THREE-DIMENSIONAL COMPUTATIONS FOR BOUNDARY OPTIMAL CONTROL PROBLEMS IN INCOMPRESSIBLE MAGNETOHYDRODYNAMICS

In this paper we present the results of some three-dimensional computations of boundary optimal control problems in incompressible Magnetohydrodynamics, obtained with the lifting function approach and with the implementation of a gradient algorithm for the solution of the optimality system. Possible applications of these problems are manifold, such as aluminum casting in metallurgy, crystal growth in semiconductor industry and liquid metal MHD pumps and generators. In these applications it is of great interest to achieve the control on the velocity profile through the Lorentz force acting on the fluid. The magnetic field on the boundary can be used as a means for steering the velocity profile to a desired one or for minimizing other quantities of interest. With the lifting functio

Coupled boundary optimal control problems in thermal fluid dynamics with lifting function approach and Vanka-type solvers

In this paper we present the results of Fluid-Structure Interaction (FSI) compu- tations of an elastic solid object and a laminar incompressible viscous flow using a penalty- projection algorithm. The mathematical problem consists of the Navier-Stokes equations cou- pled with a suitable structure model describing the behavior of the solid element. We consider two cases for the structure model. The first model deals with the equations of incompressible solid mechanics. We propose to split this saddle-point problem into a system of decoupled elliptic equations for velocity and pressure. Since stress-free boundary conditions cannot be enforced with a pure projection method, a penalty correction term is introduced. The results demonstrate that this penalty correction term is necessary in order to improve the accuracy of the computation of boundary quantities. The second model that we consider in this work consists of a simplified shell model. This approximation has many advantages since we can simulate the mechanical behavior of thin walls without suffering from the problem of locking and also with a reduction of the number of degrees of freedom. Finallly, we show and compare the results of a series of simulations in terms of accuracy using the above described models. These cases show stability and robustness of the proposed algorithm for appropriate values of the penalty parameter together with a reduction of the computational effort compared to monolithic algorithms

A Monolithic FEM Multigrid Penalty-Projection Solver for Incompressible Fluid-Structure Interaction

In this paper we present the results of computations of fluid-structure interaction of an incompressible elastic object and a laminar incompressible viscous flow with the use of the penalty-projection method. The equations of incompressible solid mechanics are based on a mixed formulation where pressure and displacement variables are coupled in a saddle-point manner, similar to the incompressible Navier-Stokes system. The discretization and numerical solution of these problems is a challenging task both from the algorithmic and from the computational point of view. We propose to reformulate the solid problem from a displacement-pressure formulation into a velocity-pressure one and to reduce the solution of this saddle-point problem into a system of decoupled elliptic equations for velocity and pressure with a projection method. In order to constrain the divergence of the solid velocity, a penalty method is implemented. This approach leads in a natural manner to a monolithic treatment of the coupling between solid and fluid. Therefore, the continuity of the velocity at the interface is ensured as well as the energy balance. This formulation is particularly attractive in the field of fluid-structure interaction, due to a large number of applications that can be found in biomechanics, such as the mechanical interaction of incompressible biological elastic tissues with blood or organic fluids. The FSI problem is solved in the Arbitrary Lagrangian-Eulerian framework with a conforming finite element method using a piecewise-quadratic function space for the velocity and piecewise linear for the pressure. The decoupled system is advanced in two steps: in a first stage the momentum balance is solved to obtain an intermediate velocity, which is corrected in the second stage by projecting it onto a divergence-free space. A multigrid algorithm is implemented together with an interface with PETSc parallel computing libraries in order to reduce the CPU time. We show the results of a series of simulations concerning some typical FSI applications, such as blood flow in a compliant vessel

A penalty-projection algorithm for incompressible fluid-structure interaction

This paper deals with boundary optimal control problems for the temperature and Navier-Stokes equations, where we propose to transform a boundary optimal control problem into a distributed problem through the lifting function approach of nonhomogeneous boundary conditions. The lifting function approach defines controls in the function spaces which are naturally associated to the volume variables, without stronger regularity requirements. For strong and robust optimization the state-adjoint system must be solved in a coupled way. To this purpose, we propose the use of domain decomposition Vanka-type solvers. With this type of solvers the problem is split into small blocks of finite element subdomains with a small number of degrees of freedom, so that an optimal solution is computed by solving the fully coupled state- adjoint system on each subdomain. We present a numerical study of a class of optimal control problems where temperature is the observed quantity and the control quantity corresponds to the boundary values of the fluid temperature in a portion of the boundary. The control region for the observed quantity is a part of the domain where it is interesting to match a desired temperature value. In a multi-physics framework the desired temperature matching can be achieved by the contribution of different physical mechanisms, involving not only boundary temperature control but also boundary velocity, the overall effect of which is studied. We consider a classical finite element method for the numerical discretization of such problems and we illustrate the results of some test cases for the state, adjoint and control solution, in order to show that candidate boundary controls can be computed in an effective manner

A Distributed Control Approach for the Boundary Optimal Control of the Steady MHD Equations

Abstract. A new approach is presented for the boundary optimal control of the MHD equations in which the boundary control problem is transformed into an extended distributed control problem. This can be achieved by considering boundary controls in the form of lifting functions which extend from the boundary into the interior. The optimal solution is then sought by exploring all possible extended functions. This approach gives robustness to the boundary control algorithm which can be solved by standard distributed control techniques over the interior of the domain