
Mathematics in Engineering, 2019, 1(2): 252280. doi: 10.3934/mine.2019.2.252
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A saddle point approach to an optimal boundary control problem for steady NavierStokes equations
1 MOXLaboratory for Modeling and Scientific Computing, Department of Mathematics, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy
2 Department of Mathematics, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy
3 Professor Emeritus of Mathematics Institute, Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 8, 1015 Lausanne, Switzerland
Received: , Accepted: , Published:
References
1. Akcelik V, Biros G, Ghattas O, et al. (2006) Parallel algorithms for PDEconstrained optimization, In: Heroux, M.A., Raghavan, P., Simon, H.D., Editors, Parallel Processing for Scientific Computing, Philadelphia: Society for Industrial and Applied Mathematics, 291–322.
2. Beneš M, Kučera P, (2012) On the Navier–Stokes flows for heatconducting fluids with mixed boundary conditions. J Math Anal Appl 389: 769–780.
3. Beneš M, Kučera P, (2016) Solutions to the Navier–Stokes equations with mixed boundary conditions in twodimensional bounded domains. Math Nachr 289: 194–212.
4. Berggren M, (1998) Numerical solution of a flowcontrol problem: vorticity reduction by dynamic boundary action. SIAM J Sci Comput 19: 829–860.
5. Biros G, Ghattas O, (2005) Parallel Lagrange–Newton–Krylov–Schur methods for PDEconstrained optimization. Part II: The Lagrange–Newton solver and its application to optimal control of steady viscous flows. SIAM J Sci Comput 27: 714–739.
6. Brezzi F, (1974) On the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO 2: 129–151.
7. Dautray R, Lions JL, (2000) Mathematical Analysis and Numerical Methods for Science and Technology. Berlin Heidelberg: SpringerVerlag.
8. Dedè L, (2007) Optimal flow control for NavierStokes equations: Drag minimization. Int J Numer Meth Fluids 55: 347–366.
9. Desai M, Ito K, (1994) Optimal controls of Navier–Stokes equations. SIAM J Control Optim 32: 1428–1446.
10. Do H, Owida AA, Morsi YS, (2012) Numerical analysis of coronary artery bypass grafts: An over view. Comput Meth Prog Bio 108: 689–705.
11. Elman H, Silvester D, Wathen A, (2005) Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics. Series in Numerical Mathematics and Scientific Computation, Oxford University Press.
12. Fursikov A, Gunzburger MD, Hou L, (1998) Boundary value problems and optimal boundary control for the Navier–Stokes system: The twodimensional case. SIAM J Control Optim 36: 852–894.
13. Fursikov A, Rannacher R, (2010) Optimal neumann control for the twodimensional steadystate NavierStokes equations, In: Fursikov, A.V., Galdi, G.P., Pukhnachev, V.V., Editors, New Directions in Mathematical Fluid Mechanics: The Alexander V. Kazhikhov Memorial Volume, Basel: Birkhäuser Basel.
14. Ghattas O, Bark J, (1997) Optimal control of two and threedimensional incompressible Navier Stokes flows. J Comput Phys 136: 231–244.
15. Girault V, Raviart PA, (1986) Finite Element Methods for NavierStokes Equations: Theory and Algorithms. Berlin and New York: SpringerVerlag.
16. Gresho P, Sani R, (1998) Incompressible Flow and the Finite Element Method: Advection Diffusion and Isothermal Laminar Flow. John Wiley & Sons.
17. Guerra T, Sequeira A, Tiago J, (2015) Existence of optimal boundary control for the Navier Stokes equations with mixed boundary conditions. Port Math 72: 267–283.
18. Gunzburger MD, (2003) Perspectives in Flow Control and Optimization. Series in Advances in Design and Control. Philadephia: Society for Industrial and Applied Mathematics.
19. Gunzburger MD, Hou L, Svobodny T, (1992) Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J Control Optim 30: 167–181.
20. Gunzburger MD, Hou L, Svobodny TP, (1991) Analysis and finite element approximation of optimal control problems for the stationary NavierStokes equations with distributed and Neumann controls. Math Comput 57: 123–151.
