Time integration; Fluid mechanics; Particle methods; CFD; Generalized-Alpha; PFEM
Abstract :
[en] Despite the increasing use of the Particle Finite Element Method (PFEM) in fluid flow simulation and the outstanding success of the Generalized-alpha time integration method, very little discussion has been devoted to their combined performance. This work aims to contribute in this regard by addressing three main aspects. Firstly, it includes a detailed implementation analysis of the Generalized-alpha method in PFEM. The work recognizes and compares different implementation approaches from the literature, which differ mainly in the terms that are alpha-interpolated (state variables or forces of momentum equation) and the type of treatment for the pressure in the time integration scheme. Secondly, the work compares the performance of the Generalized-alpha method against the Backward Euler and Newmark schemes for the solution of the incompressible Navier-Stokes equations. Thirdly, the study is enriched by considering not only the classical velocity-pressure formulation but also the displacement-pressure formulation that is gaining interest in the fluid-structure interaction field. The work is carried out using various 2D and 3D benchmark problems such as the fluid sloshing, the solitary wave propagation, the flow around a cylinder, and the collapse of a cylindrical water column.
Disciplines :
Mechanical engineering
Author, co-author :
Fernandez Sanchez, Eduardo Felipe ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS-Mécanique numérique non linéaire
Février, Simon ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > LTAS-Mécanique numérique non linéaire
Lacroix, Martin ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M)
Boman, Romain ; Université de Liège - ULiège > Aérospatiale et Mécanique (A&M)
Generalized-alpha scheme in the PFEM for velocity-pressure and displacement-pressure formulations of the incompressible Navier-Stokes equations
Publication date :
25 August 2022
Journal title :
International Journal for Numerical Methods in Engineering
ISSN :
0029-5981
eISSN :
1097-0207
Publisher :
John Wiley & Sons, Hoboken, United States - New Jersey
Peer reviewed :
Peer Reviewed verified by ORBi
Name of the research project :
ALFEWELD
Funding text :
This work was supported by the ALFEWELD project: Amélioration et modélisation du FMB (Friction Melt Bonding) pour le soudage par recouvrement de l’aluminium et de l’acier. Convention 1710162, funded by the WALInnov program of the Walloon Region of Belgium.
Oñate E, Idelsohn SR, Del Pin F, Aubry R. The particle finite element method—an overview. Int J Comput Methods. 2004;1(02):267-307.
Carbonell JM, Rodríguez J, Oñate E. Modelling 3D metal cutting problems with the particle finite element method. Comput Mech. 2020;66(3):603-624.
Cerquaglia ML, Thomas D, Boman R, Terrapon V, Ponthot JP. A fully partitioned Lagrangian framework for FSI problems characterized by free surfaces, large solid deformations and displacements, and strong added-mass effects. Comput Methods Appl Mech Eng. 2019;348:409-442.
Meduri S, Cremonesi M, Frangi A, Perego U. A Lagrangian fluid–Structure interaction approach for the simulation of airbag deployment. Finite Elem Anal Des. 2022;198:103659.
Bobach BJ, Boman R, Celentano D, Terrapon V, Ponthot JP. Simulation of the Marangoni effect and phase change using the particle finite element method. Appl Sci. 2021;11(24):11893.
Sengani F, Mulenga F. A review on the application of particle finite element methods (PFEM) to cases of landslides. Int J Geotech Eng. 2022;16(3):367-381.
Cremonesi M, Franci A, Idelsohn S, Oñate E. A state of the art review of the particle finite element method (PFEM). Arch Comput Methods Eng. 2020;27(5):1709-1735.
Idelsohn SR, Oñate E, Pin FD. The particle finite element method: a powerful tool to solve incompressible flows with free-surfaces and breaking waves. Int J Numer Methods Eng. 2004;61(7):964-989.
Idelsohn SR, Marti J, Limache A, Oñate E. Unified Lagrangian formulation for elastic solids and incompressible fluids: application to fluid–Structure interaction problems via the PFEM. Comput Methods Appl Mech Eng. 2008;197(19-20):1762-1776.
Idelsohn SR, Oñate E. The challenge of mass conservation in the solution of free-surface flows with the fractional-step method: problems and solutions. Int J Numer Methods Biomed Eng. 2010;26(10):1313-1330.
Ryzhakov P, Oñate E, Rossi R, Idelsohn S. Improving mass conservation in simulation of incompressible flows. Int J Numer Methods Eng. 2012;90(12):1435-1451.
