Keywords: blood flow, computational fluid dynamics, finite element model.

The purpose of this paper is to develop a numerical computational methodology for steady blood flow simulation using the finite element method. Due to the complexity of the cardiovascular system simplifying assumptions for the mathematical modelling process are needed. For the steady flow case Himeno [1] showed that considering blood as an incompressible non-Newtonian fluid yields no change in the essential flow characteristics when compared to the Newtonian assumption. In diseased vessels which are often the subject of interest, the arteries are less compliant, wall motion is reduced and the assumption of zero wall motion is used.

The fluid flow is governed by the incompressible Navier-Stokes equations. To impose incompressibility, penalty methods are used and pressure is eliminated as a field variable since it can be recovered. If the standard Galerkin formulation is applied it is necessary to use compatible spaces for velocity and pressure in order to satisfy the Babuska-Brezzi stability condition [2,3]. In this work reduced integration is considered for the terms related with pressure. A smoothing technique allows getting continuous fields for pressure and deviatoric stresses.

In a Galerkin formulation there is no doubt that the most difficult problem arises because of the presence of nonlinear convective terms. The numerical scheme requires a stabilization technique in order to avoid oscillations in the numerical solution. In this study the streamline upwind Petrov-Galerkin method is used.

The developed model for blood flow simulation was applied to three dimensional flows in two idealized situations. First, it was considered a severe stenotic geometry providing some understanding of the genesis of atherosclerosis and other arterial lesions [1]. In the second example blood flow in an artery with two successive bends was simulated [4]. This study is important for understanding atherosclerotic plaques formation. Numerical results such as axial velocity and pressure fields are in good agreement with those given in the literature.

The outcomes will contribute to the characterization of the physiology of the human circulatory system and to detect turbulence caused by changes in flow velocities and vessel diameters. This research will continue aiming at the simulation of real vascular morphologies.

1

R. Himeno, "Blood Flow Simulation toward Actual Application at Hospital", in "The 5th Asian Computational Fluid Dynamics", Busan, Korea, n1-n6, 2003.

2

I. Babuska, J. Osborn, J. Pitkaranta, "Analysis of mixed methods using, mesh dependent norms", Math. Comp., 35, 1039-1062, 1980. doi:10.1090/S0025-5718-1980-0583486-7

3

I. Babuska, "Error bounds for finite element method", Numer. Math., 16, 322-333, 1971. doi:10.1007/BF02165003

4

H.W. Hoogstraten, J.G. Kootstra, B. Hillen, J.K.B. Krijger, P.J.W. Wensing, "Numerical Simulation of Blood flow in an artery with two successive bends", J. Biomechanics, 29, 1075-1083, 1996. doi:10.1016/0021-9290(95)00174-3

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