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Cerebral Aneurysm  Variable Wall Thickness, High Blood Pressure
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 membrane element.
The computations highlighted here are some of the earliest cardiovascular FSI computations carried out by the T*AFSM. The numerical methods used in these computations 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 these computations 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]. The CT model of the artery approximated in these computations was reported in [9]. The inflow velocity used in the computations during the cardiac cycle is a close approximation to the one reported in [10], and is shown below. We consider normal and high blood pressure profiles (NBP and HBP), which are also shown below, in combination with uniform and variable wall thicknesses (UWT and VWT). In VWT, the wall thickness for the aneurysm is 1/3 of the wall thickness for the rest of the artery segment. The computations were carried out on the ADA system at Rice University. For more details on these computations, see [8].

Fig. 1. Arterial geometry. For details, see [8]. 


Fig. 2. Inflowvelocity and bloodpressure profiles for the cardiac cycle. For details, see [8]. 

Fig. 3. Bloodflow patterns at an instant during the cardiac cycle, obtained from the NBPUWT computation. For details, see [8]. 

Fig. 4. Bloodflow patterns at an instant during the cardiac cycle, obtained from the NBPUWT computation. For details, see [8]. 
  
Fig. 5. Arterial shape at three different instants during the cardiac cycle, obtained from the NBPUWT computation. For details, see [8]. 

Fig. 6. Deformation of the aneurysm for all four cases. The order of the cases from minimum to maximum deformation is NBPUWT, HBPUWT, NBPVWT and HBPVWT. For details, see [8]. 
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. 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.
10. 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.
