In this work, we address the numerical solution of fluid-structure interaction problems. This issue is particularly difficulty to tackle when the fluid and the solid densities are of the same order, for instance as it happens in hemodynamic applications, since fully implicit coupling schemes are required to ensure stability of the resulting method. Thus, at each time step, we have to solve a highly non-linear coupled system, since the fluid domain depends on the unknown displacement of the structure. Standard strategies for solving this non-linear problems, are fixed point based methods such as Block-Gauss-Seidel (BGS) iterations. Unfortunately, these methods are very CPU time consuming and usually show slow convergence. We propose a modified fixed-point algorithm which combines the standard BGS iterations with a transpiration formulation. Numerical experiments show the great improvement in computing time with respect to the standard BGS method.
Mots clés : fluid-structure interaction, Block-Gauss-Seidel iterations, transpiration, highly coupled non-linear problems, weak and strong coupling algorithms, partitioned procedures
@article{M2AN_2003__37_4_601_0,
author = {Deparis, Simone and Fern\'andez, Miguel Angel and Formaggia, Luca},
title = {Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {601--616},
publisher = {EDP-Sciences},
volume = {37},
number = {4},
year = {2003},
doi = {10.1051/m2an:2003050},
mrnumber = {2018432},
zbl = {1118.74315},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003050/}
}
TY - JOUR AU - Deparis, Simone AU - Fernández, Miguel Angel AU - Formaggia, Luca TI - Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2003 SP - 601 EP - 616 VL - 37 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2003050/ DO - 10.1051/m2an:2003050 LA - en ID - M2AN_2003__37_4_601_0 ER -
%0 Journal Article %A Deparis, Simone %A Fernández, Miguel Angel %A Formaggia, Luca %T Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions %J ESAIM: Modélisation mathématique et analyse numérique %D 2003 %P 601-616 %V 37 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2003050/ %R 10.1051/m2an:2003050 %G en %F M2AN_2003__37_4_601_0
Deparis, Simone; Fernández, Miguel Angel; Formaggia, Luca. Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 4, pp. 601-616. doi : 10.1051/m2an:2003050. http://www.numdam.org/articles/10.1051/m2an:2003050/
[1] , and, Approximation par éléments finis d'un modèle de coques minces géométriquement exact. Rev. Européenne Élém. Finis 4 (1995) 633-661. | Zbl
[2] , Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity. North-Holland, Amsterdam (1988). | MR | Zbl
[3] and, Block Iterative Algorithms for Non-Linear Coupled Problems. Advanced Computational Methods in Structural Mechanics, CIMNE, Barcelona (1996). | Zbl
[4] , Fluid-Structure Interaction Problems in Hemodynamics and Axisymmetric Modeling of Blood Flow. Ph.D. thesis, École Politechnique Fédérale de Lausanne (EPFL), in preparation.
[5] , An Arbitrary Lagrangian-Eulerian Finite Element Method for Transient Dynamic Fluid-Structure Interactions. Comput. Methods Appl. Mech. Engrg. 33 (1982) 689-723. | Zbl
[6] , and, Deriving Adequate Formulations for Fluide Structure Interaction Problems: from ALE to Transpiration. Rev. Européenne Élém. Finis 9 (2000) 681-708. | Zbl
[7] and, 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
[8] , Modèles Simplifiés d'Interaction Fluide-Structure. Ph.D. thesis, University of Paris IX, France (2001).
[9] and, 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).
[10] and, A Stability Analysis for the Arbitrary Lagrangian Eulerian Formulation with Finite Elements. East-West J. Numer. Math. 7 (1999) 105-131. | Zbl
[11] and, A Quasi-Newton Algorithm Based on a Reduced Model for Fluid Structure Problems in Blood Flows. ESAIM: M2AN 37 (2003) 631-647. | Numdam | Zbl
[12] ,,,,, and, Practical Design and Optimisation in Computational Fluids Dynamics, in proceedings of the AIAA 24th Fluid Dynamics Conference, Orlando, Florida (1993) 93-3111.
[13] , Numerical Methods for Nonlinear Three-Dimensional Elasticity. Handbook of Numerical Analysis, Vol. III, North-Holland, Amsterdam (1994) 465-622. | MR | Zbl
[14] and, Fluid Structure Interaction with Large Structural Displacements. Comput. Methods Appl. Mech. Engrg. 190 (2001) 3039-3067. | Zbl
[15] , On Displacement Thickness. J. Comput. Phys. 4 (1958) 383-392. | Zbl
[16] , Étude mathématique des modèles aux tensions de Reynolds et simulation numérique d'écoulements turbulents sur parois fixes et mobiles. Ph.D. thesis, University of Paris VI, France (1999).
[17] , and, Accelerated Iterative Substructuring Schemes for Instationnary Fluid-structure Interaction, in proceedings of the First MIT Conference on Computational Fluid and Solid Mechanics, Elsevier (2001) 1325-1328.
[18] , Mathematical and Numerical Analysis of Inverse and Control Problems for Fluid-Structure Interaction Systems. Ph.D. thesis, École Nationale des Ponts et Chaussées, France (2002).
[19] , Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Switzerland (2001).
[20] , and, Partitioned Procedures for the Transient Solution of Coupled Aeroelastic Problems - Part I: Model Problem, Theory and Two-Dimensional Application. Comput. Methods Appl. Mech. Engrg. 124 (1995) 79-112. | Zbl
[21] , and, Partitioned Procedures for the Transient Solution of Coupled Aeroelastic Problems - Part II: Energy Transfer Analysis and Three-Dimensional applications. Comput. Methods Appl. Mech. Engrg. 124 (1995) 79-112. | Zbl
[22] and, Mathematical Modelling and Numerical Simulation of the Cardiovascular System. To appear as chapter in Modelling of Living Systems, N. Ayache Ed., Handbook of numerical Analysis Series, Elsevier, Amsterdam (2003). | MR
[23] and, Using Surface Transpiration With an Euler Method for Cost-Effective Aerodynamic Analysis, in proceedings of the AIAA 24th Applied Aerodynamics Conference, 93-3506, Monterey, Canada (1993).
[24] , Une méthode eulérienne pour le calcul de forces fluide-élastiques. Ph.D. thesis, University of Paris VI, France (1998).
[25] and, Numerical Efficiency of Different Partitioned Methods for Fluid-Structure interaction. ZAMM, Z. Angew. Math. Mech. 80 (2000) 557-558. | Zbl
Cité par Sources :
