**6. Multi scale Modeling**

To solve the above-mentioned equation of motion, we should use mathematics like the Fourier

**a.** *f ext* =0 or free lattice: the solution is gained by superposing of normal modes called

**b.** *f ext* ≠0 : to solve this type of problems we use Green's function method that requires use

**c.** Quasi-static problem or approximation: this is a name for time-independent problems where any noticeable change of the external forcing occurs during a period of time that is much longer than the characteristic time of atomic vibrations. This leads to eliminate the

( ) *ext*

*t ft* - å*K u* <sup>=</sup> (8)

*uu u* 1 0 = + Q{ } X{ *<sup>a</sup>*} (9)

standing waves, because the lattice oscillate around its equilibrium position.

' ' ( )

*nn n n*

To solve this equation, we can use green's functions method. One of issue problems in quasistatic approximation is multiscale boundary conditions. We discuss this in a separate clause.

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

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

If *u*1 is the displacement of the pseudo-atom, *u*<sup>0</sup> is the fine grain displacement, and *ua* is the

transform or the Laplace transform.

12 Graphene - New Trends and Developments

of Unit pulse convolution.

**5. Multi scale boundary condition**

coarse grain displacement, then we can say:

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

them together.

boundary.

In terms of *f ext*, we have three types of problems:

first term in motion equation, so we have the following:

'

*n*

The philosophy of arising multiple- scale methods is that, in actuality, nano- materials are always used along with large- scale materials and we are compelled to create a method for modeling them. Atomistic methods such as MD and ab initio are not perfect to model the entire configuration because these methods are limited to time and length scales; thus, they are validated only through fine- scale parts of configuration and not for the other part. There is the same situation for continuum methods because they are validated for large time and length scale, and they are not useable for fine- scale parts of these configurations. Here is the point that the role of multiple- scale methods becomes prominent. Multiple scale methods try to blend the methods that are validated on their own scales of time and length separately.

The base of all these methods is that each scale is modeled by its special method, and their output becomes boundary condition for the other; indeed, there will be exchanges between these methods to model entire of the configuration.

Whatever method we take, it must be free of two issue problems:

