MATHICSE Technical Report : An adjoint-based method for the numerical approximation of shape optimization problems in presence of fluid-structure interaction

In this work, we propose both a theoretical framework and a numerical method to tackle shape op- timization problems related with uid dynamics applications in presence of uid-structure interactions. We present a general framework relying on the solution of a suitable adjoint problem and the character- ization of the shape gradient of the cost functional to be minimized. We show how to derive a system of (first-order) optimality conditions combining several tools from shape analysis and how to exploit them in order to set a numerical iterative procedure to approximate the optimal solution. We also show how to deal efficiently with shape deformations (resulting from both the uid-structure interaction and the optimization process). As benchmark case, we consider an unsteady Stokes ow in an elastic channel with compliant walls, whose motion under the effect of the ow is described through a linear Koiter shell model. Potential applications are related e.g. to design of cardiovascular prostheses in physiological flows or design of components in aerodynamics.