Numerical simulation of a pulsatile flow through a flexible channel

Cornel Marius Murea

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 40, Issue: 6, page 1101-1125
  • ISSN: 0764-583X

Abstract

top
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.

How to cite

top

Murea, Cornel Marius. "Numerical simulation of a pulsatile flow through a flexible channel." ESAIM: Mathematical Modelling and Numerical Analysis 40.6 (2007): 1101-1125. <http://eudml.org/doc/194346>.

@article{Murea2007,
abstract = { 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. },
author = {Murea, Cornel Marius},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Fluid-structure interaction; Navier-Stokes equations; Arbitrary Lagrangian Eulerian Method.; fluid-structure interaction; arbitrary Lagrangian Eulerian method.},
language = {eng},
month = {2},
number = {6},
pages = {1101-1125},
publisher = {EDP Sciences},
title = {Numerical simulation of a pulsatile flow through a flexible channel},
url = {http://eudml.org/doc/194346},
volume = {40},
year = {2007},
}

TY - JOUR
AU - Murea, Cornel Marius
TI - Numerical simulation of a pulsatile flow through a flexible channel
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/2//
PB - EDP Sciences
VL - 40
IS - 6
SP - 1101
EP - 1125
AB - 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.
LA - eng
KW - Fluid-structure interaction; Navier-Stokes equations; Arbitrary Lagrangian Eulerian Method.; fluid-structure interaction; arbitrary Lagrangian Eulerian method.
UR - http://eudml.org/doc/194346
ER -

References

top
  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.  
  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.  
  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.  
  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.  
  5. R. Dautray and J.L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 7, 9, Masson (1988).  
  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).  
  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: M2AN37 (2003) 601–616.  
  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.  
  10. G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique. Dunod, Paris (1972).  
  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.  
  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.  
  14. J.F. Gerbeau and M. Vidrascu, A quasi-Newton algorithm on a reduced model for fluid - structure interaction problems in blood flows. ESAIM: M2AN37 (2003) 663–680.  
  15. C. Grandmont, Existence for a three-dimensional steady state fluid-structure interaction problem. J. Math. Fluid Mech.4 (2002) 76–94.  
  16. C. Grandmont and Y. Maday, Existence for an unsteady fluid-structure interaction problem. ESAIM: M2AN34 (2000) 609–636.  
  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.  
  18. F. Hecht and O. Pironneau, A finite element software for PDE: freefem++, .  URIhttp://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).  
  20. H.P. Langtangen, Computational Partial Differential Equations: numerical methods and Diffpack programming. Springer, Berlin (1999).  
  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.  
  23. Y. Maday, B. Maury and P. Metier, Interaction de fluides potentiels avec une membrane élastique, in ESAIM Proc., Soc. Math. Appl. Indust., Paris10 (1999) 23–33.  
  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.  
  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.  
  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.  
  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).  
  29. A. Quarteroni, M. Tuveri and A. Veneziani, Computational vascular fluid dynamics: problems, models and methods. Comput. Visual. Sci.2 (2000) 163–197.  
  30. J. Steindorf and H.G. Matthies, Partioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. Comput. Struct.80 (2002) 1991–1999.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.