A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers

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陳信源(Sin-Yuan Chen) 查詢紙本館藏

畢業系所

數學系

論文名稱

(A Study for Linear Stability Analysis of Incompressible Flows on Parallel Computers)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

關鍵字(英)

★ incompressible flow
★ Navier-Stokes equations
★ bifurcation
★ parallel computing
★ linear stability analysis
★ generalized eigenvalue problem
★ pseudo transient continuation

論文目次

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

參考文獻

[1] Online CUBIT user's manual. http://cubit.sandia.gov/documentation.html.
[2] ParaView homepage. http://www.paraview.org.
[3] N. Alleborn, K. Nandakumar, H. Raszillier, and F. Durst. Further contributions on the two-dimensional flow in a sudden expansion. Journal of Fluid Mechanics, 330:169-188, 1997.
[4] S. Balay, K. Buschelman, W.D. Gropp, D. Kaushik, M.G. Knepley, L.C. McInnes, B.F. Smith, and H. Zhang. Petsc web page. http://www.mcs.anl.gov/petsc.
[5] F. Battaglia, S.J. Tavener, A.K. Kulkarni, and C.L. Merkle. Bifurcation of low Reynolds number flows in symmetric channels. AIAA Journal, 35:99-105, 1997.
[6] W. Cherdron, F. Durst, and JH Whitelaw. Asymmetric
flows and instabilities in symmetric ducts with sudden expansions. Journal of Fluid Mechanics Digital Archive, 84(01):13-31, 1978.
[7] K.A. Cli e, T.J. Garratt, and A. Spence. Eigenvalues of block matrices arising from problems in fuid mechanics. SIAM Journal on Matrix Analysis and Applications, 15:1310-1318, 1994.
[8] J. Demmel, J. Gilbert, and X. Li. Superlu homepage. http://www.cs.berkeley.edu/ dem-mel/SuperLU.html.
[9] D. Drikakis. Bifurcation phenomena in incompressible sudden expansion flows. Physics of Fluids, 9:76-87, 1997.
[10] F. Durst, J.C.F. Pereira, and C. Tropea. The plane symmetric sudden-expansion flow at low Reynolds numbers. Journal of Fluid Mechanics, 248:567-581, 1993.
[11] J. A. Elbanna, H.; Sabbagh. Interaction of two nonequal plane parallel jets. AIAA Journal, 25:12-13, 1987.
[12] R. M. Fearn, T. Mullin, and K. A. Cli e. Nonlinear flow phenomena in a symmetric sudden expansion. Journal of Fluid Mechanics, 211:595-608, 1990.
[13] L.P. Franca and S.L. Frey. Stabilized nite element methods. II: The incompressible Navier-Stokes
equations. Computer Methods in Applied Mechanics and Engineering, 99:209-233, 1992.
[14] A. Greenbaum. Iterative Methods for Solving Linear Systems. SIAM, Philadelphia, PA, 1997.
[15] S.K. Hannani, M. Stanislas, and P. Dupont. Incompressible Navier-Stokes computations with SUPG
and GLS formulations - a comparison study. Computer Methods in Applied Mechanics and Engineering, 124:153-170, 1995.
[16] T. Hawa and Z. Rusak. The dynamics of a laminar
ow in a symmetric channel with a sudden expansion. Journal of Fluid Mechanics, 436:283-320, 2001.
[17] V. Hernandez, J.E. Roman, and V. Vidal. SLEPc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software, 31:351-362, 2005.
[18] C.Y. Huang and F.N. Hwang. Parallel pseudo-transient newton-krylov-schwarz continuation algorithms for bifurcation analysis of incompressible sudden expansion
flows. Appl. Numer. Math., 60(7):738-751, 2010.
[19] M. Kadja and G. Bergeles. Numerical investigation of bifurcation phenomena occurring in flows through planar sudden expansions. Acta Mechanica, 153:47-61, 2002.
[20] G. Karypis. METIS homepage. http://cubit.sandia.gov/documentation.html.
[21] R.B. Lehoucq and A.G. Salinger. Large-scale eigenvalue calculations for stability analysis of steady flows on massively parallel computers. International Journal for Numerical Methods in Fluids, 36:309-327, 2001.
[22] K. Meerbergen, A. Spence, and D. Roose. Shift-invert and cayley transforms for the detection of eigenvalues with largest real part of nonsymmetric matrices. BIT, 34:409-423, 1994.
[23] David R. Miller and Edward W. Comings. Force-momentum elds in a dual-jet flow. Journal of Fluid Mechanics, 7:237-256, 1960.
[24] David R. Miller and Edward W. Comings Static pressure distribution in the free turbulent jet. Journal of Fluid Mechanics. Static pressure distribution in the free turbulent jet. Journal of Fluid Mechanics, 3:1-16, 1957.
[25] S. Mishra and K. Jayaraman. Asymmetric flows in planar symmetric channels with large expansion ratio. International Journal for Numerical Methods in Fluids, 38:945-962, 2002.
[26] J. Mizushima and Y. Shiotani. Structural instability of the bifurcation diagram for two-dimensional flow in a channel with a sudden expansion. Journal of Fluid Mechanics, 420:131-145, 2000.
[27] Y. Saad. Iterative Methods for Sparse Linear Systems. PWS, Boston, MA, 1996.
[28] Y. Saad and Martin H. Schultz. Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems. SIAM Journal on Scienti c and Statistical Computing, 7(3):856-869, 1986.
[29] M. Shapira, D. Degani, and D. Weihs. Stability and existence of multiple solutions for viscous flow in suddenly enlarged channels. Computing Fluids, 18(3):239-258, 1990.
[30] C.Y. Soong, P.Y. Tzeng, and C.D. Hsieh. Numerical investigation of flow structrue and bifurcation phenomena of con fined plane twin-jet flow. Physics of Fluids, 10:2909-2921, 1998.
[31] S.H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley Reading, 1994.
[32] T.E. Tezduyar. Stabilized nite element formulations for incompressible flow computations. Adv. Appl. Mech., 28:1-44, 1991.
[33] E.M. Wahba. Iterative solvers and inflow boundary conditions for plane sudden expansion flows. Appl. Math. Model., 31:2553-2563, 2007.

指導教授

黃楓南(Feng-Nan Hwang)

審核日期

2010-7-27

推文