Mathematical and Numerical Models for Boundary Optimal Control Problems Applied to Fluid-Structure Interaction

The main purpose of this work is to develop mathematical and numerical methods for the optimal control of fluid-structure interaction simulations. In particular, we focus on the Koiter shell fluid-structure model and on the adjoint formalism for the control problem. Using the Koiter approach, the dimensionality of the solid is reduced to reduce the computational cost of the fluid-structure simulations. In order to couple the fluid and the structure domains, the Koiter shell equations are embedded into the fluid equations as a Robin boundary condition. The coupling fluid-structure conditions are automatically treated in an implicit way, so the stability of the numerical scheme is preserved. This model has many applications in cases where a fluid interacts with a thin membrane that deforms mainly in the normal direction. Then, an adjoint-based optimal control theory of the presented Koiter fluid-structure model in the steady case is studied. In fact, a boundary optimal control theory is applied to the fluid-structure Koiter model, including the existence of the solution of the fluid-structure problem, the existence of the optimal solution and regularity and differentiability properties. Moreover, the fractional operators are introduced to be applied to the framework described above, in order to model properly the regularization term in the boundary optimal control problems. All the numerical simulations presented in this work, with the exception of the fractional simulations, have been simulated with the in-house multigrid finite element based code FEMuS.