Coupled systems mechanics

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Coupling non-matching finite element discretizations in small-deformation inelasticity: Numerical integration of interface variables

  • Amaireh, Layla K. (Applied Science Private University) ;
  • Haikal, Ghadir (Lyles School of Civil Engineering, Purdue University)
  • Received : 2018.08.04
  • Accepted : 2018.09.07
  • Published : 2019.02.25
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Abstract

Finite element simulations of solid mechanics problems often involve the use of Non-Confirming Meshes (NCM) to increase accuracy in capturing nonlinear behavior, including damage and plasticity, in part of a solid domain without an undue increase in computational costs. In the presence of material nonlinearity and plasticity, higher-order variables are often needed to capture nonlinear behavior and material history on non-conforming interfaces. The most popular formulations for coupling non-conforming meshes are dual methods that involve the interpolation of a traction field on the interface. These methods are subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) stability condition, and are therefore limited in their implementation with the higher-order elements needed to capture nonlinear material behavior. Alternatively, the enriched discontinuous Galerkin approach (EDGA) (Haikal and Hjelmstad 2010) is a primal method that provides higher order kinematic fields on the interface, and in which interface tractions are computed from local finite element estimates, therefore facilitating its implementation with nonlinear material models. The inclusion of higher-order interface variables, however, presents the issue of preserving material history at integration points when a increase in integration order is needed. In this study, the enriched discontinuous Galerkin approach (EDGA) is extended to the case of small-deformation plasticity. An interface-driven Gauss-Kronrod integration rule is proposed to enable adaptive enrichment on the interface while preserving history-dependent material data at existing integration points. The method is implemented using classical J2 plasticity theory as well as the pressure-dependent Drucker-Prager material model. We show that an efficient treatment of interface variables can improve algorithmic performance and provide a consistent approach for coupling non-conforming meshes in inelasticity.

