Reduced integration schemes in micromorphic computational homogenization of elastomeric mechanical metamaterials

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 a higher-order integration rules. Because evaluation of constitutive law in multiscale schemes involves an expensive solution of a non-linear boundary value problem, computational efficiency 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 the expensive mass-matrix to a cheaper stiffness-matrix equivalent are first outlined. An efficient one-point integration quadrilateral element is then introduced and proper hourglass stabilization 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.

arXiv comment21 pages, 8 figures, 3 tables, abstract shortened to fulfill 1920 character limit, small changes after review

References in corpus (3)

• Micromorphic Computational Homogenization for Mechanical Metamaterials with Patterning Fluctuation Fields
• A Newton Solver for Micromorphic Computational Homogenization Enabling Multiscale Buckling Analysis of Pattern-Transforming Metamaterials
• Extended Micromorphic Computational Homogenization for Mechanical Metamaterials Exhibiting Multiple Geometric Pattern Transformations

Cited by in corpus (3)

• A Newton Solver for Micromorphic Computational Homogenization Enabling Multiscale Buckling Analysis of Pattern-Transforming Metamaterials
• A computational approach to identify the material parameters of the relaxed micromorphic model
• A Comparative Study of Enriched Computational Homogenization Schemes Applied to Two-Dimensional Pattern-Transforming Elastomeric Mechanical Metamaterials