Skip to main content
Log in

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

  • ORIGINAL RESEARCH
  • Published:
International Journal of Material Forming Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Kobayashi S, Oh Si, Altan T (1989) Metal forming and the finite-element method. Oxford University Press

  2. Peric D, Owen D R J (2004) Computational modeling of forming processes. Wiley

  3. Babuska I, Aziz AK (1976) On the angle condition in the finite element method. SIAM J Numer Anal 13(2):214–226

    Article  MathSciNet  MATH  Google Scholar 

  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):1–16

    Article  Google Scholar 

  5. Zienkiewicz O C (1984) Flow formulation for numerical solution of forming processes. Wiley, Chichester

    Google Scholar 

  6. Hwu Y J, Lenard J G (1988) A finite element study of flat rolling. J Eng Mater Technol 110(1):22–27

    Article  Google Scholar 

  7. Belytschko T, Kennedy J M (1978) Computer models for subassembly simulation. Nucl Eng Des 49:17–38

    Article  Google Scholar 

  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):259–310

    Article  MATH  Google Scholar 

  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):855–880

    Article  MATH  Google Scholar 

  10. Belytschko T, Krongauz Y, Organ D, Fleming M, Meshless P K (1996) An overview and recent developments. Comput Methods Appl Mech Eng 139:3–47

    Article  Google Scholar 

  11. Puso M A, Solberg J (2006) A stabilized nodally integrated tetrahedral. Int J Numer Methods Eng 67(6):841–867

    Article  MathSciNet  MATH  Google Scholar 

  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):121–137

    Article  MathSciNet  MATH  Google Scholar 

  13. Krysl P, Kagey H (2012) Reformulation of nodally integrated continuum elements to attain insensitivity to distortion. Int J Numer Methods Eng 90(7):805–818

    Article  MathSciNet  MATH  Google Scholar 

  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 1455–1466

  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 223:172–181

    Article  MATH  Google Scholar 

  16. Quak W (October 2011) On meshless and nodal-based numerical methods for forming processes, PhD thesis. Enschede, The Netherlands

  17. Belytschko T, Lu Y Y, Gu L (1994) Element-free galerkin methods. Int J Numer Methods Eng 37(2):229–256

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu W K, Jun S, Yi F Z (1995) Reproducing kernel particle methods. Int J Numer Methods Fluids 20(8-9):1081–1106

    Article  MATH  Google Scholar 

  19. Huerta A, Fernández-Méndez S (2000) Enrichment and coupling of the finite element and meshless methods. Int J Numer Methods Eng 48:1615–1636

    Article  MATH  Google Scholar 

  20. Sukumar N, Moran B, Belytschko T (1998) The natural element method in solid mechanics. Int J Numer Methods Eng 43(5):839–887

    Article  MathSciNet  MATH  Google Scholar 

  21. Preparata FP, Shamos M I (1985) Computational geometry: an introduction. Springer, New York

    Book  Google Scholar 

  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):519–546

    Article  MATH  Google Scholar 

  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):1436–1465

    Article  MATH  Google Scholar 

  24. Sukumar N (2004) Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Methods Eng 61(12):2159–2181

    Article  MathSciNet  MATH  Google Scholar 

  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):2167–2202

    Article  MathSciNet  MATH  Google Scholar 

  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):868–895

    Google Scholar 

  27. Rosolen A, Arroyo M (2013) Blending isogeometric analysis and local maximum entropy meshfree approximants. Comput Methods Appl Mech Eng 264:95–107

    Article  MathSciNet  MATH  Google Scholar 

  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):723–751

    Article  MATH  Google Scholar 

  29. Millán D, Rosolen A, Arroyo M (2013) Nonlinear manifold learning for meshfree finite deformation thin shell analysis. Int J Numer Methods Eng 93:685–713

    Article  Google Scholar 

  30. Millán D, Arroyo M (2013) Nonlinear manifold learning for model reduction in finite elastodynamics. Comput Methods Appl Mech Eng 261262(0):118–131

    Article  Google Scholar 

  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 Physics

  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 Physics

  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):303–319

    Article  MathSciNet  Google Scholar 

  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):320–336

    Article  MathSciNet  MATH  Google Scholar 

  35. Rabczuk M D, Arroyo T M, Amiri F (2013) Phase-field modeling of fracture mechanics in linear thin shells. J Appl Math

  36. Quaranta G, Kunnath S K, Sukumar N (2012) Maximum-entropy meshfree method for nonlinear static analysis of planar reinforced concrete structures. Eng Struct 42:179–189

    Article  Google Scholar 

  37. Cyron CJ, Nissen K, Gravemeier V, Wall WA (2010) Stable meshfree methods in fluid mechanics based on greens functions. Comput Mech 46(2):287–300

