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A stable second-order scheme for fluid-structure interaction with strong added-mass effects. (English) Zbl 1349.76236
Summary: In this paper, we present a stable second-order time accurate scheme for solving fluid-structure interaction problems. The scheme uses so-called Combined Field with Explicit Interface (CFEI) advancing formulation based on the Arbitrary Lagrangian-Eulerian approach with finite element procedure. Although loosely-coupled partitioned schemes are often popular choices for simulating FSI problems, these schemes may suffer from inherent instability at low structure to fluid density ratios. We show that our second-order scheme is stable for any mass density ratio and hence is able to handle strong added-mass effects. Energy-based stability proof relies heavily on the connections among extrapolation formula, trapezoidal scheme for second-order equation, and backward difference method for first-order equation. Numerical accuracy and stability of the scheme is assessed with the aid of two-dimensional fluid-structure interaction problems of increasing complexity. We confirm second-order temporal accuracy by numerical experiments on an elastic semi-circular cylinder problem. We verify the accuracy of coupled solutions with respect to the benchmark solutions of a cylinder-elastic bar and the Navier-Stokes flow system. To study the stability of the proposed scheme for strong added-mass effects, we present new results using the combined field formulation for flexible flapping motion of a thin-membrane structure with low mass ratio and strong added-mass effects in a uniform axial flow. Using a systematic series of fluid-structure simulations, a detailed analysis of the coupled response as a function of mass ratio for the case of very low bending rigidity has been presented.

76M10 Finite element methods applied to problems in fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Full Text: DOI
[1] Cebral, J. R.; Lohner, R., Conservative load projection and tracking for fluid-structure problems, AIAA J., 35, 687-692, (1997) · Zbl 0895.73077
[2] Jaiman, R.; Geubelle, P.; Loth, E.; Jiao, X., Transient fluid-structure interaction with non-matching spatial and temporal discretizations, Comput. Fluids, 50, 120-135, (2011) · Zbl 1271.76242
[3] Huang, L., Flutter of cantilevered plates in axial flow, J. Fluids Struct., 2, 127-147, (1995)
[4] Peskin, C. S., The immersed boundary method, Acta Numer., 11, 479-517, (2002) · Zbl 1123.74309
[5] Jaiman, R.; Shakib, F.; Oakley, O. H.; Constantinides, Y., Fully coupled fluid-structure interaction for offshore applications, (ASME Offshore Mechanics and Arctic Engineering OMAE09-79804 CP, (2009))
[6] Jaiman, R., Advances in ALE based fluid-structure interaction modeling for offshore engineering applications, (6th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), Vienna, (2012))
[7] Arienti, M.; Hung, P.; Morano, E.; Shepherd, E., A level set approach to eulerian-Lagrangian coupling, J. Comput. Phys., 185, 213-251, (2003) · Zbl 1047.76567
[8] LeVeque, R. J.; Li, Z., Immersed interface methods for Stokes flow with elastic boundaries, SIAM J. Sci. Comput., 18, 709-735, (1997) · Zbl 0879.76061
[9] Glowinski, R.; Pan, T. W.; Periaux, J., A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 112, 133-148, (1994) · Zbl 0845.76069
[10] Fedkiw, R. P., Coupling an eulerian fluid calculation to a Lagrangian solid calculation with the ghost fluid method, J. Comput. Phys., 175, 200-224, (2002) · Zbl 1039.76050
[11] Donea, J.; Giuliani, S.; Halleux, J. P., An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl. Mech. Eng., 33, 689-723, (1982) · Zbl 0508.73063
[12] Liu, J., Simple and efficient ALE methods with provable temporal accuracy up to fifth order for the Stokes equations on time varying domains, SIAM J. Numer. Anal., 51, 743-772, (2013) · Zbl 1268.76015
[13] Hron, J.; Turek, S., A monolithic FEM/multigrid solver for an ALE formulation of fluid-structure interaction with applications in biomechanics, (2006), Springer · Zbl 1323.74086
[14] Heil, M.; Hazel, A. L.; Boyle, J., Solvers for large-displacement fluid-structure interaction problems: segregated versus monolithic approaches, Comput. Mech., 43, 91-101, (2008) · Zbl 1309.76126
