# Coupling equations and numerical resolution of fluid-structure interaction problems

Abstract : The first fourth chapters concern steady fluid-structure interaction. We study unsteady interaction in the fifth and sixth chapters. The last two chapters are dedicated to the free boundary flow with surface tension which has some similarities with fluid-structure interaction.

One of the recurrent idea of this work is to take as a control'' a part of the boundary conditions at the interface and then to observe'' if all the boundary conditions hold at the interface. The observation is treated by the least squares method and an optimal control problem is obtained. In chapter 1, we prove that the cost function is weak sequentially lower semi-continuous and consequently, one can prove that the existence of an optimal control. We prove the differentiability of the cost function and we derive the analytic gradient. Numerical results are presented. In chapter 3, we study the sensitivity of the problem and we give the analytic form of the gradient whiteout employing the adjoint state. Numerical results are performed. In chapter 4, we impose the normal velocity of the fluid and the normal stress to the interface. This is a rarely used approach for solving the Stokes equations. We try to minimize the tangential velocity of the fluid at the interface. We prove that the problem is well posed and we present numerical results. In chapter 5, we introduce an algorithm where an optimization problem have to be solved at each time step. It is an algorithm well adapted for the pulsating flow. Numerical results are presented for relatively large time steps.

A convergence result concerning an algorithm for dynamic mesh is presented in chapter 6. In chapter 7 and 8, we try to find numerically the evolution of a two-dimensional domain with applications to cell development. The fluid flow in a moving domain depends on the surface tension at the free boundary. This surface tension is proportionally to the curvature of the boundary. The have employed front-tracking'' like algorithms. Numerical results are presented.
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https://tel.archives-ouvertes.fr/tel-00167976
Contributor : Cornel Marius Murea <>
Submitted on : Thursday, August 23, 2007 - 10:35:57 PM
Last modification on : Tuesday, October 16, 2018 - 2:26:02 PM
Long-term archiving on: : Friday, April 9, 2010 - 1:04:42 AM

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• HAL Id : tel-00167976, version 1

### Citation

Cornel Marius Murea. Coupling equations and numerical resolution of fluid-structure interaction problems. Mathematics [math]. Université de Haute Alsace - Mulhouse, 2007. ⟨tel-00167976⟩

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