A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations

Andrea Manzoni; Alfio Quarteroni; Sandro Salsa; Andrea Manzoni; Alfio Quarteroni; Sandro Salsa

doi:10.3934/mine.2019.2.252

Mathematics in Engineering

2019, Volume 1, Issue 2: 252-280. doi: 10.3934/mine.2019.2.252

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Research article

A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations

1.
MOX-Laboratory 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: 16 November 2018 Accepted: 08 February 2019 Published: 20 February 2019

In this paper we propose a saddle point approach to solve boundary control problems for the steady Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions, both in two and three dimensions. We provide a comprehensive theoretical framework to address ($i$) the well posedness analysis for the optimal control problem related to this system and ($ii$) the derivation of a system of first-order optimality conditions. We take advantage of a suitable treatment of boundary Dirichlet controls (and data) realized by means of Lagrange multipliers. In spite of the fact that this approach is rather common, a detailed analysis is still missing for mixed boundary conditions. We consider the minimization of quadratic cost (e.g., tracking-type or vorticity) functionals of the velocity. A descent method is then applied for numerical optimization, exploiting the Galerkin finite element method for the discretization of the state equations, the adjoint (Oseen) equations and the optimality equation. Numerical results are shown for simplified two-dimensional fluid flows in a tract of blood vessel where a bypass is inserted; to avoid to simulate the whole bypass configuration, we represent its action by a boundary velocity control.
- optimal control,
- partial differential equations,
- fluid dynamics,
- Navier-Stokes equations,
- finite element method
Citation: Andrea Manzoni, Alfio Quarteroni, Sandro Salsa. A saddle point approach to an optimal boundary control problem for steady Navier-Stokes equations[J]. Mathematics in Engineering, 2019, 1(2): 252-280. doi: 10.3934/mine.2019.2.252

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Abstract

In this paper we propose a saddle point approach to solve boundary control problems for the steady Navier-Stokes equations with mixed Dirichlet-Neumann boundary conditions, both in two and three dimensions. We provide a comprehensive theoretical framework to address ($i$) the well posedness analysis for the optimal control problem related to this system and ($ii$) the derivation of a system of first-order optimality conditions. We take advantage of a suitable treatment of boundary Dirichlet controls (and data) realized by means of Lagrange multipliers. In spite of the fact that this approach is rather common, a detailed analysis is still missing for mixed boundary conditions. We consider the minimization of quadratic cost (e.g., tracking-type or vorticity) functionals of the velocity. A descent method is then applied for numerical optimization, exploiting the Galerkin finite element method for the discretization of the state equations, the adjoint (Oseen) equations and the optimality equation. Numerical results are shown for simplified two-dimensional fluid flows in a tract of blood vessel where a bypass is inserted; to avoid to simulate the whole bypass configuration, we represent its action by a boundary velocity control.

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