Nonlinear multi‐scale homogenization with different structural models at different scales

Abstract

We present an extension of the computational homogenisation theory to cases where different structural models are used at different scales and no energy potential can be defined at the small scale. We observe that volumetric averaging, that is not applicable in such cases unless similarities exist in the macro-and micro-scale models, is not a necessary prerequisite to carry out computational homogenisation. At each material point of the macro-model we replace the conventional representative volume element with a representative domain element (RDE). To link the large-and small-scale problems we then introduce a linear operator, mapping the smooth part of the small-scale displacement field of each RDE to the large-scale strain field, and a trace operator to impose boundary conditions in the RDE. The latter is defined based on engineering judgement, analogously to the conventional theory.A generalised Hill's condition, rather than being invoked, is derived from duality principles and is used to recover the stress measures at the large scale. For the implementation in a nonlinear finite-element analysis 'control nodes' and constraint equations are used. The effectiveness of the procedure is demonstrated for three beam-to-truss example problems, for which multi-scale convergence is numerically analysed.

An accurate and computationally efficient small-scale nonlinear FEA of flexible risers

“…This means that different structural models are used at different scales, which makes classical computational homogenisation, e.g. Geers et al (2010), not directly applicable as shown in detail by Edmans et al (2013), where an extension of that theory has been formulated. Referring to the original paper for the details of the derivation, a geometrically non-linear formulation is assumed at the large scale: in particular, it is assumed that displacements and rotations are large, while macro strains are small enough so that a geometrically linear formulation can be adopted at the small scale.…”

“…Therefore, denoting by N c the number of nodes on either end cross section,N c pairs of nodes are defined by this correspondence. Following (Edmans et al, 2013;Rahmati et al, 2016), for each one of these pairs of nodes, a new 'dummy' projected node is introduced on a plane which is parallel to the end cross section in their un-deformed configuration. This is shown in Fig.…”

“…where X n P indicates the position vector of the projected node n P with respect to the reference point and × denotes the standard cross product of two vectors. In this way, displacements and rotations of the reference point correspond to generalised strain components in the model (Edmans et al, 2013;Rahmati et al, 2016). For example, prescribing a rotation ϕ R P of the reference point about one axis in the cross section is equivalent to prescribing periodic boundary conditions corresponding to a bending curvature equal to ϕ L / R P , L being the length of the model.…”

“…cross section) of the large-scale beam model, the stress resultants corresponding to assigned generalised strains are determined through the solution of the small-scale FE problem. This requires recasting the computational homogenisation problem in a more general theory which can link different structural models at different scales (Edmans, 2012;Edmans et al, 2013).…”

“…It also explains how cyclic symmetry of the riser can be exploited by writing periodic boundary conditions based on the multiscale theory derived by Edmans et al (2013) for the case when different structural models are used at different scales. This results in periodic boundary conditions, written in terms of dummy nodes, which are different from those used in the aforementioned work by Leroy et al (2010).…”