Simulation of fluid flows of multi-materials is an intriguing topic in computational mechanics. Capturing the physics of the interface between different materials poses a challenge because of the discontinuities that may occur on the interface. Several methods have been proposed in the literature to deal with this issue. In this paper, a technique based on Nitsche’s method has been employed on a fixed mesh combined with the PFEM-2 strategy for the solution of Navier–Stokes equations on multi-fluid flows. The novelty of this technique is its capability of capturing the strong and weak discontinuities and its compatibility for the application of various types of boundary conditions on the interface.
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Osher SJ, Fedkiw RP (2006) Level set methods and dynamic implicit surfaces. Springer, New York
Rossi R, Larese A, Dadvand P, Oñate E (2013) An efficient edge-based level set finite element method for free surface flow problems. Int J Numer Methods Fluids 71(6):687–716
Cyril HW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1):201–225
Coppola-Owen AH, Codina R (2005) Improving Eulerian two-phase flow finite element approximation with discontinuous gradient pressure shape functions. Int J Numer Methods Fluids 49(12):1287–1304
Fries TP (2009) The intrinsic XFEM for two-fluid flows. Int J Numer Methods Fluids 60:437–471
Becker PA, Idelsohn SR, Oñate E (2015) A unified monolithic approach for multi-fluid flows and fluid-structure interaction using the particle finite element method with fixed mesh. Comput Mech 55:1091–1104
Idelsohn SR, Gimenez J, Marti J, Nigro N (2017) Elemental enriched spaces for the treatment of weak and strong discontinuous fields. Comput Methods Appl Mech Eng 313:535–559
Idelsohn SR, Gimenez JM, Nigro NM (2018) Multifluid flows with weak and strong discontinuous interfaces using an elemental enriched space. Int J Numer Methods Fluids 86:750–769
Marti J, Ortega E, Idelsohn SR (2017) An improved enrichment method for weak discontinuities for thermal problems. Int J Numer Methods Heat Fluid Flow 27(8):1748–764
Cruchaga M, Celentano D, Tezduyar T (2001) A moving Lagrangian interface technique for flow computations over fixed meshes. Comput Methods Appl Mech Eng 191(6):525–543
Barton PT, Obadia B, Drikakis D (2011) A conservative level-set method for compressible solid/fluid problems on fixed grids. J Comput Phys 230:7867–7890
Spelt PDM (2005) A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J Comput Phys 207(2):389–404
Zhang YL, Zou QP, Greaves D (2010) Numerical simulation of free-surface flow using the level-set method with global mass correction. Int J Numer Methods Fluids 63(6):366–396
Ausas RF, Dari EA, Buscaglia GC (2011) A geometric mass-preservating redistancing scheme for the level set function. Int J Numer Methods Fluids 65(8):989–1010
Oñate E, Idelsohn SR, Del Pin F, Romain A (2004) The particle finite element method: an overview. Int J Comput Methods 1(02):267–307
Marti J, Ryzhakov PB (2019) An explicit-implicit finite element model for the numerical solution of incompressible Navier–Stokes equations on moving grids. Comput Methods Appl Mech Eng 350:750–765
Idelsohn S, de Mier-Torrecilla M, Oñate E (2009) Multi-fluid flows with the particle finite element method. Comput Methods Appl Mech Eng 198(33):2750–2767
Mier-Torrecilla M (2010) Numerical simulation of multi-fluid flows with the particle finite element method. Ph.D. Thesis
Idelsohn SR, Mier-Torrecilla M, Marti J, Oñate E (2011) The particle finite element method for multi-fluid flows. In: Oñate E, Owen R (eds) Particle-based methods: fundamentals and applications. Springer, Dordrecht, pp 135–158
Marti J, Ryzhakov PB, Idelsohn SR, Oñate E (2012) Combined Eulerian-PFEM approach for analysis of polymers in fire situations. Int J Numer Meth Eng 92(9):782–801
Ryzhakov PB, Jarauta A (2016) An embedded approach for immiscible multi-fluid problems. Int J Numer Methods Fluids 81:357–376
