The purpose of this paper is to investigate the problem of weak parting coupling between incompressible fluids and shell structures that can develop large displacements. For this, a code computational model with formulation based on the finite element method (FEM) for analysis of incompressible flows in arbitrary Lagrangian-Eulerian description (ALE), which is coupled to an existing dynamic analysis program.
In this work a positional FEM approach for the dynamic shell modeling considering the geometric nonlinearity was coupled to an FEM based methodology for the simulation of Newtonian fluids in ALE description using quadratic order elements for velocity and linear for pressure. In addition, a coupling proposal without the need of coincidence of the nodes of the domains accompanied by a scheme of dynamic movement of the fluid network based on the use of an auxiliary mesh with cubic order elements was successfully implemented.
For the consideration of the geometric nonlinearity of shell structures, a formulation described in positions that does not interpolate rotations as degrees of freedom was employed. This technique proved to be robust and capable of simulating dynamic instability problems.
The treatment of the fluid by means of the mixed formulation, or pressure-velocity, with stabilization by means of the Streamline Upwind Petrov-Galerkin (SUPG) technique proved to be quite suitable for the simulation of laminar flows, producing satisfactory results and in accordance with the literature.
fluid-structure interaction, arbitrary lagrangian-eulerian description, incompressible flows, nonlinear geometric analysis, partitioned coupling
Ambethkar V., Kumar M. (2017). Numerical solutions of 2-D unsteady incompressible flow with heat transfer in a driven square cavity using streamfunction-vorticity formulation. International Journal of Heat and Technology, Vol. 35, No. 1, pp. 459-473. http://doi.org/10.18280/ijht.350303
Badia S., Nobile F., Vergara C. (2008). Fluid–structure partitioned procedures based on Robin transmission conditions. Journal of Computational Physics, Vol. 227, No. 14, pp. 7027-7051. http://doi.org/10.1016/j.jcp.2008.04.006
Blom F. J. (1998). A monolithical fluid-structure interaction algorithm applied to the piston problem. Computer methods in applied mechanics and engineering, Vol. 167, No. 3-4, pp. 369-391. https://doi.org/10.1016/S0045-7825(98)00151-0
Boffi D., Gastaldi L. (2004). Stability and geometric conservation laws for ALE formulations. Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 42, pp. 4717-4739. https://doi.org/10.1016/j.cma.2004.02.020
Brummelen-Van E. H. (2009). Added mass effects of compressible and incompressible flows in fluid-structure interaction. Journal of Applied Mechanics, Vol. 76, No. 2, pp. 021206. doi:10.1115/1.3059565
Causin P., Gerbeau J. F., Nobile F. (2005). Added-mass effect in the design of partitioned algorithms for fluid–structure problems. Computer Methods in Applied Mechanics and Engineering, Vol. 194, No. 42, pp. 4506-4527. https://doi.org/10.1016/j.cma.2004.12.005
Donea J., Giuliani S., Halleux J. P. (1982). An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Computer Methods in Applied Mechanics and Engineering, Vol. 33, No. 1-3, pp. 689-723. https://doi.org/10.1016/0045-7825(82)90128-1
Farhat C., Lesoinne M., Maman N. (1995). Mixed explicit/implicit time integration of coupled aeroelastic problems: Three‐field formulation, geometric conservation and distributed solution. International Journal for Numerical Methods in Fluids, Vol. 21, No. 10, pp. 807-835. https://doi.org/10.1002/fld.1650211004
Felippa C. A., Park K. C., Farhat C. (2001). Partitioned analysis of coupled mechanical systems. Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 24, pp. 3247-3270. https://doi.org/10.1016/S0045-7825(00)00391-1
Formaggia L., Nobile F. (2004). Stability analysis of second-order time accurate schemes for ALE–FEM. Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 39, pp. 4097-4116. https://doi.org/10.1016/j.cma.2003.09.028
Förster C., Wall W. A., Ramm E. (2007). Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering, Vol. 196, No. 7, pp. 1278-1293. https://doi.org/10.1016/j.cma.2006.09.002
Gerbeau J. F., Vidrascu M. (2003). A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 37, No. 4, pp. 631-647. https://doi.org/10.1051/m2an:2003049