Zhu M, Scott MH. Modeling fluid–structure interaction by the particle finite element method in OpenSees. Comput Struct. 2014;132:12-21.
Oñate E, Franci A, Carbonell JM. Lagrangian formulation for finite element analysis of quasi-incompressible fluids with reduced mass losses. Int J Numer Methods Fluids. 2014;74(10):699-731.
Franci A, Oñate E, Carbonell JM. On the effect of the bulk tangent matrix in partitioned solution schemes for nearly incompressible fluids. Int J Numer Methods Eng. 2015;102(3-4):257-277.
Franci A, Cremonesi M, Perego U, Oñate E. A Lagrangian nodal integration method for free-surface fluid flows. Comput Methods Appl Mech Eng. 2020;361:112816.
Franci A. Lagrangian finite element method with nodal integration for fluid–Solid interaction. Comput Part Mech. 2021;8(2):389-405.
Ryzhakov P, Rossi R, Idelsohn S, Onate E. A monolithic Lagrangian approach for fluid–Structure interaction problems. Comput Mech. 2010;46(6):883-899.
Ryzhakov P, Cotela J, Rossi R, Oñate E. A two-step monolithic method for the efficient simulation of incompressible flows. Int J Numer Methods Fluids. 2014;74(12):919-934.
Ryzhakov P. A modified fractional step method for fluid–structure interaction problems. Revista internacional de métodos numéricos para cálculo y diseño en ingeniería. 2017;33(1-2):58-64.
Ryzhakov P, Marti J, Idelsohn S, Oñate E. Fast fluid–Structure interaction simulations using a displacement-based finite element model equipped with an explicit streamline integration prediction. Comput Methods Appl Mech Eng. 2017;315:1080-1097.
Cremonesi M, Meduri S, Perego U, Frangi A. An explicit Lagrangian finite element method for free-surface weakly compressible flows. Comput Part Mech. 2017;4(3):357-369.
Chung J, Hulbert G. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. J Appl Mech. 1993;60:371-375.
Jansen KE, Whiting CH, Hulbert GM. A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method. Comput Methods Appl Mech Eng. 2000;190(3-4):305-319.
Dettmer W, Perić D. An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation. Comput Methods Appl Mech Eng. 2003;192(9-10):1177-1226.
Hilber HM, Hughes TJ, Taylor RL. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq Eng Struct Dyn. 1977;5(3):283-292.
Wood W, Bossak M, Zienkiewicz O. An alpha modification of Newmark's method. Int J Numer Methods Eng. 1980;15(10):1562-1566.
Graillet D, Ponthot JP. Numerical simulation of crashworthiness with an implicit finite element code. Aircr Eng Aerosp Technol. 1999;71(1):12-20.
Ponthot JP, Graillet D. Efficient implicit schemes for the treatment of the contact between deformable bodies: application to shock-absorber devices. Int J Crashworth. 1999;4(3):273-286.
Valdés Vázquez JG. Nonlinear Analysis of Orthotropic Membrane and Shell Structures Including Fluid-Structure Interaction. PhD thesis. Universitat Politècnica de Catalunya (UPC); 2007.
Bazilevs Y, Calo V, Cottrell J, Hughes T, Reali A, Scovazzi G. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng. 2007;197(1-4):173-201.
Gravemeier V, Kronbichler M, Gee MW, Wall WA. An algebraic variational multiscale-multigrid method for large-eddy simulation: generalized-α time integration, Fourier analysis and application to turbulent flow past a square-section cylinder. Comput Mech. 2011;47(2):217-233.
Lovrić A, Dettmer WG, Kadapa C, Perić D. A new family of projection schemes for the incompressible Navier–Stokes equations with control of high-frequency damping. Comput Methods Appl Mech Eng. 2018;339:160-183.
Saksono P, Dettmer W, Perić D. An adaptive remeshing strategy for flows with moving boundaries and fluid–Structure interaction. Int J Numer Methods Eng. 2007;71(9):1009-1050.
Liu J, Lan IS, Tikenogullari OZ, Marsden AL. A note on the accuracy of the generalized-α scheme for the incompressible Navier-Stokes equations. Int J Numer Methods Eng. 2021;122(2):638-651.
Radovitzky R, Ortiz M. Lagrangian finite element analysis of Newtonian fluid flows. Int J Numer Methods Eng. 1998;43(4):607-619.
