An adjoint based pressure boundary optimal control approach for fluid-structure interaction problems

In this work we investigate a new pressure boundary optimal control approach to the fluid-structure interaction problem based on Lagrangian multipliers and adjoint variables. We consider the steady FSI problem written in variational monolithic form in order to balance automatically solid and liquid forces at the interface and propose a pressure boundary optimal control method with the purpose to control the solid deformation in a well defined region by changing the fluid pressure on domain boundaries. The optimality system is obtained by imposing the first order necessary condition to the Lagrangian functional. In order to couple also the adjoint variables we must introduce a fictitious velocity field in the solid region that balances automatically interface adjoint forces as well. The system is solved in a segregated approach with different optimization schemes, such as the steepest descent and the quasi-Newton methods. We implemented the algorithms in a finite element code with mesh-moving capabilities for the study of large solid displacements. In order to support the proposed approach we perform numerical tests in two and three-dimensional spaces.