Effective thermal conductivity#
Usually, after the design of the circuitry, the empty space on chips gets filled with equally spaced small rectangles in each layer. Optically, these structures are supposed to be far enough away that their influence on the structures can be neglected. But for thermal considerations, those fill structures can have an impact on the temperature distribution on the chip and thus e.g. on the crosstalk between thermal phase shifters. As it’s computationally challenging to include all the small cuboids in the model (which is especially for the meshing a major challenge), a preferable approach is to consider the filled area as a homogenous area of higher thermal conductivity. For this, we calculate the effective thermal conductivity of the filled area by examining a single unit cell.
To have an intuitively understandable problem, we consider half of the unit cell to be filled with a highly thermally conductive material (metal/silicon) surrounded by a material with low thermal conductance (e.g. silicon dioxide) Let’s start with the mesh!
unitcell_width = 0.8
unitcell_length = 0.8
unitcell_thickness = 0.387
fill_width = 0.4
fill_length = 0.4
fill_thickness = 0.22
0.22
Show code cell source
using PyCall
np = pyimport("numpy")
shapely = pyimport("shapely")
shapely.affinity = pyimport("shapely.affinity")
OrderedDict = pyimport("collections").OrderedDict
meshwell = pyimport("meshwell")
Prism = pyimport("meshwell.prism").Prism
Model = pyimport("meshwell.model").Model
model = Model()
fill_polygon = shapely.box(
xmin = unitcell_length / 2 - fill_length / 2,
ymin = unitcell_width / 2 - fill_width / 2,
xmax = unitcell_length / 2 + fill_length / 2,
ymax = unitcell_width / 2 + fill_width / 2,
)
fill = Prism(
polygons = fill_polygon,
buffers = Dict(
unitcell_thickness / 2 - fill_thickness / 2 => 0.0,
unitcell_thickness / 2 + fill_thickness / 2 => 0.0,
),
model = model,
physical_name = "fill",
mesh_order = 1,
resolution = Dict("resolution" => 0.05, "SizeMax" => 0.2, "DistMax" => 1),
)
unitcell_polygon =
shapely.box(xmin = 0, ymin = 0, xmax = unitcell_length, ymax = unitcell_width)
unitcell_buffer = Dict(0 => 0.0, unitcell_thickness => 0.0)
unitcell = Prism(
polygons = unitcell_polygon,
buffers = unitcell_buffer,
model = model,
physical_name = "unitcell",
mesh_order = 2,
)
"""
BOUNDARIES
"""
right_polygon = shapely.box(
xmin = unitcell_length,
ymin = 0,
xmax = unitcell_length + 1,
ymax = unitcell_width,
)
right = Prism(
polygons = right_polygon,
buffers = unitcell_buffer,
model = model,
physical_name = "right",
mesh_bool = false,
mesh_order = 0,
)
left_polygon = shapely.box(xmin = -1, ymin = 0, xmax = 0, ymax = unitcell_width)
left = Prism(
polygons = left_polygon,
buffers = unitcell_buffer,
model = model,
physical_name = "left",
mesh_bool = false,
mesh_order = 0,
)
up_polygon = shapely.box(
xmin = 0,
ymin = unitcell_width,
xmax = unitcell_length,
ymax = unitcell_width + 1,
)
up = Prism(
polygons = up_polygon,
buffers = unitcell_buffer,
model = model,
physical_name = "up",
mesh_bool = false,
mesh_order = 0,
)
down_polygon = shapely.box(xmin = 0, ymin = -1, xmax = unitcell_length, ymax = 0)
down = Prism(
polygons = down_polygon,
buffers = unitcell_buffer,
model = model,
physical_name = "down",
mesh_bool = false,
mesh_order = 0,
)
above_buffer = Dict(unitcell_thickness => 0.0, unitcell_thickness + 1 => 0.0)
above = Prism(
polygons = unitcell_polygon,
buffers = above_buffer,
model = model,
physical_name = "above",
mesh_bool = false,
mesh_order = 0,
)
below_buffer = Dict(-1 => 0.0, 0 => 0.0)
below = Prism(
polygons = unitcell_polygon,
buffers = below_buffer,
model = model,
physical_name = "below",
mesh_bool = false,
mesh_order = 0,
)
"""
ASSEMBLE AND NAME ENTITIES
"""
entities = [unitcell, fill, up, down, left, right, above, below]
mesh = model.mesh(
entities_list = entities,
verbosity = 0,
filename = "mesh.msh",
default_characteristic_length = 0.2,
)
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Info : It. 0 - 0 nodes created - worst tet radius 2.5504 (nodes removed 0 0)
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Info : Optimization starts (volume = 0.0352) with worst = 0.0371703 / average = 0.763883:
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Info : No ill-shaped tets in the mesh :-)
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Info : Done optimizing mesh (Wall 0.00839889s, CPU 0.00839s)
Info : 1315 nodes 8214 elements
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Info : Writing '/tmp/tmp0ekehbno/mesh.msh'...
