Research objective: For multi-scaling, develop method for coupling atomistic and continuum simulations of solids that obeys the full set of mechanics relations, including kinematic compatibility as well as balance of momentum and energy, without ad hoc

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• Research objective For multi-scaling, develop
  method for coupling atomistic and continuum
  simulations of solids that obeys the full set of
  mechanics relations, including kinematic
  compatibility as well as balance of momentum and
  energy, without ad hoc assumptions.
• Approach Using the space-time discontinuous
  Galerkin (SDG) formalism, we maintain continuity
  and balance laws across the atomistic/continuum
  interface via flux-based coupling. The SDG method
  naturally includes space and time averaging
  through weak-coupling on boundaries between
  elements, and thereby transfers coarse-grained
  information from atomistic to continuum regions.
  We will employ statistical mechanics methods to
  repopulate modes on atomistic side that are
  unresolved on continuum side.
• Significant Results
• (1) Produced a hybrid SDG method that obeys
  inherently compatibility and energy and momentum
  balance.
• (2) Hybrid SDG coarse-grains to continuum and
  produces exact motion in 1-D. (see figure)
• (3) Method works with any atomistic potential.

For 1-D phonons we use continuum and one embedded
atomistic element, we input a traveling wave and
plot the wave in both continuum (LHS) and atomic
(RHS) regions. Colors show various times.
Solution is well represented in all elements.
In the SDG Hybrid method, we solve for unknown
tractions at atomistic boundary to maintain
continuity. Method inherently maintains balance
laws.

Broader Impact Method potentially allows
continuum finite-element calculations of
deformation, etc., based on arbitrary atomistic
interactions. Offers a consistent method to
remove spurious reflections at atomistic-continuum
interface.