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Research: Mathematics of Micromechanics

Most of the important features of materials (such as plasticity or failure) are determined by the fine (micro, nano, molecular, and quantum) structure of materials. Micromechanics, understood as a discipline aiming at predicting such properties relies mostly on the numerical simulation of the macroscopic response of a microscopic structure. However, even with modern supercomputers, the spatial and temporal scales accessible for direct molecular simulations cannot resolve the fine structure in many cases.

This makes coarse-graining in space and time necessary. Roughly speaking, spatial coarse-graining is a method of looking only at a few atoms of interest (or sometimes not looking at atoms at all), and the temporal coarse-graining is a method allowing not to track every thermal oscillation of atoms.


Interatomic Potemtials.

A significant part of my present research interests lies in interatomic potentials development.


Atomistic-to-Continuum Coupling.

Much of my past research concerned the Atomistic-to-Continuum (AtC) coupling method (also known as a quasicontinuum class of methods)—a particular form of spatial coarse-graining for crystals. [details and publications on the AtC coupling]



Another direction ("orthogonal" to the AtC coupling) is treatment of multilattices (corresponding, e.g., metallic alloys). no result found, sorry.



Study of accuracy of numerical methods for computing defects relies on the basic properties of such defects. For instance, to answer the question of how many degrees of freedom (or atoms) one needs to reach a certain accuracy, one needs to understand the regularity of defects (expressed in terms of decay of the elastic far-fields).
Submitted Articles
[1]Accuracy of computation of crystalline defects at finite temperature (M. Luskin, A.V. Shapeev). [bibtex] [arXiv]

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