PDDO-MultiPhysics Examples Heat transfer and thermo-mechanics

Effective conductivity of a fibre composite

Transverse heat conduction through the unit cell of a fibre composite: two materials, a bonded interface, and Rayleigh's formula for the effective conductivity as the reference.

fiber_composite_conduction.yaml 30,014 nodes backward-euler, 61 s

1The Problem

A square array of fibres, ten times more conductive than the matrix, occupies 12.6 % of the volume. Its unit cell is a box with one fibre through the middle; by symmetry, a temperature difference across the cell and insulated sides give the response of the whole array, and the mean heat flux gives its effective conductivity. Rayleigh's classical solution for a square array is 1.2292 times the matrix conductivity.

$$\nabla\cdot(k\,\nabla T) = 0, \qquad k = \begin{cases} 1 & \text{matrix} \\ 10 & \text{fibre}\end{cases}$$
$$[\![T]\!] = 0, \quad [\![\,k\,\nabla T\cdot\mathbf{n}\,]\!] = 0 \text{ on the interface}, \qquad k_{\text{eff}} = \overline{k\,\partial T/\partial x}$$

2The Problem File

Instead of one geometry, the domain lists regions: the matrix box and then the fibre cylinder. A later region replaces an earlier one where they overlap, and their common surface becomes an interface. A parameter can take a value per region, k: {region: {matrix: 1.0, fiber: 10.0}}. The interface conditions need no input: the engine reads the flux from the div() term of the equation and makes it continuous across the interface, along with the temperature.

The complete problem file, examples/geometry/fiber_composite_conduction.yaml 
# Two materials (regions): transverse heat conduction through a unit cell of a fiber
# composite (square array of fibers, fiber volume fraction f = pi r^2 = 0.126).
#   div(k grad T) = 0,   k = 1 (matrix), 10 (fiber)
# The cell boundaries are symmetry planes of the array: T = 0 / 1 on x- / x+, insulated
# elsewhere. The effective conductivity is the mean flux (monitor k_eff); Rayleigh's
# formula for a square array: k_eff = 1 + 2f / (1/b - f - 0.3058 b f^4),
# b = (k_f - k_m)/(k_f + k_m)  ->  1.2292.
# The fiber is listed after the matrix: it replaces the matrix where they overlap. Its
# surface becomes the interface (continuity of T and of k grad(T).n); its end faces lie on
# the cell faces z- / z+ and keep the names bottom / top.
"k">name: fiber_composite_conduction
"k">domain:
  "k">type: geometry
  "k">spacing: 0.025                # fiber radius = 8 spacings; 0.035 -> +1.6 %, 0.025 -> +0.5 %
  "k">regions:
    - "k">name: matrix
      "k">geometry: {primitive: box, size: [1, 1, 0.4]}
    - "k">name: fiber
      "k">geometry: {primitive: cylinder, radius: 0.2, height: 0.4, center: [0.5, 0.5, 0.2]}
"k">parameters:
  "k">k: {region: {matrix: 1.0, fiber: 10.0}}
"k">fields:
  "k">T: scalar
"k">relations:
  "k">q: "-k*grad(T)"                 # heat flux
"k">equations:
  "k">T: "div(k*grad(T)) = 0"
"k">boundary_conditions:
  - "k">faces: [x-]
    "k">T: {dirichlet: 0}
  - "k">faces: [x+]
    "k">T: {dirichlet: 1}
  - "k">faces: [y-, y+, z-, z+, bottom, top]
    "k">T: {neumann: 0}
"k">time:
  "k">end: 1
  "k">dt: 1
  "k">scheme: backward-euler
"k">output:
  "k">fields: [T, q]
"k">monitors:
  "k">k_eff: "mean(k*dx(T))"
The model in the GUI: the Model Builder tree on the left, the settings of the selected node in the middle, the graphics window on the right.
Figure 1. The model in the GUI: the Model Builder tree on the left, the settings of the selected node in the middle, the graphics window on the right.
The Several materials table: regions in priority order, each with its geometry.
Figure 2. The Several materials table: regions in priority order, each with its geometry.
A parameter with one value per region.
Figure 3. A parameter with one value per region.

3Discretization

With a spacing of 0.025 the cell holds 30,014 nodes. Nodes on the interface come in pairs, one for each material, with opposite normals; the families of each region stay inside it.

The point cloud coloured by region: matrix and fibre.
Figure 4. The point cloud coloured by region: matrix and fibre.

4Results

Nodes
30,014 (point cloud)
PDDO order
2, horizon 2.015 Δx
Time scheme
backward-euler
Steps
1 × Δt = 1
Newton iterations
1
Factorizations
1
Run time
61 s

The computed effective conductivity is 1.2350, +0.47 % from Rayleigh's value. With a coarser spacing of 0.035 the difference is +1.6 %: it falls with refinement, as it should.

Temperature in the cell. The fibre, ten times more conductive, is almost isothermal.
Figure 5. Temperature in the cell. The fibre, ten times more conductive, is almost isothermal.
The heat flux qx on three slices: concentrated through the fibre.
Figure 6. The heat flux qx on three slices: concentrated through the fibre.

5Run It Yourself

The problem file ships with PDDO-MultiPhysics. Open it in the graphical interface and press Compute, or run it from the command line:

python -m pddo_mp gui examples/geometry/fiber_composite_conduction.yaml
python -m pddo_mp run examples/geometry/fiber_composite_conduction.yaml

Results go to results/fiber_composite_conduction/ next to the problem file: a ParaView time series (.pvd), the monitors as CSV, and the final state. Any entry can be changed on the command line, for example --set domain.spacing=0.05 for a coarser, faster run. The user manual describes every part of the problem file and of the interface.

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