臺灣博碩士論文加值系統

　　本研究使用廣義有限差分法分析二維黏性不可壓縮流場，並配合共享記憶體多執行緒架構的平行化程式演算以加速模式計算效率。論文中以數個案例證明本研究所提出之無網格計算方法的可行性與準確性，並綜合比較分析程式平行化之後的平行效率。本研究中使用的數學控制方程式為速度—渦度方程式，此方程式為奈維爾—史托克斯方程式的另一種形式，解決了原始變數法中壓力項與未知壓力邊界條件的處理問題。對二維流體流動的問題而言，速度渦度方程式中方程式與未知數的個數皆與原始變數法相同，因此速度—渦度方程式是極具競爭力的一組控制方程式。
　　傳統上在進行流場的模擬時，大部分是使用有限差分法或有限體積法等須建立網格的方法，而在空間離散方法上，本研究採用最新發展之廣義有限差分法，此方法為一種局部化的無網格法，除了可避免建立複雜的網格及數值積分外，由於此方法的局部化特性，使得離散控制方程式後所產生的線性系統會是一個稀疏矩陣，與傳統無網格法所產生的滿矩陣特性不同，在本研究中也成功的將此方法所組成的矩陣儲存為稀疏矩陣，並以可平行化的非對稱稀疏矩陣直接法函式庫準確且快速求得數值解。除此之外，藉由移動最小二乘法的推導，任意點上的空間微分項可以表示為鄰近點物理量的線性權重累加，因此在程式撰寫上非常簡單方便，並且可以避免一般無網格法常見的病態矩陣問題。另一方面，本論文採用隱式尤拉法進行時間軸的積分，減輕對於時間步長的嚴格要求，並能穩定地獲得不同時間階層的演化過程。而在分別採用廣義有限差分法與隱式尤拉法對方程式的空間與時間軸離散之後，在每個時刻都會產生一組非線性代數方程組，本研究利用牛頓法解決此一問題，牛頓法的收斂速率是二次式，是非常有效率的演算方法。
　　本論文同時使用廣義有限差分法、隱式尤拉法與牛頓法求解以速度—渦度方程式為控制方程式的二維黏性不可壓縮流場問題，並採用數個案例驗證模式的可行性與結果的準確性，案例包含有方形穴室流場、長方形穴室流場、交錯雙方形穴室流場等，本論文所計算的結果皆與前人研究相似，因此可證明本文所提出模式求解此問題的可行性；而模式中的不同參數也用一系列的數值實驗探討其影響性，例如：不同總點數、不同子區域點數、不同時間步長等，所獲得的結果都可以證明本研究所提出模式的穩定性、準確性及一致性。
　　在本論文中，除了建立上述一套以無網格法為基礎的模擬模式之外，同時也藉由OpenMP將程式改為平行化計算，縮短在單一共享式記憶體多處理器架構的電腦上演算的時間，並測試本模式平行化後的計算效率。由測試結果可知，本研究所使用的數值模式平行化後的計算效率有明顯提升，並且以四個執行緒的計算效率最佳。

In this study, we use the generalized finite difference method (GFDM) to simulate the two-dimensional fluid fields of incompressible viscous fluid. In order to accelerate the computational speed, we parallelize our program on a multithreading computer which has shared memory architecture. To validate the feasibility, accuracy and the parallel efficiency of the meshless method proposed in this article, we provided three numerical examples. The velocity-vorticity formulation are solved in this study and it is another form of the standard Navier-Stokes equations. The problem of the pressure term and the pressure boundary condition in the primary-variable formulation of the Navier-Stokes equations can be overcame. In addition, the number of the unknown variables in the velocity-vorticity formulation is the same as the primary-variable formulation in two-dimensional fluid fields. Therefore, the velocity-vorticity formulation has great potential to be used in engineering applications.
The methods, used for flow fields, usually require grids in the computational domain, such as the finite difference method and the finite volume method. Instead, we adopt the GFDM, a novel meshless method. It can get rid of mesh generation and numerical quadrature as well as construct a sparse matrix by introducing the localized concept. Moreover, we used the parallelizable non-symmetric sparse matrix direct solver to solve the resultant linear algebraic system. The GFDM is based on the moving-least-square method to approximate the derivatives at every node in computational domain by linear summation of nodal values. Therefore, the programming by using the GFDM is quite easy and convenient and can reduce the probability of ill-conditioned matrix. On the other hand, we use the implicit Euler method on the time quadrature, so we can adopt larger time step to acquire the stable result. To apply the above methods will result in non-linear algebraic equations. We solve it by the Newton’s method because its convergence is at least quadratic.
