## 5 Examples
### Allen-Cahn Equation
However, we would instead like to make use of our previous knowledge and only use a finer grid around the vicinity of the laser path--where the actual melting takes place and a finer grid is needed to appropriately resolve the flow field. As such, there is no need to use algorithms that flag candidate cells and employ remeshing at every time step. The rest of the domain can be meshed using coarser cells, as they are primarily present in the simulation to properly account for heat conduction to surrounding solidified material. A slice of the resulting discretization is depicted in Table 4.9 The actual mesh is box-shaped, and the visible regions are colored to illustrate the different phases present as well as the varying cell sizes. One can observe that the hexahedral cells become coarser with increasing depth of the powder bed. #### 5.3.3 Computing Resources

We start the selection process for suitable numerical schemes as in the previous sections by investigating the computational hardware, as stated within process P1 in Figure 9:0 For this problem, we use the same hardware given in Figure 9:1 That is, we aim to solve this problem on the 18-core CPU machine. As given in Section 4.5, such a system of desktop scale does not, per se, necessitate the use of massively parallel hardware, as the amount of processes falls below the recommended amount of 50 to 100 workers. #### 5.3.4 Problem Scale

Using the system of PDEs as input, we can proceed with analyzing the problem length scales according to process P2 in Figure 9:2 Given the coefficients and material properties of the PDE system, we note that all physics are expected to operate on a length scale of a few micrometers. We can, thus, conclude that this particular problem does not exhibit multiscale properties that would necessitate adaptivity. Therefore, in combination with the given hardware, we can negate the first decision point D1 in Figure 5 and proceed with the field-wise analysis of the PDE system. #### 5.3.5 Classification

We begin with the field-wise classification of the PDE system, that is, decisions D2 to D5, along with the corresponding processes as follows: We consider all governing equations of the system (Equations (38)-(45)) individually for each field quantity. That is, we extract the individual equations that one needs to solve. Afterward, we follow the classification methods described by P3 and P4, whereas the latter is only relevant for hyperbolic systems, as illustrated in Figure 9:3

We first examine the phase fraction fields. As all variables are governed by the same set of equations, and, thus, the resulting classification applies to every field, we abbreviate

Figure 9:4 Slice of the computational domain for the laser powder bed fusion test case. The grid consists of hexahedral elements with varying sizing. Colors denote different phases in the initial condition. This figure is adapted from [97]. the discussion by only classifying one phase fraction \(\alpha\). The relevant governing equation then is exactly the passive advection equation that was introduced in Section 5.2. Figure 9:5

It then immediately follows from the previous discussion that the phases are governed by first-order, linear, hyperbolic systems. We consequently arrive at the discontinuous Galerkin method for these variables. Next, we will deal with the velocity vector \(\underline{u}\). Extracting all components from Equation (36) that contain differentials of \(\underline{u}\), we have

Figure 9:6

This system of equations, for each component, possesses the exact structure of the heat diffusion equation, which is known to be a model parabolic equation. Following the more rigorous approach given in Section 4.2, we would obtain for the coefficient matrix \(a_{ij}\)

\[a_{ij}^{\underline{u}}= \begin{bmatrix}0&0&0\\ 0&-\eta&0\\ 0&0&-\eta\end{bmatrix} . \tag{48}\]

The first zero-diagonal element is due to the time component only appearing in a first, but not in any second, derivative.