OPEN ACCESS
In this paper, a formulation using the Generalized Finite Element Method (GFEM) in conjunction with Lagrange Multipliers is proposed to impose the boundary condition on the interface of the Fluid-Structure Interaction (FSI) problem. The objective of this work is the development of an efficient and robust computational code for solving problems of Fluid Mechanics and FSI. We chose a formulation of Immersed Borders to allow simulations of problems involving complex movements and transformations of the structure. In problems with these characteristics, classical ALE approaches tend to lose robustness because of the need for fluid mesh reconstruction to avoid excessive distortion of the elements. Examples of future applications are biomechanics, aeroelasticity of civil works and aerospace and multiphysical structures. The numerical examples solved proved that the formulation and implementation in finite elements developed in this work are capable to solve problems of 2D flow of fluids described by the Navier-Stokes equation for incompressible flows, even in regimes with dominant convection; and, to simulate the fluid problems with mobile interfaces using the concept of boundaries immersed in two dimensions
generalized finite element method, mobile interfaces, incompressible flows
Akkerman I., Dunaway J., Kvandal J., Spinks J., Bazilevs Y. (2012). Toward free-surface modeling of planing vessels: simulation of the Fridsma hull using ALE-VMS. Computational Mechanics, Vol. 50, No. 6, pp. 719-727. https://doi.org/10.1007/s00466-012-0770-2
Bazilevs Y., Calo V. M., Hughes T. J., Zhang Y. (2008). Isogeometric fluid-structure interaction: theory, algorithms, and computations. Computational Mechanics, Vol. 43, No. 1, pp. 3-37. https://doi.org/10.1007/s00466-008-0315-x
Bazilevs Y., Gohean J., Hughes T. J. R., Moser R. D., Zhang Y. (2009). Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Computer Methods in Applied Mechanics and Engineering, Vol. 198, No. 45-46, pp. 3534-3550. https://doi.org/10.1016/j.cma.2009.04.015
Braess H, Wriggers P. (2000). Arbitrary Lagrangian Eulerian finite element analysis of free surface flow. Comput. Computer Methods in Applied Mechanics and Engineering, Vol. 190, pp. 95-109. https://doi.org/10.1016/S0045-7825(99)00416-8
Brooks A. N., Hughes T. J. R. (1982). Streamline upwind/pretrov-galerkin formulations fo convective dominated flows with particular emphasis on the incompressible navier-stokes equations. Computer Methods in Applied Mechanics and Engineering, Vol. 32, pp. 199-259. https://doi.org/10.1016/0045-7825(82)90071-8
Buscaglia G. C., Ausas R. F. (2011). Variational formulations for surface tension, capillarity and wetting. Comput. Methods Appl. Mech. Engrg. Vol. 200, pp. 3011-3025. https://doi.org/10.1016/j.cma.2011.06.002
Dettmer W., Peric D. (2006). A computational framework for free surface fluid flows accounting for surface tension. Comput. Methods Appl. Mech. Engrg. Vol. 195, pp. 3038-3071. https://doi.org/10.1016/j.cma.2004.07.057
Elias R. N., Coutinho A. L. G. A. (2009). Computational techniques for stabilized edge-based finite element simulation of nonlinear free-surface flows. J. Offshore Mech. Arct. Eng., Vol. 54, No. 6-8, pp. 965-993. https://doi.org/10.1002/fld.1475
Feng Y. T., Peric D. (2000). A time-adaptive space-time finite element method for incompressible Lagrangian flows with free surfaces: computational issues. Comput. Methods Appl. Mech. Engrg., Vol. 190, No. 5, pp. 499-518. https://doi.org/10.1016/S0045-7825(99)00425-9
Förster C., Wall W. A., Ramm E. (2007). Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Comput. Methods Appl. Mech. Engrg., Vol. 196, No. 7, pp. 1278-1293. https://doi.org/10.1016/j.cma.2006.09.002
Gerstenberger A., Wall W. A. (2010). An embedded Dirichlet formulation for 3D continua. Int. J. Numer. Meth. Engng, Vol. 82, No. 5, 537-563. https://doi.org/10.1002/nme.2755
Gerstenberger A., Wall W. A. (2008). An eXtended Finite Element Method/Lagrange multiplier based Approach for Fluid-Structure Interaction. Comput. Methods Appl. Mech. Engrg., Vol. 197, No. 19, pp. 1699-1714. https://doi.org/10.1016/j.cma.2007.07.002
Idelsohn S. R., Marti J., Limache A., Oñate E. (2008). Unified Lagrangian Formulation for Elastic Solids and Incompressible Fluids: Application to Fluid–Structure Interaction Problems via the PFEM. Comput. Methods Appl. Mech. Engrg., Vol. 197, No. 19-20, pp. 1762-1776. https://doi.org/10.1016/j.cma.2007.06.004
Idelsohn S. R., Oñate E., Pin F. D., Calvo N. (2006). Fluid–Structure Interaction using the Particle Finite Element Method. Comput. Methods Appl. Mech. Engrg., Vol. 195, pp. 2100-2123.
