Skip to main content
SpringerLink
Log in

A stabilized mixed implicit Material Point Method for non-linear incompressible solid mechanics

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this work a stabilized mixed formulation for the solution of non-linear solid mechanics problems in nearly-incompressible conditions is presented. In order to deal with high material deformation, an implicit Material Point Method is chosen. Such choice allows avoiding the classical limitations of the Finite Element Method, e.g., element tangling and extreme mesh distortion. The proposed mixed formulation, with displacement and pressure as primary variables, is tested through classical benchmarks in solid and geo-mechanics where a Neo-Hookean, a J2 and a Mohr-Coulomb plastic law are employed. Further, the stabilized mixed formulation is compared with a displacement-based formulation to demonstrate how the proposed approach gets better results in terms of accuracy, not only when incompressible materials are simulated, but also in the case of compressible ones.

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

Access this article

Log in via an institution

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22

Similar content being viewed by others

References

  1. Zienkiewicz O, Taylor R, Zhu J (eds) (2013) The finite element method: its basis and fundamentals, 7th edn. Butterworth-Heinemann, Oxford

    MATH  Google Scholar 

  2. de Souza Neto E, Perić D, Owen D (2008) Computational methods for plasticity. Wiley, New York

    Book  Google Scholar 

  3. Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Methods Eng 29(8):1595

    Article  MathSciNet  MATH  Google Scholar 

  4. Hughes TJR (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15(9):1413

    Article  MathSciNet  MATH  Google Scholar 

  5. Taylor RL, Beresford PJ, Wilson EL (1976) A non-conforming element for stress analysis. Int J Numer Methods Eng 10(6):1211

    Article  MATH  Google Scholar 

  6. Auricchio F, da Veiga LB, Lovadina C, Reali A (2005) An analysis of some mixed-enhanced finite element for plane linear elasticity. Comput Methods Appl Mech Eng 194(27):2947

    Article  MathSciNet  MATH  Google Scholar 

  7. de Souza Neto E, Perič D, Dutko M, Owen D (1996) Design of simple low order finite elements for large strain analysis of nearly incompressible solids. Int J Solids Struct 33(20):3277

    Article  MathSciNet  MATH  Google Scholar 

  8. Moran B, Ortiz M, Shih CF (1990) Formulation of implicit finite element methods for multiplicative finite deformation plasticity. Int J Numer Methods Eng 29(3):483

    Article  MathSciNet  MATH  Google Scholar 

  9. Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int j Numer Methods Eng 33(7):1413

    Article  MATH  Google Scholar 

  10. Reddy B, Simo J (1995) Stability and convergence of a class of enhanced strain methods. SIAM J Numer Anal 32:1705

    Article  MathSciNet  MATH  Google Scholar 

  11. Ortiz-Bernardin A, Hale J, Cyron C (2015) Volume-averaged nodal projection method for nearly-incompressible elasticity using meshfree and bubble basis functions. Comput Methods Appl Mech Eng 285:427

    Article  MathSciNet  MATH  Google Scholar 

  12. Sussman T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26(1):357

    Article  MATH  Google Scholar 

  13. Brink U, Stein E (1996) On some mixed finite element methods for incompressible and nearly incompressible finite elasticity. Comput Mech 19(1):105

    Article  MATH  Google Scholar 

  14. Chiumenti M, Valverde Q, de Saracibar CA, Cervera M (2002) A stabilized formulation for incompressible elasticity using linear displacement and pressure interpolations. Comput Methods Appl Mech Eng 191(46):5253

    Article  MathSciNet  MATH  Google Scholar 

  15. Cervera M, Chiumenti M, Valverde Q, de Saracibar CA (2003) Mixed linear/linear simplicial elements for incompressible elasticity and plasticity. Comput Methods Appl Mech Eng 192(49):5249

    Article  MATH  Google Scholar 

  16. Chiumenti M, Valverde Q, de Saracibar CA, Cervera M (2004) A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra. Int J Plast 20(8):1487

    Article  MATH  Google Scholar 

  17. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics: Part ii: Strain localization. Comput Methods Appl Mech Eng 199(37):2571

    Article  MathSciNet  MATH  Google Scholar 

  18. Cervera M, Chiumenti M, Benedetti L, Codina R (2015) Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: compressible and incompressible plasticity. Comput Methods Appl Mech Eng 285:752

