Problem Formulation

Heat Distribution in a Thin Metal Plate

The generalized Poisson equation for heat distribution, considering a non-constant thermal conductivity $k(x, y)$, is given by:

$$\nabla \cdot (k(x, y) \nabla u(x, y)) = -q(x, y),$$

where:

• $\nabla \cdot (k(x, y) \nabla u(x, y))$ represents the divergence of the heat flux.
• $q(x, y)$ is the heat source distribution within the plate.
• $x$ and $y$ are the spatial coordinates.
• $u(x, y)$ is the temperature distribution.

In many practical scenarios, especially with metal plates, certain simplifications can be made. Most metals used in engineering applications are both isotropic and homogeneous. This means that:

• Isotropic Thermal Conductivity: The thermal conductivity $k$ does not depend on the direction of heat flow within the material, simplifying the modeling process.
• Homogeneous Material: The material properties, including thermal conductivity, are uniform throughout the plate, meaning $k(x, y) = k$ is constant across the domain.

Given these properties, the thermal conductivity can be reasonably approximated as a constant. This leads to a simplified version of the Poisson equation:

$$\Delta u(x, y) = -\frac{q(x, y)}{k}$$

where $\Delta u(x, y)$ is the Laplacian of the temperature function, and $k$ is now treated as a constant value.

Task

Solve a 2D Poisson equation to model heat distribution in a thin plate using Neural Networks.

The problem requires capturing spatially structured features, and the solution should be smooth and continuous, reflecting the physical properties of heat distribution.

Data

Data that was used for training and validation can be found at Google Drive.

References

• This repo is based on the code kingly provided by CERFACS in their PlasmaNet repository.