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Cerebral Aneurysm  Hyperelastic Arterial Wall
One of the major computational challenges in cardiovascular fluid mechanics is accurate modeling of the fluidstructure interaction (FSI) between the blood flow and arterial walls. The blood flow depends on the arterial geometry, and the deformation of the arterial wall depends on the blood flow. The mathematical equations governing the blood flow and arterial deformations need to be solved simultaneously, with proper kinematic and dynamic conditions coupling the two physical systems.
The arterial geometry used here (see Fig. 1) is a close approximation to the computed tomography (CT) model of a singleartery segment of the middle cerebral artery of a 57 yearold male with aneurysm. The arterial wall (i.e. the structural mechanics part of the problem) is modeled with the continuum element made of hyperelastic (Fung) material.
The numerical methods used in this computation were introduced and implemented on parallel computing platforms by the T*AFSM. The set of numerical methods introduced by the T*AFSM over the years and used in this computation includes the DSD/SST formulation [14], the quasidirect FSI method [5, 6], the stabilized spacetime FSI (SSTFSI) technique [7], and special techniques for arterial FSI computations [8, 9]. The CT model of the artery approximated in this computation was reported in [10]. The inflow velocity used in the computation during the cardiac cycle is a close approximation to the one reported in [11], and is shown below. The traction condition imposed at the outflow boundary during the cardiac cycle is based on a normal blood pressure profile, which is also shown below. One of the special arterial FSI techniques used in the computation is the "estimated zeropressure arterial geometry" method introduced in [9]. The computation was carried out on the ADA system at Rice University. For more details on this computation, see [9].

Fig. 1. Arterial geometry. For details, see [9]. 


Fig. 2. Inflowvelocity and bloodpressure profiles for the cardiac cycle. For details, see [9]. 

Fig. 3. Estimated zeropressure arterial geometry. For details, see [9]. 




Fig. 4. Arterial geometry at instants shown in Fig. 2. For details, see [9]. 




Fig. 5. Bloodflow patterns at instants shown in Fig. 2. For details, see [9]. 
References
1. T.E. Tezduyar, "Stabilized Finite Element Formulations for Incompressible Flow Computations", Advances in Applied Mechanics, 28 (1992) 144.
2. T.E. Tezduyar, M. Behr and J. Liou, "A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces  The DeformingSpatialDomain/SpaceTime Procedure: I. The Concept and the Preliminary Numerical Tests", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 339351.
3. T.E. Tezduyar, M. Behr, S. Mittal and J. Liou, "A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces  The DeformingSpatialDomain/SpaceTime Procedure: II. Computation of Freesurface Flows, Twoliquid Flows, and Flows with Drifting Cylinders", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 353371.
4. T.E. Tezduyar, "Computation of Moving Boundaries and Interfaces and Stabilization Parameters", International Journal for Numerical Methods in Fluids, 43 (2003) 555575.
5. T.E. Tezduyar, S. Sathe, R. Keedy and K. Stein, "SpaceTime Techniques for Finite Element Computation of Flows with Moving Boundaries and Interfaces", Proceedings of the III International Congress on Numerical Methods in Engineering and Applied Sciences, Monterrey, Mexico, CDROM (2004).
6. T.E. Tezduyar, S. Sathe, R. Keedy and K. Stein, "SpaceTime Finite Element Techniques for Computation of FluidStructure Interactions", Computer Methods in Applied Mechanics and Engineering, 195 (2006) 20022027.
7. T.E. Tezduyar and S. Sathe, "Modeling of FluidStructure Interactions with the SpaceTime Finite Elements: Solution Techniques", International Journal for Numerical Methods in Fluids, 54 (2007) 855900.
8. T.E. Tezduyar, S. Sathe, T. Cragin, B. Nanna, B.S. Conklin, J. Pausewang and M. Schwaab, "Modeling of FluidStructure Interactions with the SpaceTime Finite Elements: Arterial Fluid Mechanics", International Journal for Numerical Methods in Fluids, 54 (2007) 901922.
9. T.E. Tezduyar, S. Sathe, M. Schwaab and B.S. Conklin, "Arterial Fluid Mechanics Modeling with the Stabilized SpaceTime FluidStructure Interaction Technique", International Journal for Numerical Methods in Fluids, 57 (2008) 601629.
10. R. Torii, M. Oshima, T. Kobayashi, K. Takagi and T.E. Tezduyar, "Influence of Wall Elasticity in PatientSpecific Hemodynamic Simulations", Computers & Fluids, 36 (2007) 160168.
11. R. Torii, M. Oshima, T. Kobayashi, K. Takagi and T.E. Tezduyar, "Computer Modeling of Cardiovascular FluidStructure Interactions with the DeformingSpatialDomain/Stabilized SpaceTime Formulation", Computer Methods in Applied Mechanics and Engineering, 195 (2006) 18851895.
