簡易檢索 / 詳目顯示

回結果列表
研究生： 戴嘉展
Dai, Jia-Zhan
論文名稱： 線性史托克問題的可混雜不連續有限元法解
Solving linear Stokes equation using HDG methods with HDG3D
指導教授： 陳旻宏
Chen, Min-Hung
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2022
畢業學年度： 110
語文別： 英文
論文頁數： 28
中文關鍵詞： 可混雜不連續有限元方法 、史托克方程 、矩陣實驗室
外文關鍵詞： Hybridizable discontinuous Galerkin, Stokes equation, Matlab, HDG3D
相關次數： 點閱：101  下載：4 
分享至:
查詢本校圖書館目錄 查詢臺灣博碩士論文知識加值系統 勘誤回報

    • 在這篇論文中我們使用可混雜不連續有限元方法解線性史托克方程。可混雜不連續有限
    元方法是不連續有限元方法的其中一種新方法，可以改善不連續有限元方法運算量過高
    的缺點。在數值實驗中我們使用 HDG3D 的矩陣實驗室的套件得到符合方法收斂性質的
    收斂階數。另外我們也證明了變數近似的存在性與唯一性。

    In this paper we apply the hybridizable discontinuous Galerkin (HDG) methods to solve the Stokes equation in polyhedral domain. The HDG methods are the new type of the discontinuous Galerkin (DG) methods. We proposed the HDG methods to deal with the shortcoming of DG methods for high computation cost. In numerical experiments we use HDG3D which is the Matlab implementation of the HDG methods and obtain the approximations of the velocity, pressure, and gradient converge with the order of p + 1 when the degree of polynomials p ě 0. We also prove the existence and uniqueness of the approximations variables.

    摘要 i Abstract ii 誌謝 iii 1 Introduction 1 2 HDG formulation of the Stokes equation 2 2.1 Notation 3 2.2 Formulation of the HDG methods 5 2.3 Weak forms of the local and global problems 6 2.4 The HDG linear system 7 2.5 Characterization of the numerical trace and the pressure mean 10 2.6 Existence and uniqueness of the numerical solution 13 3 Implement of HDG3D 17 3.1 Volume integrals 17 3.1.1 The Dubiner basis on elements 17 3.1.2 The mean of the Dubiner basis on elements 17 3.2 Surface integrals 18 3.2.1 The Dubiner basis on faces 18 3.2.2 Three types of surface matrices 18 3.3 Local solvers 19 3.3.1 Volume terms 19 3.3.2 Surface terms 19 3.3.3 Matrices related to local solvers 20 3.4 Global solbers 21 4 Numerical experiment 22 5 Conclusion 24 Bibligraphy 25 Appendix 26

    [1] Giacomini, Matteo, Ruben Sevilla, and Antonio Huerta. ”Tutorial on hybridizable dis-
    continuous Galerkin (HDG) formulation for incompressible flow problems.” Modeling
    in Engineering Using Innovative Numerical Methods for Solids and Fluids. Springer,
    Cham, 2020. 163-201.
    [2] Reed, William H., and Thomas R. Hill. Triangular mesh methods for the neutron
    transport equation. No. LA-UR-73-479; CONF-730414-2. Los Alamos Scientific Lab.,
    N. Mex.(USA), 1973.
    [3] Donea, Jean, and Antonio Huerta. Finite element methods for flow problems. John
    Wiley & Sons, 2003.
    [4] Cockburn, Bernardo, and Jayadeep Gopalakrishnan. ”The derivation of hybridizable
    discontinuous Galerkin methods for Stokes flow.” SIAM Journal on Numerical Anal-
    ysis 47.2 (2009): 1092-1125.
    [5] Sevilla, Ruben, and Antonio Huerta. ”Tutorial on Hybridizable Discontinuous
    Galerkin (HDG) for second-order elliptic problems.” Advanced finite element tech-
    nologies. Springer, Cham, 2016. 105-129.
    [6] Giacomini, Matteo, Ruben Sevilla, and Antonio Huerta. ”HDGlab: An open-source
    implementation of the hybridisable discontinuous Galerkin method in MATLAB.”
    Archives of Computational Methods in Engineering 28.3 (2021): 1941-1986.
    [7] Nguyen, Ngoc Cuong, Jaime Peraire, and Bernardo Cockburn. ”A hybridizable dis-
    continuous Galerkin method for Stokes flow.” Computer Methods in Applied Mechan-
    ics and Engineering 199.9-12 (2010): 582-597.
    [8] Fu, Zhixing, Luis F. Gatica, and Francisco-Javier Sayas. ”Matlab tools for HDG in
    three dimensions.” arXiv preprint arXiv:1305.1489 (2013).

    下載圖示 校內：立即公開
    校外：立即公開
    QR CODE