×

Investigation of fluid-structure interactions using a velocity-linked P2/P1 finite element method and the generalized- \(\alpha \) method. (English) Zbl 1246.74059

Summary: A velocity-linked algorithm for solving unsteady fluid-structure interaction (FSI) problems in a fully coupled manner is developed using the arbitrary Lagrangian-Eulerian method. The P2/P1 finite element is used to spatially discretize the incompressible Navier-Stokes equations and structural equations, and the generalized- \(\alpha \) method is adopted for temporal discretization. Common velocity variables are employed at the fluid-structure interface for the strong coupling of both equations. Because of the velocity-linked formulation, kinematic compatibility is automatically satisfied and forcing terms do not need to be calculated explicitly. Both the numerical stability and the convergence characteristics of an iterative solver for the coupled algorithm are investigated by solving the FSI problem of flexible tube flows. It is noteworthy that the generalized- \(\alpha \) method with small damping is free from unstable velocity fields. However, the convergence characteristics of the coupled system deteriorate greatly for certain Poisson’s ratios so that direct solvers are essential for these cases. Furthermore, the proposed method is shown to clearly display the advantage of considering FSI in the simulation of flexible tube flows, while enabling much larger time-steps than those adopted in some previous studies. This is possible through the strong coupling of the fluid and structural equations by employing common primitive variables.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Wootton, Fluid mechanics of vascular systems, diseases, and thrombosis, Annual Review of Biomedical Engineering 1 pp 299– (1999) · doi:10.1146/annurev.bioeng.1.1.299
[2] Yamaguchi, Computational blood flow analysis - new trends and methods, Journal of Biomechanical Science and Engineering 1 pp 29– (2006) · doi:10.1299/jbse.1.29
[3] Zhang, Investigation of two-dimensional channel flow with a partially compliant wall using finite volume-finite difference approach, International Journal for Numerical Methods in Fluids 49 pp 635– (2005) · Zbl 1075.76045 · doi:10.1002/fld.1022
[4] Figueroa, A coupled momentum method for modeling blood flow in three-dimensional deformable arteries, Computer Methods in Applied Mechanics and Engineering 195 pp 5685– (2006) · Zbl 1126.76029 · doi:10.1016/j.cma.2005.11.011
[5] Zilian, The enriched space-time finite element method (EST) for simultaneous solution of fluid-structure interaction, International Journal for Numerical Methods in Engineering 75 pp 305– (2008) · Zbl 1195.74212 · doi:10.1002/nme.2258
[6] Guidoboni, Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow, Journal of Computational Physics 228 pp 6916– (2009) · Zbl 1261.76056 · doi:10.1016/j.jcp.2009.06.007
[7] Lee, Modelling of flow and wall behaviour in a mildly stenosed tube, Medical Engineering & Physics 24 pp 575– (2002) · doi:10.1016/S1350-4533(02)00048-6
[8] Kuhl, An arbitrary Lagrangian Eulerian finite-element approach for fluid-structure interaction phenomena, International Journal for Numerical Methods in Engineering 57 pp 117– (2003) · Zbl 1062.74617 · doi:10.1002/nme.749
[9] Greenshields, A unified formulation for continuum mechanics applied to fluid-structure interaction in flexible tubes, International Journal for Numerical Methods in Engineering 64 pp 1575– (2005) · Zbl 1122.74379 · doi:10.1002/nme.1409
[10] Ishihara, A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation, International Journal for Numerical Methods in Engineering 64 pp 167– (2005) · Zbl 1104.74026 · doi:10.1002/nme.1348
[11] Bazilevs, Isogeometric fluid-structure interaction analysis with applications to arterial blood flow, Computational Mechanics 38 pp 310– (2006) · Zbl 1161.74020 · doi:10.1007/s00466-006-0084-3
[12] Liu, Immersed finite element method and its applications to biological systems, Computer Methods in Applied Mechanics and Engineering 195 pp 1722– (2006) · Zbl 1178.76232 · doi:10.1016/j.cma.2005.05.049
[13] Papadakis, A novel pressure-velocity formulation and solution method for fluid-structure interaction problems, Journal of Computational Physics 227 pp 3383– (2008) · Zbl 1329.74080 · doi:10.1016/j.jcp.2007.12.004
[14] Braun, A partitioned model for fluid-structure interaction problems using hexahedral finite elements with one-point quadrature, International Journal for Numerical Methods in Engineering 79 pp 505– (2009) · Zbl 1171.74438 · doi:10.1002/nme.2566
[15] Idelsohn, Fluid-structure interaction problems with strong added mass effect, International Journal for Numerical Methods in Engineering 80 pp 1261– (2009) · Zbl 1183.74059 · doi:10.1002/nme.2659
