Structural Simulation in Fused Filament Fabrication (FFF)#
Mathematical Model:
\[\begin{split}
\begin{align*}
&\text{Mechanical Deformation:} \quad \rho \frac{\partial^2 \mathbf{u}}{\partial t^2} - \nabla \cdot (\boldsymbol{\sigma}) = \mathbf{0} \\
&\text{Constitutive Relation:} \quad \boldsymbol{\sigma} = \mathbf{D} : \boldsymbol{\varepsilon}
\end{align*}
\end{split}\]
Weak Formulation:
\[
\begin{align*}
&\int_{\Omega} \left(\rho \frac{\partial^2 \mathbf{u}}{\partial t^2} \cdot \mathbf{v} - \nabla \cdot (\boldsymbol{\sigma}) \cdot \mathbf{v}\right) \,d\Omega = 0
\end{align*}
\]