# A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows

Jean-Frédéric Gerbeau; Marina Vidrascu

- Volume: 37, Issue: 4, page 631-647
- ISSN: 0764-583X

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topGerbeau, Jean-Frédéric, and Vidrascu, Marina. "A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 37.4 (2003): 631-647. <http://eudml.org/doc/244907>.

@article{Gerbeau2003,

abstract = {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.},

author = {Gerbeau, Jean-Frédéric, Vidrascu, Marina},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {fluid-structure interaction; quasi-Newton algorithm; added mass effect; blood flows; approximate tangent operator; shells; large displacements},

language = {eng},

number = {4},

pages = {631-647},

publisher = {EDP-Sciences},

title = {A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows},

url = {http://eudml.org/doc/244907},

volume = {37},

year = {2003},

}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2003

PB - EDP-Sciences

VL - 37

IS - 4

SP - 631

EP - 647

AB - 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.

LA - eng

KW - fluid-structure interaction; quasi-Newton algorithm; added mass effect; blood flows; approximate tangent operator; shells; large displacements

UR - http://eudml.org/doc/244907

ER -

## References

top- [1] K.J. Bathe, Finite Element Procedures. Prentice Hall (1996). Zbl0994.74001
- [2] M. Bathe and R.D. Kamm, A fluid-structure interaction finite element analysis of pulsative blood flow through a compliant stenotic artery. J. Biomech. Engng. 121 (1999) 361–369.
- [3] P.N. Brown and Y. Saad, Convergence theory of nonlinear Newton-Krylov algorithms. SIAM J. Optim. 4 (1994) 297–330. Zbl0814.65048
- [4] D. Chapelle and K.J. Bathe, The Finite Element Analysis of Shells – Fundamentals. Springer Verlag (2003). Zbl1103.74003
- [5] S. Deparis, M.A. Fernández, L. Formaggia and F. Nobile, 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). Zbl1118.74315
- [6] J. Donéa, S. Giuliani and J.P. Halleux, An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comp. Meth. Appl. Mech. Engng. (1982) 689–723. Zbl0508.73063
- [7] M.A. Fernández and M. Moubachir, 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).
- [8] 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. Comp. Meth. Appl. Mech. Engrg. 191 (2001) 561–582. Zbl1007.74035
- [9] J.-F. Gerbeau, 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] P. Le Tallec, Numerical methods for nonlinear three-dimensional elasticity, in Handbook of numerical analysis, Vol. III, North-Holland (1994) 465–622. Zbl0875.73234
- [11] P. Le Tallec and J. Mouro, Fluid structure interaction with large structural displacements. Comput. Meth. Appl. Mech. Engrg. 190 (2001) 3039–3067. Zbl1001.74040
- [12] X. Ma, G.C. Lee and S.G. Wu, Numerical simulation for the propagation of nonlinear pulsatile waves in arteries. Transactions of the ASME 114 (1992) 490–496.
- [13] H.G. Matthies and J. Steindorf, Partitioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. preprint, 2000. Zbl0951.74629
- [14] H.G. Matthies and J. Steindorf, How to make weak coupling strong, in Computational Fluid and Solid Mechanics, K.J. Bathe Ed., Elsevier (2001) 1317–1319.
- [15] D.P. Mok and W.A. Wall, 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). MR2070766
- [16] D.P. Mok, W.A. Wall and E. Ramm, 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).
- [17] D.P. Mok, W.A. Wall and E. Ramm, Accelerated iterative substructuring schemes for instationary fluid-structure interaction, in Computational Fluid and Solid Mechanics, K.J. Bathe Ed., Elsevier (2001) 1325–1328.
- [18] H. Morand and R. Ohayon, Interactions fluides-structures, Vol. 23 of Recherches en Mathématiques Appliquées. Masson, Paris (1992). Zbl0754.73071MR1180076
- [19] J. Mouro, Interactions fluide structure en grands déplacements. Résolution numérique et application aux composants hydrauliques automobiles. Ph.D. thesis, École Polytechnique, France (1996).
- [20] F. Nobile, Numerical approximation of fluid-structure interaction problems with application to haemodynamics. Ph.D. thesis, EPFL, Switzerland (2001).
- [21] M.S. Olufsen, Modeling the Arterial System with Reference to an Anesthesia Simulator. Ph.D. thesis, Roskilde University (1998).
- [22] K. Perktold and G. Rappitsch, Mathematical modeling of local arterial flow and vessel mechanics, in Computational Methods for Fluid-Structure interaction, J. Crolet and R. Ohayon Eds., Pitman (1994). Zbl0809.76098MR1424706
- [23] K. Perktold and G. Rappitsch, Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model. J. Biomech. 28 (1995) 845–856.
- [24] S. Piperno, 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. Zbl0910.76065
- [25] A. Quarteroni, M. Tuveri and A. Veneziani, Computational Vascular Fluid Dynamics: Problems, Models and Methods. Comp. Vis. Sci. 2 (2000) 163–197. Zbl1096.76042
- [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). Zbl0931.65118MR1857663
- [27] K. Rhee and S.M. Lee, Effects of radial wall motion and flow waveform on the wall shear rate distribution in the divergent vascular graft. Ann. Biomed. Eng. (1998).
- [28] S. Rugonyi and K.J. Bathe, On finite element analysis of fluid flows fully coupled with structural interactions. CMES 2 (2001).
- [29] D. Tang, J. Yang, C. Yang and D.N. Ku, A nonlinear axisymmetric model with fluid-wall interactions for steady viscous flow in stenotic elastic tubes. J. Biomech. Engng. 121 (1999) 494–501.
- [30] S.A. Urquiza, M.J. Venere, F.M. Clara and R.A. Feijóo, Finite element (one-dimensional) haemodynamic model of the human arterial system, in ECCOMAS, Barcelona (2000).
- [31] H. Zhang and K.J. Bathe, 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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