ABSTRACT
In the previous chapter, we have performed a sensitivity analysis for a coupled fluid-solid model, where the solid was driven by a basic second order differential equation. This has allowed us to simply reduce the differential order by expanding the solid state and use its velocity as an explicit state variable. Then we have been able to use the framework developed in Chapter 5 concerning the derivative of the Navier-Stokes system with respect to the velocity of the moving domain. Here, the situation is quite different since we are considering a more general solid model with a non-linear constitutive law. In this case, the parametrization of the solid by its velocity is more challenging and the straightforward use of the results obtained in Chapter 7 is not obvious. As a consequence, we choose to work with the solid displacement state variable. Then, we need to use the results obtained in Chapter 6 concerning the derivative of the Navier-Stokes using the non-cylindrical identity perturbations. The chapter is organized as follows: first of all, we introduce the mechanical problem and its mathematical description. We use the classical arbitrary Euler-Lagrange (ALE) formulation, particularly suited for problems involving moving boundaries. Then, in the second part, we state the main result of this chapter, namely the cost function gradient computation involving the solution of a linear adjoint problem. Finally, its proof is fully developed.