Avancini G, Sanches RA. A total Lagrangian position-based finite element formulation for free-surface incompressible flows. Finite Elem Anal Des. 2020;169:103348.
Hughes TJ, Franca LP, Balestra M. A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng. 1986;59(1):85-99.
Donea J, Huerta A. Finite Element Methods for Flow Problems. John Wiley & Sons; 2003.
Tezduyar TE. Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech. 1991;28:1-44.
Février S. Development of a 3D Compressible Flow Solver for PFEM Fluid Simulation. Master's thesis, University of Liege; 2020. http://hdl.handle.net/2268.2/9010
Newmark NM. A method of computation for structural dynamics. J Eng Mech Div. 1959;85(3):67-94.
Belytschko T, Liu WK, Moran B, Elkhodary K. Nonlinear Finite Elements for Continua and Structures. John wiley & sons; 2014.
Géradin M, Rixen DJ. Mechanical Vibrations: Theory and Application to Structural Dynamics. 3rd ed. John Wiley & Sons; 2014.
Franci A. Unified Lagrangian Formulation for Fluid and Solid Mechanics, Fluid-Structure Interaction and Coupled Thermal Problems Using the PFEM. PhD thesis, Polytechnic University of Catalonia, Springer; 2016.
Franci A, Cremonesi M. On the effect of standard PFEM remeshing on volume conservation in free-surface fluid flow problems. Comput Part Mech. 2017;4(3):331-343.
Meduri S. A Fully Explicit Lagrangian Finite Element Method for Highly Nonlinear Fluid-Structure Interaction Problems. PhD thesis, Politecnico di Milano, Italy; 2019. http://hdl.handle.net/10589/145705.
Cerquaglia ML. Development of a Fully-Partitioned PFEM-FEM Approach for Fluid-Structure Interaction Problems Characterized by Free Surfaces, Large Solid Deformations, and Strong Added-Mass Effects. PhD thesis. University of Liege, Belgium; 2019. https://hdl.handle.net/2268/233166
Guennebaud G, Jacob B. Eigen v3; 2010. http://eigen.tuxfamily.org
Da TKF, Loriot S, Yvinec M. 3D alpha shapes. CGAL editorial board 5.4, 2022.
OpenMP architecture review board. OpenMP application program interface version 4.5; 2008.
Geuzaine C, Remacle JF. Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int J Numer Methods Eng. 2009;79(11):1309-1331.
Fernádez E. Time integration in the PFEM [Video List]. Accessed May 1, 2022. https://youtube.com/playlist?list=PLDU2T_jKI__89gnigGPovj_Y7gahnyTjJ
Ramaswamy B. Numerical simulation of unsteady viscous free surface flow. J Comput Phys. 1990;90(2):396-430.
Staroszczyk R. Simulation of solitary wave mechanics by a corrected smoothed particle hydrodynamics method. Arch Hydro-Eng Environ Mech. 2011;58(1-4):23-45.
Idelsohn S, Nigro N, Limache A, Oñate E. Large time-step explicit integration method for solving problems with dominant convection. Comput Methods Appl Mech Eng. 2012;217:168-185.
Idelsohn SR, Nigro NM, Gimenez JM, Rossi R, Marti JM. A fast and accurate method to solve the incompressible Navier-Stokes equations. Eng Comput. 2013;30(2):197-222.
Martin JC, Moyce WJ. Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane. Philos Trans Royal Soc Lond Ser A Math Phys Sci. 1952;244(882):312-324.
Akin JE, Tezduyar TE, Ungor M. Computation of flow problems with the mixed interface-tracking/interface-capturing technique (MITICT). Comput Fluids. 2007;36(1):2-11.
Cruchaga M, Celentano D, Breitkopf P, Villon P, Rassineux A. A surface remeshing technique for a Lagrangian description of 3D two-fluid flow problems. Int J Numer Methods Fluids. 2010;63(4):415-430.
Battaglia L, Storti MA, Delia J. Simulation of free-surface flows by a finite element interface capturing technique. Int J Comput Fluid Dyn. 2010;24(3-4):121-133.
Tang B, Li JF, Wang TS. Viscous flow with free surface motion by least square finite element method. Appl Math Mech. 2008;29(7):943-952.
Lobovskỳ L, Botia-Vera E, Castellana F, Mas-Soler J, Souto-Iglesias A. Experimental investigation of dynamic pressure loads during dam break. J Fluids Struct. 2014;48:407-434.
