## 5 Examples
### Allen-Cahn Equation
Most prominently, research has shown that the accurate representation of surface tension forces due to temperature gradient plays a key role in obtaining realistic results regarding the morphology of the solidified tracks, temperature gradients, and presence of pores--all of which have shown to be important indicators for process quality [100]. The multiphase problem that arises from this application is discretized using the volume of fluid method. The equations for this model were introduced in Section 5.2. In total, we have one solid metallic, one liquid metallic, and two gaseous phases for vaporized metal and the shielding gas, yielding four phase fractions that need to be tracked. Following the previous discussion in Section 5.2, one may omit to model one of these phases using a conservation law as one can calculate it using the compatibility condition in Equation (27) as well. With the phase fractions in place, one then obtains the mixed material constants at a given degree of freedom by a rule of mixture law. For thermal conductivity \(\kappa\), for instance, the phase-averaged value at DoF \(i\) is

\[\kappa_{\text{VOF}}=\sum_{k=0}^{N_{\text{phases}}}\kappa_{k}\alpha_{k}. \tag{35}\]

We continue by enumerating the governing equations of this problem. For the melting of metallic materials, one finds that the flow field can be described by the incompressible Navier-Stokes equations in the low Reynolds number regime, plus some source terms that account for the additional physics:

\[\partial_{t}(\rho\underline{u})+\
abla\cdot\big{[}(p-p_{\text{recoil}}) \underline{l}\big{]}-\rho\underline{k}-\eta\Delta\underline{u}+\
abla\cdot \underline{\Gamma}_{\text{capillary}}=0, \tag{36}\]with the additional condition for incompressibility

Table 4.0

Here, \(\underline{k}\) is the vector of gravity and other volumetric forces. One important driving force for melt flow is the temperature-dependent surface tension force, also called the Marangoni force. We can summarize all capillary forces normal and tangential to the interface using the so-called capillary stress tensor

Table 4.1

where \(\sigma\) denotes the temperature-dependent coefficient of surface tension, \(I\) is the identity tensor, and \(n\) is the unit normal vector to the capillary surface. The divergence of this quantity represents the capillary force acting on interfaces [101]. This, in turn, needs to be computed using the phase fractions and the surface gradient operator \(\
abla_{s}\)

Table 4.2 Table 4.3

For the gaseous phase of vaporized material, we additionally consider the recoil pressure

Table 4.4

The only conserved quantity left is the total energy of the system. The corresponding balance equation that governs the evolution of temperature reads

Table 4.5

with the following (nonlinear) source terms for the laser heat input, vaporization loss, and radiation loss, respectively

Table 4.6 Table 4.7 (44) Table 4.8

The laser that provides \(Q_{\text{Laser}}\) travels along a straight line throughout the domain with constant velocity and laser power \(P\) with beam radius \(r_{0}\). Thus, this source term represents a transient heat source where its position can be easily interpolated given the current time. Further details on the physical models can, for example, be found in [97]. #### 5.3.2 Computational Domain

We solve the given problem on a grid that is overall three-dimensional and hexahedral in shape. The bounding box has dimensions \(2.0\,\mathrm{mm}\) in length, \(0.3\,\mathrm{mm}\) in width, and \(0.5\,\mathrm{mm}\) in height. In this problem setting, steep temperature gradients are to be expected due to the concentrated heat input. As a result, it makes sense to employ a finer grid at locations where the melt pool dynamics take place. One possible way to account for that is to use a discretization technique that can incorporate adaptive grids.