Numerical simulation of a pulsatile flow through a flexible channel
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, p. 1101-1125
An algorithm for approximation of an unsteady fluid-structure interaction problem is proposed. The fluid is governed by the Navier-Stokes equations with boundary conditions on pressure, while for the structure a particular plate model is used. The algorithm is based on the modal decomposition and the Newmark Method for the structure and on the Arbitrary lagrangian Eulerian coordinates and the Finite Element Method for the fluid. In this paper, the continuity of the stresses at the interface was treated by the Least Squares Method. At each time step we have to solve an optimization problem which permits us to use moderate time step. This is the main advantage of this approach. In order to solve the optimization problem, we have employed the Broyden, Fletcher, Goldforb, Shano Method where the gradient of the cost function was approached by the Finite Difference Method. Numerical results are presented.
DOI : https://doi.org/10.1051/m2an:2007003
Classification:  74F10,  75D05,  65M60
Keywords: fluid-structure interaction, Navier-Stokes equations, arbitrary lagrangian eulerian method
@article{M2AN_2006__40_6_1101_0,
     author = {Murea, Cornel Marius},
     title = {Numerical simulation of a pulsatile flow through a flexible channel},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {6},
     year = {2006},
     pages = {1101-1125},
     doi = {10.1051/m2an:2007003},
     zbl = {pre05161015},
     mrnumber = {2297106},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_6_1101_0}
}
Murea, Cornel Marius. Numerical simulation of a pulsatile flow through a flexible channel. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 6, pp. 1101-1125. doi : 10.1051/m2an:2007003. http://www.numdam.org/item/M2AN_2006__40_6_1101_0/

[1] G. Bayada, M. Chambat, B. Cid and C. Vazquez, On the existence of solution for a non-homogeneous Stokes-rod coupled problem. Nonlinear Anal. Theory Methods Appl., 59 (2004) 1-19. | Zbl 1086.74013

[2] H. Beirao Da Veiga, On the existence of strong solution to a coupled fluid structure evolution problem. J. Math. Fluid Mech. 6 (2004) 21-52. | Zbl 1068.35087

[3] P. Causin, J.F. Gerbeau, F. Nobile, Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Engrg. 194 (2005) 4506-4527. | Zbl 1101.74027

[4] A. Chambolle, B. Desjardins, M.J. Esteban, C. Grandmont, Existence of weak solutions for an unsteady fluid-plate interaction problem. J. Math. Fluid Mech. 7 (2005) 368-404. | Zbl 1080.74024

[5] R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 7, 9, Masson (1988). | MR 1016604 | Zbl 0749.35005

[6] J.E. Dennis, Jr., and R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations. Classics in Applied Mathematics, 16, Society for Industrial and Applied Mathematics, Philadelphia, PA (1996). | MR 1376139 | Zbl 0847.65038

[7] S. Deparis, Numerical Analysis of Axisymmetric Flows and Methods for Fluid-Structure Interaction Arising in Blood Flow Simulation, Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Switzerland (2004).

[8] S. Deparis, M.A. Fernandez and L. Formaggia, Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions. ESAIM: M2AN 37 (2003) 601-616. | Numdam | Zbl 1118.74315

[9] B. Desjardins, M. Esteban, C. Grandmont and P. Le Tallec, Weak solutions for a fluid-elastic structure interaction model. Rev. Mat. Complut. 14 (2001) 523-538. | Zbl 1007.35055

[10] G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972). | MR 464857 | Zbl 0298.73001

[11] C. Farhat and M. Lesoinne, Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems, Comput. Methods Appl. Mech. Engrg. 182 (2000) 499-515. | Zbl 0991.74069

[12] M.A. Fernandez and M. Moubachir, A Newton method using exact jacobians for solving fluid-structure coupling. Comput. Struct. 83 (2005) 127-142.

[13] L. Formaggia, J.F. Gerbeau, F. Nobile and A. Quarteroni, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engrg. 191 (2001), 561-582. | Zbl 1007.74035

[14] J.F. Gerbeau and M. Vidrascu, A quasi-Newton algorithm on a reduced model for fluid - structure interaction problems in blood flows. ESAIM: M2AN 37 (2003) 663-680. | Numdam | Zbl 1070.74047

[15] C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem. J. Math. Fluid Mech. 4 (2002) 76-94. | Zbl 1009.76016

[16] C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem. ESAIM: M2AN 34 (2000) 609-636. | Numdam | Zbl 0969.76017

[17] J.-L. Guermond and L. Quartapelle, On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207-238. | Zbl 0914.76051

[18] F. Hecht and O. Pironneau, A finite element software for PDE: freefem++, http://www.freefem.org.

[19] C.T. Kelley, Solving nonlinear equations with Newton's method. Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003). | Zbl 1031.65069

[20] H.P. Langtangen, Computational Partial Differential Equations: numerical methods and Diffpack programming. Springer, Berlin (1999). | MR 1690649 | Zbl 0929.65098

[21] P. Le Tallec, Introduction à la dynamique des structures, Cours École Polytechnique, Ellipses (2000).

[22] P. Le Tallec and J. Mouro, Fluid-structure interaction with large structural displacements. Comput. Methods Appl. Mech. Engrg. 190 (2001) 3039-3067. | Zbl 1001.74040

[23] Y. Maday, B. Maury and P. Metier, Interaction de fluides potentiels avec une membrane élastique, in ESAIM Proc., Soc. Math. Appl. Indust., Paris 10 (1999) 23-33. | Zbl 1013.74021

[24] C. Murea, The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries. Comput. Math. Appl. 49 (2005) 171-186. | Zbl 1067.92032

[25] C. Murea and C. Vazquez, Sensitivity and approximation of the coupled fluid-structure equations by virtual control method. Appl. Math. Optim. 52 (2005) 357-371. | Zbl 1136.74319

[26] F. Nobile, Numerical approximation of fluid-structure interaction problems with application to haemodynamics. Ph.D. thesis, EPFL, Lausanne (2001).

[27] O. Pironneau, Conditions aux limites sur la pression pour les équations de Stokes et Navier-Stokes. C. R. Acad. Sc. Paris, 303 (1986) 403-406. | Zbl 0613.76028

[28] A. Quarteroni and L. Formaggia, Mathematical Modelling and Numerical Simulation of the Cardiovascular System. Chapter in Modelling of Living Systems, N. Ayache Ed., Handbook of Numerical Analysis Series, Vol. XII, P.G. Ciarlet Ed., Elsevier, Amsterdam (2004). | MR 2087609

[29] A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci. 2 (2000) 163-197. | Zbl 1096.76042

[30] J. Steindorf and H.G. Matthies, Partioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. Comput. Struct. 80 (2002) 1991-1999.