Example 1: Porous Material

An OpenBTE simulation is specified by a combination of a material, geometry and solver. We begin with creating a material. To this end, we load previously computed first-principled calculations on Si at room temperature

from openbte import load_rta

rta_data = load_rta('Si_rta')

Next step is to perform MFP interpolation

from openbte import RTA2DSym

mat = RTA2DSym(rta_data)

The RTA2DSym model is for simulation domains which have translational symmetry along the z axis. To create a geometry, we instantiate an object of the class Geometry

from openbte import Geometry

G = Geometry(0.1)

where \(0.1\) is the carhacteristic size (in nm) of the mesh. In this example, we create a porous material with porosity \(0.2\) and rectangular aligned pores. Given the periodicity of the system, we simulate only a unit-cell to which we apply periodic boundary conditions. To define the unit-cell, we use the add_shape method

from openbte import rectangle

L        = 10 #nm

G.add_shape(rectangle(area = L*L))

To add the hole in the middle, we use the add_hole method

porosity = 0.2

area = porosity*L*L

G.add_hole(rectangle(area = area,x=0,y=0))

To apply boundary conditions, we need to assign a name to sides and refer to them in the solver section. Sides are selected with selector. In this case, we assign all internal sides the name Boundary

G.set_boundary_region(selector = 'inner',region = 'Boundary')

To apply periodic boundary conditions along both axes, we use the set_periodicity method

G.set_periodicity(direction = 'x',region      = 'Periodic_x')

G.set_periodicity(direction = 'y',region      = 'Periodic_y')

At this point, we are ready to save the mesh on disk

G.write_geo()

If everything went smoothly, you should see mesh.geo in your current directory. You can open them with GMSH to check that the geometry has been created correctly. To create a meshed geometrym we use the function get_mesh()

from openbte import get_mesh

mesh = get_mesh()

Before setting up the solvers, we need to specify boundary conditions and perturbation. In this case, we apply a difference of temperature of \(\Delta T_{\mathrm{ext}} = 1\) K along x

from openbte.objects import BoundaryConditions

boundary_conditions = BoundaryConditions(periodic={'Periodic_x': 1,'Periodic_y':0},diffuse='Boundary')

Note that we also specifies diffuse boundary conditions along the region Boundary. In this example, we are interested in the effective thermal conductivity along x

from openbte.objects import EffectiveThermalConductivity

effective_kappa = EffectiveThermalConductivity(normalization=-1,contact='Periodic_x')

where normalization (\(\alpha\)) is used in the calculation of the effective thermal conductivity \(\kappa_{\mathrm{eff}} = \alpha\int_{-L/2}^{L/2}\mathbf{J}(L/2,y)\cdot \mathbf{\hat{n}}dy\). For rectangular domain, \(\alpha =-L_x/L_y/\Delta T_{\mathrm{ext}}\).

To run BTE calculations, we first solve standard heat conduction

from openbte import Fourier

fourier     = Fourier(mesh,mat.kappa,boundary_conditions,\
                       effective_thermal_conductivity=effective_kappa)

Finally, using fourier as first guess, we solve the BTE

from openbte import BTE_RTA

bte     = BTE_RTA(mesh,mat,boundary_conditions,fourier=fourier,\
          effective_thermal_conductivity=effective_kappa)

Before plotting the results, we group together Fourier and BTE results

from openbte.objects import OpenBTEResults

results =  OpenBTEResults(mesh=mesh,material = mat,solvers=[fourier,bte])

Lastly, the temperature and heat flux maps can be obtained with

results.show()

GMSH