一個好的模擬工具是可根據患者特殊解剖結構和生理狀況，在臨床上被使用於幫助醫師或學者們研究血管疾病以提高診斷並且計劃手術做法。在本論文中，我們著重於開發平行區域分解演算法，為解一描述流體在血管中的方程所離散化後的非線性系統，其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。特別地，在每個 time step 是用 Newton-Krylov-Schwarz algorithm 來解這樣一個非線性系統。我們使用 PETSc 套件來實現流體模擬工具的平行化 並且將它與其他軟體合併成為一個平行化的血流模擬系統，包括 Cubit 是用來產生網格、ParMETIS 是做網格分割、而 ParaView 則作為視覺化的工具。我們利用 a straight artery model 和 an end-to-side graft model 來驗證我們平行化的程式碼正確性並且研究其演算法的平行化處理效能。

A good simulation tool based on patient-specific anatomy and physiological conditions can be clinically used to help physicians or researchers to study vascular diseases, to enhance diagnoses, as well as to plan surgery procedures. In this paper, we focus on developing parallel domain decomposition algorithms for solving nonlinear systems arising from the discretization of blood flow model equations, where a stabilized finite element method is used for the spatial discretization, while an implicit backward Euler finite difference method for the temporal discretization. In particular, at each time step, the resulting system solved by the Newton-Krylov-Schwarz algorithm. We implement the parallel fluid solver using PETSc and integrate it with other software packages into a parallel blood flow simulation system, including Cubit, ParMETIS and ParaView for mesh generation, mesh partitioning, and visualization, respectively. We validated our parallel code and investigated the parallel performance of our algorithms for both a straight artery model and an end-to-side graft model.

[1] Online cubit users manual, 2009. http://cubit.sandia.gov/documentation.html.
[2] Paraview homepage, 2009. http://www.paraview.org/.
[3] S. Balay, K. Buschelman, W. D. Gropp, D. Kaushik, M.G. Knepley, L. Curfman
McInnes, B. F. Smith, and H. Zhang. PETSc Web page, 2009.
http://www.mcs.anl.gov/petsc.
[4] C.G. Caro, J.M. Fitz-Gerald, and R.C. Schroter. Atheroma and arterial wall shear
observation, correlation and proposal of a shear dependent mass transfer mechanism
for atherogenesis. Proceedings of the Royal Society of London. Series B, Biological
Sciences, 177:109–133, 1971.
[5] 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–
1284, 1980.
[6] J.E. Dennis and R.B. Schnabel. Numerical Methods for Unconstrained Optimization
and Nonlinear Equations. Society for Industrial and Applied Mathematics, 1996.
[7] 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.
[8] D.L. FRY. Certain histological and chemical responses of the vascular interface to
acutely induced mechanical stress in the aorta of the dog. Circulation Research,
24(1):93–108, 1969.
[9] K. Schloegel V. Kumar S. Shekhar G. Karypis, R. Aggarwal. Metis home page,
2009. http://wwwusers.cs.umn.edu/karypis/metis/.
[10] M.D. Gunzburger. Finite Element Methods for Viscous Incompressible Flows: A
Guide to Theory, Practice, and Algorithms. Academic Press, 1989.
[11] C. Johnson and C. Johnson. Numerical Solution of Partial Differential Equations by
the Finite Element Method. Cambridge University Press, 1987.
[12] 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, 113:458–463, 1991.
[13] 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.
[14] 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.
[15] 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, 123:80–87, 2001.
[16] R. Lohner, J. Cebral, O. Soto, P. Yim, and J.E. Burgess. Applications of patientspecific
CFD in medicine and life sciences. International Journal for Numerical
Methods in Fluids, 43:637–650, 2003.
[17] F. Loth. Velocity and wall shear measurements inside a vascular graft model under
steady and pulsatile flow conditions. PhD thesis, Georgia Institute of Technology,
1993.
[18] 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, 119:187–194,
1997.
[19] J. Nocedal and S.J. Wright. Numerical Optimization. Springer, 1999.
[20] J.N. Reddy and D.K. Gartling. The Finite Element Method in Heat Transfer and
Fluid Dynamics. CRC press, 2001.
[21] 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.
[22] 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.
[23] S.S. White, C.K. Zarins, D.P. Giddens, H. Bassiouny, F. Loth, S.A. 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, 115:104–111, 1993.
[24] M. Zamir. The Physics of Pulsatile Flow. Springer, 2000.