Computational Mechanics

, Volume 54, Issue 3, pp 803–820 | Cite as

A multiscale overlapped coupling formulation for large-deformation strain localization

  • WaiChing SunEmail author
  • Alejandro Mota
Original Paper


We generalize the multiscale overlapped domain framework to couple multiple rate-independent standard dissipative material models in the finite deformation regime across different length scales. We show that a fully coupled multiscale incremental boundary-value problem can be recast as the stationary point that optimizes the partitioned incremental work of a three-field energy functional. We also establish inf-sup tests to examine the numerical stability issues that arise from enforcing weak compatibility in the three-field formulation. We also devise a new block solver for the domain coupling problem and demonstrate the performance of the formulation with one-dimensional numerical examples. These simulations indicate that it is sufficient to introduce a localization limiter in a confined region of interest to regularize the partial differential equation if loss of ellipticity occurs.


Domain coupling Variational principle Energy based coupling method Multiscale modeling 



We thank James W. Foulk III for providing us with the analytical solution of the singular bar problem. Thanks are also due to Micheal L. Parks for suggesting the one-dimensional patch test. Support for this work was received through the U.S. Department of Energy’s Advanced Simulation and Computing (ASC) Program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.


  1. 1.
    Abellan Marie-Angèle, de Borst René (2006) Wave propagation and localisation in a softening two-phase medium. Comput Methods Appl Mech Eng 195(37):5011–5019CrossRefzbMATHGoogle Scholar
  2. 2.
    Aubertin P, Réthoré J, de Borst R (2009) Energy conservation of atomistic/continuum coupling. Int J Numer Methods Eng 78(11):1365–1386Google Scholar
  3. 3.
    Babus̆ka I (1973) The finite element method with lagrangian multipliers. Numerische Mathematik 20:179–192. ISSN 0029–599XGoogle Scholar
  4. 4.
    Badia S, Parks M, Bochev P, Gunzburger M, Lehoucq R (2008) On atomistic-to-continuum coupling by blending. Multiscale Model Simul 7(1):381–406CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bathe K-J (2001) The inf-sup condition and its evaluation for mixed finite element methods. Comput Struct 79(2):243–252. ISSN 0045–7949Google Scholar
  6. 6.
    Bauman PT, Ben Dhia H, Elkhodja N, Oden J, Prudhomme S (2008) On the application of the arlequin method to the coupling of particle and continuum models. Comput Mech 42:511–530Google Scholar
  7. 7.
    Bauman PT, Oden JT, Prudhomme S (2009) Adaptive multiscale modeling of polymeric materials with arlequin coupling and goals algorithms. Comput Methods Appl Mech Eng 198(5):799– 818CrossRefzbMATHGoogle Scholar
  8. 8.
    Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119–1149CrossRefGoogle Scholar
  9. 9.
    Belytschko T, Fish J, Engelmann BE (1988) A finite element with embedded localization zones. Comput Methods Appl Mech Eng 70(1):59–89CrossRefzbMATHGoogle Scholar
  10. 10.
    Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New YorkzbMATHGoogle Scholar
  11. 11.
    Belytschko T, Loehnert S, Song J-H (2008) Multiscale aggregating discontinuities: A method for circumventing loss of material stability. Int J Numer Methods Eng 73(6):869–894Google Scholar
  12. 12.
    Ben Dhia H (1998) Multiscale mechanical problems: the arlequin method. Compt Rendus de Acad des Sci Ser IIB 326(12):899– 904Google Scholar
  13. 13.
    Ben Dhia H (2008) Further insights by theoretical investigations of the multiscale arlequin metho. Int J Multiscale Comput Eng 6(3):215–232CrossRefGoogle Scholar
  14. 14.
    Ben Dhia H, Rateau G (2005) The arlequin method as a flexible engineering design tool. Int J Numer Methods Eng 62(11):1442–1462Google Scholar
  15. 15.
    Borja RI (2000) A finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation. Comput Methods Appl Mech Eng 190(11–12):1529–1549Google Scholar
  16. 16.
    Borja RI (2002) Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a petrov-galerkin multiscale formulation. Comput Methods Appl Mech Eng 191(27–28):2949–2978Google Scholar
