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.
