Heat solid

Strong formulation

Heat transfer in solids is a thermal conduction process that is governed by the equation

\begin{equation} \rho C_p \frac{\partial T}{\partial t} = -\nabla \cdot \boldsymbol{q} + Q, \end{equation}

where

$\rho$ is the material density ( $kg/m^3$ )
$C_p$ is specific heat capacity of the material ( $J/kg \cdot K$ )
$\boldsymbol{q}$ is the heat flux density ( $W/m^2$ )
$Q$ is the volumetric heat source ( $W/m^3$ )
$T$ is the Temperature field ( $K$ ).

Constitutive equation

Heat flux density due to conduction is given by Fick’s law of diffusion

\begin{equation} \boldsymbol{q} = -\kappa \nabla T, \end{equation}

where $\kappa$ is the thermal conductivity of the material ( $W/m^2 K$ ).

The governing equation $(1)$ then becomes

\begin{equation} -\rho C_p \frac{\partial T}{\partial t} + \nabla \cdot (\kappa \nabla T) + Q = 0. \end{equation}

Weak formulation

The weak form of solid mechanics is obtained by multiplying the partial differential equation in $(3)$ with the test function of Temperature field $T^\prime$ , and integrating over the whole domain $\Omega$ to get

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; \nabla \cdot (\kappa \nabla T) \; T^{\prime} \; d\Omega + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Applying Leibniz rule on the divergence term, we get

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Omega}\; \nabla \cdot (\kappa \nabla T \; T^{\prime}) \; d\Omega + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Applying Divergence theorem on the divergence term, we get

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Gamma}\; (\kappa \nabla T \cdot \boldsymbol{n}) \; T^{\prime} \; d\Gamma + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Substituting $(2)$ into the boundary term, we get the final weak formulation

\int_{\Omega}\; -\rho C_p \frac{\partial T}{\partial t} \; T^{\prime} \; d\Omega + \int_{\Omega}\; -\kappa \nabla T \cdot \nabla T^{\prime} \; d\Omega + \int_{\Gamma}\; (-\boldsymbol{q} \cdot \boldsymbol{n}) \; T^{\prime} \; d\Gamma + \int_{\Omega}\; Q \; T^{\prime} \; d\Omega = 0.

Available Interactions

Constraint

Applies a fixed temperature $T$ to a selected region in Kelvins. This enforces a Dirichlet boundary condition for the heat equation, keeping the temperature constant at the specified value. Use this to setup fixed-temperature boundaries for the model.

Heat source

Applies a heat source $Q$ to a selected region, allowing you to model energy input or removal in a heat transfer simulation. This enforces a Neumann boundary condition in the context of finite element formulation. Zero heat flux boundary condition is automatically applied at the boundaries.

Periodicity

Imposes periodic boundary conditions on scalar field $T$ . Reduces computational domain size for symmetric problems.

Convection

Lump T/Φ

Couplings to Other Physics

This formulation supports the following couplings:

Joule heating (Current flow or Magnetism H)
Models resistive heat generation caused by electrical currents.