    Article  MathSciNet  MATH  Google Scholar 

  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):1189–1214

    Article  MATH  Google Scholar 

  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 Processing

    Article  Google Scholar 

  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):270–276

    Article  Google Scholar 

  41. Wu C T , Chen JS, Pan C, Roque C (1998) A lagrangian reproducing kernel particle method for metal forming analysis. Comput Mech, 289

  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, 642–646

  43. Yoon S, Chen J S (2002) Accelerated meshfree method for metal forming simulation. Finite Elem Anal Des 38(10):937–948

    Article  MATH  Google Scholar 

  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):6661–6675

    Article  MATH  Google Scholar 

  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):3–43

    Article  MATH  Google Scholar 

  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):4269–4286

    Article  MATH  Google Scholar 

  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):597–600

    Article  MATH  Google Scholar 

  48. Cueto E, Chinesta F (2013) Meshless methods for the simulation of material forming. Int J Mater Form, 1–19

  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, Spain

    Google Scholar 

  50. Chenot J L, Bellet M (1992) Numerical Modelling of Material Deformation Processes. In: Hartley P, Pillinger I, Sturgess C (eds). Springer, London, pp 179–224

  51. Dolbow J, Belytschko T (1999) Volumetric locking in the element free Galerkin method. Int J Numer Methods Eng 46(6):925–942

    Article  MathSciNet  MATH  Google Scholar 

  52. González D, Cueto E, Doblaré M (2004) Volumetric locking in natural neighbour Galerkin methods. Int J Numer Methods Eng 61(4):611–632

    Article  Google Scholar 

  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):129–151

    MathSciNet  MATH  Google Scholar 

  54. Babuka I (1973) The finite element method with lagrangian multipliers. Numerische Mathematik 20(3):179–192

    Article  Google Scholar 

  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):37–51

    Article  MathSciNet  MATH  Google Scholar 

  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):1859–1871

    Article  MathSciNet  MATH  Google Scholar 

  57. Arnold D N, Brezzi F, Fortin M (1984) A stable finite element for the stokes equations. CALCOLO 21(4):337–344

    Article  MathSciNet  MATH  Google Scholar 

  58. Ortiz A, Puso MA, Sukumar N (2011) Maximum-entropy meshfree method for incompressible media problems. Finite Elem Anal Des 47(6):572–585

    Article  MathSciNet  Google Scholar 

  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):1605–1632

    Article  MathSciNet  MATH  Google Scholar 

  60. Rosolen A, Millán D, Arroyo M (2012) Second order convex maximum entropy approximants with applications to high order PDE. Int J Numer Methods Eng

  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):741–764

    MATH  Google Scholar 

  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 221–222:83–103

    Article  MathSciNet  Google Scholar 

  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):416–446

    Article  MathSciNet  MATH  Google Scholar 

  64. Codina R (1998) Comparison of some finite element methods for solving the diffusion-convection-reaction equation. Comput Methods Appl Mech Eng 156(14s):185–210

    Article  MathSciNet  MATH  Google Scholar 

  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):1585–1607

    Article  MathSciNet  MATH  Google Scholar 

  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):1189–1207

    Article  MathSciNet  MATH  Google Scholar 

  67. Peco C, Rosolen A, Arroyo M (2013) Estabilización de las ecuaciones de stokes con aproximantes locales de máxima entropía. Submitted to RIMNI

  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 11–19

  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):1123–1149

    Article  MathSciNet  Google Scholar 

  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):85–99

    Article  MathSciNet  Google Scholar 

  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):153–170

    Article  MathSciNet  MATH  Google Scholar 

  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):157–192

    Article  MathSciNet  MATH  Google Scholar 

  73. Wriggers P (2003) Computational contact mechanics. Comput Mech 32:141–141

    Article  Google Scholar 

  74. Hormann K, Sukumar N (2008) Maximum entropy coordinates for arbitrary polytopes. In: Proceedings of SGP 2008

  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):551–559

    Article  MathSciNet  MATH  Google Scholar 

  76. Kalpakjian S (1992) Manufacturing Processes for Engineering Materials, 5/e (New Edition). Pearson Education

  77. Quak W., Boogaard A.H., Hutink J (2009) Meshless methods and forming processes. Int J Mater Form 2(1):585–588

    Article  Google Scholar 

  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):187–190

    Article  Google Scholar 

  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):695–718

    Article  Google Scholar 

Download references

Acknowledgments

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).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to F. Greco.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Greco, F., Filice, L., Peco, C. et al. A stabilized formulation with maximum entropy meshfree approximants for viscoplastic flow simulation in metal forming. Int J Mater Form 8, 341–353 (2015). https://doi.org/10.1007/s12289-014-1167-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12289-014-1167-x

Keywords

Navigation