# A Fluid-Structure Interaction Problem Using Fictitious Domain Approach with Penalization

Andrei Halanay^{*}, Cornel Murea and Dan Tiba

In the present paper, a penalization of the Stokes equation is used to obtain approximate solutions of the boundary problem in a larger domain, including the domain occupied by the structure, when a fluid-structure interaction problem is studied. This yields a fictitious fluid equation in the structure domain.The coefficients of the fluid problem, excepting the penalizing term, are constant and independent of the deformation of the structure. Subtracting the structure equations from the fictitious fluid equations in the structure domain and using the Green's formula, we obtain a weak formulation where the continuity of the stress tensor at the interface does not appear explicitly. The problem under study is a free boundary problem and a fundamental difficulty is to find the free interface between the fluid and the structure, that is not known and has to be identified together with the solution of the given system of equations. Approximate solutions will be founded as fixed points of a specific contractive operator. An algorithm is developed and numerical simulations of a the deformation of a tall building under wind action are illustrating the theory.

Mathematics Subject Classification: 76D05 35Q30

Keywords: Stokes equation;stress tensor;free boundary;penalization

Minisymposion: Shape Optimization and Free Boundaries