Idelsohn S. R., Storti M. A., Oñate E. (2001). Lagrangian formulations to solve free surface incompressible inviscid fluid flows. Comput. Methods Appl. Mech. Engrg., Vol. 191, pp. 583-593. https://doi.org/10.1016/S0045-7825(01)00303-6
Kulasegaram S., Bonet J., Lewis R. W., Profit M. A. (2004). A variational formulation based contact algorithm for rigid boundaries in two-dimensional SPH applications. Comput. Mech., Vol. 33, pp. 316–325. https://doi.org/10.1007/s00466-003-0534-0
Küttler U., Wall W. A. (2008). Fixed-point fluid–structure interaction solvers with dynamic relaxation. Comput. Mech., Vol. 43, pp. 61-72. https://doi.org/10.1007/s00466-008-0255-5
Legay A., Chessa J., Belytschko T. (2006). An Eulerian-Lagrangian method for fluid-structure interaction based on level sets. Comput. Methods Appl. Mech. Engrg., Vol. 195, pp. 2070 -2087. https://doi.org/10.1016/j.cma.2005.02.025
Lew A. J., Buscaglia G. C. (2008). A discontinuous-Galerkin-based immersed boundary method. Int. J. Numer. Meth. Engng, Vol. 76, pp. 427-454. https://doi.org/10.1051/m2an/2010069
Lins E. F., Elias R. N., Rochinha F. A., Coutinho A. L. G. A. (2010). Residual-based variational multiscale simulation of free surface flows. Comput. Mech., Vol. 46, pp. 545–557. https://doi.org/10.1007/s00466-010-0495-z
Löhner R., Yang C., Oñate E. (2006). On the simulation of flows with violent free surface motion. Comput. Methods Appl. Mech. Engrg. Vol. 195, pp. 5597–5620. https://doi.org/10.1016/j.cma.2005.11.010
Maier A., Gee M. W., Reeps C., Pongratz J., Eckstein H. H., Wall. W. A. (2010). A comparison of diameter, wall stress, and rupture potential index for abdominal aortic aneurysm rupture risk prediction. Annals of Biomedical Engineering, Vol. 38, pp. 3124–3134. https://doi.org/10.1007/s10439-010-0067-6
Marrone S., Antuono M., Colagrossi A., Colicchio G., Le-Touzé D., Graziani G. (2011). δ-SPH model for simulating violent impact flows. Comput. Methods Appl. Mech. Engrg. Vol. 200, pp. 1526-1542. https://doi.org/10.1016/j.cma.2010.12.016
Moës N., Béchet E., Tourbier M. (2006). Imposing Dirichlet boundary conditions in the extended finite element method. Int. J. Numer. Methods Engrg, Vol. 67, pp. 1641-1669. https://doi.org/10.1002/nme.1675
Oñate E., García J. (2001). A finite element method for fluid-structure interaction with surface waves using a finite calculus formulation. Comput. Methods Appl. Mech. Engrg. Vol. 191, pp. 635-660. https://doi.org/10.1016/S0045-7825(01)00306-1
Oñate E., Idelsohn S. R., Celigueta M. A., Rossi R. (2008). Advances in the Particle Finite Element Method for the Analysis of Fluid–Multibody Interaction and Bed Erosion in Free Surface Flows. Comput. Methods Appl. Mech. Engrg. Vol. 197, pp. 1777-1800. https://doi.org/10.1016/j.cma.2007.06.005
Rüberg T., Cirak F. (2012). Subdivision-stabilised immersed b-spline finite elements for moving boundary flows. Comput. Methods Appl. Mech. Engrg. Vol. 209–212, pp. 266–283. Retrieved from https://doi.org/10.1016/j.cma.2011.10.007
Sanders J. D., Laursen T. A., Puso M. A. (2012). A Nitsche embedded mesh method. Comput. Mech. Vol. 49, pp. 243–257. https://doi.org/10.1007/s00466-011-0641-2
