Discussion Forum

Hi,

I'd like to use COMSOL to solve for the nonconserved dynamics of the phase field crystal (PFC) model:

du/dt = - ( \epsilon + ( 1 + \nabla^2 )^2 ) u - \tau u/2 - u/3,

where u = u(t,x,y) is the dependent variable, \epsilon and \tau are constants and \nabla^2 is the Laplacian.

Using the coefficient form for PDEs the time derivative and the linear term are easy, the second- and third-order terms I can specify in the source term and there's also a coefficient for a term with the Laplacian.

But the PFC model incorporates also fourth-order gradient terms:

d^4 u / dx^4
d^4 u / dy^4
d^4 u / ( dx^2 dy^2 ) (if I'm not mistaken...)

How can I implement these in COMSOL? It doesn't recognize variables and functions such as uxxxx, uxxyy, grad(), div(), del(), nabla() or Laplacian(). I googled extensively, watched tutorials and read documentation, but haven't yet figured this out.

My primary motivation is to be able to solve the model on curved surfaces (I really don't need COMSOL for solving flat systems, but I guess I have to figure out how to use COMSOL for that first), so tips regarding gradients in this context are also welcome.

A big thanks!

Edit: I think that I'll also have to set the shape functions of the elements to fourth order at least for COMSOL to be able to evaluate the fourth-order gradients, right?