Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions
ESAIM: Modélisation mathématique et analyse numérique, Volume 37 (2003) no. 4, pp. 601-616.

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.

DOI: 10.1051/m2an:2003050
Classification: 65M60, 65B99, 74F10
Keywords: fluid-structure interaction, Block-Gauss-Seidel iterations, transpiration, highly coupled non-linear problems, weak and strong coupling algorithms, partitioned procedures
Deparis, Simone ; Fernández, Miguel Angel ; Formaggia, Luca 1

1 Politecnico di Milano, MOX, Piazza L. da Vinci 32, 20133 Milano, Italy.
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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, Volume 37 (2003) no. 4, pp. 601-616. doi : 10.1051/m2an:2003050. http://archive.numdam.org/articles/10.1051/m2an:2003050/

[1] M. Carrive-Bédouani, P. Le Tallec and J. Mouro, Approximation par éléments finis d'un modèle de coques minces géométriquement exact. Rev. Européenne Élém. Finis 4 (1995) 633-661. | Zbl

[2] P.G. Ciarlet, Mathematical Elasticity, Vol. I, Three-Dimensional Elasticity. North-Holland, Amsterdam (1988). | MR | Zbl

[3] R. Codina and M. Cervera, Block Iterative Algorithms for Non-Linear Coupled Problems. Advanced Computational Methods in Structural Mechanics, CIMNE, Barcelona (1996). | Zbl

[4] S. Deparis, Fluid-Structure Interaction Problems in Hemodynamics and Axisymmetric Modeling of Blood Flow. Ph.D. thesis, École Politechnique Fédérale de Lausanne (EPFL), in preparation.

[5] J. Donea, An Arbitrary Lagrangian-Eulerian Finite Element Method for Transient Dynamic Fluid-Structure Interactions. Comput. Methods Appl. Mech. Engrg. 33 (1982) 689-723. | Zbl

[6] T. Fanion, M.A. Fernández and P. Le Tallec, Deriving Adequate Formulations for Fluide Structure Interaction Problems: from ALE to Transpiration. Rev. Européenne Élém. Finis 9 (2000) 681-708. | Zbl

[7] 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. | Zbl

[8] M.A. Fernández, Modèles Simplifiés d'Interaction Fluide-Structure. Ph.D. thesis, University of Paris IX, France (2001).

[9] 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).

[10] L. Formaggia and F. Nobile, A Stability Analysis for the Arbitrary Lagrangian Eulerian Formulation with Finite Elements. East-West J. Numer. Math. 7 (1999) 105-131. | Zbl

[11] J.F. Gerbeau and M. Vidrascu, 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] W.P. Huffman, R.G. Melvin, D.P. Young, F.T. Johnson, J.E. Bussoletti, M.B. Bieterman and C.L. Hilmes, Practical Design and Optimisation in Computational Fluids Dynamics, in proceedings of the AIAA 24th Fluid Dynamics Conference, Orlando, Florida (1993) 93-3111.

[13] P. Le Tallec, Numerical Methods for Nonlinear Three-Dimensional Elasticity. Handbook of Numerical Analysis, Vol. III, North-Holland, Amsterdam (1994) 465-622. | MR | Zbl

[14] P. Le Tallec and J. Mouro, Fluid Structure Interaction with Large Structural Displacements. Comput. Methods Appl. Mech. Engrg. 190 (2001) 3039-3067. | Zbl

[15] M.J. Lighthill, On Displacement Thickness. J. Comput. Phys. 4 (1958) 383-392. | Zbl

[16] G. Medic, 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] D.P. Mok, W.A. Wall and E. Ramm, 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] M. Moubachir, 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] F. Nobile, Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, Switzerland (2001).

[20] S. Piperno, C. Farhat and B. Larrouturou, 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] S. Piperno, C. Farhat and B. Larrouturou, 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] A. Quarteroni and L. Formaggia, 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] P. Raj and B. Harris, 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] J.Y. Renou, Une méthode eulérienne pour le calcul de forces fluide-élastiques. Ph.D. thesis, University of Paris VI, France (1998).

[25] J. Steindorf and H.G Matthies, Numerical Efficiency of Different Partitioned Methods for Fluid-Structure interaction. ZAMM, Z. Angew. Math. Mech. 80 (2000) 557-558. | Zbl

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