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The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries. (English) Zbl 1067.92032
Summary: Using the Arbitrary Lagrangian Eulerian coordinates and the least squares method, a two-dimensional steady fluid structure interaction problem is transformed into an optimal control problem. Sensitivity analysis is presented. The Broyden, Fletcher, Goldfarb, Shano (BFGS) algorithm gives satisfactory numerical results even when we use a reduced number of discrete controls.

MSC:
92C35 Physiological flow
76M25 Other numerical methods (fluid mechanics) (MSC2010)
92C50 Medical applications (general)
Software:
FreeFem++
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[1] 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, 6-7, 561-582, (2001) · Zbl 1007.74035
[2] Gerbeau, J.F.; Vidrascu, M., A quasi-Newton algorithm on a reduced model for fluid-structure interaction problems in blood flows, M2AN math. model. numer. anal., 37, 4, 631-647, (2003) · Zbl 1070.74047
[3] Chen, H.; Sheu, T., Finite-element simulation of incompressible fluid flow in an elastic vessel, Int. J. numer. meth. fluids, 42, 131-146, (2003) · Zbl 1056.76044
[4] Farhat, C.; Geuzaine, Ph.; Grandmont, C., The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids, J. comput. phys., 174, 2, 669-694, (2001) · Zbl 1157.76372
[5] Grandmont, C., Existence for a three-dimensional steady state fluid-structure interaction problem, J. math. fluid mech., 4, 1, 76-94, (2002) · Zbl 1009.76016
[6] Bayada, G.; Chambat, M.; Cid, B.; Vazquez, C., On the existence of solution for a non-homogeneus Stokes-rod coupled problem, Equipe d’analyse numérique Lyon-saint etienne, 362, (2003), (UMR 5585)
[7] Grandmont, C.; Maday, Y., Existence for an unsteady fluid-structure interaction problem, M2AN math. model. numer. anal., 34, 3, 609-636, (2000) · Zbl 0969.76017
[8] Desjardins, B.; Esteban, M.; Grandmont, C.; LeTallec, P., Weak solutions for a fluid-elastic structure interaction model, Rev. mat. complut., 14, 2, 523-538, (2001) · Zbl 1007.35055
[9] Beirao da Veiga, H., On the existence of strong solution to a coupled fluid structure evolution problem, J. math. fluid mech., 6, 21-52, (2004) · Zbl 1068.35087
[10] Fernandez, M.A.; Moubachir, M., Sensitivity analysis for an incompressible aeroelastic system, Math. models methods appl. sci., 12, 8, 1109-1130, (2002) · Zbl 1041.76016
[11] Bayada, G.; Durany, J.; Vázquez, C., Existence of a solution for a lubrication problem in elastic journal bearings with thin bearing, Math. meth. appl. sci., 18, 455-466, (1995) · Zbl 0820.35110
[12] Murea, C.; Maday, Y., Existence of an optimal control for a nonlinear fluid-cable interaction problem, ()
[13] C. Murea and C. Vázquez, Sensitivity and approximation of fluid-structure coupled by virtual control method, (submitted). · Zbl 1136.74319
[14] Lewis, R.M., A nonlinear programming perspective on sensitivity calculations for systems governed by state equations, (), no. 12
[15] Murea, C., Optimal control approach for the fluid-structure interaction problems, (), 442-450 · Zbl 1033.35013
[16] Le Tallec, P.; Mouro, J., Fluid-structure interaction with large structural displacements, Comput. methods appl. mech. engrg., 190, 24-25, 3039-3067, (2001) · Zbl 1001.74040
[17] Nobile, F., Numerical approximation of fluid-structure interaction problems with application to haemodynamics, (2001), Ecole Polytechnique Féderale de Lausanne
[18] Deparis, S.; Fernandez, M.A.; Formaggia, L., Acceleration of a fixed-point algorithm for fluid-structure interaction using transpiration conditions, M2AN math. model. numer. anal., 37, 4, 601-616, (2003) · Zbl 1118.74315
[19] J. Steindorf and H.G. Matthies, Partioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction (to appear). · Zbl 1312.74009
[20] Bernadou, M., Formulation variationnelle, approximation et implementation de problemes de barres et de poutres bi- et tri-dimensionnelles. partie B : barres et poutres bidimensionnelles, (1987)
[21] Osses, A., A rotated multiplier applied to the controllability of waves, elasticity and tangential Stokes control, SIAM J. control optim., 40, 3, 777-800, (2001) · Zbl 0997.93013
[22] Girault, V.; Raviart, P.-A., Springer series in computational mathematics, (1986), Springer-Verlag Switzerland
[23] Gurtin, M.E., An introduction to continuum mechanics, (1981), Academic Press Berlin · Zbl 0559.73001
[24] Adams, R., Sobolev spaces, (1975), Academic Press · Zbl 0314.46030
[25] Schmidt-Nielsen, K., Physiologie animale, ()
[26] Hecht, F.; Pironneau, O., A finite element software for PDE: freefem++, (2003), May
[27] Thiriet, M., A three-dimensional numerical method of a critical flow in a collapsed tube, (), No 3867
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