Periodic Boundary Conditions in Computational Homogenization

[thero...@gmail.com]

Dear all,

I want to use periodic boundary condition for computational homogenization, essentially I would like to have the following BC:
u_2(x_2) - u_1(x_1) = F*(x_2-x_1), where F is some 2x2 matrix, u_1 and u_2 are state variables on opposite boundaries, and x_2 and x_1 are the coordinates.

I found this. I looked at the tests all_periodic_trans.i and trapezoid.i, but I do not really understand how the BC works and if it is what I am looking for.

If I understand correctly, in all_periodic_trans.i the PBC only enforce u_2 - u_1 = 0 by having u(x_2) = u(x_1+10), but I would rather like to have u_2(x_2) - u_1(x_1) = F*(x_2-x_1). Is that possible? Can some help please?

Thank you.

[thero...@gmail.com]

Anyone can help me with this?

[SudiptaBiswas]

From your description, the problem is not entirely clear to me. However, you might find the strain periodicity approach useful here. Here is the paper that describes it, 

https://www.sciencedirect.com/science/article/pii/S0168874X20301165

[thero...@gmail.com]

Thank you for the suggestion. Yep, I am looking for something like this, but with the standard periodic displacement for now (strain periodicity definitely looks interesting). Has it been implemented inside of moose?

[SudiptaBiswas]

Yes, here is the associated documentation, https://www.mooseframework.org/syntax/Modules/TensorMechanics/GlobalStrain/index.html.