Possibilities of the particle finite element method for fluid-structure interaction problems with free surface waves
Keywords:
particle finite element method, finite element method, fluid-structure interaction, finite calculusAbstract
We present a general formulation for analysis of fluid-structure interaction problems using the particle finite element method (PFEM). The key feature of the PFEM is the use of a Lagrangian description to model the motion of nodes (particles) in both the fluid and the structure domains. Nodes are thus viewed as particles which can freely move and even separate from the main analysis domain representing, for instance, the effect of water drops. A mesh connects the nodes defining the discretized domain where the governing equations, expressed in an integral from, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility condition in the fluid is introduced via the finite calculus (FIC) method. A fractional step scheme for the transient coupled fluid-structure solution is described. Examples of application of the PFEM method to solve a number of fluid-structure interaction problems involving large motions of the free surface and splashing of waves are presented.
Downloads
References
[AUB 04] AUBRY R., IDELSOHN S.R., OÑATE E., “Particle finite element method in fluid
mechanics including thermal convection-diffusion”, Computer & Structures, submitted,
[CHO 67] CHORIN, A.J., “A numerical solution for solving incompressible viscous flow problems”,
J. Comp. Phys., vol. 2, 1967, p. 12–26.
[COD 02] CODINA R., “Stabilized finite element approximation of transient incompressible
flows using orthogonal subscales”, Comp. Meth. Appl. Mech. Engng., vol. 191, 2002,
p. 4295–4321.
[COD 98] CODINA R., VAZQUEZ M., ZIENKIEWICZ O.C., “A general algorithm for compressible
and incompressible flow - Part III. The semi-implicit form”, Int. J. Num. Meth.
in Fluids, vol. 27, 1998, p. 13–32.
[COD 00] CODINA R., BLASCO J., “Stabilized finite element method for the transient Navier-
Stokes equations based on a pressure gradient operator”, Comp. Meth. in Appl. Mech.
Engng., vol. 182, 2000, p. 277–301.
[COD 02B] CODINA R., ZIENKIEWICZ O.C., “CBS versus GLS stabilization of the incompressible
Navier-Stokes equations and the role of the time step as stabilization parameter”,
Communications in Numerical Methods in Engineering, vol. 18, 2002, p. 99–112.
[CRU 97] CRUCHAGA M.A., OÑATE E., “A finite element formulation for incompressible
flow problems using a generalized streamline operator”, Comp. Meth. Appl. Mech.
Engng., vol. 143, 1997, p. 49–67.
[CRU 99] CRUCHAGA M.A., OÑATE E., “A generalized streamline finite element approach
for the analysis of incompressible flow problems including moving surfaces”, Comp.
Meth. Appl. Mech. Engng., vol. 173, 1999, p. 241–255.
[DON 03] DONEA J., HUERTA A., Finite element method for flow problems. J. Wiley, 2003.
[EDE 99] EDELSBRUNNER H., MUCKE E.P., “Three-dimensional alpha shapes”, ACM Trans.
Graphics, vol. 13, 1999, p. 43–72.
[FRA 92] FRANCA L.P., FREY S.L., “Stabilized finite element methods: II. The incompressible
Navier-Stokes equations”, Comp. Meth. Appl. Mech. Engng., vol. 99, 1992, p.
–233.
[GAR 03] GARCÍA J., OÑATE E., “An unstructured finite element solver for ship hydrodynamic
problems”, J. Appl. Mech., vol. 70, 2003, p. 18–26.
[GID 04] “GID. The personal pre/postprocessor”, CIMNE, Barcelona, www.gidhome.com,
[HAN 90] HANSBO P., SZEPESSY A., “A velocity-pressure streamline diffusion finite element
method for the incompressible Navier-Stokes equations”, Comp. Meth. Appl. Mech.
Engng., vol. 84, 1990, p. 175–192.
[HUG 86] HUGHES T.J.R., FRANCA L.P., BALESTRA M., “A new finite element formulation
for computational fluid dynamics. V Circumventing the Babuska-Brezzi condition:
A stable Petrov-Galerkin formulation of the Stokes problem accomodating equal order
interpolations”, Comp. Meth. Appl. Mech. Engng., vol. 59, 1986, p. 85–89.
[HUG 89] HUGHES T.J.R., FRANCA L.P., HULBERT G.M., “A new finite element formulation
for computational fluid dynamics: VIII. The Galerkin/least-squares method for
advective-diffusive equations”, Comp. Meth. Appl. Mech. Engng., vol. 73, 1989, p.
–189.
[HUG 94] HUGHES T.J.R., HAUKE G., JANSEN K., “Stabilized finite element methods in
fluids: Inspirations, origins, status and recent developments”, in: Recent Developments
in Finite Element Analysis. A Book Dedicated to Robert L. Taylor, T.J.R. Hughes, E.
Oñate and O.C. Zienkiewicz (Eds.), CIMNE, Barcelona, Spain, p. 272–292, 1994.