    Article  MathSciNet  MATH  Google Scholar 

  19. Simo J, Taylor R, Pister K (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51(1):177

    Article  MathSciNet  MATH  Google Scholar 

  20. Brezzi F (1974) On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers. ESAIM: Math Model Numer Anal 8(R2):129

    MathSciNet  MATH  Google Scholar 

  21. Babuška I (1972/73) The finite element method with lagrangian multipliers. Numerische Mathematik 20:179

  22. Babuška I (1973) The finite element method with penalty. Math Comput 27:221

  23. Fortin Michel (1977) An analysis of the convergence of mixed finite element methods. RAIRO Anal numér 11(4):341

    Article  MathSciNet  MATH  Google Scholar 

  24. Hughes TJ, Franca LP, Balestra M (1986) 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 59(1):85

    Article  MATH  Google Scholar 

  25. Hughes T, Franca L, Hulbert G (1989) A new finite element formulation for computational fluid dynamics: Viii. the galerkin/least-squares method for advective-diffusive equations. Comput Methods Appl Mech Eng 73(2):173

    Article  MathSciNet  MATH  Google Scholar 

  26. Hughes TJ (1995) Multiscale phenomena: Green’s functions, the dirichlet-to-neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput Methods Appl Mech Eng 127(1):387

    Article  MathSciNet  MATH  Google Scholar 

  27. Oñate E (1998) Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems. Comput Methods Appl Mech Eng 151(1):233 Containing papers presented at the Symposium on Advances in Computational Mechanics

    Article  MathSciNet  MATH  Google Scholar 

  28. Oñate E (2000) A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation. Comput Methods Appl Mech Eng 182(3):355

    Article  MathSciNet  MATH  Google Scholar 

  29. Codina R (2000) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput Methods Appl Mech Eng 190(13):1579

    Article  MathSciNet  MATH  Google Scholar 

  30. Codina R, Blasco J (2000) Stabilized finite element method for the transient navier stokes equations based on a pressure gradient projection. Comput Methods Appl Mech Eng 182(3):277

    Article  MathSciNet  MATH  Google Scholar 

  31. Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191(39):4295

    Article  MathSciNet  MATH  Google Scholar 

  32. Mast C, Mackenzie-Helnwein P, Arduino P, Miller G, Shin W (2012) Mitigating kinematic locking in the material point method. J Comput Phys 231(16):5351

    Article  MathSciNet  Google Scholar 

  33. Kularathna S, Soga K (2017) Implicit formulation of material point method for analysis of incompressible materials. Comput Methods Appl Mech Eng 313:673

    Article  MathSciNet  Google Scholar 

  34. Chorin AJ (1968) Numerical solution of the navier–stokes equations. Math Comput 22(104):745

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang F, Zhang X, Sze KY, Lian Y, Liu Y (2017) Incompressible material point method for free surface flow. J Comput Phys 330:92

    Article  MathSciNet  MATH  Google Scholar 

  36. Iaconeta I, Larese A, Rossi R, Oñate E (2017) An implicit material point method applied to granular flows. In:Proceedings of the 1st international conference on the material point method (MPM 2017), Procedia Engineering 175:226

  37. Dadvand P (2007) A framework for developing finite element codes for multi-disciplinary applications. (Ph.D. thesis: Universidad Politécnica de Cataluña)

  38. Dadvand P, Rossi R, Oñate E (2010) An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 17:253

    Article  MATH  Google Scholar 

  39. Iaconeta I, Larese A, Rossi R, Guo Z (2017) Comparison of a material point method and a galerkin meshfree method for the simulation of cohesive-frictional materials. Materials 10:10

    Article  Google Scholar 

  40. Dohrmann CR, Bochev PB (2004) A stabilized finite element method for the stokes problem based on polynomial pressure projections. Int J Numer Methods Fluids 46(2):183

    Article  MathSciNet  MATH  Google Scholar 

  41. Rodriguez J, Carbonell J, Cante J, Oliver J (2015) The particle finite element method (PFEM) in thermo-mechanical problems. Int J Numer Methods Eng

  42. Monforte L, Carbonell JM, Arroyo M, Gens A (2016) Performance of mixed formulations for the particle finite element method in soil mechanics problems. Comput Particle Mech pp 1–16

  43. Wriggers P (2006) Computational contact mechanics. Springer, New York

    Book  MATH  Google Scholar 

  44. Simo J, Hughes T (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  45. Simo JC (1988) A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part I. Continuum formulation. Comput Methods Appl Mech Eng 66(2):199

    Article  MATH  Google Scholar 

  46. Clausen J, Damkilde L, Andersen L (2006) Efficient return algorithms for associated plasticity with multiple yield planes. Int J Numer Methods Eng 66(6):1036