Heil M., Hazel A. L., Boyle J. (2008). Solvers for large-displacement fluid–structure interaction problems: Segregated versus monolithic approaches. Computational Mechanics, Vol. 43, No. 1, pp. 91-101. https://doi.org/10.1007/s00466-008-0270-6
Hou G., Wang J., Layton A. (2012). Numerical methods for fluid-structure interaction—a review. Communications in Computational Physics, Vol. 12, No. 2, pp. 337-377. https://doi.org/10.4208/cicp.291210.290411s
Hron J., Mádlík M. (2007). Fluid-structure interaction with applications in biomechanics. Nonlinear Analysis: Real World Applications, Vol. 8, No. 5, pp. 1431-1458. https://doi.org/10.1016/j.nonrwa.2006.05.007
Hübner B., Walhorn E., Dinkler D. (2004). A monolithic approach to fluid–structure interaction using space–time finite elements. Computer Methods in Applied Mechanics and Engineering, Vol. 193, No. 23, pp. 2087-2104. https://doi.org/10.1016/j.cma.2004.01.024
Ike C. C. (2018). Flexural analysis of rectangular kirchhoff plate on winkler foundation using galerkin-vlasov variational method. Mathematical Modelling of Engineering Problems, Vol. 5, No. 2, pp. 83-92. https://doi.org/10.18280/mmep.050205
Jha B. K., Yusuf T. S. (2018). Transient pressure driven flow in an annulus partially filled with porous material: Azimuthal pressure gradient. Mathematical Modelling of Engineering Problems, Vol. 5, No. 3, pp. 260-267. https://doi.org/10.18280/mmep.050320
Kanchi H., Masud A. (2007). A 3D adaptive mesh moving scheme. International Journal for Numerical Methods in Fluids, Vol. 54, No. 6, pp. 923-944. https://doi.org/10.1002/fld.1512
Koobus B., Farhat C. (1999). Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes. Computer Methods in Applied Mechanics and Engineering, Vol. 170, No. 1, pp. 103-129. https://doi.org/10.1016/S0045-7825(98)00207-2
Lesoinne M., Farhat C. (1996). Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Computer Methods in Applied Mechanics and Engineering, Vol. 134, No. 1, pp. 71-90. https://doi.org/10.1016/0045-7825(96)01028-6
MOK D. P. (2001). Partitioned approaches in structural dynamics and fluid-structure interaction. Thesis (Doctorate)- Institute of Structural Engineering, University of Stuttgart, Stuttgart.
Morton S. A., Melville R. B., Visbal M. R. (1998). Accuracy and coupling issues of aeroelastic Navier-Stokes solutions on deforming meshes. Journal of Aircraft, Vol. 35, No. 5, pp. 798-805. https://doi.org/10.2514/2.2372
Piperno S. (1997). Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations. International Journal for Numerical Methods in Fluids, Vol. 25, No. 10, pp. 1207-1226. 10.1002/(SICI)1097-0363(19971130)25:103.0.CO;2-R
Roux F. X., Garaud J. D. (2009). Domain decomposition methodology with Robin interface matching conditions for solving strongly coupled fluid-structure problems. International Journal for Multiscale Computational Engineering, Vol. 7, No. 1. https://doi.org/10.1007/978-3-540-69387-1_34
Sanches R. A. K. (2011). On the fluid-shell coupling using the finite element method. Thesis (Doctorate) - SET-EESC-USP, São Paulo, Brazil.
Sanches R. A. K., Coda H. B. (2010b). Fluid-structure interaction using an arbitrary Lagrangian-Eulerian fluid solver coupled to a positional Lagrangian shell solver. Applied Mathematical Modelling, Vol. 38, No. 14, pp. 3401-3418. https://doi.org/10.1016/j.apm.2013.11.025
Sanches R. A., Coda H. B. (2010a). Mecánica Computacional. Vol. XXIX. No. 48. Mathematical Foundations of MEF and Meshless Methods (C).
Tallec Le P., Mouro J. (2001). Fluid structure interaction with large structural displacements. Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 24, pp. 3039-3067. https://doi.org/10.1016/S0045-7825(00)00381-9
Teixeira P. R. D. F., Awruch A. M. (2005). Numerical simulation of fluid–structure interaction using the finite element method. Computers & Fluids, Vol. 34, No. 2, pp. 249-273. https://doi.org/10.1016/j.compfluid.2004.03.006
Thomas P. D., Lombard C. K. (1979). Geometric conservation law and its application to flow computations on moving grids. AIAA Journal, Vol. 17, No. 10, pp. 1030-1037. https://doi.org/10.2514/3.61273
Vázquez J. G. V. (2007). Nonlinear Analysis of Orthotropic Membrane and Shell Structures Including Fluid-Structure Interaction (Doctoral dissertation, Universitat Politècnica de Catalunya). Barcelona, Espanha.
Wall W. A., Ramm E. (1998). Fluid-structure interaction based upon a stabilized (ALE) finite element method. Barcelona. In IV World Congress on Computational Mechanics, Vol. 5.
Warburton G. B. (1976). The dynamical behaviour of structures. 2. ed. Oxford: Pergamon Press.