Jacob Fish

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 © 2010 John Wiley & Sons, Ltd.

An adaptive two-scale homogenization procedure combining a reduced order and a direct mathematical homogenization method is developed. A simple strain-based indicator is used to guide the construction of a nearly optimal model consisting of either a direct or a reduced order unit cell model. Consistent interscale transfer operators are developed to map the information between the two homogenization models. Verification and validation studies are conducted on a composites plate with a center crack, metal-polyurea sandwich plate, and fabric composites circular tube.

A hierarchical model reduction approach aimed at reducing computational complexity of non‐linear homogenization at multiple scales is developed. The method consists of the following salient features: (1) formulation of non‐linear unit cell problems at multiple scales in terms of eigendeformation modes that a priori satisfy equilibrium equations at multiple scales and thus eliminating the need for costly solution of discretized non‐linear equilibrium, (2) the ability to control the discretization of the eigendeformation modes at multiple scales to maintain desired accuracy, and (3) hierarchical solution strategy that requires sequential solution of single‐scale problems.A two‐scale formulation is verified against an one‐dimensional model problem for which an analytical solution can be obtained and a three‐scale formulation is validated against tube crash experiments. Copyright © 2009 John Wiley & Sons, Ltd.

We present a generalization of the classical mathematical homogenization theory aimed at accounting for finite unit cell distortions, which gives rise to a nonperiodic asymptotic expansion. We introduce an auxiliary macro‐deformed configuration, where the overall Cauchy stress is defined, and nonperiodic boundary conditions. Verification studies against a direct numerical simulation demonstrate the versatility of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd.

A new method for propagating arbitrary failure modes is presented. Arbitrary failure modes are resolved on a refined local patch of elements and then embedded into the coarse grid using partition of unity method. Strong discontinuities are propagated by means of element erosion in the superimposed patch of elements only. The method, coined as the rs‐version of the finite element method (or reduced order s‐method), has been integrated in ABAQUS and verified on several test problems. Copyright © 2007 John Wiley & Sons, Ltd.

We present a computational homogenization approach for linear and non‐linear solid mechanics problems, which is fully compatible with conventional finite element code architecture. A seamless implementation in ABAQUS is presented including Python script, validation problems and a web‐link where script files, user‐defined subroutines and input files can be accessed. For linear problems, we demonstrate how to utilize ABAQUS existing facilities to develop analysis attributes required for solving a unit‐cell problem. For non‐linear problems, a Python script invoked by a coarse‐scale stress update procedure is introduced to carry out the scale bridging. The purpose of this paper is twofold: (i) to motivate practitioners to adopt the computational homogenization as an integral part of their analysis and design process; and (ii) to encourage commercial code vendors to seamlessly integrate the architectures proposed in their legacy codes. Copyright © 2007 John Wiley & Sons, Ltd.