在本篇論文中，我們研究如何利用有限元素法來模擬流固耦合問題，該類問題描述黏性流體流經彈性結構體後產生的交互作用，並且使用有限元素法的工具包FreeFem++來進行數值模擬。我們的研究重點為二維空間變數下的穩態流固耦合問題。流體部份的行為是由黏性不可壓縮的史托克或那維爾-史托克方程來掌控，而彈性體的部份則是利用彈性的勒梅系統來描述。我們提出了一個直接的有限元素迭代演算法來處理該耦合問題，其中彈性體的部分使用了標準的有限元素法，而流體的部分則是利用穩定化有限元素法來進行數值模擬。本文所提供的幾個數值實例驗證了該直接迭代有限元素演算法的效率。

In this thesis, we study the finite element approximations to a fluid-structure interaction problem that describes the viscous fluid flow interacting with an elastic structure by using the finite element package FreeFem++. We focus on the steady-state fluid-structure interaction problem in two dimensions. The fluid motion is governed by the viscous, incompressible Stokes or the Navier-Stokes equations, while the elastic solid is modeled by the Lame system of elasticity. We propose a direct iterative finite element algorithm to solve the coupling system, where the structure part is solved by the standard finite element method and the fluid part is solved by a stabilized finite element method. Numerical simulations of several examples are presented to illustrate the effectiveness of the proposed direct iterative finite element algorithm.

[1] G. R. Barrenechea and M. A. Fernandez, A stabilized ¯nite element method for the linearized
Navier-Stokes equations with dominating reaction, European Congress on Computational
Methods in Applied Sciences and Engineering, July 24-28, 2004.
[2] P. B. Bochev, C. R. Dohrmann, and M. D. Gunzburger, Stabilization of low-order mixed
¯nite elements for the Stokes equations, SIAM Journal on Numerical Analysis, 44 (2006),
pp. 82-101.
[3] P. Causin, J. F. Gerbeau, and F. Nobile, Added-mass e®ect in the design of partitioned
algorithms for °uid-structure problems, Computer Methods in Applied Mechanics and En-
gineering, 194 (2005), pp. 4506-4527.
[4] Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee, Analysis of a linear °uid-structure inter-
action problem, Discrete and Continuous Dynamical Systems, 9 (2003), pp. 633-650.
[5] Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee, Semidiscrete ¯nite element approximations
of a linear °uid-structure interaction problem, SIAM Journal on Numerical Analysis, 42
(2004), pp. 1-29.
[6] L. P. Franca, S. L. Frey, and T. J. R. Hughes, Stabilized ¯nite element methods: I. application
to the advective-di®usive model, Computer Methods in Applied Mechanics and Engineering,
95 (1992), pp. 253-276.
[7] L. P. Franca and S. L. Frey, Stabilized ¯nite element methods: II. the incompressible Navier-
Stokes equations, Computer Methods in Applied Mechanics and Engineering , 99 (1992), pp.
209-233.
[8] M. A. Fernandez, J.-F. Gerbeau, and C. Grandmont, A projection semi-implicit scheme for
the coupling of an elastic structure with an incompressible °uid, International Journal for
Numerical Methos in Engineering , 69 (2007), pp. 794-821.
[9] L. Formaggia, J. F. Gerbeau, F. Nobile, and A. Quarteroni, On the coupling of 3D and
1D Navier-Stokes equations for °ow problems in compliant vessels, Computer Methods in
Applied Mechanics and Engineering, 191 (2001), pp. 561-582.
[10] L. P. Franca, T. J. R. Hughes, and R. Stenberg, Stabilized ¯nite element methods for the
Stokes problem, Chapter 4 in Incompressible Computational Fluid Dynamics, pp. 87-107,
M. Gunzburger and R. A. Nicolaides (Eds), Cambridge University Press, 1993.
[11] F. Hecht, FreeFem++, Third Edition, Version 3.9-1, Laboratoire Jacques-Louis Lions, Uni-
versite Pierre et Marie Curie, Paris.
[12] E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, and T. Coupez, Stabilized ¯nite element
method for incompressible °ows with high Reynolds number, Journal of Computational
Physics, 229 (2010), pp. 8643-8665.
[13] J. Janela, A. Moura, and A. Sequeira, A 3D non-Newtonian °uid-structure interaction model
for blood °ow in arteries, Journal of Computational and Applied Mathematics, 234 (2010),
pp. 2783-2791.
[14] U. Kuttler and W. A. Wall, Fixed-point °uid-structure interaction solvers with dynamic
relaxation, Computational Mechanics, 43 (2008), pp. 61-72.
[15] R. Rannacher, Finite element methods for the incompressible Navier-Stokes equations,
Short Course on Theoretical and Numerical Fluid Mechanics, Vancouver, British Columbia,
Canada, July 27-28, 1996.