A simulation on vertically shaken granular layers
  A hybrid model of molecular dynamics and continuum mechanics is introduced to study a system of vertically shaken granular layers. Despite the simplicity the model shows pattern formation in the granular layers due to the formation of heaplets. We show from a simple analysis that the onset of pattern formation is density dependent and this result is justified by the subsequent computer simulations. Our simulations also show that the heaping process can be divided into three stages: an early stationary stage, an intermediate growing stage and a late-time saturated stage. In the early stage, the average volume of the heaplets remains almost unchanged until the system crosses over to the intermediate growing stage. The length of time that system remains in the early stage defines the onset time of the instabilities, $k_0$ which depends on the shaking intensity, $\gamma$. In the growing stage, the average volume of the heaplets grows with time and can be approximated by a power law with a shaking intensity dependent growth exponent, $z$. From our simulation, we show that the growth exponent, $z$ depends linearly on the shaking intensity, $\gamma$. There is a critical shaking intensity $\gamma_c \simeq 14$, such that for shaking intensities greater than $\gamma_c$, $z$ cannot be properly defined. The late-time saturated stage is where most of the particles are trapped in a big heap and this big heap is in equilibrium with the surrounding granular gas. 
Y. K. Goh and R. L. Jacobs