Sawada T., Tezuka A. (2010). High-order Gaussian quadrature in X-FEM with the Lagrange Multiplier for fluid-structure coupling. Int. J. Numer. Meth. Fluids Vol. 64, pp. 1219-1239. https://doi.org/10.1002/fld.2343
Schäfer M., Turek S. (1996). Benchmark computations of laminar flow around a cylinder. Notes Numer. Fluid Mech. Vol. 52, pp. 547-566. https://doi.org/10.1007/978-3-322-89849-4_39
Takizawa K., Yabe T., Tsugawa Y., Tezduyar T., Mizoe H. (2007). Computation of free-surface flows and fluid–object interactions with the CIP method based on adaptive meshless soroban grids. Comput. Mech. Vol. 40, pp. 167–183. https://doi.org/10.1007/s00466-006-0093-2
Takizawa K., Christopher J., Tezduyar T. E., Sathe S. (2010). Space-time finite element computation of arterial fluid-structure interactions with patient-specific data. Int. J. Numer. Meth. Biomed. Engng. Vol. 26, pp. 101-116. https://doi.org/10.1002/cnm.1241
Tezduyar T. E., Osawa Y. (2000). Finite Element Stabilization Parameters Computed from Element Matrices and Vectors. Comput. Methods Appl. Mech. Engrg. Vol. 190, pp. 411-430. https://doi.org/10.1016/S0045-7825(00)00211-5
Tezduyar T. E., Takizawa K., Brummer T., Chen P. R. (2011). Space–time fluid–structure interaction modeling of patient-specific cerebral aneurysms. Int. J. Numer. Meth. Biomed. Engng. Vol. 27, pp. 1665-1710. https://doi.org/10.1007/978-94-007-7769-9_2
Torii R., Oshima M., Kobayashi T., Takagi K., Tezduyar T. (2011). Influencing factors in image-based fluid–structure interaction computation of cerebral aneurysms. Int. J. Numer. Meth. Fluids, Vol. 65, pp. 324–340. https://doi.org/10.1002/fld.2448
Torii R., Oshima M., Kobayashi T., Takagi K., Tezduyar T. (2007). Numerical investigation of the effect of hypertensive blood pressure on cerebral aneurysm—Dependence of the effect on the aneurysm shape. Int. J. Numer. Meth. Fluids, Vol. 54, pp. 995-1009. https://doi.org/10.1002/fld.1497
Torii R., Oshima M., Kobayashi T.,Takagi K., Tezduyar T. (2009). Fluid–structure interaction modeling of blood flow and cerebral aneurysm: Significance of artery and aneurysm shapes. Comput. Methods Appl. Mech. Engrg. Vol. 198, pp. 3613-3621. https://doi.org/10.1016/j.cma.2008.08.020
Turek S., Hron J., Razzaq M., Wobker H., Schäfer M (2010). Numerical Benchmarking of Fluid-Structure Interaction: A comparison of different discretization and solution approaches. Lecture Notes in Computational Science and Engineering, Vol. 73, pp. 413-424. https://doi.org/10.1007/978-3-642-14206-2_15
Zienkiewicz O. C., Taylor R. L. (2009). The finite element method. fluid dynamics. Vol. 3.
Zhaosheng Y. (2005). A DLM/FD method for fluid/flexible-body interactions. J. Comput. Phys., Vol. 207, pp. 1-27.
Zilian A., Fries T. P. (2009). A localized mixed-hybrid method for imposing interfacial constraints in the extended finite element method (XFEM). Int. J. Numer. Meth. Engng, Vol. 79, pp. 733-752. https://doi.org/10.1002/nme.2596
Zilian A., Legay A. (2008). The enriched space–time finite element method (EST) for simultaneous solution of fluid–structure interaction. Int. J. Numer. Meth. Engng, Vol. 75, pp. 305-334. https://doi.org/10.1002/nme.2258