Heat solid

Strong formulation

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

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

where

• $\boldsymbol{q}$ 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

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

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

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

Applying Leibniz rule on the divergence term, we get

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

Applying Divergence theorem on the divergence term, we get

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

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

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