# Sintering of Two Viscoelastic Particles: A Computational Approach
## 4 Numerical Method  
### Moving Domain  
To capture the motion of the moving domain \(\Omega\), the position of the surface \(\Gamma\) is predicted from previous time steps using:  
\[\hat{\mathbf{x}}_{\Gamma}^{n+1}=\mathbf{x}_{\Gamma}^{n}, \tag{22}\]  
for the first time step and:  
\[\hat{\mathbf{x}}_{\Gamma}^{n+1}=2\mathbf{x}_{\Gamma}^{n}-\mathbf{x}_{\Gamma}^{n-1}, \tag{23}\]  
for all subsequent time steps. Herein, \(\hat{\mathbf{x}}_{\Gamma}^{n+1}\) is the prediction of the surface position for time \(t^{n+1}\), and \(\mathbf{x}_{\Gamma}^{n}\) and \(\mathbf{x}_{\Gamma}^{n-1}\) are the surface positions at time \(t^{n}\) and \(t^{n-1}\), respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Symbol** & **Value** & **References** \\ \hline Density & \(\rho\) & \(\mathcal{O}(840)\) kg/m\({}^{3}\) & [23; 24] \\ Viscosity & \(\eta_{0}\) & \(\mathcal{O}(400)\) Pa\(\cdot\)s & [25] \\ Surface tension & \(\gamma\) & \(\mathcal{O}(0.03)\) N/m & [24; 25] \\ Relaxation time & \(\lambda\) & \(\mathcal{O}(0.05)\) s & [25] \\ Particle radius & \(R\) & \(\mathcal{O}(3\times 10^{-5})\) m & [25] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of polyamide 12 (PA12) powder for SLS at \(T=175\,^{\circ}\mathrm{C}\) (melt). Next, the displacement of the fluid mesh is calculated from the surface displacement using a Laplace equation [26]. For the first time step, the mesh velocity in each node is calculated using a first-order backwards differencing scheme,  
\[\mathbf{u}_{\text{m}}^{n+1}=\frac{\mathbf{x}_{\text{m}}^{n+1}-\mathbf{x}_{\text{m}}^{n}}{ \Delta t}, \tag{24}\]  
whereas for subsequent time steps, a second-order backwards differencing scheme is used:  
\[\mathbf{u}_{\text{m}}^{n+1}=\frac{\frac{3}{2}\mathbf{x}_{\text{m}}^{n+1}-2\mathbf{x}_{ \text{m}}^{n}+\frac{1}{2}\mathbf{x}_{\text{m}}^{n-1}}{\Delta t}. \tag{25}\]  
Herein, \(\mathbf{u}_{\text{m}}^{n+1}\) is the mesh velocity at time \(t^{n+1}\), and \(\mathbf{x}_{\text{m}}^{n+1}\), \(\mathbf{x}_{\text{m}}^{n}\) and \(\mathbf{x}_{\text{m}}^{n-1}\) are the mesh coordinates at time \(t^{n+1}\), \(t^{n}\) and \(t^{n-1}\), respectively. Subsequently, the governing equations and boundary conditions are discretized on the newly-defined domain \(\Omega\). The momentum and mass balance are multiplied with the test functions \(\mathbf{v}\) and \(q\) in the appropriate function spaces:  
\[(\mathbf{v},-\
abla\cdot\mathbf{\sigma}) =\mathbf{0},\quad\text{for all}\,\mathbf{v}, \tag{26}\] \[(q,\
abla\cdot\mathbf{u}) =0,\quad\text{for all}\,q. \tag{27}\]  
Note that \((\cdot,\cdot)\) defines the inner product on \(\Omega\). Using partial integration and Gauss' theorem, we obtain:  
\[\left((\
abla\mathbf{v})^{\text{T}},\mathbf{\sigma}\right) =\left(\mathbf{v},\mathbf{\sigma}\cdot\mathbf{n}\right)_{\Gamma},\quad\text{ for all}\,\mathbf{v}, \tag{28}\] \[(q,\
abla\cdot\mathbf{u}) =0,\quad\text{for all}\,q. \tag{29}\]  
Herein, \(\left(\cdot\cdot\right)_{\Gamma}\) defines the inner product on \(\Gamma\). The right-hand side of Equation (28) can be filled using Equation (9), which is rewritten using partial integration and [PERSON]'s surface divergence theorem [27],  
\[\left(\mathbf{v},\mathbf{\sigma}\cdot\mathbf{n}\right)_{\Gamma}=-\left((\
abla_{\text{s} }\mathbf{v})^{\text{T}},\gamma\mathbf{I}_{\text{s}}\right)_{\Gamma}+\left((\
abla_{ \text{s}}\cdot\mathbf{n})\mathbf{n},\gamma\mathbf{I}_{\text{s}}\cdot\mathbf{v}\right)_{ \Gamma}+\left(\mathbf{b},\gamma\mathbf{I}_{\text{s}}\cdot\mathbf{v}\right)_{\partial \Gamma},\quad\text{for all}\,\mathbf{v}, \tag{30}\]  
where \(\mathbf{b}=\mathbf{t}\times\mathbf{n}\) is the binormal and \(\left(\cdot,\cdot\right)_{\partial\Gamma}\) defines the inner product on the boundary of the surface \(\partial\Gamma\), i.e., the locations where the surface \(\Gamma\) meets the symmetry axis \(\Gamma_{\text{sym}}\). Since the surface tension always acts tangential to the surface, \(\mathbf{n}\cdot\gamma\mathbf{I}_{\text{s}}=0\).