[15] Robinson-Mosher, A.; Schroeder, C.; Fedkiw, R., A symmetric positive definite formulation for monolithic fluid structure interaction, J. Comput. Phys., 230, 1547-1566, (2011) · Zbl 1390.74054
[16] Gibou, F.; Min, C., Efficient symmetric positive definite second-order accurate monolithic solver for fluid/solid interactions, J. Comput. Phys., 231, 3246-3263, (2012) · Zbl 1400.76049
[17] Felippa, C. A.; Park, K. C.; Farhat, C., Partitioned analysis of coupled mechanical systems, Comput. Methods Appl. Mech. Eng., 190, 3247-3270, (2001) · Zbl 0985.76075
[18] Matthies, H. G.; Niekamp, R.; Steindorf, J., Algorithms for strong coupling procedures, Comput. Methods Appl. Mech. Eng., 195, 2028-2049, (2006) · Zbl 1142.74050
[19] Jaiman, R.; Geubelle, P.; Loth, E.; Jiao, X., Combined interface condition method for unsteady fluid-structure interaction, Comput. Methods Appl. Mech. Eng., 200, 27-39, (2011) · Zbl 1225.74091
[20] Farhat, C.; van der Zee, K. G.; Geuzaine, P., Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear computational aeroelasticity, Comput. Methods Appl. Mech. Eng., 195, 1973-2001, (2006) · Zbl 1178.76259
[21] Blom, F. J., A monolithical fluid-structure interaction algorithm applied to the piston problem, Comput. Methods Appl. Mech. Eng., 167, 369-391, (1998) · Zbl 0948.76046
[22] Michler, C.; van Brummelen, E. H.; Hulshoff, S. J.; de Borst, R., The relevance of conservation for stability and accuracy of numerical methods for fluid-structure interaction, Comput. Methods Appl. Mech. Eng., 192, 4195-4215, (2003) · Zbl 1181.74156
[23] Piperno, S.; Farhat, C., Partitioned procedures for the transient solution of coupled aeroelastic problems part 1: model problem, theory, and two-dimensional application, Comput. Methods Appl. Mech. Eng., 124, 79-112, (1995) · Zbl 1067.74521
[24] Causin, P.; Gerbeau, J. F.; Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Comput. Methods Appl. Mech. Eng., 194, 4506-4527, (2005) · Zbl 1101.74027
[25] van Brummelen, E. H., Added mass effects of compressible and incompressible flows in fluid-structure interaction, J. Appl. Mech., 76, 021206, (2009)
[26] Liu, J., Combined field formulation and a simple stable explicit interface advancing scheme for fluid structure interaction, manuscript, available at
[27] Unverdi, S. O.; Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100, 25-37, (1992) · Zbl 0758.76047
[28] Fernández, M. A.; Gerbeau, J.-F.; Grandmont, C., A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid, Int. J. Numer. Methods Eng., 69, 794-821, (2007) · Zbl 1194.74393
[29] Masud, A.; Hughes, T. J.R., A space-time Galerkin/least-square finite element formulation of the Navier-Stokes equations for moving domain problems, Comput. Methods Appl. Mech. Eng., 146, 91-126, (1997) · Zbl 0899.76259
[30] Jaiman, R.; Jiao, X.; Geubelle, P.; Loth, E., Assessment of conservative load transfer on fluid-solid interface with nonmatching meshes, Int. J. Numer. Methods Eng., 64, 2014-2038, (2005) · Zbl 1122.74544
[31] Temam, R., Navier-Stokes equations. theory and numerical analysis, (2001), AMS Chelsea Publishing · Zbl 0981.35001
[32] Weinan, E.; Liu, Jian-Guo, Projection method III: spatial discretization on the staggered grid, Math. Comput., 71, 237, 27-47, (2002) · Zbl 1058.76043
[33] Liu, J.-G.; Liu, J.; Pego, R. L., Stable and accurate pressure approximation for unsteady incompressible viscous flow, J. Comput. Phys., 229, 3428-3453, (2010) · Zbl 1307.76029
[34] Quarteroni, A.; Valli, A., Numerical approximation of partial differential equations, (1994), Springer-Verlag Berlin · Zbl 0803.65088
[35] Brenner, S. C.; Scott, R., The mathematical theory of finite element methods, (2002), Springer-Verlag · Zbl 1012.65115
[36] Chen, L.; Zhang, C.-S., A coarsening algorithm on adaptive grids by newest vertex bisection and its application, J. Comput. Math., 28, 767-789, (2010) · Zbl 1240.65350
[37] Liu, J., Open and traction boundary conditions for the incompressible Navier-Stokes equations, J. Comput. Phys., 228, 7250-7267, (2009) · Zbl 1386.76114
[38] Jaiman, R.; Parmar, M. K.; Gurugubelli, P. S., Added mass and aeroelastic stability of a flexible plate interacting with mean flow in a confined channel, J. Appl. Mech., 81, (2013)
[39] Watanabe, Y.; Suzuki, S.; Sughihara, M.; Sueoka, Y., An experimental study of paper flutter, J. Fluids Struct., 16, 529-542, (2002)
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