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Wall Shear Stress and Oscillatory Shear Index Calculations With Refined Meshes

We report the wall shear stress (WSS) and oscillatory shear index (OSI) calculations we carried out with different levels of fluid mechanics mesh refinement, using the special techniques we developed for WSS and OSI calculations. The lumen geometry used here was provided by Dr. Ryo Torii (University College London). 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], a multiscale sequentially-coupled arterial FSI technique [12], and special techniques for WSS and OSI calculations [13]. The lumen geometry was used earlier in simulations reported in [14, 15]. The inflow volumetric flow rate is calculated (see [13]) from the maximum-velocity waveform used in [14]. 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. The zero-pressure configuration and the wall thickness normalized by the wall thickness at the inflow can be seen at Cerebral Aneurysm -- High-Resolution Wall Shear Stress. The structural mechanics meshes are shown in Fig. 2. 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]). In the computations reported here, three fluid mechanics meshes are used: a coarse mesh, a medium mesh with such layered, high-refinement zones, and a fine mesh with higher refinement also over the arterial wall. The coarse and medium meshes have the same triangular mesh at the arterial wall. All three fluid mechanics meshes at the inflow plane are shown in Fig. 3. Fluid-structure interface for the fluid mechanics meshes is shown in Figs. 4 and 5. The layered, high-refinement meshes near the arterial wall give us higher-resolution WSS values (see Figs. 6-9). For more details on these computations, see [13].


Fig. 1. Volumetric flow rate and outflow pressure profile, with marks where the snapshots shown in Fig. 9 were taken. For details, see [13].


Fig. 2. Coarse and fine structural mechanics meshes when the outflow pressure is maximum. For details, see [13].


Fig. 3. Coarse, medium and fine fluid mechanics meshes at the inflow plane. For details, see [13].


Fig. 4. Fluid-structure interface for the coarse and medium fluid mechanics meshes. For details, see [13].


Fig. 5. Fluid-structure interface for the fine fluid mechanics mesh. For details, see [13].


Fig. 6. WSS for the coarse, medium and fine meshes when the volumetric flow rate is maximum. For details, see [13].


Fig. 7. Time-averaged WSS for the coarse, medium and fine meshes. For details, see [13].


Fig. 8. OSI for the coarse, medium and fine meshes. For details, see [13].


Fig. 9. Streamlines computed with the fine mesh art instants shown in Fig. 1. For details, see [13].

References

1. T.E. Tezduyar, "Stabilized Finite Element Formulations for Incompressible Flow Computations", Advances in Applied Mechanics, 28 (1992) 1-44, doi: 10.1016/S0065-2156(08)70153-4.

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, doi: 10.1016/0045-7825(92)90059-S.

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, doi: 10.1016/0045-7825(92)90060-W.

4. T.E. Tezduyar, "Computation of Moving Boundaries and Interfaces and Stabilization Parameters", International Journal for Numerical Methods in Fluids, 43 (2003) 555-575, doi: 10.1002/fld.505.

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, doi: 10.1016/j.cma.2004.09.014.

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, doi: 10.1002/fld.1430.

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, doi: 10.1002/fld.1443.

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, doi: 10.1002/fld.1633.

10. T.E. Tezduyar, M. Schwaab and S. Sathe, "Sequentially-Coupled Arterial Fluid-Structure Interaction (SCAFSI) Technique", Computer Methods in Applied Mechanics and Engineering, 198 (2009) 3524-3533, 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", International Journal for Numerical Methods in Biomedical Engineering, 26 (2010) 101-116, doi: 10.1002/cnm.1241.

12. T.E. Tezduyar, K. Takizawa, C. Moorman, S. Wright and J. Christopher, "Multiscale Sequentially-Coupled Arterial FSI Technique", Computational Mechanics, 46 (2010) 17-29, doi: 10.1007/s00466-009-0423-2.

13. K. Takizawa, C. Moorman, S. Wright, J. Christopher and T.E. Tezduyar, "Wall Shear Stress Calculations in Space-Time Finite Element Computation of Arterial Fluid-Structure Interactions", Computational Mechanics, 46 (2010) 31-41, doi: 10.1007/s00466-009-0425-0.

14. 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, doi: 10.1002/fld.1497.

15. 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, doi: 10.1007/s00466-008-0325-8.