  17. 17.
    Brezzi F, Marini LD (2005) The three-field formulation for elasticity problems. GAMM Mitteilungen 28:124–153CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Brezzi F, Douglas J, Marini LD (1985) Two families of mixed finite elements for second order elliptic problems. Numerische Mathematik 47:217–235Google Scholar
  19. 19.
    Chapelle D, Bathe KJ (1993) The inf-sup test. Comput Struct 47(4–5):537–545Google Scholar
  20. 20.
    Chen Yi-Chao (1991) On strong ellipticity and the legendre-hadamard condition. Arch Ration Mech Anal 113(2):165–175CrossRefzbMATHGoogle Scholar
  21. 21.
    Feyel F(2003) A multilevel finite element method (fe2) to describe the response of highly non-linear structures using generalized continua. Comput Methods Appl Mech Eng 192(28–30):3233–3244.Google Scholar
  22. 22.
    Fleck NA, Hutchson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49(10):2245–2271Google Scholar
  23. 23.
    Guidault PA, Belytschko T (2007) On the l2 and the h1 couplings for an overlapping domain decomposition method using lagrange multipliers. Int J Numer Methods Eng 70(3):322–350Google Scholar
  24. 24.
    Han F, Lubineau G (2012) Coupling of nonlocal and local continuum models by the Arlequin approach. Int J Numer Methods Eng 89(6):671–685Google Scholar
  25. 25.
    Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, West SussexGoogle Scholar
  26. 26.
    Kouznetsova VG, Geers MGD, Brekelmans WAM (2004) Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Methods Appl Mech Eng 193(48–51):5525–5550Google Scholar
  27. 27.
    McAuliffe C, Waisman H (2013) Mesh insensitive formulation for initiation and growth of shear bands using mixed finite elements. Comput Mech 2:1–17MathSciNetGoogle Scholar
  28. 28.
    Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46:131–150CrossRefzbMATHGoogle Scholar
  29. 29.
    Mota A, Foulk JW, Ostien JT (2013) Variational nonlocal regularization in finite-deformation inelasticity. In: Proceedings of the 12th U.S. National Congress on Computational Mechanics, Raleigh, NCGoogle Scholar
  30. 30.
    Prudhomme S, Ben Dhia H, Bauman PT, Elkhodja N, Oden JT (2008) Computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method. Comput Methods Appl Mech Eng 197(41–42):3399–3409Google Scholar
  31. 31.
    Rudnicki JW, Rice JR (1975) Conditions for the localization of deformation in pressure-sensitive dilatant materials. J Mech Phys Solids 23(6):371–394Google Scholar
  32. 32.
    Sun W, Andrade JE, Rudnicki JW (2011) Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability. Int J Numer Methods Eng 88(12):1260–1279Google Scholar
  33. 33.
    Sun W, Chen Q, Ostien JT (2013a) Modeling hydro-mechanical responses of strip and circular footings on saturated collapsible geomaterials. Acta Geotech 5:189–198. doi: 10.1007/s11440-013-0276-x
  34. 34.
    Sun W, Ostien JT, Salinger AG (2013b) A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain. Int J Numer Anal Methods Geomech 37(16):2755–2788Google Scholar
  35. 35.
    Toh K-C, Phoon K-K, Chan S-H (2004) Block preconditioners for symmetric indefinite linear systems. Int J N Methods Eng 60(8):1361–1381Google Scholar
  36. 36.
    White JA, Borja RI (2011) Block-preconditioned newton-krylov solvers for fully coupled flow and geomechanics. Comput Geosci 15:647–659.Google Scholar
  37. 37.
    White JA, Borja RI, Fredrich JT (2006) Calculating the effective permeability of sandstone with multiscale lattice boltzmann/finite element simulations. Acta Geotechn 1:195–209Google Scholar
  38. 38.
    Yang Q, Mota A, Ortiz M (2005) A class of variational strain-localization finite elements. Int J Numer Methods Eng 62(8):1013–1037Google Scholar
  39. 39.
    Zhang HW, Schrefler BA, Wriggers P (2003) Interaction between different internal length scales for strain localisation analysis of single phase materials. Comput Mech 30(3):212–219CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Civil Engineering and Engineering MechanicsColumbia University in the City of New YorkNew YorkUSA
  2. 2.Mechanics of Materials DepartmentSandia National LaboratoriesLivermoreUSA

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