Abstract
We present a general formulation for modeling bed erosion in free surface flows using the particle finite element method (PFEM). The key feature of the PFEM is the use of an updated Lagrangian description to model the motion of nodes (particles) in domains containing fluid and solid subdomains. Nodes are viewed as material points (called particles) which can freely move and even separate from the fluid and solid subdomains representing, for instance, the effect of water drops or soil/rock particles. A mesh connects the nodes defining the discretized domain in the fluid and solid regions where the governing equations, expressed in an integral form, are solved as in the standard FEM. The necessary stabilization for dealing with the incompressibility of the fluid is introduced via the finite calculus (FIC) method. An incremental iterative scheme for the solution of the nonlinear transient coupled fluid-structure problem is described. The erosion mechanism is modeled by releasing the material adjacent to the bed surface according to the frictional work generated by the fluid shear stresses. The released bed material is subsequently transported by the fluid flow. Examples of application of the PFEM to solve a number of bed erosion problems involving large motions of the free surface and splashing of waves are presented.
Similar content being viewed by others
References
Archard JF (1953) Contact and rubbing of flat surfaces. J Appl Phys 24(8):981–988
Aubry R, Idelsohn SR, Oñate E (2005) Particle finite element method in fluid mechanics including thermal convection-diffusion. Comput Struct 83(17–18):1459–1475
Codina R, Zienkiewicz OC (2002) CBS versus GLS stabilization of the incompressible Navier–Stokes equations and the role of the time step as stabilization parameter. Commun Num Meth Eng 18(2):99–112
Darby S, Thorne C (1996) Numerical simulation of widening and bed deformation of straight sand-bed rivers. J Hydr Eng ASCE 122(4):184–193
Donea J, Huerta A (2003) Finite element method for flow problems. Wiley, New York
Edelsbrunner H, Mucke EP (1999) Three dimensional alpha shapes. ACM Trans Graph 13:43–72
Fell R, Wan CF, Cyganiewics J, Foster M (2003) Time for development of internal erosion and piping in embankment dams. J Geotech Geoenviron Eng 129:307–314
García J, Oñate E (2003) An unstructured finite element solver for ship hydrodynamic problems. J Appl Mech 70:18–26
Idelsohn SR, Oñate E, Del Pin F, Calvo N (2002) Lagrangian formulation: the only way to solve some free-surface fluid mechanics problems. In: Mang HA, Rammerstorfer FG, Eberhardsteiner J (eds) 5th World congress on computational mechanics, July 7–12, Vienna, Austria
Idelsohn SR, Oñate E, Calvo N, Del Pin F (2003) The meshless finite element method. Int J Num Meth Eng 58(6):893–912
Idelsohn SR, Oñate E, Del Pin F (2003) A Lagrangian meshless finite element method applied to fluid–structure interaction problems. Comput Struct 81:655–671
Idelsohn SR, Calvo N, Oñate E (2003) Polyhedrization of an arbitrary point set. Comput Meth Appl Mech Eng 192(22–24):2649–2668
Idelsohn SR, Oñate E, Del Pin F (2004) The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int J Num Meth Eng 61:964–989
Kovacs A, Parker G (1994) A new vectorial bedload formulation and its application to the time evolution of straight river channels. J Fluid Mech 267:153–183
Oñate E (1998) Derivation of stabilized equations for advective–diffusive transport and fluid flow problems. Comput Meth Appl Mech Engng 151:233–267
Oñate E, (2000) A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation. Comput Meth Appl Mech Eng 182(1–2):355–370
Oñate E (2004) Possibilities of finite calculus in computational mechanics. Int J Num Meth Eng 60(1):255–281
Oñate E, Idelsohn SR (1998) A mesh free finite point method for advective–diffusive transport and fluid flow problems. Comput Mech 21:283–292
Oñate E, García J (2001) A finite element method for fluid–structure interaction with surface waves using a finite calculus formulation. Comput Meth Appl Mech Eng 191:635–660
Oñate E, Sacco C, Idelsohn SR (2000) A finite point method for incompressible flow problems. Comput Visual Sci 2:67–75
Oñate E, Idelsohn SR, Del Pin F (2003) Lagrangian formulation for incompressible fluids using finite calculus and the finite element method. In: Kuznetsov Y, Neittanmaki P, Pironneau O (eds) Numerical methods for scientific computing variational problems and applications. CIMNE, Barcelona
Oñate E, García J, Idelsohn SR (2004) Ship hydrodynamics. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics. Wiley, New York
Oñate E, Idelsohn SR, Del Pin F, Aubry R (2004) The particle finite element method. An overview. Int J Comput Methods 1(2):267-307
Oñate E, Rojek J (2004) Combination of discrete element and finite element method for dynamic analysis of geomechanic problems. Comput Meth Appl Mech Eng 193:3087–3128
Oñate E, Idelsohn SR, Celigueta MA (2006) Lagrangian formulation for fluid–structure interaction problems using the particle finite element method. Bugeda G et al (eds.) CIMNE, Barcelona
Parker DB, Michel TG, Smith JL (1995) Compaction and water velocity effects on soil erosion in shallow flow. J Irrigation Drainage Eng 121:170–178
Phillips BC, Sutherland AJ (1989) Spatial lag effects in bed load sediment transport. J Hydr Res 24(1):45–56
Rahuel JL, Holly FM, Belleudy PJ, Yang G (1989) Modeling of riverbed evolution for bedload sediment mixtures. J Hydr Engrg ASCE 115(1):1521–1542
Sekine M, Parker G (1992) Bed-load transport on transverse slope. Int J Hydr Eng ASCE 118(4):513–535
Struiksma N, Olesen KW, Flokstra C, Vriend HJ (1985) Bed deformation in curved alluvial channels. J Hydr Res 23:57–79, Delft
van Rijn LC (1984) Sediment transport. Part III: bed forms and alluvial roughness. J Hydr Engrg ASCE 110(12):1733–1754
van Rijn LC (1986) Mathematical modeling of suspended sediment in nonuniform flow. J Hydr Eng ASCE 112(6):433–455
Wan CF and Fell R (2004) Investigation of erosion of soils in embankment dams. J Geotech Geoenviron Eng 130:373–380
Wu W, Rodi W, Wenka T (1997) Three-dimensional calculation of river flow. In: Proceedings of 27th IAHR congress, international association for hydraulic research, Delft
Zienkiewicz OC, Taylor RL, Nithiarasu P (2006) The finite element method for fluid dynamics. Elsevier, Amsterdam
Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics. Elsevier, Amsterdam
Acknowledgments
Thanks are given to Dr. R. Rossi and Ms M. de Mier for useful suggestions. This research was partially supported by the FP6 Programme of the European Commission on Global Change and Ecosystems through Project RAMWAS, Project no. FP6-037081. Dr. S.R. Idelsohn is an ICREA Professor at CIMNE.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Oñate, E., Celigueta, M.A. & Idelsohn, S.R. Modeling bed erosion in free surface flows by the particle finite element method. Acta Geotech. 1, 237–252 (2006). https://doi.org/10.1007/s11440-006-0019-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11440-006-0019-3