Reduced Order Computational Continua

INTRODUCTION The paper presents a new multiscale framework that is both mathematically rigorous and practical in the sense that it has been successfully applied in aerospace, automotive and civil engineering industries. The “rigor” aspect of the method is provided by recently developed computational continua (C) formulation (Fish and Kuznetsov, 2009), which is endowed with fine-scale details, introduces no scale separation, makes no assumption about infinitesimality of the fine-scale structure, does not require higher order continuity, introduces no new degrees-of-freedom and is free of higher order boundary conditions. The “practicality” aspect of the proposed method is inherited from the reduced order homogenization (Yuan and Fish, 2009, Fish and Yuan, 2008) approach, which constructs residual free-fields that eliminate the bottleneck of satisfying fine-scale equilibrium equations and is endowed with a hierarchical model improvement capability where the cost of the most inexpensive member of the sequence is comparable to that of semianalytical or phenomenological methods. Blending of the two methods into a single cohesive computational framework, hereafter to be referred to as the Reduced order Computational Continua or simply RC, that inherits the underlying characteristics of its two ingredients, is the main objective of the present manuscript. We conclude the manuscript with a brief summary and discussion of future research directions. In the present manuscript we consider a heterogeneous body formed by a repetition of a fine structure (unit cells) occupying an open, bounded domain 3 ζ Ω ⊂ . The unit cell domain denoted as 3 Θ ⊂ is assumed to be finite, i.e. unlike in the homogenization theories it is not infinitesimally small compared to the coarse-scale domain. The following governing equations on ζ ∈Ω x are stated at the fine-scale of interest ( ) ( ) ( ) ( ) ( ) ( ) ( )