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A novel approach to model the flow of generalized Newtonian fluids with the finite pointset method


Several numerical meshless methods have been proposed to try to solve some of the limitations of traditional mesh-based methods. Among those, the finite pointset method, which has been applied in several physical problems, shows a great potential. This work presents an extension of the available numerical approaches based on the finite pointset method to allow dealing with generalized Newtonian fluids, which possess a high viscosity. The developed finite pointset method solver is verified through the comparison of its predictions with analytical solutions, for simple flows, and with results provided by the well-established and validated open-source computational library OpenFOAM, for more complex cases studies. The excellent results obtained on the verification case studies prove the proper implementation of the new finite pointset method solver.

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  1. 1.

    Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Comput Methods Appl Mech Engrg 139(1–4):3–47

    Article  Google Scholar 

  2. 2.

    Carneiro OS, Nóbrega JM (2012) Design of extrusion forming tools. Smithers Rapra, London

    Google Scholar 

  3. 3.

    Chorin AJ (1968) Numerical solution of the Navier–Stokes equations. Math Comput 22(104):745–762

    MathSciNet  Article  Google Scholar 

  4. 4.

    Costa R, Nóbrega JM, Clain S, Machado GJ (2019) Very high-order accurate polygonal mesh finite volume scheme for conjugate heat transfer problems with curved interfaces and imperfect contacts. Comput Methods Appl Mech Eng 357:112560

    MathSciNet  Article  Google Scholar 

  5. 5.

    Fasshauer GE (2006) Meshfree methods. In: Rieth M, Schommers W (eds) Handbook of theoretical and computational nanotechnology, vol 2. American Scientific Publishers, pp 33–97

    Google Scholar 

  6. 6.

    Gonçalves N, Carneiro O, Nóbrega J (2013) Design of complex profile extrusion dies through numerical modeling. J Non Newtonian Fluid Mech 200:103–110

    Article  Google Scholar 

  7. 7.

    Gonçalves N, Teixeira P, Ferrás L, Afonso A, Nóbrega J, Carneiro O (2015) Design and optimization of an extrusion die for the production of wood-plastic composite profiles. Polym Eng Sci 55(8):1849–1855

    Article  Google Scholar 

  8. 8.

    Jasak H, Jemcov A, Tukovic Z et al (2007) Openfoam: A c++ library for complex physics simulations. In: International workshop on coupled methods in numerical dynamics, IUC Dubrovnik Croatia, vol 1000, pp 1–20

  9. 9.

    Kuhnert J (1999) General smoothed particle hydrodynamics. PhD thesis, Technische Universität Kaiserslautern

  10. 10.

    Kuhnert J (2003) An upwind finite pointset method (fpm) for compressible Euler and Navier–Stokes equations. In: Meshfree methods for partial differential equations, pp 239–249

  11. 11.

    Kuhnert J (2014) Meshfree numerical schemes for time dependent problems in fluid and continuum mechanics. In: Advances in PDE modeling and computation, pp 119–136

  12. 12.

    Kuhnert J, Michel I, Mack R (2017) Fluid structure interaction (FSI) in the meshfree finite pointset method (FPM): theory and applications. In: International workshop on meshfree methods for partial differential equations, pp 73–92

  13. 13.

    Libreros J, Trujillo M (2021) Effects of mesh generation on modelling aluminium anode baking furnaces. Fluids 6(4):140

    Article  Google Scholar 

  14. 14.

    Michel I, Seifarth T, Kuhnert J, Suchde P (2021) A meshfree generalized finite difference method for solution mining processes. Comput Part Mech 8(3):561–574

    Article  Google Scholar 

  15. 15.

    Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul 79(3):763–813

    MathSciNet  Article  Google Scholar 

  16. 16.

    Oñate E, Idelsohn S (1998) A mesh-free finite point method for advective-diffusive transport and fluid flow problems. Comput Mech 21(4):283–292

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Oñate E, Idelsohn S, Zienkiewicz O, Taylor R (1996) A finite point method in computational mechanics. Applications to convective transport and fluid flow. Int J Numer Methods Eng 39(22):3839–3866

    MathSciNet  Article  Google Scholar 

  18. 18.

    Oñate E, Idelsohn S, Zienkiewicz O, Taylor R, Sacco C (1996) A stabilized finite point method for analysis of fluid mechanics problems. Comput Methods Appl Mech Eng 139(1–4):315–346

    MathSciNet  Article  Google Scholar 

  19. 19.

    Reséndiz-Flores EO, Saucedo-Zendejo FR (2015) Two-dimensional numerical simulation of heat transfer with moving heat source in welding using the finite pointset method. Int J Heat Mass Transf 90:239–245

    Article  Google Scholar 

  20. 20.

    Reséndiz-Flores EO, Saucedo-Zendejo FR (2018) Meshfree numerical simulation of free surface thermal flows in mould filling processes using the finite pointset method. Int J Therm Sci 127:29–40

    Article  Google Scholar 

  21. 21.

    Saucedo-Zendejo FR, Reséndiz-Flores EO (2017) A new approach for the numerical simulation of free surface incompressible flows using a meshfree method. Comput Methods Appl Mech Eng 324:619–639

    MathSciNet  Article  Google Scholar 

  22. 22.

    Saucedo-Zendejo FR, Reséndiz-Flores EO (2020) Meshfree numerical approach based on the finite pointset method for static linear elasticity problems. Comput Methods Appl Mech Eng 372:113367

    MathSciNet  Article  Google Scholar 

  23. 23.

    Saucedo-Zendejo FR, Reséndiz-Flores EO, Kuhnert J (2019) Three-dimensional flow prediction in mould filling processes using a GFDM. Comput Part Mech 6(3):411–425

    Article  Google Scholar 

  24. 24.

    Spanjaards MM, Hulsen MA, Anderson PD (2021) Die shape optimization for extrudate swell using feedback control. J Non Newtonian Fluid Mech 293:104552

    MathSciNet  Article  Google Scholar 

  25. 25.

    Suchde P (2021) A meshfree Lagrangian method for flow on manifolds. Int J Numer Methods Fluids 93(6):1871–1894

    MathSciNet  Article  Google Scholar 

  26. 26.

    Suchde P, Kuhnert J, Tiwari S (2018) On meshfree GFDM solvers for the incompressible Navier–Stokes equations. Comput Fluids 165:1–12

    MathSciNet  Article  Google Scholar 

  27. 27.

    Tiwari S, Kuhnert J (2002) A meshfree method for incompressible fluid flows with incorporated surface tension. Rev Eur Elem 11(7–8):965–987

    MATH  Google Scholar 

  28. 28.

    Tiwari S, Kuhnert J (2007) Modeling of two-phase flows with surface tension by finite pointset method (FPM). J Comput Appl Math 203:376–386

    MathSciNet  Article  Google Scholar 

  29. 29.

    Tomé M, Araujo M, Evans J, McKee S (2019) Numerical solution of the Giesekus model for incompressible free surface flows without solvent viscosity. J Non Newtonian Fluid Mech 263:104–119

    MathSciNet  Article  Google Scholar 

  30. 30.

    Xu R, Stansby P, Laurence D (2009) Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach. J Comput Phys 228(18):6703–6725

    MathSciNet  Article  Google Scholar 

  31. 31.

    Zhang L, Ademiloye A, Liew K (2019) Meshfree and particle methods in biomechanics: prospects and challenges. Arch Comput Methods Eng 26(5):1547–1576

    MathSciNet  Article  Google Scholar 

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Financial support for this work was provided by the Government of the State of Coahuila and the Council of Science and Technology of the State of Coahuila (COECYT) through the project FONCYT COAH-2020-C14-B005. J. Miguel Nóbrega also acknowledges the funding by FEDER funds through the COMPETE 2020 Programme and National Funds through FCT (Portuguese Foundation for Science and Technology) under the projects UID-B/05256/2020, UID-P/05256/2020. The authors would also like to acknowledge the Minho Advanced Computing Center (MACC) for providing HPC resources that contributed to the research results reported within this paper.

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Correspondence to Felix R. Saucedo-Zendejo.

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Saucedo-Zendejo, F.R., Nóbrega, J.M. A novel approach to model the flow of generalized Newtonian fluids with the finite pointset method. Comp. Part. Mech. (2021).

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  • Finite pointset method
  • Computational rheology
  • Generalized Newtonian fluids
  • Polymer processing