruban and J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations, J. Math. Pures Appl., 92 (2009), 113-162. [25] P. Plotnikov and J. Sokolowski, Domain dependence of solutions to compressible Navier-Stokes equations, SIAM Journal on Control and Optimization, 45 (2006), 1165-1197. [26] P. Plotnikov and J. Sokolowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. [27] P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization Monografie Matematyczne, 73 Birkhauser, 2012. doi: 978-3-0348-0366-3. [28] P. Plotnikov, J. Sokolowski and A. Zochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics MMAR'09 (2009), 4 pages. [29] S. Schmidt and V. Schulz, Shape derivatives for general objective functions and the incompressible Navier-Stokes equations, Control and Cybernetics, 39 (2010), 677-713. [30] J. Simon, Domain variation for drag in Stokes flow, in: Control Theory of Distributed Parameter Systems and Applications (Shanghai, 1990), Lecture Notes in Control and Inform. Sci. , 159, Springer, Berlin, (1991), 28-42. [31] T. Slawig, A formula for the derivative with respect to domain variations in Navier-Stokes flow based on an embedding domain method, SIAM J. Control Optim., 42 (2003), 495-512. [32] T. Slawig, An explicit formula for the derivative of a class of cost functionals with respect to domain variations in Stokes flow, SIAM J. Control Optim., 39 (2000), 141-158.

show all references

##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] H. Beirão da Veiga, Stationary motions and the incompressible limit for compressible viscous fluids, Houston J. Math., 13 (1987), 527-544. [3] H. Beirão da Veiga, An $L^p$-theory for the $n$-dimensional, stationary, compressible Navier-Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions, Comm. Math. Phys., 109 (1987), 229-248. [4] H. Beirão da Veiga, Existence results in Sobolev spaces for a transport equation, Ricerche Mat., 36 (1987), 173-184. [5] V. Berinde, Iterative Approximation of Fixed Points Lecture Notes in Mathematics 1912, Springer Verlag, 2007. [6] C. Brandenburg, F. Lindemann, M. Ulbrich and S. Ulbrich, Advanced numerical methods for PDE constrained optimization with application to optimal design in Navier Stokes flow, Constrained Optimization and Optimal Control for partial Differential Equations, in Series: Internat. Ser. Numer. Math. , Birkhäuser/Springer Basel AG, Basel, 160 (2012), 257-275. [7] E. Casas, An optimal control problem governed by the evolution Navier-Stokes equations. In S. S. Sritharan, editor, Optimal Control of Viscous Flows, SIAM, Philadelphia, (1998), 79-95. [8] R. Eymard, T. Gallouet, R. Herbin and J. Latche, A convergent finite element-finite volume scheme for the compressible Stokes problem; part Ⅱ -The isentropic case, Mathematics of Computation, 79 (2010), 649-675. [9] E. Feireisl, T. Karper and M. Pokorný, Mathematical Theory of Compressible Viscous Fluids, Birkhäuser, Basel, 2016. [10] T. Gallouet, R. Herbin and J. Latche, A convergent finite element-finite volume scheme for the compressible Stokes problem; part â… -The isothermal case, Mathematics of Computation, 78 (2009), 1333-1352. [11] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001. [12] T. Kondoh, T. Matsumori and A. Kawamoto, Drag minimization and lift maximization in laminar flows via topology optimization employing simple objective function expressions based on body force integration, Struct. Multidiscip. Optim., 45 (2012), 693-701. [13] B. Mohammadi and O. Pironneau, Shape optimization in fluid mechanics, Ann. Rev. Fluid Mech., 36 (2004), 255-279. [14] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids, 2$^{nd}$ edition, Oxford Univ. Press, Oxford, 2010. [15] S. Novo, Compressible Navier-Stokes model with inflow-outflow boundary conditions, J. Math. Fluid Mech., 7 (2005), 485-514. [16] A. Novotný, About steady transport equation. Ⅰ. $L^p$-approach in domains with smooth boundaries, Comment. Math. Univ. Carolin., 37 (1996), 43-89. [17] A. Novotný, About steady transport equation. Ⅱ. Schauder estimates in domains with smooth boundaries, Portugal. Math., 54 (1997), 317-333. [18] A. Novotný and M. Padula, Existence and uniqueness of stationary solutions for viscous compressible heat conductive fluid with large potential and small non-potential external forces, Sibirsk. Mat. Zh. , 34 (1993), 120-146 (in Russian). [19] A. Novotný and M. Padula, $L^p$-approach to steady flows of viscous compressible fluids in exterior domains, Arch. Ration. Mech. Anal., 126 (1994), 243-297. [20] A. Novotný and M. Padula, Physically reasonable solutions to steady compressible Navier-Stokes equations in $3D$-exterior domains $(v_{âˆž}\ne 0)$, Math. Ann., 308 (1997), 439-489. [21] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Univ. Press, Oxford, 2004. [22] M. Padula, Existence and uniqueness for viscous steady compressible motions, Arch. Ration. Mech. Anal., 97 (1987), 89-102. [23] P. I. Plotnikov, E. V. Ruban and J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier-Stokes equations: well-posedness and sensitivity analysis, SIAM J. Math. Anal., 40 (2008), 1152-1200. [24] P. I. Plotnikov, E. V. Ruban and J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier-Stokes and transport equations, J. Math. Pures Appl., 92 (2009), 113-162. [25] P. Plotnikov and J. Sokolowski, Domain dependence of solutions to compressible Navier-Stokes equations, SIAM Journal on Control and Optimization, 45 (2006), 1165-1197. [26] P. Plotnikov and J. Sokolowski, Shape derivative of drag functional, SIAM J. Control Optim., 48 (2010), 4680-4706. [27] P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization Monografie Matematyczne, 73 Birkhauser, 2012. doi: 978-3-0348-0366-3. [28] P. Plotnikov, J. Sokolowski and A. Zochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics MMAR'09 (2009), 4 pages. [29] S. Schmidt and V. Schulz, Shape derivatives for general objective functions and the incompressible Navier-Stokes equations, Control and Cybernetics, 39 (2010), 677-713. [30] J. Simon, Domain variation for drag in Stokes flow, in: Control Theory of Distributed Parameter Systems and Applications (Shanghai, 1990), Lecture Notes in Control and Inform. Sci. , 159, Springer, Berlin, (1991), 28-42. [31] T. Slawig, A formula for the derivative with respect to domain variations in Navier-Stokes flow based on an embedding domain method, SIAM J. Control Optim., 42 (2003), 495-512. [32] T. Slawig, An explicit formula for the derivative of a class of cost functionals with respect to domain variations in Stokes flow, SIAM J. Control Optim., 39 (2000), 141-158.
Computational domain $\Omega = B\setminus S$
On the left: the plot of $J$ versus number of steps; on the right: the final shape of the obstacle $\Gamma$ after minimization of the drag
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