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Coupling strategies for the numerical simulation of blood flow in deformable arteries by 3D and 1D models. (English) Zbl 1171.76436

Summary: The fluid structure interaction mechanism in vascular dynamics can be described by either 3D or 1D models, depending on the level of detail of the flow and pressure patterns needed for analysis. A successful strategy that has been proposed in the past years is the so-called geometrical multiscale approach, which consists of coupling both 3D and 1D models so as to use the former only in those regions where details of the fluid flow are needed and describe the remaining part of the vascular network by the simplified 1D model.In this paper we review recently proposed strategies to couple the 3D and 1D models, and within the 3D model, to couple the fluid and structure sub-problems. The 3D/1D coupling strategy relies on the imposition of the continuity of flow rate and total normal stress at the interface. On the other hand, the fluid-structure coupling strategy employs Robin transmission conditions. We present some numerical results and show the effectiveness of the new approaches.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
92C35 Physiological flow
76Z05 Physiological flows
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[1] Badia, S.; Nobile, F.; Vergara, C., Fluid-structure partitioned procedures based on Robin transmission conditions, J. Comput. Phys., 227, 14, 7027-7051 (2008) · Zbl 1140.74010
[2] Causin, P.; Gerbeau, J-F.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Engrg., 194, 4506-4527 (2005) · Zbl 1101.74027
[3] Deparis, S.; Discacciati, M.; Fourestey, G.; Quarteroni, A., Fluid-structure algorithms based on Steklov-Poincaré operators, Comput. Methods Appl. Mech. Engrg., 195, 41-43, 5797-5812 (2006) · Zbl 1124.76026
[4] Fernández, M. A.; Gerbeau, J.-F.; Grandmont, C., A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid, Internat. J. Numer. Meth. Engrg., 69, 4, 794-821 (2007) · Zbl 1194.74393
[5] Fernández, M. A.; Moubachir, M., A Newton method using exact Jacobian for solving fluid-structure coupling, Comput. Struct., 83, 2-3, 127-142 (2005)
[6] Formaggia, L.; Gerbeau, J. F.; Nobile, F.; Quarteroni, A., On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comput. Methods Appl. Mech. Engrg., 191, 561-582 (2001) · Zbl 1007.74035
[7] Formaggia, L.; Moura, A.; Nobile, F., On the stability of the coupling of 3D and 1D fluid-structure interaction models for blood flow simulations, ESAIM: Math. Mod. Numer. Anal. (M2AN), 41, 4, 743-769 (2007) · Zbl 1139.92009
[8] Formaggia, L.; Nobile, F., A stability analysis for the Arbitrary Lagrangian Eulerian formulation with finite elements, East-West J. Numer. Math., 7, 105-132 (1999) · Zbl 0942.65113
[9] L. Formaggia, A. Veneziani, Reduced and multiscale models for the human cardiovascular system, Lecture notes VKI, Lecture Series 2003-07, Brussels, 2003. Available as MOX Report 21; L. Formaggia, A. Veneziani, Reduced and multiscale models for the human cardiovascular system, Lecture notes VKI, Lecture Series 2003-07, Brussels, 2003. Available as MOX Report 21
[10] A. Moura, The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling 3D and 1D FSI models, Ph.D. Thesis, Politecnico di Milano, 2007; A. Moura, The Geometrical Multiscale Modelling of the Cardiovascular System: Coupling 3D and 1D FSI models, Ph.D. Thesis, Politecnico di Milano, 2007
[11] Nobile, F.; Vergara, C., An effective fluid-structure interaction formulation for vascular dynamics by generalized robin conditions, SIAM J. Sci. Comput., 30, 2, 731-763 (2008) · Zbl 1168.74038
[12] Perktold, K.; Resch, M.; Florian, H., Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model, J. Biomech. Eng., 113, 464-475 (1991)
[13] Quaini, A.; Quarteroni, A., A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method, Math. Models Methods Appl. Sci., 17, 6, 957-983 (2007) · Zbl 1388.74041
[14] Veneziani, A.; Vergara, C., Flow rate defective boundary conditions in haemodinamics simulations, Internat. J. Numer. Methods Fluids, 47, 803-816 (2005) · Zbl 1134.76748
[15] Vignon-Clementel, I. E.; Figueroa, C. A.; Jansen, K. E.; Taylor, C. A., Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries, Comput. Methods Appl. Mech. Engrg., 195, 29-32, 3776-3796 (2006) · Zbl 1175.76098
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