  The most essential concept in concurrent multiscale methods involving
atomistic-continuum coupling is how to define the relation between atomistic
and continuum regions. A well-known coupling method that has been frequently
employed in different concurrent multiscale methods such as Quasicontinuum is
the strong compatibility coupling (SCC). Although the SCC is a highly accurate
coupling method, it constrains the mesh generation and restricts the reduction
of computational cost. In this paper, first, we explain the notion of coupling
models through the interface in the context of continuum mechanics for
quasi-static problems. Then, the SCC is relaxed to overcome the downsides in
the following approaches: a surface approximation approach and a force
approximation approach. Based on the latter, we develop a brand new coupling
method called consistent linear coupling (CLC). Overall, six different coupling
schemes are introduced in line with the surface and force approximation
approaches. The schemes are compared in terms of solving quasi-static 3D
elastic nanoscale contact between an aluminum substrate and a diamond
semi-sphere. Numerical results reveal that the CLC-based schemes are more
accurate when compared with the surface approximation schemes. Furthermore, we
demonstrate that by decreasing the size of interface elements, the accuracy of
the CLC converges to and even exceeds the accuracy of SCC, but nevertheless
with significantly less computational cost. In other words, in contrast to the
SCC, the accuracy of the CLC is tunable, and it can be optimized proportional
to the computational cost and accuracy. Finally, we sketch the idea of how the
CLC can make the consistency near the atomistic-continuum interface by
eliminating ghost forces. The energy of a one-dimensional atomic chain with a
finite range of interaction is relaxed to validate the CLC method.