In this paper, we analyze the two-dimensional flow fields of incompressible viscous fluid, described by the velocity-vorticity formulation, by using the GFDM, the implicit Euler method and the Newton’s method. To verify the accuracy and the stability of our model, we simulated some examples include cavity flow, rectangular cavity flow, and staggered double cavity flow. The results of these examples are very similar to previous work, so we can deduce that our model has the ability for accurately solving flow-field problems. We also examine the influence of the variables in our model, such as, the number of total nodes, the number of nodes in the sub-domain and the length of time steps. All these results and comparisons can verify the accuracy, consistency, and stability of our model.
We also parallelized the developed model by OpenMP and test the efficiency on a shared memory computer. According to the results, our parallelized model provided a satisfying efficiency when four threads are used.

摘要 I
Abstract II
目錄 IV
圖目錄 VI
表目錄 X
第一章 導論 1
1.1研究目的 1
1.2文獻回顧 1
1.2.1速度—渦度方程式 1
1.2.2廣義有限差分法 3
1.2.3平行計算 4
1.3 參考文獻 6
第二章 物理問題與控制方程式 14
2.1奈維爾—史托克斯方程式 14
2.2速度—渦度方程式 17
2.3控制方程式之無因次化 18
2.4邊界條件 19
2.5 參考文獻 20
第三章 數值方法 22
3.1廣義有限差分法 22
3.1.1有限差分法 22
3.1.2廣義有限差分法 24
3.2隱式尤拉法 29
3.2.1隱式尤拉法 29
3.2.2牛頓法 30
3.3 LU分解法 32
3.3.1 LU分解法 32
3.3.2 Multithreaded SuperLU 34
3.4 參考文獻 37
第四章 平行演算法 39
4.1 OpenMP程式架構與常用指令 39
4.2 OpenMP指令應用範例 41
4.3平行化運算流程 43
4.3.1 第一部分：廣義有限差分法權重計算與行壓縮稀疏矩陣預處理 43
4.3.2 第二部分：速度—渦度方程式數值演算 44
4.4參考文獻 44
第五章 數值算例結果與比較 53
5.1方形穴室流場 53
5.2長方形穴室流場 55
5.3交錯雙方形穴室流場 56
5.4參考文獻 58
第六章 平行化效率 95
6.1二維簡單例題測試 95
6.2方形穴室流場 96
6.3長方形穴室流場 97
6.4交錯雙方形穴室流場 97
6.5參考文獻 98
第七章 結論與建議 120
7.1結論 120
7.2建議 121
7.3參考文獻 121
個人簡歷 122

J. H. Ferziger and M. Peric, Computational Methods for Fluid Dynamics, 3rd ed. Springer-Verlag, New York, 1996.
X. S. Li, J. W. Demmel, J. R. Gilbert, L. Grigori, M. Shao and I. Yamazaki, SuperLU Users’ Guide, 2011.
C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, New York, 1997.
丁婉玉(2014)，以廣義有限差分法求解二維奈維爾—史托克斯方程式及強制熱對流問題，國立臺灣海洋大學河海工程學系碩士學位論文。
A. L. F. Lima E Silva, A. Silveira-Neto and J. J. R. Damasceno, Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method, Journal of Computational Physics, vol. 189, pp. 351-370, 2003.
E. Erturk, Numerical solutions of 2-D steady incompressible flow over a backward-facing step, Part I: High Reynolds number solutions, Computers and Fluids, vol. 37, pp. 633-655, 2008.
D. Kim and H. Choi, A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids, Journal of Computational Physics, vol. 162, pp. 411-428, 2000.
G. X. Wu and Z. Z. Hu, Numerical simulation of viscous flow around unrestrained cylinders, Journal of Fluids and Structures, vol. 22, pp. 371-390, 2006.
C. H. Liu and D. Y. C. Leung, Development of a finite element solution for the unsteady Navier–Stokes equations using projection method and fractional-θ-scheme, Computer Methods in Applied Mechanics and Engineering, vol. 190, pp. 4301-4317, 2001.
A. Masud and R. A. Khurram, A multiscale finite element method for the incompressible Navier–Stokes equations, Computer Methods in Applied Mechanics and Engineering, vol. 195 pp.1750-1777, 2006.
B. Klein, F. Kummer and M. Oberlack, A SIMPLE based discontinuous Galerkin solver for steady incompressible flows, Journal of Computational Physics, vol. 237, pp. 235-250, 2013.
