## 3 Numerical Experiments
### Cases  
The first case is a cube with side length 80 with chamfered edges on the top (see Figure 11). The case serves as an example for almost arbitrary design spaces of the optimization method. However, it can be compared at the same time with the classic example of the box without chamfering from the literature. The box is supported at four bearing points on the underside. The displacement of the vertices of one of these supports is restricted in all three spatial directions (\(u_{x}=u_{y}=u_{z}=0\)). The vertices of the other supports can move freely in the plane but not in the z-direction (\(u_{z}=0\)). The square load application and constraint surfaces are slightly protruding. The first element layer adjacent to the surfaces is always considered as a non-design space in the optimization and cannot be modified.  
Figure 10: Pre-processing step for remeshing the blue design area. Determination of the convex hull (orange), offset calculation of the convex hull (gray) with subsequent meshing.  
The second case is a cantilever clamped on one side (see Figure 12), which is subjected to a force \(F\) in the negative z-direction on the front surface. Two square surfaces with a side length of 15 length units are used for clamping. The vertices of the upper surface are completely fixed in all three spatial directions (\(u_{x}=u_{y}=u_{z}=0\)) and the vertices of the lower surface are only fixed in the x and y directions (\(u_{x}=u_{y}=0\)). Only half of the design domain is considered with an additional symmetry boundary condition so that the result of the optimization is a symmetrical design. The dimensions of the design domain can be taken from Figure 12. As with the box, the load application and constraint surfaces are slightly protruding, and the first element layer is a non-design space.  
For both the single and multi-material cases, \(v_{min}=-0.8\) and \(v_{max}=0.8\) are used. A normal velocity of \(-0.2\) is specified for the non-design spaces described above. The filter parameter is selected as 1.25 times the average edge length, and the target edge length is \(l_{tar}=2.0\). The target volume fraction is set to 15% of the design space. In the multi-material case, this is divided equally between the two materials so that \(f_{V1}\leq 0.075\) and \(f_{V2}\leq 0.075\) applies.  
To determine the convergence of the optimization, a convergence criterion is evaluated after the optimization has been performed. This takes into account the five previous iterations in the sense of a moving average and uses this filtered value to calculate the per  
Figure 11: Design space of a box with chamfered edges, which is supported at four locations in the z-plane and has a load application location on the surface with a force in the negative z-direction in the amount of \(F\).  
Figure 12: Symmetrical design space of the cantilever beam with a fixed support (\(u_{x}=u_{y}=u_{z}=0\)), a floating support (\(u_{x}=u_{y}=0\)) and a load application point on the front side with a force in the negative \(z\)-direction in the amount of \(F\).  
centage change between two points. In the single-material case, the limit for the percentage change is set at \(1\times 10^{-4}\) and \(1\times 10^{-3}\) in the multi-material case.