Computational continua

We develop a new coarse-scale continuum formulation hereafter referred to as computational continua. By this approach, the coarse-scale governing equations are stated on a so-called computational continua domain consisting of disjoint union of computational unit cells, positions of which are determined to reproduce the weak form of the governing equations on the fine scale. The label 'computational' is conceived from both theoretical and computational considerations. From a theoretical point of view, the computational continua is endowed with fine-scale details; it introduces no scale separation; and makes no assumption about infinitesimality of the fine-scale structure. From computational point of view, the computational continua does not require higher-order continuity; introduces no new degrees-of-freedom; and is free of higher-order boundary conditions. The proposed continuum description features two building blocks: the nonlocal quadrature scheme and the coarse-scale stress function. The nonlocal quadrature scheme, which replaces the classical Gauss (local) quadrature, allows for nonlocal interactions to extend over finite neighborhoods and thus introduces nonlocality into the two-scale integrals employed in various homogenization theories. The coarse-scale stress function, which replaces the classical notion of coarse-scale stress being the average of fine-scale stresses, is constructed to express the governing equations in terms of coarse-scale fields only. Perhaps the most interesting finding of the present manuscript is that the coarse-scale continuum description that is consistent with an underlying fine-scale description depends on both the coarse-scale discretization and fine-scale details. As a prelude to introducing the computational continua framework, we unveil the relation between the generalized continua and higher-order mathematical homogenization theory and point out to their limitations. This serves as motivation to the main part of the manuscript, which is the computational continua formulation. Copyright (C) 2010 John Wiley & Sons, Ltd.