M. Ramšak, L. Škerget, M. Hriberšek and Z. Žunič, A multidomain boundary element method for unsteady laminar flow using stream function-vorticity equations, Engineering Analysis with Boundary Elements, vol. 29, pp.1-14, 2005.
K. Kovářík, J. Mužík and D. Sitányiová, A fractional step local boundary integral element method for unsteady two-dimensional incompressible flow, Engineering Analysis with Boundary Elements, vol. 44, pp. 55-63, 2014.
H. Ding, C. Shu, K.S. Yeo and D. Xu, Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method, Computer Methods in Applied Mechanics and Engineering, vol. 193, pp. 727-744, 2004.
Y. Kim, D. W. Kim, S. Jun and J. H. Lee, Meshfree point collocation method for the stream-vorticity formulation of 2D incompressible Navier–Stokes equations. Computer Methods in Applied Mechanics and Engineering, vol. 196, pp. 3095-3109, 2007.
M. Lashckarbolok and E. Jabbari, Collocated discrete least squares (CDLS) meshless method for the streamfunction-vorticity formulation of 2D incompressible Navier-Stokes equations, Scientia Iranica A, vol. 19(6), pp. 1422-1430, 2012.
C. A. Bustamante, H. Powe and W. F. Florez, A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations, Computers and Mathematics with Applications vol. 65, pp. 1939-1955, 2013.
楊啟宏(2014)，以局部化徑向基底函數配點法求解二維柏格斯方程式、速度—渦度方程式及自然熱對流問題，國立臺灣海洋大學河海工程學系碩士學位論文。
H. Fasel, Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations, Journal of Fluid Mechanics, vol. 78, pp. 355-383, 1976.
S. C. R. Dennis and J. D. Hudson, Methods of solution of the velocity-vorticity formulation of the Navier-Stokes equations, Journal of Computational Physics, vol. 122, pp. 300-306, 1995.
W. E and J. G. Liu, Vorticity boundary condition and related issues for finite difference schemes, Journal of Computational Physics, vol. 124, pp. 368-382, 1996.
W. E and J. G. Liu, Finite difference methods for 3D viscous incompressible flows in the vorticity-vector potential formulation on nonstaggered grids, Journal of Computational Physics, vol. 138, pp. 57-82, 1997.
H. L. Meitz and H. F. Fasel, A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation, Journal of Computational Physics, vol. 157, pp. 371-403, 2000.
G. Guevremont, W. G. Habashi and M. M. Hafez, Finite element solution of the Navier-Stokes equations by a velocity-vorticity method, International Journal for Numerical Methods in Fluids, vol. 10, pp. 461-475, 1990.
G. Guevremont, W. G. Habashi, P. L. Kotiuga and M. M. Hafez, Finite element solution of the 3D compressible Navier-Stokes equations by a velocity-vorticity method, Journal of Computational Physics, vol. 107, pp. 176-187, 1993.
D. C. Lo and D. L. Young, Two-dimensional incompressible flows by velocity-vorticity formulation and finite element method, Chinese Journal of Mechanical Engineering, vol. 17, pp. 13-20, 2001.
D. C. Lo and D. L. Young, Arbitrary Lagrangian-Eulerian finite element analysis of free surface flow using a velocity-vorticity formulation, Journal of Computational Physics, vol. 195, pp. 175-201, 2004.
G. C. Bourantas, E. D. Skouras, V. C. Loukopoulos and G. C. Nikiforidis, Meshfree point collocation schemes for 2D steady state incompressible Navier-Stokes equations in velocity-vorticity formulation for high values of Reynolds number, Computer Modeling in Engineering and Sciences, vol. 59, pp. 31-63, 2010.
C. A. Wang, H. Sadat and C. Prax, A new meshless approach for three dimensional fluid flow and related heat transfer problems, Computers &;Fluids, vol. 69, pp. 136-146, 2012.
A. Bogomolny, Fundamental solutions method for elliptic boundary-value problems, SIAM Journal on Numerical Analysis, vol. 22, no. 4, pp. 644-669, 1985.
K. Murugesan and H. Okamoto, The collocation points of the fundamental solutions method for the potential problems, Computers and Mathematics with Applications, vol. 31, pp. 123-137, 1996.
A. Poullikkas, A. Karageorghis and G. Georgiou, The numerical solution of three-dimensional Signorini problems with the method of fundamental solutions, Engineering Analysis with Boundary Elements, vol. 25, pp. 221-227, 2001.
D. L. Young, C. W. Chen, C. M. Fan, K. Murugesan and C. C. Tsai, The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders, European Journal of Mechanics B/Fluids, vol. 24, pp. 703-716, 2005.
