Multiscale science and engineering have so far been largely driven by needs of specific applications, and lack of a mathematical foundation and systematic approach has limited progress. To develop a general purpose theoretical framework and to transform it into a useful design system, this project addresses the following fundamental issues: (1) What is the validity limit of homogenization and how to extend it to boundary layers at a reasonable computational cost? (2) How to deal with the issue of lack of periodicity that naturally arises due to high gradients even in a periodic heterogeneous medium? (3) What should be the size of the representative volume element and how it is affected by large distortions? (4) How to introduce multiple temporal scales arising in mechanical/thermal fatigue problems? (5) How to account for multiple physical processes inherent in environmental degradation problems involving coupled mechanical-diffusion-reaction processes? These issues will be studied and the resulting design system will be validated against experimental data provided by industrial partners.

The Multiscale Design System that will grow out of this research program will advance a wide array of fields requiring light weight structural components in aerospace, printed circuit boards, robotic arms, medical prostheses, implants, valves, seals, electronic devices and radiological equipment. This general purpose multiscale design system offers a transformational approach that may lead to life-saving improvements in medical devices, more resilient infrastructure, energy-saving measures that will increase the efficiency of environmental processes, and a variety of more cost-effective products. The education program will extend from undergraduate to graduate work with outreach to women and minorities. Technology transfer to industrial partners from GE, Boeing, GM and Renegade Materials has a potentially of impacting their design processes.

Project Report

PI: Jacob Fish Awardee: Columbia University NSF Award Number: 1127810 This research effort was concerned with development of both predictive and practical computational tools for composite materials structures. The salient features of the computational framework developed are as follows: (i) systematic information reduction methodology that reduces complex coupled multiphysics fine-scale information having hundreds of thousands of finite elements to a manageable number of deformation modes and state variables, (ii) scale-separation free methodology that requires no new degrees of freedom and is free of higher order boundary conditions, (iii) stochastic multiscale methodology that translates geometrical and material uncertainties into component level uncertainties, (iv) physics-based fatigue life prediction methodology that is not based on Paris-law like formulations currently used for metals, and (v) regularization techniques based on the staggered nonlocal formulation and rescaling of constitutive equations at multiple scales that allow propagation of strong discontinuities consistent with fracture mechanics and insensitive to mesh density. The stochastic capabilities developed were based on stochastic collocation approach that has been found to be considerably more efficient than the Monte-Carlo approach; the reduced order scale-separation-free capabilities were based on computational continua formulation; and the fatigue capabilities developed were based on multiple space-time formulation. The multiple temporal scales were introduced due to rapidly oscillatory loading on one hand and slow degradation of material properties on the other. The formulations developed were validated against multiphysics and fatigue experiments performed by Rolls-Royce and NASA at various environmental conditions. This work culminated in the first multiscale textbook recently published by Wiley[1]. We believe that with the publication of the textbook and with successful feasibility studies of utilization of multiscale technologies in practice there will be a widespread adoption of multiscale methods in academia, government and industry. [1] J. Fish, Practical Multiscaling, Wiley 2013

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Columbia University
New York
United States
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