Numerical Method:
The finite element method determines an approximate normal velocity of interface that satisfies a weak statement. In our work, an interface is modeled by an assembly of straight line to characterize linear geometries and create the interface with nodal points.
The variation of total free energy associated with the virtual motion of single element is given as:
The left-hand side of weak statement can be expressed in terms of variations of nodal positions, velocities of nodal points, and angle theta.
Where viscosity matrix H is given as:
For n nodal points, the weak statement becomes:
And because the above weak statement is valid for arbitrary virtual change, it follows
Since the viscosity matrix and the force column are dependent on the nodal positions, the above equation is a set of non-linear ordinary differential equations that should be solved numerically.
