International Journal of Material Forming

, Volume 8, Issue 3, pp 341–353 | Cite as

A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming

  • F. Greco
  • L. Filice
  • C. Peco
  • M. Arroyo


The finite element method is the reference technique in the simulation of metal forming and provides excellent results with both Eulerian and Lagrangian implementations. The latter approach is more natural and direct but the large deformations involved in such processes require remeshing-rezoning algorithms that increase the computational times and reduce the quality of the results. Meshfree methods can better handle large deformations and have shown encouraging results. However, viscoplastic flows are nearly incompressible, which poses a challenge to meshfree methods. In this paper we propose a simple model of viscoplasticity, where both the pressure and velocity fields are discretized with maximum entropy approximants. The inf-sup condition is circumvented with a numerically consistent stabilized formulation that involves the gradient of the pressure. The performance of the method is studied in some benchmark problems including metal forming and orthogonal cutting.


Maximum entropy Metal forming Viscoplasticity Stabilization 



Francesco Greco acknowledges the travel research fellowship awarded by the Fondo Sociale Europeo. Marino Arroyo and Christian Peco acknowledge the support of the European Research Council under the European Community’s 7th Framework Programme (FP7/2007-2013)/ERC grant agreement nr 240487, and of the Ministerio de Ciencia e Innovacion (DPI2011-26589). MA acknowledges the support received through the prize “ICREA Academia” for excellence in research, funded by the Generalitat de Catalunya. CP acknowledges FPI-UPC Grant and FPU Ph.D. Grant (Ministry of Science and Innovation, Spain).


  1. 1.
    Kobayashi S, Oh Si, Altan T (1989) Metal forming and the finite-element method. Oxford University PressGoogle Scholar
  2. 2.
    Peric D, Owen D R J (2004) Computational modeling of forming processes. WileyGoogle Scholar
  3. 3.
    Babuska I, Aziz AK (1976) On the angle condition in the finite element method. SIAM J Numer Anal 13(2):214–226MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zienkiewicz O C, Godbole P N (1974) Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes. Int J Numer Methods Eng 8(1):1–16CrossRefGoogle Scholar
  5. 5.
    Zienkiewicz O C (1984) Flow formulation for numerical solution of forming processes. Wiley, ChichesterGoogle Scholar
  6. 6.
    Hwu Y J, Lenard J G (1988) A finite element study of flat rolling. J Eng Mater Technol 110(1):22–27CrossRefGoogle Scholar
  7. 7.
    Belytschko T, Kennedy J M (1978) Computer models for subassembly simulation. Nucl Eng Des 49:17–38CrossRefGoogle Scholar
  8. 8.
    Liu W K, Herman C, Jiun-Shyan C, Ted B (1988) Arbitrary lagrangian-eulerian petrov-galerkin finite elements for nonlinear continua. Comput Methods Appl Mech Eng 68(3):259–310CrossRefzbMATHGoogle Scholar
  9. 9.
    Yu-Kan H, Liu W K (1993) An ale hydrodynamic lubrication finite element method with application to strip rolling. Int J Numer Methods Eng 36(5):855–880CrossRefzbMATHGoogle Scholar
  10. 10.
    Belytschko T, Krongauz Y, Organ D, Fleming M, Meshless P K (1996) An overview and recent developments. Comput Methods Appl Mech Eng 139:3–47CrossRefGoogle Scholar
  11. 11.
    Puso M A, Solberg J (2006) A stabilized nodally integrated tetrahedral. Int J Numer Methods Eng 67(6):841–867MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Quak W, Boogaard A H, González D , Cueto E (2011) A comparative study on the performance of meshless approximations and their integration. Comput Mech 48(2):121–137MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Krysl P, Kagey H (2012) Reformulation of nodally integrated continuum elements to attain insensitivity to distortion. Int J Numer Methods Eng 90(7):805–818MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Greco F, Filice L, Alfaro I, Cueto E (2011) On the performances of different nodal integration techniques and their stabilization. In: Proceedings Computational Plast XI - Fundamentals and Application Conference, pp 1455–1466Google Scholar
  15. 15.
    Greco F, Umbrello D, Di Renzo S, Filice L, Alfaro I, Cueto E (2011) Application of the nodal integrated finite element method to cutting, A preliminary comparison with the traditional fem approach. Adv Mater Res 223:172–181CrossRefzbMATHGoogle Scholar
  16. 16.
    Quak W (October 2011) On meshless and nodal-based numerical methods for forming processes, PhD thesis. Enschede, The NetherlandsGoogle Scholar
  17. 17.
    Belytschko T, Lu Y Y, Gu L (1994) Element-free galerkin methods. Int J Numer Methods Eng 37(2):229–256MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Liu W K, Jun S, Yi F Z (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8-9):1081–1106CrossRefzbMATHGoogle Scholar
  19. 19.
    Huerta A, Fernández-Méndez S (2000) Enrichment and coupling of the finite element and meshless methods. Int J Numer Methods Eng 48:1615–1636CrossRefzbMATHGoogle Scholar
  20. 20.
    Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43(5):839–887MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Preparata FP, Shamos M I (1985) Computational geometry: an introduction. Springer, New YorkCrossRefGoogle Scholar
  22. 22.
    Cueto E, Doblaré M, Gracia L (2000) Imposing essential boundary conditions in the natural element method by means of density-scaled ?-shapes. Int J Numer Methods Eng 49(4):519–546CrossRefzbMATHGoogle Scholar
  23. 23.
    Alfaro I, Yvonnet J, Chinesta F, Cueto E (2007) A study on the performance of natural neighbour-based galerkin methods. Int J Numer Methods Eng 71(12):1436–1465CrossRefzbMATHGoogle Scholar
  24. 24.
    Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Methods Eng 61(12):2159–2181MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Arroyo M, Ortiz M (2006) Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int J Numer Methods Eng 65(13):2167–2202MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rosolen A, Millán D, Arroyo M (2010) On the optimum support size in meshfree methods: a variational adaptivity approach with maximum entropy approximants. Int J Numer Methods Eng 82(7):868–895Google Scholar
  27. 27.
    Rosolen A, Arroyo M (2013) Blending isogeometric analysis and local maximum entropy meshfree approximants. Comput Methods Appl Mech Eng 264:95–107MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Millán D, Rosolen A, Arroyo M (2011) Thin shell analysis from scattered points with maximum-entropy approximants. Int J Numer Methods Eng 85(6):723–751CrossRefzbMATHGoogle Scholar
  29. 29.
    Millán D, Rosolen A, Arroyo M (2013) Nonlinear manifold learning for meshfree finite deformation thin shell analysis. Int J Numer Methods Eng 93:685–713CrossRefGoogle Scholar
  30. 30.
    Millán D, Arroyo M (2013) Nonlinear manifold learning for model reduction in finite elastodynamics. Comput Methods Appl Mech Eng 261262(0):118–131CrossRefGoogle Scholar
  31. 31.
    Millán D, Arroyo M, Hashemian B (2013) B. Biasing molecular dynamics simulations with smooth and nonlinear data-driven collective variables. Submitted to Journal of Chemical PhysicsGoogle Scholar
  32. 32.
    Arroyo P A I, M Abdollahi A (2013) Effect of flexoelectricity on the electromechanical response of nano cantilever beams. Submitted to Journal of Computational PhysicsGoogle Scholar
  33. 33.
    Rosolen A, Peco C, Arroyo M (2013) An adaptive meshfree method for phase-field models of biomembranes Part I: approximation with maximum-entropy basis functions. J Comput Phys 249(0):303–319MathSciNetCrossRefGoogle Scholar
  34. 34.
    Peco C, Rosolen A, Arroyo M (2013) An adaptive meshfree method for phase-field models of biomembranes Part II: a lagrangian approach for membranes in viscous fluids. Chin J Comput Phys 249(0):320–336MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rabczuk M D, Arroyo T M, Amiri F (2013) Phase-field modeling of fracture mechanics in linear thin shells. J Appl MathGoogle Scholar
  36. 36.
    Quaranta G, Kunnath S K, Sukumar N (2012) Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures. Eng Struct 42:179–189CrossRefGoogle Scholar
  37. 37.
    Cyron CJ, Nissen K, Gravemeier V, Wall WA (2010) Stable meshfree methods in fluid mechanics based on greens functions. Comput Mech 46(2):287–300MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Bonet J, Kulasegaram S (2000) Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int J Numer Methods Eng 47(6):1189–1214CrossRefzbMATHGoogle Scholar
  39. 39.
    Guo Y M, Nakanishi K (2003) A backward extrusion analysis by the rigidplastic integrallessmeshless method. J Mater Process Technol 13(140):19–24. Proceedings of the 6th Asia Pacific Conference on materials ProcessingCrossRefGoogle Scholar
  40. 40.
    Wen H, Dong X, Yan C, Ruan X (2007) Three dimension profile extrusion simulation using meshfree method. Int J Adv Manuf Tech 34(3-4):270–276CrossRefGoogle Scholar
  41. 41.
    Wu C T , Chen JS, Pan C, Roque C (1998) A lagrangian reproducing kernel particle method for metal forming analysis. Comput Mech, 289Google Scholar
  42. 42.
    Chen J S, Roque CMOL, Chunhui P, Button S T (1998) Analysis of metal forming process based on meshless method. J Mater Process Technol, 642–646Google Scholar
  43. 43.
    Yoon S, Chen J S (2002) Accelerated meshfree method for metal forming simulation. Finite Elem Anal Des 38(10):937–948CrossRefzbMATHGoogle Scholar
  44. 44.
    Alfaro I, Yvonnet J, Cueto E, Chinesta F, Doblaré M (2006) Meshless methods with application to metal forming. Comput Methods Appl Mech Eng 195(489):6661–6675CrossRefzbMATHGoogle Scholar
  45. 45.
    Alfaro I, González D, Bel D, Cueto E, Doblar M, Chinesta F (2006) advances in the meshless simulation of aluminium extrusion and other related forming processes. Archives Comput Methods Eng 13(1):3–43CrossRefzbMATHGoogle Scholar
  46. 46.
    Alfaro I, Bel D, Cueto E, Doblaré M, Chinesta F (2006) Three-dimensional simulation of aluminium extrusion by the -shape based natural element method. Comput Methods Appl Mech Eng 195(33-36):4269–4286CrossRefzbMATHGoogle Scholar
  47. 47.
    Alfaro I, Gagliardi F, Olivera J, Cueto E, Filice L, Chinesta F (2009) Simulation of the extrusion of hollow profiles by natural element methods. Int J Mater Form 2(Supplement 1):597–600CrossRefzbMATHGoogle Scholar
  48. 48.
    Cueto E, Chinesta F (2013) Meshless methods for the simulation of material forming. Int J Mater Form, 1–19Google Scholar
  49. 49.
    Quak W, Gonzalez D, Cueto E, van den Boogaard A H (2009) On the use of local max-ent shape functions for the simulation of forming processes. In: Onate E, Owen D R J (eds) X International Conference on Computational Plasticity, COMPLAS X. CIMNE, Barcelona, SpainGoogle Scholar
  50. 50.
    Chenot J L, Bellet M (1992) Numerical Modelling of Material Deformation Processes. In: Hartley P, Pillinger I, Sturgess C (eds). Springer, London, pp 179–224Google Scholar
  51. 51.
    Dolbow J, Belytschko T (1999) Volumetric locking in the element free Galerkin method. Int J Numer Methods Eng 46(6):925–942MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    González D, Cueto E, Doblaré M (2004) Volumetric locking in natural neighbour Galerkin methods. Int J Numer Methods Eng 61(4):611–632CrossRefGoogle Scholar
  53. 53.
    Brezzi F (1974) On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. ESAIM: Math Model Numer Anal Modél Math Anal Numér 8(R2):129–151MathSciNetzbMATHGoogle Scholar
  54. 54.
    Babuka I (1973) The finite element method with lagrangian multipliers. Numerische Mathematik 20(3):179–192CrossRefGoogle Scholar
  55. 55.
    Li J, He Y, Chen Z (2009) Performance of several stabilized finite element methods for the stokes equations based on the lowest equal-order pairs. Computing 86(1):37–51MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Ortiz A, Puso M A, Sukumar N (2010) Maximum-entropy meshfree method for compressible and near-incompressible elasticity. Comput Methods Appl Mech Eng 199(25–28):1859–1871MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Arnold D N, Brezzi F, Fortin M (1984) A stable finite element for the stokes equations. CALCOLO 21(4):337–344MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Ortiz A, Puso MA, Sukumar N (2011) Maximum-entropy meshfree method for incompressible media problems. Finite Elem Anal Des 47(6):572–585MathSciNetCrossRefGoogle Scholar
  59. 59.
    Cyron CJ, Arroyo M, Ortiz M (2009) Smooth, second order, non-negative meshfree approximants selected by maximum entropy. Int J Numer Methods Eng 79(13):1605–1632MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Rosolen A, Millán D, Arroyo M (2012) Second order convex maximum entropy approximants with applications to high order PDE. Int J Numer Methods EngGoogle Scholar
  61. 61.
    González D, Cueto E, Doblaré M (2010) A higher-order method based on local maximum entropy approximation. Int J Numer Methods Eng 83(6):741–764zbMATHGoogle Scholar
  62. 62.
    Bompadre A, Perotti L E, Cyron C, Ortiz M (2012) Convergent meshfree approximation schemes of arbitrary order and smoothness. Comput Methods Appl Mech Eng 221–222:83–103MathSciNetCrossRefGoogle Scholar
  63. 63.
    Puso M A, Chen J S, Zywicz E, Elmer W (2008) Meshfree and finite element nodal integration methods. Int J Numer Methods Eng 74(3):416–446MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Codina R (1998) Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput Methods Appl Mech Eng 156(14s):185–210MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Barth T, Bochev P, Gunzburger M, Shadid J (2004) A taxonomy of consistently stabilized finite element methods for the stokes problem. SIAM J Sci Comput 25(5):1585–1607MathSciNetCrossRefzbMATHGoogle Scholar
  66. 66.
    Bochev P, Gunzburger M (2004) An absolutely stable pressure-poisson stabilized finite element method for the stokes equations. SIAM J Numer Anal 42(3):1189–1207MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Peco C, Rosolen A, Arroyo M (2013) Estabilización de las ecuaciones de stokes con aproximantes locales de máxima entropía. Submitted to RIMNIGoogle Scholar
  68. 68.
    Brezzi F, Pitkaranta J (1984) On the stabilization of finite element approximations of the Stokes equations. Notes on Numerical Fluid Mechanics, Efficient Solutions of Elliptic Systems, vol 10. Viewig, Braunschweig, pp 11–19Google Scholar
  69. 69.
    Greco F, Sukumar N (2013) Derivatives of maximum-entropy basis functions on the boundary: theory and computations. Int J Numer Methods Eng 94(12):1123–1149MathSciNetCrossRefGoogle Scholar
  70. 70.
    Hughes T J R, Franca L P, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. circumventing the babuka-brezzi condition: a stable Petrov-Galerkin formulation of the stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59(1):85–99MathSciNetCrossRefGoogle Scholar
  71. 71.
    Jansen K E, Collis S S, Whiting C, Shakib F (1999) A better consistency for low-order stabilized finite element methods. Comput Methods Appl Mech Eng 174(1-2):153–170MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Harari I, Hughes T J R (1992) What are C and h?: Inequalities for the analysis and design of finite element methods. Comput Methods Appl Mech Eng 97(2):157–192MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Wriggers P (2003) Computational contact mechanics. Comput Mech 32:141–141CrossRefGoogle Scholar
  74. 74.
    Hormann K, Sukumar N (2008) Maximum entropy coordinates for arbitrary polytopes. In: Proceedings of SGP 2008Google Scholar
  75. 75.
    Edelsbrunner H, Kirkpatrick D, Seidel R (1983) On the shape of a set of points in the plane. IEEE Trans Inform Theory 29(4):551–559MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Kalpakjian S (1992) Manufacturing Processes for Engineering Materials, 5/e (New Edition). Pearson EducationGoogle Scholar
  77. 77.
    Quak W., Boogaard A.H., Hutink J (2009) Meshless methods and forming processes. Int J Mater Form 2(1):585–588CrossRefGoogle Scholar
  78. 78.
    Ceretti E, Taupin E, Altan T (1997) Simulation of metal flow and fracture applications in orthogonal cutting, blanking, and cold extrusion. CIRP Ann-Manuf Technol 46(1):187–190CrossRefGoogle Scholar
  79. 79.
    Arrazola P J, zel T, Umbrello D, Davies M, Jawahir I S (2013) Recent advances in modelling of metal machining processes. CIRP Ann-Manuf Technol 62(2):695–718CrossRefGoogle Scholar

Copyright information

© Springer-Verlag France 2014

Authors and Affiliations

  1. 1.Department of Mechanical, Energy and Management EngineeringUniversity of CalabriaRendeItaly
  2. 2.LaCàN, Universitat Politècnica de Catalunya (UPC)BarcelonaSpain

Personalised recommendations