沈浸邊界法於流固耦合問題之有限元素分析

Abstract

本論文的主旨在於建立一套有限元素模式，用來模擬研究流體與移動結構物之間的相互作用行為。這樣的數值模式通常包括幾個要件：一、可靠而有效率的運算核心。二、處理複雜幾何和移動邊界的能力。三、準確的描述流體在移動物體上的作用力。在運算核心部分，我們分別嘗試異階元素方程式與運算子拆解法來求解原始變數型態的奈維爾-史托克斯方程式。在複雜幾何問題上，我們考慮非結構性元素和結合沈浸邊界法的結構性網格。為了處理移動邊界，首先我們建立一套混合卡式沈浸邊界模式，並利用差分的技巧來處理移動邊界與流體網格不重合的問題；其次，我們進一步提出一套結合移動網格的全新數值方法，在處理此類問題能得到更佳的準確度。最後，為了精準的描述移動物體在流固耦合問題中所受到的作用力，我們改良了先前所提出的移動網格技巧。所有數值計算的結果和文獻資料比較都相當吻合，而許多有趣的物理現象也詳實的紀錄在數值實驗過程中。

The thesis is concerned with developing a finite element procedure based on the immersed boundary technique to allow the numerical investigations of the interaction between the flow and the moving objectives. Such a developed model must be composed of a reliable and efficient flow solver, capability of handling the complex geometry and moving boundary, as well as accurate prediction of the fluid force acting on the moving objective. For establishing the flow solver, the mixed order formulation and operator splitting scheme are first presented for solving the primitive variable form of the Navier- Stokes equations. Next, the unstructured element and the structured gird with immersed boundary technique are respectively employed for dealing with the complex geometry. As far as the moving boundary is concerned, we develop a hybrid Cartesian/immersed boundary model with a robust interpolation scheme, and further propose a novel methodology including a moving grid process to reduce the numerical diffusion near the immersed boundary. Finally, the moving grid process is improved for accurately calculating the fluid force acting on the moving objective so as to describe the fluid-structure interaction problems. All the numerical results are compared favorably with the reference data, and several interesting phenomena, such as lock-in behavior, are well captured by our developed model.