2024-05-06 by Fritz Crusius
Project means to fill a gap in publications solving elliptic PDEs in
steady state without heat generation. Finite-difference examples I found
assumed homogeneous materials, whereas the challenge in civil engineering
is to model thermal bridge effects of component inhomogeneities.
This project serves as as compendium to a German blog article on heat bridges
https://energieoekonom.de/waermeleitung-finite-difference/.
minimal_system.py
The minimal system example reflects the explanation of setting up a
finite-difference system of linear equations, and solving it for node temperatures.
verification_system.py
For verification of correctness and accuracy of results, a comparison of
simulated and theoretical values across a wall of 10 cm thickness consisting
of two layers of 4 cm of solid brick and an in between insulation layer of
2 cm polyurethane.
This project also demonstrates how heat conductivities reflected in a
finite-difference-grid should be scaled to take account of the actual
dimensions of a modeled component. Furthermore, the project shows how
to model the inner room and outer environment air.
png_model_2d.py
Creates a heat conductivity map from an input PNG grayscale image and a mapping
CSV file, giving luminosities a heat conductivity. Was used to create the
cross-section2.*.png result images.
Images in root directory were created using the following command line:
png_model_2d.py -pixel_step 10 cross_sections/cross-section2.png
For the low resolution gradient field image, a wider pixel step of 50 was used:
png_model_2d.py -pixel_step 50 cross_sections/cross-section2.png