[IDE 02] IDELSOHN S.R., OÑATE E., DEL PIN F., CALVO N., “Lagrangian formulation: the
only way to solve some free-surface fluid mechanics problems”, Fith World Congress on
Computational Mechanics, Mang HA, Rammerstorfer FG and Eberhardsteiner J. (eds),
Vienna, Austria, 2002.
[IDE 03] IDELSOHN S.R., OÑATE E., CALVO N., DEL PIN F., “The meshless finite element
method”, Int. J. Num. Meth. Engng., vol. 58, 2003, p. 893–912.
[IDE 03B] IDELSOHN S.R., OÑATE E., DEL PIN F., “A lagrangian meshless finite element
method applied to fluid-structure interaction problems”, in Computer and Structures,
vol. 81, 2003, p. 655–671.
[IDE 03C] IDELSOHN S.R., CALVO N., OÑATE E., “Polyhedrization of an arbitrary point
set”, Comp. Meth. Appl. Mech. Engng., vol. 192, 2003, p. 2649–2668.
[IDE 04] IDELSOHN S.R., OÑATE E., DEL PIN F., “The particle finite element method, a
powerful tool to solve incompressible flows with free-surfaces and breaking waves”, Int.
J. Num. Meth. Engng., submitted, 2004.
[KOS 96] KOSHIZUKA S., OKA Y., “Moving particle semi-implicit method for fragmentation
of incompressible fluid”, Nuclear Engng. Science, vol. 123, 1996, p. 421–434.
[LAI 60] LAITONE E.V., “The second approximation to cnoidal waves”, J. Fluids Mech., vol.
, 1960, p. 430.
[ONA 98] Oñate E., “Derivation of stabilized equations for advective-diffusive transport and
fluid flow problems”, Comp. Meth. Appl. Mech. Engng., vol. 151, 1998, p. 233–267.
[ONA 98B] OÑATE E., IDELSOHN S.R., “A mesh free finite point method for advectivediffusive
transport and fluid flow problems”, Comp. Mech., vol. 21, 1998, p. 283–292.
[ONA 00] OÑATE E., “A stabilized finite element method for incompressible viscous flows
using a finite increment calculus formulation”, Comp. Meth. Appl. Mech. Engng., vol.
, 2000, p. 355–370.
[ONA 00B] OÑATE E., SACCO C., IDELSOHN S.R., “A finite point method for incompressible
flow problems”, Comput. and Visual. in Science, vol. 2, 2000, p. 67–75.
[ONA 01] OÑATE E., GARCÍA J., “A finite element method for fluid-structure interaction with
surface waves using a finite calculus formulation”, Comp. Meth. Appl. Mech. Engng.,
vol. 191, 2001, p. 635–660.
[ONA 03] OÑATE E., IDELSOHN S.R., DEL PIN F., “Lagrangian formulation for incompressible
fluids using finite calculus and the finite element method”, in Numerical Methods
for Scientific Computing Variational Problems and Applications, Y. Kuznetsov, P. Neittanmaki
and O. Pironneau (Eds.), CIMNE, Barcelona, 2003.
[ONA 04] OÑATE E., “Possibilities of finite calculus in computational mechanics”, Int. J.
Num. Meth. Engng., vol. 60, 2004, p. 255-281.
[ONA 04B] OÑATE E., ROJEK J., TAYLOR R.L ZIENKIEWICZ O.C., “Finite calculus formulation
for incompressible solids using linear triangles and tetraedra”, Int. J. Num. Meth.
Engng., vol. 59, 2004, p. 1473-1500.
[ONA 04C] OÑATE E., GARCÍA J., IDELSOHN S.R., “Ship hydrodynamics”, In Encyclopedia
of Computational Mechanics, E. Stein, R. de Borst and T.J.R. Hughes (Eds), J. Wiley,
[RAD 98] RADOVITZKI R., ORTIZ M., “Lagrangian finite element analysis of Newtonian
flows”, Int. J. Num. Meth. Engng., vol. 43, 1998, p. 607–619.
[SHE 96] SHENG C., TAYLOR L.K., WHITFIELD D.L.,“Implicit lower-upper / approximatefactorization
schemes for incompressible flows”, J. Comp. Phys., vol. 128, 1996, p.
–42.
[STO 95] STORTI M., NIGRO N., IDELSOHN S.R., “Steady state incompressible flows using
explicit schemes with an optimal local preconditioning”, Comp. Meth. Appl. Mech.
Engng., vol. 124, 1995, p. 231–252.
[TEZ 92] TEZDUYAR T.E., MITTAL S., RAY S.E., SHIH R.,“Incompressible flow computations
with stabilized bilinear and linear equal order interpolation velocity–pressure elements”,
Comput. Methods Appl. Mech. Engng., vol. 95, 1992, p. 221–242.
[ZIE 00] ZIENKIEWICZ O.C., TAYLOR R.L. The finite element method, 2000, 5th Edition, 3
Volumes, Butterworth–Heinemann.
