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A mathematical approach in the design of arterial bypass using unsteady Stokes equations. (English) Zbl 1158.76452

Summary: We present an approach for the study of Aorto-Coronaric bypass anastomoses configurations using unsteady Stokes equations. The theory of optimal control based on adjoint formulation is applied in order to optimize the shape of the zone of the incoming branch of the bypass (the toe) into the coronary according to several optimality criteria.

MSC:

76Z05 Physiological flows
76D07 Stokes and related (Oseen, etc.) flows
76D55 Flow control and optimization for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
92C35 Physiological flow
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[1] Agoshkov V.I. (2003). Optimal Control Approaches and Adjoint Equations in the Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow · Zbl 1236.49002
[2] Agoshkov, V. I., Quarteroni, A., and Rozza, G. (2005). Shape design in aorto coronaric bypass using perturbation theory. To appear in SIAM J. Numer. Anal. · Zbl 1120.49037
[3] Cole J.S., Watterson J.K., O’Reilly M.J.G. (2002). Numerical investigation of the haemodynamics at a patched arterial bypass anastomosis. Med. Eng. Phys 24:393–401 · doi:10.1016/S1350-4533(02)00038-3
[4] Lions, J. L. (1971). Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag. · Zbl 0203.09001
[5] Lions, J. L., Magenes, E. (1972). Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag. · Zbl 0227.35001
[6] Marchuk G.I. (1989). Methods of Numerical Mathematics. Nauka, Moscow · Zbl 0696.65001
[7] Prud’homme C., Rovas D., Veroy K., Maday Y., Patera A.T., and Turinici G. (2002). Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluid. Eng. 172:70–80 · doi:10.1115/1.1448332
[8] Quarteroni A., and Rozza G. (2003). Optimal control and shape optimization in aorto-coronaric bypass anastomoses. M3AS Math. Mod. Meth. Appl. Sci. 13(12):1801–23 · Zbl 1063.49029 · doi:10.1142/S0218202503003124
[9] Quarteroni A., and Formaggia L. (2004). Mathematical modelling and numerical simulation of the cardiovascular system. In: Ciarlet P.G., and Lions J.L. (eds). Modelling of Living Systems, Handbook of Numerical Analysis Series. Elsevier, Amsterdam
[10] Quarteroni A., Tuveri M., and Veneziani A. (2000). Computational vascular fluid dynamics: problems, models and methods. Comput. Visual Sci 2:163–197 · Zbl 1096.76042 · doi:10.1007/s007910050039
[11] Quarteroni A., and Valli A. (1994). Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin · Zbl 0803.65088
[12] Quarteroni A., Sacco R., and Saleri F. (2000). Numerical Mathematics. Springer, New York
[13] Tikhonov A.N., and Arsenin V.Ya. (1974). Methods for Solving Ill-posed Problems. Nauka, Moscow
[14] Rozza G. (2005). On optimization, control and shape design of an arterial bypass. Int. J. Numer. Meth. Fluid 47(10–11): 1411–1419 · Zbl 1155.76439 · doi:10.1002/fld.888
[15] Rozza, G. (2005). Real time reduced basis techniques for arterial bypass geometries. Bathe, K. J. (ed.), Computational Fluid and Solid Mechanics, Elsevier, pp. 1283–1287.
[16] Vainikko G.M., Veretennikov A.Yu. (1986). Iterative Procedures in Ill-posed problems. Nauka, Moscow
[17] Van Dyke M. (1975) Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford · Zbl 0329.76002
[18] Vasiliev F.P. (1981). Methods for Solving the Extremum Problems. Nauka, Moscow
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