Keywords

References

  1. Arnorld, D.N., Brezzi, F., Cockburn, B. and Marini, L.D. (2002), "Unified analysis of discontinuous Galerkin methods for elliptic problems", SIAM J. Numer. Analy., 39(5), 1749-1779. https://doi.org/10.1137/S0036142901384162
  2. Bavestrello, H., Avery, P. and Farhat, C. (2007), "Incorporation of linear multipoint constraints in substructure based iterative solvers. Part II: Blending FETI-DP and mortar methods and assembling floating substructures", Comput. Meth. Appl. Mech. Eng., 196, 1347-1368. https://doi.org/10.1016/j.cma.2006.03.024
  3. Bayat, H., Kramer, J., Wunderlich, L., Wulfinghoff, S., Wohlmuth, B., Resse, S. and Wieners, C. (2018), "Numerical evaluation of discontinuous and nonconforming finite element methods in nonlinear solid mechanics," Comput. Mech., 1-15.
  4. Becker, R., Hansbo, P. and Stenberg, R. (2003), "A finite element method for domain decomposition with non-matching grids", Math. Modell. Numer. Analy., 37(2), 209-225. https://doi.org/10.1051/m2an:2003023
  5. Belgacem, F.B., Hild, P. and Laborde, P. (1999), "Extension of the mortar finite element method to a variational inequality modeling unilateral contact", Math. Mod. Meth. Appl. Sci., 9(2), 287-303. https://doi.org/10.1142/S0218202599000154
  6. Bernardi, C., Maday, Y. and Patera, A.T. (1992), "A new nonconforming approach to domain decomposition: The mortar element method", Nonlin. Part. Different. Equat. Their Appl., 13-51.
  7. Bernardi, C., Rebello, C., Vera, C. and Coronil, C. (2009), "A posteriori error analysis for two nonoverlapping domain decomposition techniques", Comput. Meth. Appl. Mech. Eng., 59(6), 1214-1236.
  8. Bernardi, C., Rebello, T.C. and Vera, E.C. (2008), "A FETI method with a mesh independent condition number for the iteration matrix", Comput. Meth. Appl. Mech. Eng., 197(13-16), 1410-1429. https://doi.org/10.1016/j.cma.2007.11.019
  9. Bitencourt, L.A.G., Jr, Manzoli, O.L., Prazeres, P.G.C., Rodrigues, E.A. and Bittencourt, T.N. (2015), "A coupling technique for non-matching finite element meshes", Comput. Meth. Appl. Mech. Eng., 290, 19-44. https://doi.org/10.1016/j.cma.2015.02.025
  10. Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics.
  11. Di Pietro, D.A. and Nicaise, S. (2013), "A locking-free discontinuous Galerkin method for linear elasticity in locally nearly incompressible heterogeneous media", Appl. Numer. Math., 63, 105-116. https://doi.org/10.1016/j.apnum.2012.09.009
  12. Dickopf, T. and Krause, R. (2009), "Efficient simulation of multi-body contact problems on complex geometries: A flexible decomposition approach using constrained minimization", Int. J. Numer. Meth. Eng., 77(13), 1834-1862. https://doi.org/10.1002/nme.2481
  13. Dolbow, J.E. and Harari, I. (2009), "An efficient finite element method for embedded interface problems", Int. J. Numer. Meth. Eng., 78(2), 229-252. https://doi.org/10.1002/nme.2486
  14. Farhat, C. (1991), "A method for finite element tearing and interconnecting and its parallel solution algorithm", Int. J. Numer. Meth. Eng., 32(6), 1205-1227. https://doi.org/10.1002/nme.1620320604
  15. Farhat, C., Lacour, C. and Rixen, D. (2007), "Incorporation of linear multipoint constraints in substructure based iterative solvers. Part I: A numerically scalable algorithm", Int. J. Numer. Meth. Eng., 43(6), 997-1016. https://doi.org/10.1002/(SICI)1097-0207(19981130)43:6<997::AID-NME455>3.0.CO;2-B
  16. Fischer, K.A. and Wriggers, P. (2005), "Frictionless 2D contact formulations for finite deformations based on the mortar method", Comput. Mech., 36(3), 226-244. https://doi.org/10.1007/s00466-005-0660-y
  17. Grieshaber, B., McBride, A. and Reddy, B. (2015), "Uniformly convergent interior penalty methods using multilinear approximations for problems in elasticity", SIAM J. Numer. Analy., 53(5), 2255-2278. https://doi.org/10.1137/140966253
  18. Haikal, G. and Hjelmstad, K.D. (2010), "An enriched discontinuous Galerkin formulation for the coupling of non-conforming meshes", Fin. Elem. Analy. Des., 46(6), 496-503. https://doi.org/10.1016/j.finel.2009.12.008
  19. Haikal, G. (2009), "A stabilized finite element formulation of non-smooth contact", Ph.D. Dissertation, University of Illinois-Urbana Champaign, Illinois, U.S.A.
  20. Hansbo, A. and Hansbo, P. (2002), "An unfitted finite element method, based on Nitsche's method, for elliptic interface problems", Comput. Meth. Appl. Mech. Eng., 191(47-48), 5537-5552. https://doi.org/10.1016/S0045-7825(02)00524-8
  21. Hansbo, P. and Larsson, F. (2016) "The nonconforming linear strain tetrahedron for a large deformation elasticity problem", Computat. Mech., 58(6), 929-935. https://doi.org/10.1007/s00466-016-1323-x
