有限元素法與新式網格重置演算法於多顆粒運動問題的探討

本研究之整體架構旨在於應用有限元素法搭配任意拉格朗日-歐拉的描述，與目前提出的創新網格重置演算法來求解多顆粒移動邊界的問題。針對流場解析部分，直接引用運算子系統拆解法將奈維爾-史托克斯方程式中的主要變數分別作處理；另外，空間的離散部份基於葛勒金法會獲得積分型態的有限元離散式，配合目前計算上所採用的線性三角和四面體元素於二維與三維問題，可採用局部加密獲得更精準的近似值，而時間離散部份則引用後項多步法並選用二階截尾誤差去提高計算準度，且分別在對流與擴散項採用顯性與隱性作處理以增快整體計算效率與穩定性。關於移動邊界問題部份，根據任意拉格朗日-歐拉的描述，對流項會重新定義且此處採用平衡張量係數去增進其穩定性，最後將以上數值模式搭配目前創新的網格重置演算法，希冀能在多顆粒移動邊界問題的應用上取得初步的成果。經由初步選用的二維範例，和相關文獻加以驗證其模擬的結果，皆可得到令人滿意的近似參考值；因此，本研究基於二維的基礎更進一步發展並提出三維的演算法，詳細的數值模式架構搭配演算法與其結果驗證皆論述於本論文中。

關鍵字

有限元素法；任意拉格朗日-歐拉；網格重置演算法；移動邊界問題

並列摘要

The finite element method with arbitrary Lagrangian-Eulerian scheme (ALE) and a novel mesh regeneration algorithm is applied to solve the moving boundary problem in the present study. An operator-splitting scheme is introduced to decompose the equations of Navier-Stokes with regard to both velocity and pressure variables. Based on a Galerkin weak statement, the finite element discrete integral form for spatial discretization can be obtained in which triangular and tetrahedral bases will be chosen to deal with 2D and 3D problems. Furthermore, the mixed explicit-implicit stiffly stable scheme with second order truncation error is introduced on the time discretization. In addition, both convection and diffusion terms are solved separately with the explicit and implicit scheme to enhance the stability and accuracy. According to the Lagrangian method, the convective term under the simulation of moving boundary problem will be redefined with balance tensor diffusivity. Lastly, present integrated numerical model is combined with a novel mesh regeneration algorithm for applying to the multi-body interaction problem. As expected, we obtain the good feasibility of the present integrated numerical model via a series of validations by referring to numerical and experimental literatures.

並列關鍵字

finite element method ； arbitrary Lagrangian-Eulerian ； mesh regeneration algorithm ； moving boundary problem

參考文獻

[1] R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1943) 1-23.

[2] A. J. Baker, D. W. Pepper, Finite Elements 1-2-3, in: B. J. Clark, D. A. Damstra (Eds.), New York, 1991, pp. 130-136.

[3] H. A. van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Comput. 13 (1992) 631-644.

[7] J. Kim, P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys. 59 (1985) 308-323.

[8] G. E. Karniadakis, M. Israeli, S. A. Orszay, High-order splitting methods for the incompressible Navier Stokes equations, J. Comput. Phys. 97 (1991) 414-443.

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有限元素法與新式網格重置演算法於多顆粒運動問題的探討

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