Computational continua revisited

Abstract

SUMMARYIn the recent paper, Fish and Kuznetsov introduced the so-called computational continua .C 2 / approach, which is a variant of the higher order computational homogenization that does not require higher order continuity, introduces no new degrees of freedom, and is free of higher order boundary conditions. In a follow-up paper on reduced order computational continua, the C 2 formulation has been enhanced with a model reduction scheme based on construction of residual-free fields to yield a computationally efficient framework coined as RC 2 . The original C 2 formulations were limited to rectangular and box elements. The primary objectives of the present manuscript is to revisit the original formulation in three respects: (i) consistent formulation of boundary conditions for unit cells subjected to higher order coarse scale fields, (ii) effective solution of the unit cell problem for lower order approximation of eigenstrains, and (iii) development of nonlocal quadrature schemes for general two-dimensional (quad and triangle) and three-dimensional (hexahedral and tetrahedral) elements.

Dispersive computational continua

“…One of the salient features of the C 2 formulation is nonlocal quadrature [36,35,37], which is a variant of the Gauss quadrature aimed at accounting for unit cells of a finite size comparable with a characteristic coarse-scale element size. By this approach, integrals over a composite domain   in (3) …”

“…In some cases, as will be discussed later, we will employ reduced integration. The weights aI v and locations ˆa I  of either two or three quadrature points in the parent element domain are given by [37,36,35]   …”

“…The primary objectives of the present study are to extend the computational continua (C 2 ) formulation [35,36,37] to a high-frequency transient regime and analyze its dispersive properties. The C 2 formulation has in principle the potential to capture dispersive behavior, since, unlike various homogenization theories, it does not explicitly assume scale separation and at the same time considers unit cells of a finite size.…”

“…Studies of nonlinear dispersive formulations are rare (see [23,25,38,39,40,41]). Although the focus of the present manuscript is on dispersion analysis of linear elastic materials, the C 2 approach has been extensively studied for nonlinear steady state long-wavelength problems [35][36][37].…”

Homogenization Methods and Multiscale Modeling: Nonlinear Problems

“…The homogenization allows to extract the Cauchy stress tensor, along with higher order stress tensor and all accompanying tangents (Geers et al, 2001(Geers et al, , 2003Kouznetsova et al, 2002Kouznetsova et al, , 2004aKouznetsova, 2002;Kaczmarczyk et al, 2008Kaczmarczyk et al, , 2010Bacigalupo and Gambarotta, 2011). The computational continua approach (Fish and Kuznetsov, 2010;Fish et al, 2015) is a variant that relaxes the constraints on higher order continuity and boundary conditions.…”