How can I simulate heat diffusion with different materials?

[marcobernardes]

I was wondering if it would be possible to simulate 2D heat diffusion with different materials.

from pde import DiffusionPDE, ScalarField, UnitGrid
grid = UnitGrid([32, 32]) # generate grid
state = ScalarField(grid) # generate initial condition
# set boundary conditions `bc` for all axes
bc_x_left = {"value": 0}
bc_x_right = {"value": 100}
bc_x = [bc_x_left, bc_x_right]
#bc_y_top = {"value": 0}
#bc_y_bottom = {"value": 0}
bc_y_top = {"derivative": 0}
bc_y_bottom = {"derivative": 0}
bc_y = [bc_y_bottom, bc_y_top]
eq = DiffusionPDE(10,bc=[bc_x, bc_y])
result = eq.solve(state, t_range=200, dt=0.005)
result.plot()

I think the best way to do this would be to set up the diffusivity values independently. How can I do this?

Many thanks!

[david-zwicker]

There are at least two alternatives: Either you write your own class to implement the PDE or you use the predefined class PDE to specify a diffusion equation with variable coefficients. In the latter case, you can use spatially dependent coefficients to describe your material properties.

[marcobernardes]

Thank you David!

I could implement the Heat Diffusion equation but I don't know how to set customized diffusivity.
How can I do that, please?

Many thanks,

look:

from pde import PDE, DiffusionPDE, ScalarField, UnitGrid, CartesianGrid, MemoryStorage, plot_kymograph

grid = CartesianGrid([[0, 1], [0, 1]], [32, 32]) # generate grid
state = ScalarField(grid, data=0)

set boundary conditions bc for all axes

bc_x_left = {"value": 0}
bc_x_right = {"value": 100}
bc_x = [bc_x_left, bc_x_right]
bc_y_top = {"value": 0}
bc_y_bottom = {"value": 0}
bc_y = [bc_y_bottom, bc_y_top]

Expanded definition of the PDE

diffusivity = "0.01"

field = ScalarField(grid, data=0.1)
field.add_interpolated([0.5, 0.5], 0.5)
#field.plot()

eq = PDE({"c": f"{diffusivity} * laplace(c)"}, bc=[bc_x, bc_y])

result = eq.solve(state, t_range=10, dt=0.001)
result.plot()

[david-zwicker]

Could you please specify the exact equation that you're trying to solve?

[marcobernardes]

2d heat diffusion

[david-zwicker]

This is not what I meant – please give the exact mathematical formulation with boundary conditions. So far, I do not understand where your problem lies.

[marcobernardes]

for instance:

                     T = 1
   
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
   %                                        %
   %                                        %
   %    %%%%%%%%%                           %
   %    %       %                           %
   %    %  k1   %           k3              %
   %    %       %                           %
   %    %%%%%%%%%                           %
   %                                        % T = 0
   %                                        %
   %                                        %
   %                        %%%%%%%%%       %
   %                        %       %       %
   %                        %  k2   %       %
   %                        %       %       %
   %                        %%%%%%%%%       %
   %                                        %
   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

                    T = 0