We propose a quasi-Newton algorithm for solving fluid-structure interaction problems. The basic idea of the method is to build an approximate tangent operator which is cost effective and which takes into account the so-called added mass effect. Various test cases show that the method allows a significant reduction of the computational effort compared to relaxed fixed point algorithms. We present 2D and 3D fluid-structure simulations performed either with a simple 1D structure model or with shells in large displacements.
Mots clés : fluid-structure interaction, quasi-Newton algorithm, added mass effect, blood flows
@article{M2AN_2003__37_4_631_0, author = {Gerbeau, Jean-Fr\'ed\'eric and Vidrascu, Marina}, title = {A {quasi-Newton} algorithm based on a reduced model for fluid-structure interaction problems in blood flows}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {631--647}, publisher = {EDP-Sciences}, volume = {37}, number = {4}, year = {2003}, doi = {10.1051/m2an:2003049}, mrnumber = {2018434}, zbl = {1070.74047}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2003049/} }
TY - JOUR AU - Gerbeau, Jean-Frédéric AU - Vidrascu, Marina TI - A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 631 EP - 647 VL - 37 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003049/ DO - 10.1051/m2an:2003049 LA - en ID - M2AN_2003__37_4_631_0 ER -
%0 Journal Article %A Gerbeau, Jean-Frédéric %A Vidrascu, Marina %T A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 631-647 %V 37 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003049/ %R 10.1051/m2an:2003049 %G en %F M2AN_2003__37_4_631_0
Gerbeau, Jean-Frédéric; Vidrascu, Marina. A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 4, pp. 631-647. doi : 10.1051/m2an:2003049. http://www.numdam.org/articles/10.1051/m2an:2003049/
[1] Finite Element Procedures. Prentice Hall (1996). | Zbl
,[2] A fluid-structure interaction finite element analysis of pulsative blood flow through a compliant stenotic artery. J. Biomech. Engng. 121 (1999) 361-369.
and ,[3] Convergence theory of nonlinear Newton-Krylov algorithms. SIAM J. Optim. 4 (1994) 297-330. | Zbl
and ,[4] The Finite Element Analysis of Shells - Fundamentals. Springer Verlag (2003). | Zbl
and ,[5] Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions, in Second MIT Conference on Computational Fluid and Solid Mechanics, Elsevier (2003).
, , and ,[6] An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comp. Meth. Appl. Mech. Engng. (1982) 689-723. | Zbl
, and ,[7] An exact block-newton algorithm for the solution of implicit time discretized coupled systems involved in fluid-structure interaction problems, in Second MIT Conference on Computational Fluid and Solid Mechanics, Elsevier (2003).
and ,[8] On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comp. Meth. Appl. Mech. Engrg. 191 (2001) 561-582. | Zbl
, , and ,[9] A quasi-newton method for a fluid-structure problem arising in blood flows, in Second MIT Conference on Computational Fluid and Solid Mechanics, Elsevier (2003).
,[10] Numerical methods for nonlinear three-dimensional elasticity, in Handbook of numerical analysis, Vol. III, North-Holland (1994) 465-622. | Zbl
,[11] Fluid structure interaction with large structural displacements. Comput. Meth. Appl. Mech. Engrg. 190 (2001) 3039-3067. | Zbl
and ,[12] Numerical simulation for the propagation of nonlinear pulsatile waves in arteries. Transactions of the ASME 114 (1992) 490-496.
, and ,[13] Partitioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. preprint, 2000.
and ,[14] How to make weak coupling strong, in Computational Fluid and Solid Mechanics, K.J. Bathe Ed., Elsevier (2001) 1317-1319.
and ,[15] Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures, in Trends in computational structural mechanics CIMNE, K. Schweizerhof, W.A. Wall and K.U. Bletzinger Eds., Barcelona (2001). | MR
and ,[16] Partitioned analysis approach for the transient, coupled response of viscous fluids and flexible structures, in Proceedings of the European Conference on Computational Mechanics. ECCM'99, W. Wunderlich Ed., TU Munich (1999).
, and ,[17] Accelerated iterative substructuring schemes for instationary fluid-structure interaction, in Computational Fluid and Solid Mechanics, K.J. Bathe Ed., Elsevier (2001) 1325-1328.
, and ,[18] Interactions fluides-structures, Vol. 23 of Recherches en Mathématiques Appliquées. Masson, Paris (1992). | MR | Zbl
and ,[19] Interactions fluide structure en grands déplacements. Résolution numérique et application aux composants hydrauliques automobiles. Ph.D. thesis, École Polytechnique, France (1996).
,[20] Numerical approximation of fluid-structure interaction problems with application to haemodynamics. Ph.D. thesis, EPFL, Switzerland (2001).
,[21] Modeling the Arterial System with Reference to an Anesthesia Simulator. Ph.D. thesis, Roskilde University (1998).
,[22] Mathematical modeling of local arterial flow and vessel mechanics, in Computational Methods for Fluid-Structure interaction, J. Crolet and R. Ohayon Eds., Pitman (1994). | MR | Zbl
and ,[23] Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. J. Biomech. 28 (1995) 845-856.
and ,[24] Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations. Int. J. Numer. Method Fluid 25 (1997) 1207-1226. | Zbl
,[25] Computational Vascular Fluid Dynamics: Problems, Models and Methods. Comp. Vis. Sci. 2 (2000) 163-197. | Zbl
, and ,[26] Alfio Quarteroni and Alberto Valli, Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation. The Clarendon Press Oxford University Press, Oxford Science Publication (1999). | MR | Zbl
[27] Effects of radial wall motion and flow waveform on the wall shear rate distribution in the divergent vascular graft. Ann. Biomed. Eng. (1998).
and ,[28] On finite element analysis of fluid flows fully coupled with structural interactions. CMES 2 (2001).
and ,[29] A nonlinear axisymmetric model with fluid-wall interactions for steady viscous flow in stenotic elastic tubes. J. Biomech. Engng. 121 (1999) 494-501.
, , and ,[30] Finite element (one-dimensional) haemodynamic model of the human arterial system, in ECCOMAS, Barcelona (2000).
, , and ,[31] Direct and iterative computing of fluid flows fully coupled with structures, in Computational Fluid and Solid Mechanics, K.J. Bathe Ed., Elsevier (2001) 1440-1443.
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