Seamless multiscale modeling via dynamics on fiber bundles

We present a general mathematical framework for modeling the macroscale behavior of a multiscale system using only microscale models, by formulating the effective macroscale models as dynamic models on the underlying fiber bundles. This framework allows us to carry out seamless multiscale modeling using traditional numerical techniques. At the same time, they give rise to an interesting mathematical structure and new interesting mathematical problems. We discuss several examples from homogenization problems, continuum modeling of solids based on atomistic or electronic structure models, macroscale behavior of interacting diffusion, and continuum modeling of complex fluids based on kinetic and Brownian dynamics models.

Keywords

multiscale modeling, seamless, fiber bundle

2010 Mathematics Subject Classification

Primary 65J05. Secondary 35B27, 74B20, 74Q05, 76M25.

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Published 1 January 2007