當一些像雷諾數 (Reynolds number) 的物理參數被改變時，通常可以觀察到分歧現象 (Bifurcation) 所造成的過渡和不穩定的模態 (modes)。這篇論文的目的在利用平行化的 pseudo-transient continuation 演算法來預測在管壁突然性擴張的水流上所產生的對稱破壞性分歧的臨界點。為此，我們利用穩定化有限元素法 (stabilized finite element method) 對不可壓縮的 Navier-Stokes 方程做空間離散化並得到非線性的偏微分代數方程式 (PDAE's)。其中一種檢測 PDAE's 常態解 (stationary solution) 之穩定性的方法，是應用 pseudo-transient continuation 去討論受到干擾 (perturbation) 的常態解在一段時間後是否會回到原本的狀態。本文利用穩定性 backward Euler's 方法當作 time integrator，在每個 time step 所產生的非線性系統是由完全平行化的 Newton-Krylov-Schwarz 演算法來解。 從平行電腦所得到的數值結果顯示，我們的平行化 pseudo-transient continuation 演算法是非常有效率的，並且我們所觀察到的分歧現象和文獻上的實驗和數值結果趨勢相符。此外，我們在擴張比例 (expansion ratio) 較小時，會注意到一個有趣的現象—imperfect pitchfork bifurcations，進而延誤 bifurcation point 的發生，我們推測此現象乃因數值模擬時所使用的不對稱且非結構的網格。 In fluid dynamics, bifurcations phenomena, which provide the modes of transitions and instability when some physical parameter such as the Reynolds number is varied, are commonly observed. The aim of this thesis is to study numerically some parallel pseudo-transient continuation algorithm for detecting the critical points of symmetry-breaking bifurcation in sudden expansion flows. For this purpose, the resulting nonlinear partial differential algebraic equations (PDAE's) are obtained by employing a stabilized finite element method for unsteady incompressible Navier-Stokes equations as the spatial discretization. One of classical approaches for examining the stability of a stationary solution to the PDAE's is to apply the so-called pseudo-transient continuation, which can be interpreted as the context of a method-of-line (MOL) approach, beginning with some perturbed stationary solution to PDAE's and then to investigate its time-dependent response to see whether the solution returns back to original state or not after a certain time step. In current study, after employing unconditionally stable backward Euler's method as a time integrator, at each time step, the resulting nonlinear system is solved by a fully parallel Newton-Krylov-Schwarz algorithm, where inexact Newton with backtracking as a nonlinear solver and an additive Schwarz preconditioned Krylov subspace type method such as GMRES is used to solve the corresponding Jacobian systems. While the time accuracy is not our concerns, the adaptability of time step size is a key ingredient for the success of the algorithm to speed up the time-marching process. Our numerical results obtained from a parallel machine shows that our parallel pseudo-transient continuation algorithm is very robust and efficient and also confirmed qualitatively the bifurcation with the numerical and numerical results found by other researchers. Furthermore, it is interesting to note that imperfect pitchfork bifurcations were observed especially for the case with a small expansion ratio, in which the happening of bifurcation points is delayed due to asymmetric unstructured meshes used for the numerical simulation.
