Thermal Simulation in Powder Bed Fusion (PBF)

Mathematical Model:

\[\begin{split} \begin{align*} &\text{Heat Conduction:} \quad \rho c \frac{\partial T}{\partial t} - \nabla \cdot (k \nabla T) = Q \\ &\text{Melting and Solidification:} \quad \rho_l L \frac{\partial \phi}{\partial t} = H \nabla \cdot \left(\frac{\nabla T}{\lVert \nabla T \rVert}\right) \end{align*} \end{split}\]

Weak Formulation:

\[\begin{split} \begin{align*} &\int_{\Omega} \left(\rho c \frac{\partial T}{\partial t} \phi + k \nabla T \cdot \nabla \phi - Q \phi\right) \,d\Omega = 0 \\ &\int_{\Omega} \left(\rho_l L \frac{\partial \phi}{\partial t} - H \nabla \cdot \left(\frac{\nabla T}{\lVert \nabla T \rVert}\right) \phi\right) \,d\Omega = 0 \end{align*} \end{split}\]