**5. Multi scale boundary condition**

Each method has its own limitations to use. These limitations are only in time and size scale. The multi scale method tries by dividing the model in different areas in terms of size and time scale (coarse-fine grain) and relating them together generates an assimilated method. In other words, we may be able to solve one part of problem with atomistic modeling and the other part with continuum; the multi scale method uses both of them concurrently and then couples them together.

To couple the methods, we define a region of pseudo-atoms called handshake or pad. The position of pseudo-atoms is determined by the finite element method. The handshake has a duty to absorb the fine grain excitation and transfer the effects of coarse grain surrounding boundary.

If *u*1 is the displacement of the pseudo-atom, *u*<sup>0</sup> is the fine grain displacement, and *ua* is the coarse grain displacement, then we can say:

$$
\mu\_1 = \Theta\left\{\mu\_0\right\} + \Xi\left\{\mu\_a\right\} \tag{9}
$$

where *Θ* and **Ξ** are unknown operators.
