#Example of snowflake

#Developed by Yang Xia, Department of Material Science and Engineering, Shanghai Jiao Tong University, Shanghai

#Model from Ryo Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Physica D: Nonlinear Phenomena,Volume 63, Issues 3–4, 15 March 1993, Pages 410–423

#This example shows a process of dendritic crystal growth, which is an anisotropic nucleation process. This is a simple phase field model, in which one component melt growth is presented in 2D condition. 

#The PDE in this process is:

#dw/dt = L*eps*eps1*grad[F(w)] + L*grad*eps*eps*grad(w) + L*w(1-w)(w-0.5+m)
#dT/dt = grad^2(T) + K*dw/dt
#F(w) = (-dw/dy, dw/dx)
#L = 3333.33
#K = -1.8
#w is the phase field parameter, which equals to 0 when it is liquid and 1 when it is solid
#T is the dimensionless temperature, for the melting temperature, it equals to 1


#The parameter in PDE can be expressed as:

#eps = 0.01*[1+0.02*cos(4*angle)]   
#angle is the angle between grad(w) and x axis
#eps1 = d(eps)/d(angle)
#m = 0.9/3.14 * tan[10(1-T)]

#The residual can be expressed as:

#R1 = <dw/dt, _test> + <L*eps*eps1*F(w), grad_test> + <L*eps*eps*grad(w), grad_test> -<L*w(1-w)(w-0.5+m), _test>

#These four terms of residual correspond to TimeDerivative, anisoACInterface1, anisoACInterface2 and ACParsed in Kernels respectively.

#R2 = <dT/dt, _test> + <grad(T), grad_test> - <K*dw/dt, _test>

#These three terms of residual correspond to TimeDerivative, CoefDiffusion and CoefCoupleTimeDerivative in Kernels respectively.