[16] Wang, Fluid-structure interaction by the discontinuous-Galerkin method for large deformations, International Journal for Numerical Methods in Engineering 77 pp 30– (2009) · Zbl 1195.74204 · doi:10.1002/nme.2396
[17] Barker, Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling, Journal of Computational Physics 229 pp 642– (2010) · Zbl 1253.76137 · doi:10.1016/j.jcp.2009.10.001
[18] Kim, A new coupling strategy for fluid-solid interaction problems by using the interface element method, International Journal for Numerical Methods in Engineering 81 pp 403– (2010) · Zbl 1183.74286
[19] Jog, A monolithic strategy for fluid-structure interaction problems, International Journal for Numerical Methods in Engineering 85 pp 429– (2011) · Zbl 1217.74124 · doi:10.1002/nme.2976
[20] Zhang, Studies of the strong coupling and weak coupling methods in FSI analysis, International Journal for Numerical Methods in Engineering 60 pp 2013– (2004) · Zbl 1070.74049 · doi:10.1002/nme.1034
[21] Jaiman, Assessment of conservative load transfer for fluid-solid interface with non matching meshes, International Journal for Numerical Methods in Engineering 64 pp 2014– (2005) · Zbl 1122.74544 · doi:10.1002/nme.1434
[22] Badia, On some fluid-structure iterative algorithms using pressure segregation methods. Application to aeroelasticity, International Journal for Numerical Methods in Engineering 72 pp 46– (2007) · Zbl 1194.74361 · doi:10.1002/nme.1998
[23] Kollmannsberger, Fixed-grid fluid-structure interaction in two dimensions based on a partitioned Lattice Boltzmann and p-FEM approach, International Journal for Numerical Methods in Engineering 79 pp 817– (2009) · Zbl 1171.74340 · doi:10.1002/nme.2581
[24] Michler, A monolithic approach to fluid-structure interaction, Computers & Fluids 33 pp 839– (2004) · Zbl 1053.76042 · doi:10.1016/j.compfluid.2003.06.006
[25] Causin, Added-mass effect in the design of partitioned algorithms for fluid-structure problems, Computer Methods in Applied Mechanics and Engineering 194 pp 4506– (2005) · Zbl 1101.74027 · doi:10.1016/j.cma.2004.12.005
[26] Gresho, On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory, International Journal for Numerical Methods in Fluids 11 pp 587– (1990) · Zbl 0712.76035 · doi:10.1002/fld.1650110509
[27] Ramaswamy, Some recent trends and developments in finite element analysis for incompressible thermal flows, International Journal for Numerical Methods in Engineering 35 pp 671– (1992) · Zbl 0764.76038 · doi:10.1002/nme.1620350405
[28] Choi, A fractional four-step finite element formulation of the unsteady incompressible Navier-Stokes equations using SUPG and linear equal-order element methods, Computer Methods in Applied Mechanics and Engineering 143 pp 333– (1997) · Zbl 0896.76036 · doi:10.1016/S0045-7825(96)01156-5
[29] Codina, The intrinsic time for the streamline upwind/Petrov-Galerkin formulation using quadratic elements, Computer Methods in Applied Mechanics and Engineering 94 pp 239– (1992) · Zbl 0748.76082 · doi:10.1016/0045-7825(92)90149-E
[30] Chung, A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized- {\(\alpha\)} method, Journal of Applied Mechanics 60 pp 371– (1993) · Zbl 0775.73337 · doi:10.1115/1.2900803
[31] Jansen, A generalized- {\(\alpha\)} method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Computer Methods in Applied Mechanics and Engineering 190 pp 305– (2000) · Zbl 0973.76048 · doi:10.1016/S0045-7825(00)00203-6
[32] Kuhl, Energy-conserving and decaying algorithms in non-linear structural dynamics, International Journal for Numerical Methods in Engineering 45 pp 569– (1999) · Zbl 0946.74078 · doi:10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.CO;2-A
[33] Lee SH Choi HG Yoo JY Finite element simulation of blood flow in a flexible carotid artery bifurcation Proceedings of ASME-JSME-KSME Joint Fluids Engineering Conference 2011 2011
[34] Wiggert, Fluid transients and fluid-structure interaction in flexible liquid-filled piping, Applied Mechanics Reviews 54 pp 455– (2001) · doi:10.1115/1.1404122
[35] Tijsseling, Water hammer with fluid-structure interaction in thick-walled pipes, Computers & Structures 85 pp 844– (2007) · doi:10.1016/j.compstruc.2007.01.008
[36] Erlicher, The analysis of the Generalized- {\(\alpha\)} methods for non-linear dynamic problems, Computational Mechanics 28 pp 83– (2002) · Zbl 1146.74327 · doi:10.1007/s00466-001-0273-z
[37] van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing 13 pp 631– (1992) · Zbl 0761.65023 · doi:10.1137/0913035
[38] Nam, AILU preconditioning for the finite element formulation of the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 191 pp 4323– (2002) · Zbl 1015.76048 · doi:10.1016/S0045-7825(02)00369-9
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.