  22. Hansbo, P., Lovadina, C., Perugia, I. and Angalli, G. (2005), "A lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes", Numeris. Mathemat., 100(1), 91-115. https://doi.org/10.1007/s00211-005-0587-4
  23. Jin, S., Sohn, D., Lim, J.H. and Im, S. (2015), "A node-to-node scheme with the aid of variable-node elements for elasto-plastic contact analysis", Int. J. Numer. Meth. Eng., 102(12), 1761-1783. https://doi.org/10.1002/nme.4862
  24. Kim, H.G. (2002), "Interface element method (IEM) for a partitioned system with non-matching interfaces", Comput. Meth. Appl. Mech. Eng., 191(29-30), 3165-3194. https://doi.org/10.1016/S0045-7825(02)00255-4
  25. Kim, H.G. (2003), "Interface element method: Treatment of non-matching nodes at the ends of interfaces between partitioned domains", Comput. Meth. Appl. Mech. Eng., 192(15), 1841-1858. https://doi.org/10.1016/S0045-7825(03)00205-6
  26. Laurie, D. (1997), "Calculation of gauss-kronrod quadrature rules", Math. Comput. Am. Math. Soc., 66(219), 1133-1145. https://doi.org/10.1090/S0025-5718-97-00861-2
  27. Le Tallec, P. and Sassi, T. (1995) "Domain decomposition with nonmatching grids: Augmented lagrangian approach", Math. Comput., 64(212), 1367-1396. https://doi.org/10.1090/S0025-5718-1995-1308457-5
  28. Liu, R., Wheeler, M.F. and Yotov, I. (2013), "On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity", Comput. Meth. Appl. Mech. Eng., 253, 219-236. https://doi.org/10.1016/j.cma.2012.07.015
  29. Liu, R., Wheeler, M. and Dawson, C. (2009), "A three-dimensional nodal-based implementation of a family of discontinuous Galerkin methods for elasticity problems", Comput. Struct., 87(3-4), 141-150. https://doi.org/10.1016/j.compstruc.2008.11.009
  30. Lloberas-Valls, O., Cafiero, M., Cante, J., Ferrer, A. and Oliver, J. (2017), "The domain interface method in non-conforming domain decomposition multifield problems", Comput. Mech., 59(4), 579-610. https://doi.org/10.1007/s00466-016-1361-4
  31. Masud, A., Truster, T. and Bergman, L.A. (2012), "A unified formulation for interface coupling and frictional contact modeling with embedded error estimation", Int. J. Numer. Meth. Eng., 92(2), 144-177.
  32. Montero, J. and Haikal, G. (2018), "Modeling beam-solid finite element interfaces: A stabilized formulation for contact and coupled systems", Int. J. Appl. Mech., Imper. Colleg. Press, 10(9), 1850095.
  33. Nitsche, J. (1971), "U ber ein variationsprinzip zur losung von dirichlet problemen bei verwendung von teilraumen, die keinen randbedingungen unterworfen sind", Abhandlungen in der Mathematik an der Universitat Hamburg, 36(1), 9-15. https://doi.org/10.1007/BF02995904
  34. Popp, A. and Wall, W.A. (2014), "Dual mortar methods for computational contact mechanics-overview and recent developments", GAMM-Mitteilungen, 37(1), 66-84. https://doi.org/10.1002/gamm.201410004
  35. Puso, M.A. and Laursen, T.A. (2004), "A mortar segment-to-segment contact method for large deformation solid mechanics", Comput. Meth. Appl. Mech. Eng., 193(6-8), 601-629. https://doi.org/10.1016/j.cma.2003.10.010
  36. Riviere, B., Wheeler, M.F. and Girault, V. (1999), "Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems I", Comput. Geosci., 3(3-4), 337-360. https://doi.org/10.1023/A:1011591328604
  37. Sanders, J., Dolbow, J.E. and Laursen, T.A. (2009), "On methods for stabilizing constraints over enriched interfaces in elasticity", Int. J. Numer. Meth. Eng., 78(9), 1009-1036. https://doi.org/10.1002/nme.2514
  38. Simo, J.C. and Hughes, T.J.R. (1998), Computational Inelasticity, Springer.
  39. Solberg, J.M. and Papadopoulos, P. (2005), "An analysis of dual formulations for the finite element solution of two-body contact problems", Comput. Meth. Appl. Mech. Eng., 194(25-26), 2734-2780. https://doi.org/10.1016/j.cma.2004.06.045
  40. Solberg, J.M., Jones, R.E. and Papadopoulos, P. (2007), "A family of simple two-pass dual formulations for the finite element solution of contact problems", Comput. Meth. Appl. Mech. Eng., 196(4-6), 782-802. https://doi.org/10.1016/j.cma.2006.05.011
  41. Wihler, T. (2006), "Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems", Math. Comput., 75(255), 1087-1102. https://doi.org/10.1090/S0025-5718-06-01815-1
  42. Wohlmuth, B.I. (2000), "A mortar finite element method using dual spaces for the lagrange multiplier", SIAM J. Numer. Analy., 38(3), 989-1012. https://doi.org/10.1137/S0036142999350929
  43. Wriggers, P., Rust, W.T. and Reddy, B.D. (2016), "A virtual element method for contact", Comput. Mech., 58(6), 1039-1050. https://doi.org/10.1007/s00466-016-1331-x
  44. Yang, B., Laursen, T.A. and Xiaonong, M. (2005), "Two-dimensional mortar contact methods for large deformation frictional sliding", Int. J. Numer. Meth. Eng., 62(9), 1183-1225. https://doi.org/10.1002/nme.1222