模擬血液在血管裡的行為有助於醫療人員或研究學者對血管疾病掌握更多資訊，並降低手術時的風險。在這篇論文中，我們使用Power-law, Bingham, Carreau-Yasuda 模型來模擬非牛頓流體在二維的幾何圖形Backward-facing step、Four-to-One Contraction、Rotational eccentric annulus flow，與三維的幾何圖形A long straight artery、An end-to-side graft，以及針對個別病患所造出多分支的血管上流體的行為。在離散化方面，對空間上的離散是使用stabilized finite element method，而時間上的離散則是使用implicit backward Euler finite difference method，在每個time step 是用Newton-Krylov-Schwarz algorithm 來解這樣一個非線性系統。而為了幫助我們模擬更複雜的幾何形狀與加快其計算的時間，採用 Two-level methods。最後，我們還計算的壁上的剪應力，以便於醫學上的應用。

The simulation of the behavior of the blood in the arteries help medical personnel or researchers to acquire more information on vascular disease and reduce the risk of surgery. In this paper, we use the Power-law, Bingham, Carreau-Yasuda model to simulate non-Newtonian fluid in a two-dimensional geometry of Backward-facing step, Four-to-One Contraction, Rotational eccentric annulus flow， and three-dimensional geometry of A long Straight ARTERY, an end-to-side Graft, as well as for the individual patient create multiple branching vascular fluid behavior. In the discretization, where a stabilized finite element method is used for the spatial discretization, while an implicit backward Euler finite difference method for the temporal discretization. At each time step, the resulting system solved by the Newton-Krylov-Schwarz algorithm. In order to help us to simulate more complex geometry and speed up the calculation time, Two-level methods.Finally, we also calculated the wall of the shear stress, so that medical applications.

[1] C.A. Taylor, T.J.R. Hughes, and C.K. Zarins. Finite element modeling of blood flow in arteries. Computer Methods in Applied Mechanics and Engineering, 158(1-2):155–196, 1998.
[2] F. Loth, S.A. Jones, D.P. Giddens, H.S. Bassiouny, S. Glagov, and C.K. Zarins. Measurements of velocity and wall shear stress inside a PTFE vascular graft model under steady flow conditions. Journal of Biomechanical Engineering, 119:187, 1997.
[3] M. Lei, D.P. Giddens, F. Jones, S.A .and Loth, and H. Bassiouny. Pulsatile flow in an
End-to-Side Vascular Graft Model: Comparison of computations with experimental
data. Journal of Biomechanical Engineering, 123:80, 2001.
[4] M.R.Sadeghi A. Razavi, E.Shirani. Numerical simulation of blood pulsatile flow in a stenosed carotid artery using different rheological models. 2011.
[5] Christopher K. Zarins David N. Ku, Don P. Giddens and Seymour Glagov. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Positive correlation between plaque location and low oscillating shear stress. Arteriosclerosis, Thrombosis, and Vascular Biology, 5:293–302, 1985.
[6] DA Knoll and DE Keyes. Jacobian-free Newton–Krylov methods: a survey of approaches and applications. Journal of Computational Physics, 193(2):357–397, 2004.
[7] J.E. Dennis and R.B. Schnabel. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Society for Industrial Mathematics, 1996.
[8] J. Nocedal and S.J. Wright. Numerical Optimization. Springer, 1999.
[9] Online cubit users manual, 2009. http://cubit.sandia.gov/documentation.html.
[10] K. Schloegel V. Kumar S. Shekhar G. Karypis, R. Aggarwal. Metis home page,
2009. http://wwwusers.cs.umn.edu/karypis/metis/.
[11] Paraview homepage, 2009. http://www.paraview.org/.
85
[12] Rebecca Brannon. Mohrs circle and more circles. 2003.
[13] L.P. Franca and S.L. Frey. Stabilized finite element methods. II: The incompressible
Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering,
99(2-3):209–233, 1992.
[14] Y. Saad and M.H. Schultz. GMRES: A generalized minimal residual algorithm for
solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7(3):856–869,
1986.
[15] A. Klawonn and L.F. Pavarino. Overlapping Schwarz methods for mixed linear
elasticity and Stokes problems. Computer Methods in Applied Mechanics and Engineering,
165(1-4):233–245, 1998.
[16] Satish Balay, Kris Buschelman, William D. Gropp, Dinesh Kaushik, Matthew G. Knepley, Lois Curfman McInnes, Barry F. Smith, and Hong Zhang. PETSc Web page, 2009. http://www.mcs.anl.gov/petsc.
[17] Abaqus file format, 2009. http://www.simulia.com/.
[18] Visualization toolkit (vtk) homepage, 2009. http://www.vtk.org/.
[19] H.M. Crawshaw, W.C. Quist, E. Serrallach, C.R. Valeri, and F.W. LoGerfo. Flow disturbance at the distal end-to-side anastomosis: effect of patency of the proximal outflow segment and angle of anastomosis. Archives of Surgery, 115(11):1280, 1980.
[20] SS White, CK Zarins, DP Giddens, H. Bassiouny, F. Loth, SA Jones, and S. Glagov. Hemodynamic patterns in two models of end-to-side vascular graft anastomoses: effects of pulsatility, flow division, Reynolds number, and hood length. Journal of biomechanical engineering, 115:104, 1993.
[21] RS Keynton, SE Rittgers, and MCS Shu. The effect of angle and flow rate upon
hemodynamics in distal vascular graft anastomoses: an in vitro model study. Journal of biomechanical engineering, 113:458, 1991.
[22] F.N. van de Vosse F.J.H. Gijsen and J.D. Janssen. Wall shear stress in backwardfacing step flow of a red blood cell suspension. Biorheology, page 263V279, 1998. 86
[23] R.C.; Hassager O. Bird, R.B.; Armstrong. Dynamics of polymeric liquids. John Wiley and Sons Inc.,New York, NY, 1987.
[24] J.D. Janssen F.J.H. Gijsen, F.N. van de Vosse. The influence of the non-newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation
model. Journal of Biomechanics, pages 601–608, 1999.
[25] F. Loth. Velocity and wall shear measurements inside a vascular graft model under steady and pulsatile flow conditions. 1993.