C. S. Liu, A modified Trefftz method for two-dimensional Laplace equation considering the domain’s characteristic length, Computer Modeling in Engineering and Sciences, vol. 21, pp. 53-65, 2007.
C. S. Liu, W. Yeih and S. N. Atluri, On Solving the ill-conditioned system Ax=b: general-purpose conditioners obtained from the boundary-collocation solution of the Laplace equation, using Trefftz expansions with multiple length scales, Computer Modeling in Engineering and Sciences, vol. 44, pp. 281-311, 2009.
W. Yeih, C. S. Liu, C. L. Kuo and S. N. Atluri, On solving the direct/inverse Cauchy problems of Laplace equation in a multiply connected domain, using the generalized multiple-source-point boundary-collocation Trefftz method &; characteristic lengths, Computers Materials &; Continua, vol. 17, pp. 275-302, 2010.
C. M. Fan and H. H. Li, Solving the inverse Stokes problems by the modified collocation Trefftz method and Laplacian decomposition, Applied Mathematics and Computation, vol. 219, pp. 6520-6535, 2013.
C. M. Fan and H. F. Chan, The modified collocation Trefftz method and exponentially convergent scalar homotopy algorithm for the inverse boundary determination problem for the biharmonic equation, Journal of Mechanics, vol. 29, pp. 363-372, 2013.
S. M. Wong, Y.C. Hon and T.S. Li, A meshless multilayer model for a costal system by radial basis function, Computers and Mathematics with Applications, vol. 43, 585-605, 2002.
C. K. Lee, X. Liu and S. C. Fan, Local multiquadric approximation for solving boundary value problems, Computational Mechanics, vol. 30, pp. 396-409, 2003.
R. Vertnik and B. Šarler, Solution of incompressible turbulent flow by a mesh-free method, Computer Modeling in Engineering and Sciences, vol.44, no.1, pp.65-95, 2009.
G. Kosec and B. Šarler, Local RBF collocation method for Darcy flow, Computer Modeling in Engineering &; Sciences, vol. 25, pp. 197-207, 2008.
E. Divo and A. J. Kassab, An efficient localized radial basis function meshless method for fluid flow and conjugate heat transfer, ASME Journal of Heat Transfer, vol. 129, pp. 124-136, 2007.
H. F. Chan and C. M. Fan, The local radial basis function collocation method for solving two-dimensional inverse Cauchy problems, Numerical Heat Transfer, Part B, vol. 63, pp. 284-303, 2013.
C. M. Fan, C. S. Chien, H. F. Chan and C. L. Chiu, The local RBF collocation method for solving the double-diffusive natural convection in fluid-saturated porous media, International Journal of Heat and Mass Transfer, vol. 57, pp. 500-503, 2013.
C. M. Fan, C. H. Yang and M. H. Gu, Applications of the local RBF collocation method and the fictitious time integration method for Burgers’ equations, Procedia Engineering, vol. 79, pp. 569-574, 2014.
J. J. Benito, F. Urena and L. Gavete, Influence of several factors in the generalized finite difference method, Applied Mathematical Modelling, vol. 25, pp. 1039-1053, 2001.
L. Gavete, M. L. Gavete and J. J. Benito, Improvements of generalized finite difference method and comparison with other meshless method, Applied Mathematical Modelling, vol. 27, pp. 831-847, 2003.
J. J. Benito, F. Urena, L. Gavete and B. Alonso, Application of the generalized finite difference method to improve the approximated solution of pdes, Computer Modeling in Engineering and Sciences, vol. 38, pp. 39-58, 2008.
F. Urena, E. Salete, J. J. Benito and L. Gavete, Solving third and fourth order partial differential equations using GFDM: application to solve problems of plates, International Journal of Computer Mathematics, vol. 89, pp. 366-376, 2012.
H. F. Chan, C. M. Fan and C. W. Kuo, Generalized finite difference method for solving two-dimensional nonlinear obstacle problem, Engineering Analysis with Boundary Elements, vol. 37, pp. 1189-1196, 2013.
C. M. Fan and P. W. Li, Generalized finite difference method for solving two-dimensional Burgers’ equations, Procedia Engineering, vol. 79, pp. 55-60, 2014.
C. M. Fan, P. W. Li and W. Yeih, Generalized finite difference method for solving two-dimensional inverse Cauchy problems, Inverse Problems in Science and Engineering, vol. 23, pp. 737-759, 2015.
