Principles of Coarse-Graining and Coupling Using the Atom-to-Continuum Method

In this chapter we develop a basis for atom-continuum coupling simulation: (a) consistent upscaling/coarse-graining of the atomic state, (b) multiscale governing balances, (c) consistent continuum surrogate models, and (d) atomistic control schemes. In particular, from the established continuum balances and a general formulation for coarse-graining, we derive estimators for mechanical fields such as stress and displacement, thermal quantities such as heat flux and temperature, and electrostatic fields such as polarization. In addition, we formulate mechanical and thermal coupling schemes including specific algorithms for atomic regulation to achieve field consistency and flux conservation. Lastly, we demonstrate these concepts with a range of example simulations.