Heat Conduction Module

The heat conduction module solves the following model

\[C\frac{\partial T}{\partial t} - \nabla \cdot \kappa\nabla T = Q\]

where \(C(\mathbf{r},t)\) is the heat capacity, \(\kappa(T,t)\) is the thermal conductivity tensor, \(T(\mathbf{r},t)\) is the temperature, and \(Q(\mathbf{r},t)\) is the heat source. Boundary conditions include:

• Dirichlet boundary conditions, where the temperature is fixed at the boundary.

\[T = f(t)\]

• Neumann boundary conditions, where the heat flux is fixed at the boundary.

\[\mathbf{J}\cdot \mathbf{\hat{n}} = f(T,t)\]

• Robin boundary conditions, where a linear combination of temperature and heat flux is fixed at the boundary.

\[\mathbf{J}\cdot \mathbf{\hat{n}} = h(t)\left[T - T_0(t)\right]\]

where \(h(t)\) is the heat conductance and \(T_0(t)\) is the reference temperature.

Nonlinear boundary conditions are also supported, where the boundary condition depends on the temperature at the boundary. For example, thermal radiation can be modeled as:

\[\mathbf{J}\cdot \mathbf{\hat{n}} = \sigma(T^4 - T_0^4)\]

where \(\sigma\) is the Stefan-Boltzmann constant, and \(T_0\) is the reference temperature. The module support linear and nonlinear heat conduction, as well as steady state and transient transport.

Note

The model is fully differentiable, and enables to take gradients against any variable listed above, including time-dependent and nonlinear boundary conditions.