# Multi-Material Topology Optimization Taking into Account the Position of Material Interfaces in 3D
## 4 Numerical Experiments
### Bracket  
The second case is the bracket shown in Figure 7, which is often used in the topology optimization literature (see [39]). As a load, a force acting in the negative y-direction is applied via the inner surface of the lower left perforated cylinder. The inner surface of the upper right cylinder serves as the support. The following applies to the nodes of the inner surface: \(u_{x}=u_{y}=u_{z}=0\). The drilled-through cylinders on the inner surfaces of which the load is applied and the support is provided are considered as non-design space in the context of the optimization. The dimensions can be taken from Figure 7.  
Analogous to the cantilever, the comparison between pure multi-material topology optimization and multi-material topology optimization with consideration of the ITED is shown below. The parameters of the two optimizations can be found in Table 3.  
Figure 8a shows the result of the pure multi-material topology optimization and in Figure 8b that with consideration of the ITED. Analogous to the cantilever, it can again be seen that the second material is distributed over the cross-section (y-direction) in the pure multi-material topology optimization and that the second material is primarily located in the upper area when the ITED is taken into account. It is also noticeable that the second material consists of two components in the optimization with consideration of the ITED compared to one in the pure multi-material topology optimization.  
Figure 9 again shows the normal and shear stress density distributions on the interfaces of the two optimization results. Analogous to the cantilever, it can again be seen that the tensile stress on the interfaces decreases when the ITED is taken into account. On the one hand, the maximum normal stress decreases (maximum positive value) and the majority of the density distribution is shifted to the left. Consequently, in the case of the bracket, the developed method also fulfills requirement 1.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(\overline{f}_{V1}\) & \(\overline{f}_{V2}\) & \(\beta\) & \(w_{c}\) & \(w_{ted}\) & \(w_{ri}\) & \(w_{rs}\) & \(c(MMA)\) & \(l_{tar}\) \\ \hline
0.25 & 0.2 & \(-\) & 0.88 & \(-\) & 0.02 & 0.1 & 50.0 & 1.5 \\  & 0.25 & 0.78 & 0.1 & 0.02 & 0.1 & 50.0 & 1.5 \\ \hline \end{tabular}
\end{table}
Table 3: Parameters for the bracket. Pure multi-material topology optimization (top). With consideration of ITED (bottom).  
Figure 7: Design space of the bracket with a fixed support (\(u_{x}=u_{y}=u_{z}=0\)) at the top right and a load application at the lower left in the negative \(y\)-direction with the amount of \(F\). The inner radius of the two cylinders is 16 and the outer radius is 20.  
The shear stress diagrams also show that the maximum shear stress is also minimized. The density distribution of shear stress \(\sigma_{ns}\) must be considered in particular, as its values are higher than \(\sigma_{nt}\). The maximum value of the density distribution of \(\sigma_{ns}\) decreases when ITED is taken into account compared to pure multi-material topology optimization. In addition, a reduction in the variance can be observed when considering the ITED compared to the pure multi-material topology optimization. Consequently, the developed method also fulfills requirement 2 in the case of the bracket.  
Figure 8: Optimization results of the bracket case. (**a**) Multi-material optimization (iteration 162). (**b**) Multi-material optimization with the ITED and \(\beta=0.25\) (iteration 217). Material 2 (blue) in the optimization with consideration of the ITED consists of two components, whereas it consists of one in the other case.  
Figure 9: Density plots of the normal \(\sigma_{nn}\) and shear stresses \(\sigma_{ns}\) as well as \(\sigma_{nt}\) of the interface for the bracket case. When taking the ITED into account, both the tensile stress and the maximum shear stresses on the interface are reduced.