On atomistic-to-continuum couplings without ghost forces in three   dimensions
  In this paper we construct energy based numerical methods free of ghost forces in three dimensional lattices arising in crystalline materials. The analysis hinges on establishing a connection of the coupled system to conforming finite elements. Key ingredients are: (i) a new representation of discrete derivatives related to long range interactions of atoms as volume integrals of gradients of piecewise linear functions over bond volumes, and (ii) the construction of an underlying globally continuous function representing the coupled modeling method. 
Charalambos Makridakis, Dimitrios Mitsoudis and Phoebus Rosakis
math.NA math-ph math.MP