    Article  MathSciNet  MATH  Google Scholar 

  47. Simo J (1992) Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory. Comput Methods Appl Mech Eng 99(1):61

    Article  MathSciNet  MATH  Google Scholar 

  48. Simo J (1998) Numerical analysis and simulation of plasticity. Handbook Numer Anal 6:183

    MathSciNet  MATH  Google Scholar 

  49. Harlow F (1964) The particle-in-cell computing method for fluid dynamics. Methods Comput Phys 3:319

    Google Scholar 

  50. Sulsky D, Chen Z, Schreyer H (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118(1–2):179

    Article  MathSciNet  MATH  Google Scholar 

  51. Sulsky D, Zhou SJ, Schreyer HL (1995) Application of a particle-in-cell method to solid mechanics. Comput Phys Commun 87(1–2):236

    Article  MATH  Google Scholar 

  52. Wieckowski Z (2004) The material point method in large strain engineering problems. Comput Methods Appl Mech Eng 193(39–41):4417

    Article  MATH  Google Scholar 

  53. Sołowski W, Sloan S (2015) Evaluation of material point method for use in geotechnics. Int J Numer Anal Methods Geomech 39(7):685

    Article  Google Scholar 

  54. Bardenhagen S, Kober E (2004) The generalized interpolation material point method. CMES Comput Model Eng Sci 5(6):477

    Google Scholar 

  55. Sadeghirad A, Brannon R, Burghardt J (2011) A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. Int J Numer Methods Eng 86(12):1435

    Article  MathSciNet  MATH  Google Scholar 

  56. Sadeghirad A, Brannon R, Guilkey J (2013) Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces. Int J Numer Methods Eng 95(11):928

    Article  MathSciNet  MATH  Google Scholar 

  57. Steffen M, Wallstedt P, Guilkey J, Kirby R, Berzins M (2008) Examination and analysis of implementation choices within the material point method (MPM). Comput Model Eng Sci 31(2):107

    Google Scholar 

  58. Steffen M, Kirby RM, Berzins M (2008) Analysis and reduction of quadrature errors in the material point method (MPM). Int J Numer Methods Eng 76(6):922

    Article  MathSciNet  MATH  Google Scholar 

  59. Motlagh YG, Coombs WM (2017) An implicit high-order material point method. In: Proceedings of the 1st international conference on the material point method (MPM 2017), Procedia Engineering 175:8

  60. Cook R (1974) Improved two-dimensional finite element. J Struct Div 100:1851

    Google Scholar 

  61. Franci A (2015) Unified Lagrangian formulation for fluid and solid mechanics, fluid-structure interaction and coupled thermal problems using the PFEM (PhD thesis: Universitat Politécnica de Catalunya)

  62. Cervera M, Chiumenti M, Codina R (2010) Mixed stabilized finite element methods in nonlinear solid mechanics: part I: formulation. Comput Methods Appl Mech Eng 199(37):2559

    Article  MathSciNet  MATH  Google Scholar 

  63. Borja RI, Sama KM, Sanz PF (2003) On the numerical integration of three-invariant elastoplastic constitutive models. Comput Methods Appl Mech Eng 192(9):1227

    Article  MATH  Google Scholar 

  64. Nazem M, Sheng D, Carter JP (2006) Stress integration and mesh refinement for large deformation in geomechanics. Int J Numer Methods Eng 65(7):1002

    Article  MATH  Google Scholar 

  65. Kardani M, Nazem M, Carter J, Abbo A (2014) Efficiency of high-order elements in large-deformation problems of geomechanics. Int J Geomech 15(6):040014101

    Google Scholar 

  66. Silva MD, Krabbenhoft K, Lyamin A, Sloan S (2011) Rigid-plastic large-deformation analysis of geotechnical penetration problems. In: Proceeding of the 13th IACMAG conference. Computer methods for geomechanics: frontiers and new applications vol 1

Download references

Acknowledgements

The research was supported by the Research Executive Agency through the T-MAPPP project (FP7 PEOPLE 2013 ITN-G.A.n607453). The Spanish Ministry of Economy and Competitiveness (Ministerio de Economía y Competitividad, MINECO) through the projects HIRMA (RTC-2016-4967-5), PRECISE (BIA2017-83805-R) is also greatly acknowledged. Finally Dr. Larese gratefully acknowledges the support of the Spanish ministry through her Juan de la Cierva Incorporacion (IJCI-2015-26484).

Author information

Authors and Affiliations

  1. International Center for Numerical Methods in Engineering (CIMNE), Technical University of Catalonia (UPC), Barcelona, Spain

    I. Iaconeta, A. Larese, R. Rossi & E. Oñate

Authors
  1. I. Iaconeta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Larese
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. E. Oñate
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. Iaconeta.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iaconeta, I., Larese, A., Rossi, R. et al. A stabilized mixed implicit Material Point Method for non-linear incompressible solid mechanics. Comput Mech 63, 1243–1260 (2019). https://doi.org/10.1007/s00466-018-1647-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-018-1647-9

Keywords

Search

Navigation