# Multi-Material Topology Optimization Taking into Account the Position of Material Interfaces in 3D
## 4 Numerical Experiments
### Cantilever  
As a first case, we consider the cantilever shown in Figure 4. This is loaded on the front surface with a load \(F\) in negative z-direction and is supported on the rear surface. The load application and the support surfaces are square and protrude slightly. They are defined as non-design space for the optimization. For the nodes of the upper support surface, \(u_{x}=u_{y}=u_{z}=0\) and for the nodes of the lower support surfaces, \(u_{x}=u_{y}=0\). As the aim is to obtain symmetrical designs, only half of the design space is simulated and a symmetry boundary condition is defined on the symmetry plane (see Figure 4). Two different optimizations are carried out below. One multi-material topology optimization without taking the ITED into account and one with it. The parameters for the two optimizations can be found in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(E_{1}\) & \(E_{2}\) & \(\
u_{1}\) & \(\
u_{2}\) & \(v_{min}\) & \(v_{max}\) & \(v_{nds}\) & \(\alpha\) \\ \hline
70,000 & 35,000 & 0.3 & 0.3 & \(-0.3\) & 0.3 & \(-0.2\) & 0.01 \\ \hline \end{tabular}
\end{table} 
Table 1: General parameters for all cases. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\overline{f}_{V1}\) & \(\overline{f}_{V2}\) & \(\beta\) & \(w_{c}\) & \(w_{ited}\) & \(w_{ri}\) & \(w_{rs}\) & \(c(MMA)\) & \(l_{tar}\) \\ \hline
0.075 & 0.075 & \(-\) & 0.8 & \(-\) & 0.15 & 0.05 & 450.0 & 2.0 \\ \end{tabular}
\end{table} 
Table 2: Parameters for the cantilever. Pure multi-material topology optimization (top). With consideration of ITED (bottom). Figure 5a shows the optimization result of the pure multi-material topology optimization and in Figure 5b the optimization result with consideration of the ITED. It is clearly visible that the pure multi-material topology optimization result (see Figure 5a) distributes the second material (lower modulus of elasticity) over the entire cross-section (along the z-direction). The second material is present both in the tensile and compressive areas. The same applies to the first material. Consequently, the interfaces are also present both in the compression area and, in particular, in the tensile area (upper interfaces). As described above, the latter has a negative effect on the component strength. The optimization result with consideration of the ITED (see Figure 5b) shifts the second material into the compressive area of the beam, whereby all three existing interfaces are primarily subjected to pressure, which corresponds to the intended behavior. The latter is also illustrated by Figure 6. This shows the density distribution of the normal and shear stresses on the interfaces for the two optimizations. In the first step, the interface surfaces of the tetrahedrons were derived, the three stresses were determined on them and the diagrams were created from this. In each of these, the stress values are plotted on the x-axis and the respective occurrences on the y-axis. Figure 4: Design space of the cantilever beam with a symmetry plane. It is fixed (\(u_{x}=u_{y}=u_{z}=0\)) at the top on the back and is floating (\(u_{x}=u_{y}=0\)) at the bottom on the back. The force \(F\) in the negative \(z\)-direction is applied on the front surface [15]. Figure 5: Optimization results of the cantilever case. (**a**) Multi-material optimization (iteration 777). (**b**) Multi-material optimization with the ITED and \(\beta=0.25\) (iteration 959). The optimization with consideration of the ITED (**b**) shifts interfaces out of high-tension and high-shear regions. The diagram of the normal stress \(\sigma_{nn}\) clearly shows that the interfaces are subjected to greater compressive stress when the ITED is taken into account (density distribution shifts from the positive area to the negative area). In addition, the maximum tensile stress (maximum value in the positive range) is reduced compared to pure multi-material topology optimization. Consequently, in the case of the cantilever, the developed method fulfills requirement 1.