21. Gunzburger MD, Hou LS, Svobodny TP, (1991) Analysis and finite element approximation of optimal control problems for the stationary NavierStokes equations with Dirichlet controls. ESAIM: Math Model Numer Anal 25: 711–748.
22. Gunzburger MD, Manservisi S, (1999) The velocity tracking problem for NavierStokes flows with bounded distributed controls. SIAM J Control Optim 37: 1913–1945.
23. Gunzburger MD, Manservisi S, (2000) Analysis and approximation of the velocity tracking problem for NavierStokes flows with distributed control. SIAM J Numer Anal 37: 1481–1512.
24. Gunzburger MD, Manservisi S, (2000) The velocity tracking problem for NavierStokes flows with boundary control. SIAM J Control Optim 39: 594–634.
25. Heinkenschloss M, (1998) Formulation and analysis of a sequential quadratic programming method for the optimal Dirichlet boundary control of NavierStokes flow, In: Hager, W.W.,
26. Pardalos, P.M., Authors, Optimal Control: Theory, Algorithms, and Applications, Springer US, 178–203.
27. Herzog R, Kunisch K, (2010) Algorithms for PDEconstrained optimization. GAMM Mitteilungen 33: 163–176.
28. Hinze M, Pinnau R, Ulbrich M, et al. (2009) Optimization with PDE Constraints, Series in Mathematical Modelling: Theory and Applications, Springer Netherlands.
29. Hou L, Ravindran SS, (1999) Numerical approximation of optimal flow control problems by a penalty method: Error estimates and numerical results. SIAM J Sci Comput 20: 1753–1777.
30. Jameson A, (1988) Aerodynamic design via control theory. J Sci Comput 3: 233–260.
31. Jameson A, (1995) Optimum aerodynamic design using CFD and control theory. In: Proceedings of the 12th AIAA Computational Fluid Dynamics Conference 1995, 926–949.
32. Kim H, (2006) A boundary control problem for vorticity minimization in timedependent 2D NavierStokes equations. Korean J Math 23: 293–312.
33. Kim H, Kwon O, (2006) On a vorticity minimization problem for the stationary 2D Stokes equations. J Korean Math Soc 43: 45–63.
34. Koltukluoğlu T, Blanco P, (2018) Boundary control in computational haemodynamics. J Fluid Mech 847: 329–364.
35. Kračmar S, Neustupa J, (2001) A weak solvability of a steady variational inequality of the Navier Stokes type with mixed boundary conditions. Nonlin Anal 47: 4169–4180.
36. Kračmar S, Neustupa J, (2018) Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality. Math Nachr 291: 1801–1814.
37. Kučera P, Skalák Z, (1998) Local solutions to the Navier–Stokes equations with mixed boundary conditions. Acta Appl Math 54: 275–288.
38. Kunisch K, Vexler B, (2007) Optimal vortex reduction for instationary flows based on translation invariant cost functionals. SIAM J Control Optim 46: 1368–1397.
39. Lassila T, Manzoni A, Quarteroni A, et al. (2013) Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty. ESAIMModel Num 47: 1107–1131.
40. Lassila T, Manzoni A, Quarteroni A, et al. (2013) A reduced computational and geometrical framework for inverse problems in haemodynamics. Int J Numer Meth Bio 29: 741–776.
41. Lei M, Archie J, Kleinstreuer C, et al. (1997) Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis. J Vasc Surg 25: 637–646.
42. Maz'ya V, Rossmann J, (2009) Mixed boundary value problems for the stationary Navier–Stokes system in polyhedral domains. Arch Ration Mech Anal 194: 669–712.
43. Migliavacca F, Dubini G, (2005) Computational modeling of vascular anastomoses. Biomech Model Mechanobiol 3: 235–250.
44. Prudencio EE, Byrd R, Cai XC, (2006) Parallel full space SQP Lagrange–Newton–Krylov Sschwarz algorithms for PDEconstrained optimization problems. SIAM J Sci Comput 27: 1305– 1328.
45. Quarteroni A, Valli A, (1994) Numerical Approximation of Partial Differential Equations. Berlin Heidelberg: SpringerVerlag.
46. Sankaran S, Marsden A, (2010) The impact of uncertainty on shape optimization of idealized bypass graft models in unsteady flow. Phys Fluids 22: 121902.
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