Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-02T23:04:28.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulation of a pulsatile flow through a flexible channel

Published online by Cambridge University Press:  15 February 2007

Cornel Marius Murea
Cornel Marius Murea*
Affiliation:
Laboratoire de Mathématiques, Informatique et Applications, Université de Haute-Alsace, 4, rue des Frères Lumière, 68093 Mulhouse Cedex, France; [email protected]
Get access

Abstract

An algorithm for approximation of an unsteady fluid-structure interaction problem isproposed. The fluid is governed by the Navier-Stokes equations with boundary conditionson 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 theLeast Squares Method. At each time step we have to solve an optimization problemwhich 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 wherethe gradient of the cost function was approached by the Finite Difference Method.Numerical results are presented.

Keywords

Fluid-structure interactionNavier-Stokes equationsArbitrary Lagrangian Eulerian Method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 6 , November 2006 , pp. 1101 - 1125
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Beirao da, H. Veiga, On the existence of strong solution to a coupled fluid structure evolution problem. J. Math. Fluid Mech. 6 (2004) 21–52.
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.
Chambolle, A., Desjardins, B., Esteban, M.J., Grandmont, C., Existence of weak solutions for an unsteady fluid-plate interaction problem. J. Math. Fluid Mech. 7 (2005) 368–404. CrossRef
R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 7, 9, Masson (1988).
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).
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).
Deparis, S., Fernandez, M.A. and Formaggia, L., Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions. ESAIM: M2AN 37 (2003) 601–616. CrossRef
Desjardins, B., Esteban, M., Grandmont, C. and Le Tallec, P., Weak solutions for a fluid-elastic structure interaction model. Rev. Mat. Complut. 14 (2001) 523–538.
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).
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.
Fernandez, M.A. and Moubachir, M., Newton, A method using exact jacobians for solving fluid-structure coupling. Comput. Struct. 83 (2005) 127–142.
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.
Gerbeau, J.F. and Vidrascu, M., A quasi-Newton algorithm on a reduced model for fluid - structure interaction problems in blood flows. ESAIM: M2AN 37 (2003) 663–680.
Grandmont, C., Existence for a three-dimensional steady state fluid-structure interaction problem. J. Math. Fluid Mech. 4 (2002) 76–94. CrossRef
Grandmont, C. and Maday, Y., Existence for an unsteady fluid-structure interaction problem. ESAIM: M2AN 34 (2000) 609–636. CrossRef
Guermond, J.-L. and Quartapelle, L., On the approximation of the unsteady Navier-Stokes equations by finite element projection methods. Numer. Math. 80 (1998) 207–238. CrossRef
F. Hecht and O. Pironneau, A finite element software for PDE: freefem++, http://www.freefem.org.
C.T. Kelley, Solving nonlinear equations with Newton's method. Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2003).
H.P. Langtangen, Computational Partial Differential Equations: numerical methods and Diffpack programming. Springer, Berlin (1999).
P. Le Tallec, Introduction à la dynamique des structures, Cours École Polytechnique, Ellipses (2000).
Le Tallec, P. and Mouro, J., Fluid-structure interaction with large structural displacements. Comput. Methods Appl. Mech. Engrg. 190 (2001) 3039–3067. CrossRef
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.
Murea, C., The BFGS algorithm for a nonlinear least squares problem arising from blood flow in arteries. Comput. Math. Appl. 49 (2005) 171–186. CrossRef
Murea, C. and Vazquez, C., Sensitivity and approximation of the coupled fluid-structure equations by virtual control method. Appl. Math. Optim. 52 (2005) 357–371.
F. Nobile, Numerical approximation of fluid-structure interaction problems with application to haemodynamics. Ph.D. thesis, EPFL, Lausanne (2001).
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.
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).
Quarteroni, A., Tuveri, M. and Veneziani, A., Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci. 2 (2000) 163–197. CrossRef
Steindorf, J. and Matthies, H.G., Partioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. Comput. Struct. 80 (2002) 1991–1999.