Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials

Ondřej Rokoš,Petr Krysl,Martin Doškář,Jan Zeman

doi:10.1186/s40323-020-00152-7

Abstract

Exotic behaviour of mechanical metamaterials often relies on an internal transformation of the underlying microstructure triggered by its local instabilities, rearrangements, and rotations. Depending on the presence and magnitude of such a transformation, effective properties of a metamaterial may change significantly. To capture this phenomenon accurately and efficiently, homogenization schemes are required that reflect microstructural as well as macro-structural instabilities, large deformations, and non-local effects. To this end, a micromorphic computational homogenization scheme has recently been developed, which employs the particular microstructural transformation as a non-local mechanism, magnitude of which is governed by an additional coupled partial differential equation. Upon discretizing the resulting problem it turns out that the macroscopic stiffness matrix requires integration of macro-element basis functions as well as their derivatives, thus calling for higher-order integration rules. Because evaluation of a constitutive law in multiscale schemes involves an expensive solution of a non-linear boundary value problem, computational efficiency of the micromorphic scheme can be improved by reducing the number of integration points. Therefore, the goal of this paper is to investigate reduced-order schemes in computational homogenization, with emphasis on the stability of the resulting elements. In particular, arguments for lowering the order of integration from expensive mass-matrix to a cheaper stiffness-matrix equivalent are outlined first. An efficient one-point integration quadrilateral element is then introduced and a proper hourglass stabilization is discussed. Performance of the resulting set of elements is finally tested on a benchmark bending example, showing that we achieve accuracy comparable to the full quadrature rules, whereas computational cost decreases proportionally to the reduction in the number of quadrature points used.

Highlights

Mechanical metamaterials have recently received a great amount of attention in the engineering literature, aiming at applications ranging from acoustics to soft robotics [1,2,3,4]
We demonstrate that a one-integration point quadrilateral, which suffers from the well-known hourglassing, see e.g. [14] and Fig. 1b, can be enhanced with standard stabilization procedures and used in micromorphic computational homogenization
It has been shown that the micromorphic part of the stiffness matrix necessitates a mass-matrix equivalent integration rule, this requirement can be relaxed with only a negligible drop in accuracy and no artificial spurious modes

Summary

Introduction

Mechanical metamaterials have recently received a great amount of attention in the engineering literature, aiming at applications ranging from acoustics to soft robotics [1,2,3,4]. One of the available options is computational homogenization [6], which replaces the effective constitutive behaviour of the macroscopic material with a mechanical system, specified by a Representative Volume Element (RVE). Advantages of such an approach are that all microstructural features are taken into account, including microstructural morphology, non-linear material behaviour, or local microstructural buckling. The macroscopic continuum is treated as local, i.e. it ignores any communication among neighbouring points (a consequence of the assumption of the formulation on separation of scales) To alleviate this limitation, various enriched theories have been proposed in the literature, including higher-order and micromorphic computational homogenization schemes, see e.g. Multiple approaches can be used to reduce the computational effort, including reduced-order modelling at the level of each RVE or considering equivalent surrogate models, see e.g. [10,11,12]

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