Interface coupling method for the global–local analysis of heterogeneous models: A second-order homogenization-based strategy

Abstract This paper proposes a coupling strategy between a homogeneous macroscopic description of a structure and a description at a lower scale of some local details. In order to facilitate its implementation in industrial software, an interface coupling technique using a separation between micro and macro quantities of the unknowns is proposed. The approach leads to local solutions equivalent to those that would be obtained by second-order homogenization techniques. Its interest is that it does not require neither the construction of a second-order homogenized model nor a localization stage. The approach is validated against direct numerical simulations at microscopic scale on periodic structures under uniform and linear macroscopic strains loads.

[1]  W. Brekelmans,et al.  Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling , 1998 .

[2]  Alberto Salvadori,et al.  A multiscale framework for localizing microstructures towards the onset of macroscopic discontinuity , 2014 .

[3]  Sergio Oller,et al.  Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method , 2016, Archives of Computational Methods in Engineering.

[4]  V. Kouznetsova,et al.  Multi‐scale constitutive modelling of heterogeneous materials with a gradient‐enhanced computational homogenization scheme , 2002 .

[5]  Laurent Champaney,et al.  A multiscale extended finite element method for crack propagation , 2008 .

[6]  F. Brezzi.  On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[7]  David Dureisseix,et al.  A micro–macro and parallel computational strategy for highly heterogeneous structures , 2001 .

[8]  M. Guerich,et al.  A multi‐scale modeling method for heterogeneous structures without scale separation using a filter‐based homogenization scheme , 2016 .

[9]  Iwona M Jasiuk,et al.  Couple-stress moduli and characteristics length of a two-phase composite , 1999 .

[10]  J. Schröder,et al.  Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials , 1999 .

[11]  Olivier Allix,et al.  Nonintrusive coupling of 3D and 2D laminated composite models based on finite element 3D recovery , 2014, 1501.01933.

[12]  Francesca Rapetti,et al.  Basics and some applications of the mortar element method , 2005 .

[13]  J. Yvonnet,et al.  Nonlocal/coarse graining homogenization of linear elastic media with non-separated scales using least-square polynomial filters , 2014 .

[14]  O. Allix,et al.  Non-intrusive and exact global/local techniques for structural problems with local plasticity , 2009 .

[15]  S. Forest.  Mechanics of generalized continua: construction by homogenizaton , 1998 .

[16]  Barbara Wohlmuth,et al.  Isogeometric mortar methods , 2014, 1407.8313.

[17]  N. Kikuchi,et al.  A class of general algorithms for multi-scale analyses of heterogeneous media , 2001 .

[18]  Stein K. F. Stoter,et al.  The non-symmetric Nitsche method for the parameter-free imposition of weak boundary and coupling conditions in immersed finite elements , 2016 .

[19]  J. Willis,et al.  Variational and Related Methods for the Overall Properties of Composites , 1981 .

[20]  Sven Klinkel,et al.  The weak substitution method – an application of the mortar method for patch coupling in NURBS‐based isogeometric analysis , 2015 .

[21]  J. Oden,et al.  Analysis and adaptive modeling of highly heterogeneous elastic structures , 1997 .

[22]  Olivier Allix,et al.  On the computation of plate assemblies using realistic 3D joint model: a non-intrusive approach , 2016, Adv. Model. Simul. Eng. Sci..

[23]  Mickaël Duval,et al.  Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition , 2016 .

[24]  Robin Bouclier,et al.  Development of a new, more regular, mortar method for the coupling of NURBS subdomains within a NURBS patch: Application to a non-intrusive local enrichment of NURBS patches , 2017 .

[25]  V Varvara Kouznetsova,et al.  Computational homogenization for the multi-scale analysis of multi-phase materials , 2002 .

[26]  D. Jeulin,et al.  Determination of the size of the representative volume element for random composites: statistical and numerical approach , 2003 .

[27]  N. Kikuchi,et al.  Simulation of the multi-scale convergence in computational homogenization approaches , 2000 .

[28]  V. Kouznetsova,et al.  Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy , 2004 .

[29]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[30]  F. Feyel.  A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua , 2003 .

[31]  Y. Maday,et al.  Two different approaches for matching nonconforming grids: The Mortar Element method and the Feti Method , 1997 .

[32]  John E. Dolbow,et al.  A robust Nitsche’s formulation for interface problems , 2012 .

[33]  Mgd Marc Geers,et al.  Computational homogenization for heterogeneous thin sheets , 2010 .

[34]  Olivier Allix,et al.  On a multiscale strategy and its optimization for the simulation of combined delamination and buckling , 2012, 1208.6495.

[35]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[36]  Julien Yvonnet,et al.  A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales , 2014 .

[37]  Jean-Charles Passieux,et al.  A Nitsche-based non-intrusive coupling strategy for global/local isogeometric structural analysis , 2018, Computer Methods in Applied Mechanics and Engineering.

[38]  P. Ladevèze,et al.  A multiscale computational approach for contact problems , 2002 .

[39]  Fpt Frank Baaijens,et al.  An approach to micro-macro modeling of heterogeneous materials , 2001 .

[40]  V. Kouznetsova,et al.  Size of a representative volume element in a second-order computational homogenization framework , 2004 .

[41]  M. Fortin,et al.  Approximation of Saddle Point Problems , 1991 .

[42]  Faker Ben Belgacem,et al.  The Mortar finite element method with Lagrange multipliers , 1999, Numerische Mathematik.

[43]  Isaac Harari,et al.  An efficient finite element method for embedded interface problems , 2009 .

[44]  J. Willis,et al.  The overall elastic response of composite materials , 1983 .

[45]  Chris J. Pearce,et al.  Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization , 2008 .

[46]  Robin Bouclier,et al.  Multiscale analysis of complex aeronautical structures using robust non-intrusive coupling , 2018, Advanced Modeling and Simulation in Engineering Sciences.

[47]  W. Brekelmans,et al.  Overall behaviour of heterogeneous elastoviscoplastic materials: effect of microstructural modelling , 2000 .

[48]  Vinh Phu Nguyen,et al.  Nitsche’s method for two and three dimensional NURBS patch coupling , 2013, 1308.0802.