## 3 Numerical Experiments
### Single-Material  
For the two single-material cases, we used the parameters described below in addition to those listed above. A Young's modulus of \(E=210{,}000\) and \(\
u=0.3\) is used for the material. The weighting is selected as \(w_{c}=0.6\), \(w_{r}=0.4\) for the box case and \(w_{c}=0.5\), \(w_{r}=0.5\) for the cantilever. The regularization term is activated after iteration 30 to ensure that it does not dominate the optimization at the beginning. The level set function is redistanced every 5 iterations. The \(c\) parameter of the MMA algorithm is increased from a starting value of 1.0 to 100.0 within 150 iterations by means of a linear ramp. The latter results in a slow and smooth volume reduction, which gives the optimization method sufficient time to shape the design according to the objective function.  
Figure 13a shows the initial hole structure used to initiate the level set function, and Figure 13b shows the corresponding optimization result at convergence at iteration 107 of the box in the single-material case. It is evident that four main struts are formed to transfer the loads during the optimization process. These transfer the load directly from the point of application to the support points. In addition, cross struts develop in the form of a cross, which prevent the three floating supports from sliding sideways. The optimization result is in line with the state of the research (see [46]). However, it should be noted that depending on the optimization parameter (e.g., the \(c\) parameter of the MMA algorithm), the optimization sometimes reaches a different local optimum. This results in a rectangular cross strut structure between the four main struts instead of the cross-shaped cross strut structure.  
In addition to comparing the optimization results with the current state of research, further validation activities were carried out. Unit tests were developed for the individual program building blocks. In addition, tests were implemented and then visualized and compared using Paraview. Furthermore, the FEM was compared with the validated FEM software Abaqus ([https://www.3 ds.com/products/simulia/abaqus](https://www.3 ds.com/products/simulia/abaqus)). Abaqus was also used to validate the compliance objective function and the volume constraint.  
Figure 14a,b show the filtered history of the objective function and the volume constraint, respectively. In order to filter out the numerical fluctuation introduced by the remeshing process, a moving-average filter is applied that takes into account the five past iterations. The history of the volume constraint shows the uniform decrease in volume over time.  
Figure 15a shows the initial hole structure of the cantilever in the single-material case and Figure 15b the resulting optimization result at iteration 60. The corresponding curves of the filtered objective function and volume constraint are shown in Figure 16a and Figure 16b, respectively.  
The optimization result shows the usual truss structure consisting of triangles and agrees with the results from the state-of-the-art research (see [13]).  
Figure 13: (**a**) Initial hole structure of the box in the single-material case. (**b**) Optimization result for the box at iteration 107.  
In summary, it can be said that the developed optimization method provides plausible results in the single-material case, which are also consistent with the results from the current state of research. As a result, the developed method is considered validated for the single-material case.