Jarauta A, Ryzhakov PB, Secanell M, Waghmare P, Pons-Prats J (2016) Numerical study of droplet dynamics in a polymer electrolyte fuel cell gas channel using an embedded Eulerian–Lagrangian approach. J Power Sources 323:201–212
Marti J, Idelsohn SR, Oñate E (2018) A finite element model for the simulation of the UL-94 burning test. Fire Technol 54(6):1783–1805
Idelsohn SR, Nigro N, Limache A, Oñate E (2012) Large time-step explicit integration method for solving problems with dominant convection. Comput Methods Appl Mech Eng 217–220:68–185
Ryzhakov PB, Marti J, Idelsohn SR, Oñate E (2017) Fast fluid-structure interaction simulations using a displacement-based finite element model equipped with an explicit streamline integration prediction. Comput Methods Appl Mech Eng 315:1080–1097
Hansbo P, Hansbo A (2002) An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput Methods Appl Mech Eng 191(47–48):5537–5552
Idelsohn SR, Marti J, Becker PA, Oñate E (2014) Analysis of multifluid flows with large time steps using the particle finite element method. Int J Numer Methods Fluids 75:621–644
Idelsohn SR, Nigro N, Gimenez JM, Rossi R, Marti J (2013) A fast and accurate method to solve the incompressible Navier–Stokes equations. Eng Comput 30:197–222
Becker P (2015) An enhanced particle finite element method with special emphasis on landslides and debris flows. Ph.D. thesis. Universitat Politècnica de Catalunya
Bathe KJ (2001) The inf-sup condition and its evaluation for mixed finite element methods. Comput Struct 79:243–252
Hughes TJR (1995) Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized. Comput Methods Appl Mech Eng 73:387–401
Ryzhakov PB, Cotela J, Rossi R, Oñate E (2014) A two-step monolithic method for the efficient simulation of incompressible flows. Int J Numer Methods Fluids 74(12):919–934
Nitsche J (1971) Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 36:9–15
Cremonesi M, Ferri F, Perego U (2017) A basal slip model for Lagrangian finite element simulations of 3D landslides. Int J Numer Anal Methods Geomech 41:30–53
Dadvand P, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Archiv Comput Methods Eng 17(3):253–297
Dadvand P, Rossi R, Gil M, Martorell X, Cotela J, Juanpere E, Idelsohn SR, Oñate E (2013) Migration of a generic multi-physics framework to HPC environments. Comput Fluids 80:301–309
Ganesan S, Tobiska L (2006) Computations on flows with interfaces using arbitrary Lagrangian Eulerian method. In: European conference on computational fluid dynamics. ECCOMAS CFD
Gimenez JM, Horacio JA, Idelsohn SR, Nigro NM (2019) A second-order in time and space particle-based method to solve flow problems on arbitrary meshes. J Comput Phys 380:295–310
Hrvoje J, Jemcov A, Tukovic Z (2007) OPEN FOAM: A C++ library for complex physics simulations. In: International workshop on coupled methods in numerical dynamics. IUC, Dubrovnik, Croatia
The research that has been presented in this publication and the results obtained have been conducted and achieved with the support of the Ministerio de Economía y Competitividad (MINECO) from Spain and its funding program Ayudas para Contratos Predoctorales para la Formación de Doctores (ref. BES-2014-070613). Author Deniz C. Tanyildiz would like to express special thanks to the project: PARFLOW (ref. BIA2013-49007-C2-1-R). Dr. Riccardo Rossi would like to express special thanks to the project: EXAQUTE (ref. 800898). Moreover, the authors would like to express their gratitude to Dr. Joan Baiges from Polytechnical University of Catalonia, for his substantial help on Nitsche’s method.
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Tanyildiz, D.C., Marti, J. & Rossi, R. Solution of Navier–Stokes equations on a fixed mesh using conforming enrichment of velocity and pressure. Comp. Part. Mech. 7, 71–86 (2020). https://doi.org/10.1007/s40571-019-00285-6
- Lagrangian particles
- Nitsche’s method
- Fixed mesh