P. W. Li, C. M. Fan, C. Y. Chen and C. Y. Ku, Generalized finite difference method for numerical solutions of density-driven groundwater flows, Computer Modeling in Engineering and Sciences, vol. 101(5), pp. 319-350, 2014.
洪銘澤(2003)，CPU平行粒子群最佳化應用於平面桁架結構最佳化設計，國立交通大學土木工程系碩士論文。
林俊宏(2012)，以有限體積法分析二維淺水波方程式並評估其平行化效能，國立臺灣海洋大學河海工程學系碩士學位論文。
J. Hoeflinger, P. Alavilli, T. Jackson and B. Kuhn, Producing scalable performance with OpenMP: Experiments with two CFD applications, Parallel Computing, vol. 27, pp. 391-413, 2001.
W. Wei, O. al-Khayat and X. Cai, An OpenMP-enabled parallel simulator for particle transport in fluid flows, Procedia Computer Science, vol. 4, pp. 1475-1484, 2011.
R. D. Sandberg, Direct numerical simulations for flow and noise studies, Procedia Engineering, vol. 61, pp. 356-362, 2013.
R. Rossi, A. Larese, P. Dadvand and E. Oñate, An efficient edge-based level set finite element method for free surface flow problems, International Journal for Numerical Methods in Fluids, vol. 71, pp. 687-716, 2013
C. Couder-Castañeda, J. C. Ortiz-Alemán, M. G. Orozco-del-Castillo and M. Nava-Flores, Forward modeling of gravitational fields on hybrid multi-threaded cluster, Geofísica Internacional, vol. 54, pp. 31-48, 2015.
G. Kosec and M. Depolli, A. Rashkovska, R. Trobec, Super linear speedup in a local parallel meshless solution of thermo-fluid problems, Computer and Structures, vol. 133, pp. 30-38, 2014.
R. W. Fox, P. J. Pritchard and A. T. McDonald, Introduction to Fluid Mechanics, 7th ed. John Wiley &; Sons, Inc. 2008.
T. C. Papanastasiou, G. C. Georgiou and A. N. Alexandrou, Viscous Fluid Flow, CRC Press, United States of America.
彭國倫(2001)，FORTRAN95程式設計，碁峰資訊，台北市。
W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in Fortran 77: the art of scientific computing, Cambridge University Press, United States of America, 1986.
R. Bronson, G. B. Costa and J. T. Saccoman, Linear Algebra 3rd ed., Elsevier, United States of America, 2014.
J. W. Demmel, J. R. Gilbert and X. S. Li, An asynchronous parallel supernodal algorithm for sparse Gaussian elimination. SIAM J. Matrix Analysis and Applications, vol. 20(4), pp. 915-952, 1999.
M. Hermanns, Parallel programming in Fortran 95 using OpenMP, School of Aeronautical Engineering, Spain, 2002.
D. C. Lo, D. L. Young and K. Murugesan, GDQ method for natural convection in a square cavity using velocity-vorticity formulation, Numerical Heat Transfer, Part B, vol. 47, pp. 321-341, 2005.
C. Shu, H. Ding and K. S. Yeo, Computation of incompressible Navier-Stokes equations by local RBF-based differential quadrature method, Computer Modeling in Engineering &; Sciences, vol. 7, pp. 195-205, 2005.
Y. Shang, Y. He, D. W. Kim and X. Zhou, A new parallel finite element algorithm for the stationary Navier–Stokes equations, Finite Elements in Analysis and Design, vol. 47, pp. 1262-1279, 2011.
M. M. Gupta and J. C. Kalita, A new paradigm for solving Navier–Stokes equations: streamfunction–velocity formulation, Journal of Computational Physics, vol. 207, Issue 1, pp. 52-68, 2005.
賴威翔(2015)，以局部化徑向基底函數配點法模擬流場相關問題與工程應用，國立臺灣海洋大學河海工程學系碩士學位論文。
Y. C. Zhou, B. S. V. Patnaik, D. C. Wan and G. W. Wei, DSC solution for flow in a staggered double lid driven cavity, International Journal for Numerical Methods in Fluids, vol. 57, pp.211-234, 2003.
S. H. Meraji, A. Ghaheri1 and P. Malekzadeh, An efficient algorithm based on the differential quadrature method for solving Navier–Stokes equations, International Journal for Numerical Methods in Fluids, vol. 71, pp.422-445, 2012.
C. C. Douglas, G. Haase and U. Langer, A Tutorial on Elliptic PDE Solvers and Their Parallelization, Society for Industrial and Applied Mathematics, Philadelphia, United States of America, 2003.