浸潤參考映射技術於流固耦合力學解析研究

Abstract

本論文研究提出一新穎浸潤參考映射技術，建立出全歐拉式（Full-Eulerian）流固耦合力學解析與計算架構，於此新式研究架構中，是以參考映射技術（Reference Map Technique）為固體力學解析基礎以及計算流體動力學（Computational Fluid Dynamics）為流場求解核心，並藉由水平集（Level Sets）函數精準判定固體與流體材料邊界位置，融合了參考映射技術解析所得固體變形場域分佈與計算流體力學求解所得流體場域資訊以及基於浸潤式有限元素理論（Immersed Finite Element Method）解析所得相互干涉之流固耦合力場，且能與以固定格點觀察流體運動之歐拉守恒描述，具有完全的相容性。基於上述各連體力學理論，本論文研究實作了相應的應用數學方法，包含：水平集函數耦合流場演進、流體力學演算法、徑向基函數插值（Radial Basis Function Interpolation）與所需共軛梯度與介面面積比例解方法。 論文內容方面：第一章回顧流體與固體耦合力學的研究發展背景與相關歷程；第二章深入介紹參考映射技術的核心概念，將連體力學理論描述於歐拉空間中，以解析固體系統控制方程式與材料組成方程式，更進一步精準求解於歐拉固定格點之固體變形場域分佈，並提供線性彈性固體之靜態與擬靜態計算範例與相關數值準確性與收斂性的驗證；第三章為本論文之核心內容，提出本研究發展用以解析流固耦合力學之新式浸潤參考映射技術，以及融合水平集函數耦合流場演進、計算流體動力學、徑向基函數插值、共軛梯度與面積比例求解等應用數學方法，以建立全歐拉式流固耦合計算力學架構；第四章為應用此新式流固耦合力學理論與計算方法，透過一系列的計算例，系統性地探討流固耦合力中的解析組成對於流體場域與固體變形分佈相互牽制的影響，包含環境重力、系統慣性與材料彈性三個項目；第五章為論文結論與未來研究展望。

This thesis research presents a novel immersed reference map technique for developing full-Eulerian analytical and computational frameworks of fluid-structure interactions. The reference map technique is utilized in the proposed frameworks as the analytical kernel of solid mechanics analysis. Concerning the extensive physics of surrounding fluids, numerics of computational fluid dynamics serves as a modeling engine of flow fields. The level-set functions are deeply applied to accurately determine the locations of interfaces between solid and fluid materials. It is important that mutually interfered deformation distributions of solids and flow fields of fluids obtained by the reference map technique and the computational fluid dynamics can be integrated with the resolved fluid-structure interaction force information based on the theory of immersed finite element method.Consequently, such full-Eulerian frameworks are entirely compatible with the descriptions of fixed grids commonly adopted by various numerical solvers of fluid mechanics. According to the continuum mechanics theories aforementioned above, this research implements several corresponding applied mathematics methods for validation studies, including the evolutions of level sets coupling with flow fields, the algorithms of fluid dynamics, the interpolations using radial basis functions, the conjugate gradient method, and the interface area ratio method. The thesis is organized as follows: The background and research history of the fluid-structure interactions are thoroughly reviewed in Chapter 1. Chapter 2 introduces the essential concepts of the reference mapping technique, delineating governing equations of continuum mechanics in Eulerian descriptions and further solving deformation distributions of solid materials at such fixed grids. Static and quasi-static equilibrium examples of linear elastic solids are also given for validation study of numerical accuracy and convergence associated with computation grid sizes.Chapter 3 is the core of this study, embracing the theoretical derivations of the proposed immersed reference map technique and the computational integration of related powerful scheme implementations, including the evolutions of level sets coupling with flow fields, the algorithms of fluid dynamics, the interpolations using radial basis functions, the conjugate gradient method, and the interface area ratio method. In Chapter 4, the immersed reference map technique is implemented in a series of computational examples to systematically investigate the influences of fluid-structure interaction forces over mutually interfered fluid flows and solid deformations, including environmental gravity, physical inertia, and material elasticity. Chapter 5 wraps the conclusion and future work of the thesis.