La méthode du Lagrangien augmenté pour une probleme d'interaction fluide-structure

The augmented Lagrangian method for a fluid-structure interaction problem

Abstract

In [3, p. 91] is presented a not-conditionally stable numerical method for solving a time-depend fluid structure interaction problem. This method consists in solving at each time step the mixed hybrid system (1) and to find out: \(v\) the velocity of the fluid, \(\nu \) the velocity of the structure, \(p\) the pressure of the fluid and \(\lambda \) the force on the fluid-structure interface.

In order to solve the system (1), we can use many algorithms just like: Uzawa's algorithm or the Augmented Lagrangien Method, but these algorithms don't permit to solve the fluid-structure problem in a decoupled way (via partitioned procedures), more exactly the fluid and structure problems are not solved separately.

Consequently, we can't use the well established theories and software for the fluid and respectively for the structure.  Alternatively, we can use the iterative method presented in [4] in order to solve the fluid-structure linear system via partitioned procedures. Unfortunely, this method converges slowly.

Based on the method used in [4], we present in this paper an augmented algorithm, where the continuity of the
fluid and structure velocities on the contact surface is penalized in order to improve the convergence rate.
Numerically, the continuity of the fluid and structure velocities on the contact surface has the form \(B_{21}v + B_{22}\nu =0\).

The convergence of the method is proved in the third section.