“NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering 260, 2013, DOI: 10.1016/j.cma.2013.03.024
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[en] When considering problems of dimensions close to the characteristic length of the material, the size e ects can not be neglected and the classical (so–called first–order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second–order multiscale computational homogenization (SMCH) scheme. This second–order scheme uses the classical continuum at the micro–scale while considering second–order continuum at the macro–scale. Although the theoretical background of the second–order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high–order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro–scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak formulations allowing for inter-element discontinuities either at the C0 level or at the C1 level, and it can thus be used to constrain weakly the C1 continuity at the macro–scale. The C0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C0 displacement–based finite elements. Therefore, two formulations can be used at the macro–scale: (i) the full–discontinuous Galerkin formulation (FDG) with weak C0 and C1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high–order term enrichment into the conventional C0 finite element framework. The micro–problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non–linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and e cient way.
ARC 09/14-02 BRIDGING - From imaging to geometrical modelling of complex micro structured materials: Bridging computational engineering and material science
Funders :
Communauté française de Belgique : Direction Générale de l'Enseignement Non Obligatoire et de la Recherche Scientifique - DGENORS CÉCI - Consortium des Équipements de Calcul Intensif [BE]
Michel J., Moulinec H., Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Comput. Methods Appl. Mech. Engrg. 1999, 172(1-4):109-143. http://www.sciencedirect.com/science/article/pii/S0045782598002278, 10.1016/S0045-7825(98)00227-8.
Terada K., Hori M., Kyoya T., Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 2000, 37(16):2285-2311. http://www.sciencedirect.com/science/article/B6VJS-3YDFYCH-4/2/788d085d620c6c5ae804d8c4e4f8ae9b, 10.1016/S0020-7683(98)00341-2.
Miehe C. Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int. J. Numer. Methods Engrg. 2002, 55(11):1285-1322. http://dx.doi.org/10.1002/nme.515.
Miehe C., Koch A. Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch. Appl. Mech. 2002, 72(4):300-317. http://dx.doi.org/10.1007/s00419-002-0212-2, 10.1007/s00419-002-0212-2.
Kouznetsova V., Brekelmans W.A.M., Baaijens F.P.T. An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 2001, 27(1):37-48. http://dx.doi.org/10.1007/s004660000212.
Kouznetsova V., Geers M.G.D., Brekelmans W.A.M. Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int. J. Numer. Methods Engrg. 2002, 54(8):1235-1260. http://dx.doi.org/10.1002/nme.541.
Kouznetsova V.G., Geers M.G.D., Brekelmans W.A.M. Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput. Methods Appl. Mech. Engrg. 2004, 193(48-51):5525-5550. (Advances in Computational Plasticity). http://www.sciencedirect.com/science/article/B6V29-4D4D4JV-3/2/6fd8ac06299b9a26d8a17438871de868, 10.1016/j.cma.2003.12.073.
Kaczmarczyk L., Pearce C.J., Bicanic N. Scale transition and enforcement of rve boundary conditions in second-order computational homogenization. Int. J. Numer. Methods Engrg. 2008, 74(3):506-522. http://dx.doi.org/10.1002/nme.2188.
D. Peric, E.A. de Souza Neto, R.A. Feijóo, M. Partovi, A.J.C. Molina, On micro-to-macro transitions for multi-scale analysis of non-linear heterogeneous materials: unified variational basis and finite element implementation, Int. J. Numer. Meth. Engng. 87 (1-5) (2011) 149-170. http://dx.doi.org/10.1002/nme.3014.
Geers M.G.D., Kouznetsova V.G., Brekelmans W.A.M. Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. 2010, 234(7):2175-2182. (Fourth International Conference on Advanced Computational Methods in ENgineering (ACOMEN 2008)). http://www.sciencedirect.com/science/article/B6TYH-4X1J73B-8/2/ee8d9b69133503eaf14b00ddbe1bd8f5, 10.1016/j.cam.2009.08.077.
V.G. Kouznetsova, Computational homogenization for the multi-scale analysis of multi-phase materials, Ph.D. thesis, Technische Universiteit Eindhoven, 2002.
McBride A., Mergheim J., Javili A., Steinmann P., Bargmann S. Micro-to-macro transitions for heterogeneous material layers accounting for in-plane stretch. J. Mech. Phys. Solids 2012, 60(6):1221-1239. http://www.sciencedirect.com/science/article/pii/S0022509612000117, 10.1016/j.jmps.2012.01.003.
Forest S., Barbe F., Cailletaud G. Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials. Int. J. Solids Struct. 2000, 37(46-47):7105-7126. http://www.sciencedirect.com/science/article/pii/S0020768399003303, 10.1016/S0020-7683(99)00330-3.
Mindlin R.D. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 1964, 16:51-78.
Mindlin R. Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1965, 1:417-438. http://isn-csm.mit.edu/literature/1965-IJSS-Mindlin.pdf.
Fleck N., Hutchinson J. Strain gradient plasticity. Advances in Applied Mechanics 1997, vol. 33. Elsevier, pp. 295-361. http://www.sciencedirect.com/science/article/pii/S0065215608703880, 10.1016/S0065-2156(08)70388-0.
Peerlings R., de Borst R., Brekelmans W., Ayyapureddi S. Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Engrg. 1996, 39:3391-3403.
Hirschberger C., Kuhl E., Steinmann P. On deformational and configurational mechanics of micromorphic hyperelasticity - theory and computation. Comput. Methods. Appl. Mech. Engrg. 2007, 196:4027-4044.
Papanicolopulos S.-A., Zervos A. A method for creating a class of triangular c1 finite elements. Int. J. Numer. Methods Engrg. 2012, 89(11):1437-1450. http://dx.doi.org/10.1002/nme.3296, 10.1002/nme.3296.
Papanicolopulos S.-A., Zervos A., Vardoulakis I. A three-dimensional c1 finite element for gradient elasticity. Int. J. Numer. Methods Engrg. 2009, 77(10):1396-1415. http://dx.doi.org/10.1002/nme.2449, 10.1002/nme.2449.
Amanatidou E., Aravas N. Mixed finite element formulations of strain-gradient elasticity problems. Comput. Methods Appl. Mech. Engrg. 2002, 191(15-16):1723-1751. http://www.sciencedirect.com/science/article/pii/S004578250100353X, 10.1016/S0045-7825(01)00353-X.
Shu J.Y., King W.E., Fleck N.A. Finite elements for materials with strain gradient effects. Int. J. Numer Methods Engrg. 1999, 44(3):373-391. (eng.). http://www.refdoc.fr/Detailnotice?idarticle=11668682.
Engel G., Garikipati K., Hughes T., Larson M., Mazzei L., Taylor R. Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg. 2002, 191(34):3669-3750. http://www.sciencedirect.com/science/article/pii/S0045782502002864, 10.1016/S0045-7825(02)00286-4.
R. Bala Chandran, Development of discontinuous Galerkin method for nonlocal linear elasticity, Master thesis, Massachusetts Institute of Technology, 2007.. http://dspace.mit.edu/handle/1721.1/41730.
Noels L., Radovitzky R. A general discontinuous Galerkin method for finite hyperelasticity. Formulation and numerical applications. Int. J. Numer. Methods Engrg. 2006, 68(1):64-97. http://dx.doi.org/10.1002/nme.1699, 10.1002/nme.1699.
Ten Eyck A., Lew A. Discontinuous Galerkin methods for non-linear elasticity. Int. J. Numer. Methods Engrg. 2006, 67(9):1204-1243. http://dx.doi.org/10.1002/nme.1667, 10.1002/nme.1667.
Hansbo P., Larson M. A discontinuous Galerkin method for the plate equation. Calcolo 2002, 39:41-59. http://dx.doi.org/10.1007/s100920200001, 10.1007/s100920200001.
Noels L., Radovitzky R. A new discontinuous Galerkin method for kirchhofflove shells. Comput. Methods Appl. Mech. Engrg. 2008, 197(33-40):2901-2929. http://www.sciencedirect.com/science/article/pii/S0045782508000376, 10.1016/j.cma.2008.01.018.
Wells G.N., Garikipati K., Molari L. A discontinuous Galerkin formulation for a strain gradient-dependent damage model. Comput. Methods Appl. Mech. Engrg. 2004, 193(33-35):3633-3645. http://www.sciencedirect.com/science/article/pii/S0045782504000921, 10.1016/j.cma.2004.01.020.
Djoko J., Ebobisse F., McBride A., Reddy B. A discontinuous Galerkin formulation for classical and gradient plasticity part 1: formulation and analysis. Comput. Methods Appl. Mech. Engrg. 2007, 37-40:3881-3897. (Special Issue Honoring the 80th Birthday of Professor Ivo Babuska). http://www.sciencedirect.com/science/article/pii/S0045782507001144, 10.1016/j.cma.2006.10.045.
McBride A., Reddy B. A discontinuous Galerkin formulation of a model of gradient plasticity at finite strains. Comput. Methods Appl. Mech. Engrg. 2009, 198(21-26):1805-1820. (Advances in Simulation-Based Engineering Sciences, Honoring J. Tinsley Oden). http://www.sciencedirect.com/science/article/pii/S0045782509000152, 10.1016/j.cma.2008.12.034.
Abdulle A. Multiscale method based on discontinuous Galerkin methods for homogenization problems. Comptes Rendus Mathematique 2008, 346(1-2):97-102. http://www.sciencedirect.com/science/article/pii/S1631073X07005158, 10.1016/j.crma.2007.11.029.
Abdulle A. Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales. Math. Comp. 2012, 81:687-713. http://dx.doi.org/10.1090/S0025-5718-2011-02527-5.
Becker G., Geuzaine C., Noels L. A one field full discontinuous Galerkin method for kirchhofflove shells applied to fracture mechanics. Comput. Methods Appl. Mech. Engrg. 2011, 200(45-46):3223-3241. http://www.sciencedirect.com/science/article/pii/S0045782511002490, 10.1016/j.cma.2011.07.008.
L. Wu, D. Tjahjanto, G. Becker, A. Makradi, A. Jérusalem, L. Noels, A micro-meso-model of intra-laminar fracture in fiber-reinforced composites based on a discontinuous Galerkin/cohesive zone method, Engineering Fracture Mechanics. doi:10.1016/j.engfracmech.2013.03.018.
Nguyen V.-D., Béchet E., Geuzaine C., Noels L. Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation. Comput. Mater. Sci. 2012, 55:390-406. http://www.sciencedirect.com/science/article/pii/S0927025611005866, 10.1016/j.commatsci.2011.10.017.
Geuzaine C., Remacle J.-F. Gmsh: A 3-d finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Engrg. 2009, 79(11):1309-1331. http://dx.doi.org/10.1002/nme.2579.
Mark Ainsworth Essential boundary conditions and multi-point constraints in finite element analysis. Comput. Methods Appl. Mech. Engrg. 2001, 190(48):6323-6339. http://www.sciencedirect.com/science/article/pii/S0045782501002365, 10.1016/S0045-7825(01)00236-5.
Zervos A. Finite elements for elasticity with microstructure and gradient elasticity. Int. J. Numer. Methods Engrg. 2008, 73(4):564-595. http://dx.doi.org/10.1002/nme.2093, 10.1002/nme.2093.