For more information:
tezduyar@gmail.com
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Cerebral Aneurysm -- High-Resolution Wall Shear Stress
The lumen geometry used here was provided by Dr. Ryo Torii (Imperial College) and Professor Marie Oshima (University of Tokyo). It was extracted from the computed tomography model of a bifurcating segment of the middle cerebral artery of a 67 year-old female 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 [1-4], the quasi-direct FSI method [5, 6], the stabilized space-time FSI (SSTFSI) technique [7], and special techniques for arterial FSI computations [8-11]. The lumen geometry was used earlier in simulations reported in [12, 13]. The inflow volumetric flow rate is calculated (see [11]) from the maximum-velocity waveform used in [12]. The inflow volumetric flow rate is shown in Fig. 1. The inflow velocity profile (in space and time) is calculated (see [11]) based on this volumetric flow rate by using a special mapping technique introduced in [11]. The traction condition imposed at the outflow boundary during the cardiac cycle is based on a normal blood pressure profile, which is calculated (see [11]) by using the Windkessel model and the inflow volumetric flow rate.
The pressure profile is shown in Fig. 1. We use a variation of the "estimated zero-pressure arterial geometry" method that was introduced in [9]. In this variation (see [11]), we use variable wall thickness for the artery. Fig. 2 shows the zero-pressure configuration and the wall thickness normalized by the wall thickness at the inflow. The structural mechanics mesh is shown in Fig. 3. One of the special arterial FSI techniques used in the computation is generating layered, high-refinement meshes near the arterial walls to improve the boundary layer resolution (see [10, 11]). For the purpose of comparison, we also compute with a coarser mesh. The coarse and fine meshes have the same triangular mesh at the arterial wall, which is shown in Fig. 4. The coarse and fine fluid mechanics meshes at the inflow plane are shown in Fig. 5. The layered, high-refinement meshes near the arterial wall give us higher-resolution wall shear stress values (see Figs. 6-8). The computations were carried out on the ADA system at Rice University. For more details on these computations, see [11].
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| Fig. 1. Volumetric flow rate and outflow pressure profile, with the maximum value marked. For details, see [11]. |
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| Fig. 2. Zero-pressure surface configuration colored with normalized wall thickness. For details, see [11]. |
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| Fig. 3. Structural mechanics mesh. For details, see [11]. |
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| Fig. 4. Fluid mechanics mesh at the arterial wall. For details, see [11]. |
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| Fig. 5. Coarse and fine fluid mechanics meshes at the inflow plane. For details, see [11]. |
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| Fig. 6. Wall shear stress for the coarse mesh when the volumetric flow rate is maximum (left) and when the outflow pressure is maximum (right). For details, see [11]. |
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| Fig. 7. Wall shear stress for the fine mesh when the volumetric flow rate is maximum (left) and when the outflow pressure is maximum (right). For details, see [11]. |
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| Fig. 8. Time-averaged wall shear stress for the coarse (left) and fine (right) meshes. For details, see [11]. |
References
1. T.E. Tezduyar, "Stabilized Finite Element Formulations for Incompressible Flow Computations", Advances in Applied Mechanics, 28 (1992) 1-44.
2. T.E. Tezduyar, M. Behr and J. Liou, "A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces -- The Deforming-Spatial-Domain/Space-Time Procedure: I. The Concept and the Preliminary Numerical Tests", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 339-351.
3. T.E. Tezduyar, M. Behr, S. Mittal and J. Liou, "A New Strategy for Finite Element Computations Involving Moving Boundaries and Interfaces -- The Deforming-Spatial-Domain/Space-Time Procedure: II. Computation of Free-surface Flows, Two-liquid Flows, and Flows with Drifting Cylinders", Computer Methods in Applied Mechanics and Engineering, 94 (1992) 353-371.
4. T.E. Tezduyar, "Computation of Moving Boundaries and Interfaces and Stabilization Parameters", International Journal for Numerical Methods in Fluids, 43 (2003) 555-575.
5. T.E. Tezduyar, S. Sathe, R. Keedy and K. Stein, "Space-Time 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, CD-ROM (2004).
6. T.E. Tezduyar, S. Sathe, R. Keedy and K. Stein, "Space-Time Finite Element Techniques for Computation of Fluid-Structure Interactions", Computer Methods in Applied Mechanics and Engineering, 195 (2006) 2002-2027.
7. T.E. Tezduyar and S. Sathe, "Modeling of Fluid-Structure Interactions with the Space-Time Finite Elements: Solution Techniques", International Journal for Numerical Methods in Fluids, 54 (2007) 855-900.
8. T.E. Tezduyar, S. Sathe, T. Cragin, B. Nanna, B.S. Conklin, J. Pausewang and M. Schwaab, "Modeling of Fluid-Structure Interactions with the Space-Time Finite Elements: Arterial Fluid Mechanics", International Journal for Numerical Methods in Fluids, 54 (2007) 901-922.
9. T.E. Tezduyar, S. Sathe, M. Schwaab and B.S. Conklin, "Arterial Fluid Mechanics Modeling with the Stabilized Space-Time Fluid-Structure Interaction Technique", International Journal for Numerical Methods in Fluids, 57 (2008) 601-629.
10. T.E. Tezduyar, M. Schwaab and S. Sathe, "Sequentially-Coupled Arterial Fluid-Structure Interaction (SCAFSI) Technique", Computer Methods in Applied Mechanics and Engineering, published online, July 2008, DOI: 10.1016/j.cma.2008.05.024.
11. K. Takizawa, J. Christopher, T.E. Tezduyar and S. Sathe, "Space-Time Finite Element Computation of Arterial Fluid-Structure Interactions with Patient-Specific Data", Communications in Numerical Methods in Engineering, published online, March 2009, DOI: 10.1002/cnm.1241.
12. R. Torii, M. Oshima, T. Kobayashi, K. Takagi and T.E. Tezduyar, "Numerical Investigation of the Effect of Hypertensive Blood Pressure on Cerebral Aneurysm -- Dependence of the Effect on the Aneurysm Shape", International Journal for Numerical Methods in Fluids, 54 (2007) 995-1009.
13. R. Torii, M. Oshima, T. Kobayashi, K. Takagi and T.E. Tezduyar, "Fluid-Structure Interaction Modeling of a Patient-Specific Cerebral Aneurysm: Influence of Structural Modeling", Computational Mechanics, 43 (2008) 151-159.
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