Numerical Implementation

3D Finite Element Method:   
Please refer to this pdf file for details.

Fourth order Runge-Kutta Method:
http://en.wikipedia.org/wiki/Runge-Kutta

Stress Field Calculation:
We are using a similar method with previous project to calculate the stress field.

Notes: We applied constant biaxial forces on the sides of the thin film, which creates an equivalent stress field due to lattice misfit. The approximation holds as long as the substrate is much thicker than the thin film. 

The following plot shows strain energy associated with the 3D quantum dot growth initial condition. This plot was obtained from using Abaqus 6.7-1. The perturbation is small, on the order of 0.1, and the thickness of the substrate is 10. So we can hardly see them in the Abaqus result.

Top View:



Side View:



The following plot shows the scaled perturbation that could be hardly seen in the Abaqus result. This plot was obtained from using Matlab R2008a.