A numerical two-scale homogenization scheme: the FE2-method

Abstract

A wide class of micro-heterogeneous materials is designed to satisfy the advanced challenges of modern materials occurring in a variety of technical applications. The effective macroscopic properties of such materials are governed by the complex interaction of the individual constituents of the associated microstructure. A sufficient macroscopic phenomenological description of these materials up to a certain order of accuracy can be very complicated or even impossible. On the contrary, a whole resolution of the fine scale for the macroscopic boundary value problem by means of a classical discretization technique seems to be too elaborate. Instead of developing a macroscopic phenomenological constitutive law, it is possible to attach a representative volume element (RVE ) of the microstructure at each point of the macrostructure; this results in a two-scale modeling scheme. A discrete version of this scheme performing finite element (FE) discretizations of the boundary value problems on both scales, the macro-and the micro-scale, is denoted as the FE 2 -method or as the multilevel finite element method. The main advantage of this procedure is based on the fact that we do not have to define a macroscopic phenomenological constitutive law; this is replaced by suitable averages of stress measures and deformation tensors over the microstructure. Details concerning the definition of the macroscopic quantities in terms of their microscopic counterparts, the definition/construction of boundary conditions on the RVE as well as the consistent linearization of the macroscopic constitutive equations are discussed in this contribution. Furthermore, remarks concerning stability problems on both scales as well as their interactions are given and representative numerical examples for elasto-plastic microstructures are discussed.

Modeling of granular solids with computational homogenization: Comparison with Biot׳s theory

“…To include in the model size-effects and highly microstructural heterogeneities, second order terms were introduced by Kouznetsova et al [23,24] in which a length scale was derived from micromechanical computations by adding the gradient of the macroscopic deformation tensor to the boundary conditions of the REV (see also [25] and [26]). A comprehensive review about the computational homogenization method for monophasic materials can be found in Schröder [27].…”

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

“…This idea has been further developed in Refs. [222] and [352][353][354][355][356][357][358][359][360][361][362][363]. Mo€ es et al [364] presented an extended version of the classical finite element method, referred to as XFEM, to solve microproblems involving complex geometries [365].…”

“…See Refs. [363] and [414] for further discussions about computation of macro Piola tangent in the discrete formulation. Once M A is calculated, all the ingredients to perform the full FE 2 simulation are provided.…”

The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE2 method

“…The FE 2 method has its origins in solid mechanics, [31], [46], [47], [48], [28], [27], [55], and has found considerable interest in academia and industry; as a versatile method FE 2 has been used in non-linear problems of elasticity and inelasticity. For recent, comprehensive overviews of the FE 2 method we refer to [30], [61] and [56]. In order to account for size-dependency observed in materials science, Kouznetsova et al [41], [42] have introduced a second-order homogenization into FE 2 .…”