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研究生： 陳信源
Sin-Yuan Chen
論文名稱：
A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers
指導教授： 黃楓南
Feng-Nan Hwang
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 98
語文別： 英文
論文頁數： 46
外文關鍵詞： incompressible flow, Navier-Stokes equations, bifurcation, parallel computing, linear stability analysis, generalized eigenvalue problem, pseudo transient continuation
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    • 此研究主要是想探討不可壓縮流其平衡解的穩定性與分歧現象兩者間的關係，並且偵測流體發生對稱性破壞的臨界點。首先， 使用穩定化有限元素法對二維 Navier-Stokes 方程組施行空間離散化來描述非穩定、具黏滯性的、不可壓縮之流體。我們使用兩種方法來描述流體粒子的運動行為。其一，先引入 backward Euler''s method 對二維 Navier-Stokes 方程組施行時間離散化，接著進行時間序列的數值模擬。第二，對流體之平衡解作線性的穩定性分析；在此採用 implicit Arnoldi method 結合 Cayley transformation 來求解一個大型廣義特徵值問題之特徵根。此外，如何選取 Cayley transformation 的參數使得相對應的線性系統擁有良好的收斂性亦是非常重要的議題。最後，我們將舉例使用 SuperLU 求解線性系統，並且展示其平行效能。

    In this study, we focus in investigating the relation between the (linear) stability of stationary solutions and pitchfork bifurcations of incompressible flows, and detect the critical points of symmetry-breaking phenomena. First, a stabilized finite element method is used to discretize the 2D Navier-Stokes equations on the spatial domain for the unsteady, viscous, incompressible flow problem. There are two approaches used to determine the behavior of the solution. One is via numerical time integration. Another is to locate the steady-state solutions and then to make the linear stability analysis by computing eigenvalues of a corresponding generalized eigenvalue problem, for which an implicit Arnoldi method with the Cayley transformation is used. In addition, it is also an important issue that how to choose the parameters of the Cayley transformation such that the convergence of the linear system would be better. Finally, we show a parallel performance of SuperLU, a great parallelable algorithm which is used to solve the linear system.

    1 Introduction . . . . . . . . . . . . . . . . . . . . . 1 2 A brief review of bifurcation theory . . . . . . . . . 2 2.1 Some definitions . . . . . . . . . . . . . . . . .. 2 2.2 Pitchfork bifurcation . . . . . . . . . . . . . . . 3 3 Application to incompressible flows . . . . . . . .. . 4 3.1 Problem statement . . . . . . . . . . . . . . . . 4 3.2 Governing equation and semi-discrete formulation . 6 3.3 Numerical tools for detecting bifurcation points . 9 3.3.1 Pseudo-transient Newton-Krylov-Schwarz method 10 3.3.2 Linear stability analysis . . . . . . . . .. 11 4 Numerical results . . . . . . . . . . . . . . . . . . 13 4.1 Setup of Numerical experiments . . . . . . . . . 13 4.2 Grid resolution testing and parallel code validation 15 4.3 Predictions of pitchfork bifurcation . . . . . .. 18 4.4 The parallel performance of SuperLU factorization 30 5 Conclusions and future works . . . . . . . . . . . . 33 Bibliography . . . . . . . . . . . . . . . . . . . . . . 37

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