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PyObject <meshio mesh object>
Number of points: 1315
Number of cells:
triangle: 162
triangle: 98
triangle: 98
triangle: 98
triangle: 98
triangle: 162
triangle: 128
triangle: 128
triangle: 128
triangle: 128
triangle: 330
triangle: 330
tetra: 1554
tetra: 4572
Cell sets: up___unitcell, down___unitcell, left___unitcell, right___unitcell, above___unitcell, below___unitcell, fill___unitcell, up___None, down___None, left___None, right___None, above___None, below___None, fill___None, unitcell___None, up, down, left, right, above, below, fill, unitcell, gmsh:bounding_entities
Point data: gmsh:dim_tags
Cell data: gmsh:physical, gmsh:geometrical
Field data: up___unitcell, down___unitcell, left___unitcell, right___unitcell, above___unitcell, below___unitcell, fill___unitcell, up___None, down___None, left___None, right___None, above___None, below___None, fill___None, unitcell___None, up, down, left, right, above, below, fill, unitcell
Show code cell source
using Gridap
using GridapGmsh
using Gridap.Geometry
using GridapMakie, CairoMakie
using Femwell.Maxwell.Electrostatic
using Femwell.Thermal
For simplicity, we define the thermal conductivity of the unitcell to be 1 and the thermal conductivity of the fill to be 100, which is almost negligible in comparison.
thermal_conductivities = ["unitcell" => 1.4, "fill" => 400]
model = GmshDiscreteModel("mesh.msh")
Ω = Triangulation(model)
dΩ = Measure(Ω, 1)
labels = get_face_labeling(model)
tags = get_face_tag(labels, num_cell_dims(model))
τ = CellField(tags, Ω)
thermal_conductivities =
Dict(get_tag_from_name(labels, u) => v for (u, v) in thermal_conductivities)
ϵ_conductivities(tag) = thermal_conductivities[tag]
Info : Reading 'mesh.msh'...
Info : 54 entities
Info : 1315 nodes
Info : 8014 elements
Info : Done reading 'mesh.msh'
ϵ_conductivities (generic function with 1 method)
We define the temperatures at both sides of the unitcell to define the direction in which we want to estimate the thermal conductivity. In other directions the boundaries are considered to be insulating/mirroring.
We start with calculating the temperature distribution. From this, we calculate, analog to how we calculate resistances for electrical simulations first the integral $\int σ \left|\frac{\mathrm{d}T}{\mathrm{d}\vec{x}}\right|^2 dA which gives an equivalent of the power and calculate from this the effective thermal conductivity. We expect the thermal conductivity almost twice as high as the thermal conductivity of the material with lower thermal conductivity.
boundary_temperatures = Dict("left___unitcell" => 100.0, "right___unitcell" => 0.0)
T0 = calculate_temperature(
ϵ_conductivities ∘ τ,
CellField(x -> 0, Ω),
boundary_temperatures,
order = 2,
)
temperature_difference = abs(sum(values(boundary_temperatures) .* [-1, 1]))
power = abs(
sum(
∫(gradient(temperature(T0)) ⋅ (ϵ_conductivities ∘ τ) ⋅ gradient(temperature(T0)))dΩ,
),
)
println(
"Effective thermal conductivity: ",
(power / temperature_difference^2) /
((unitcell_thickness * unitcell_width) / unitcell_length),
)
boundary_temperatures = Dict("above___unitcell" => 100.0, "below___unitcell" => 0.0)
T0 = calculate_temperature(
ϵ_conductivities ∘ τ,
CellField(x -> 0, Ω),
boundary_temperatures,
order = 2,
)
temperature_difference = abs(sum(values(boundary_temperatures) .* [-1, 1]))
power = abs(
sum(
∫(gradient(temperature(T0)) ⋅ (ϵ_conductivities ∘ τ) ⋅ gradient(temperature(T0)))dΩ,
),
)
println(
"Effective thermal conductivity: ",
(power / temperature_difference^2) /
((unitcell_length * unitcell_width) / unitcell_thickness),
)
Effective thermal conductivity: 2.2581927994100313
Effective thermal conductivity:
2.06495405819836