# Shape Optimization of Structures by Biological Growth Method
## 3 The 3D Structural Element Optimization
### Topological Optimization of a 3D Structural Element (SKO Method)  
The goal of the topological optimization of the above example (Figure 10) was to minimize the deformation energy of the element, which would lead to an increase in its structural stiffness. Only one optimization constraint was set: the reduction in the design volume by 50%. In this example, three geometric constraints were also defined: 1. A so-called immobile region whose nodes would not change their position during the topological optimization, i.e., a region for which we were not looking for an optimal topology (the white part of the geometry in Figure 10a); 2. Demold control would ensure obtaining a simpler geometry (creating fewer partitions in the optimized region), which caused it to be more likely that the obtained optimized design would be manufacturable; 3. Symmetry along the plane x-y in Figure 10b. The result of the topological optimization of the structural element in the computational cycle \(n=15\) is shown in Figure 12a. The mesh of the optimized model was extracted for elements with material density above 20%. For further computations, the directly extracted mesh with predefined dimen  
Figure 11: Field of equivalent stresses according to von [PERSON] theory [Mpa] for the initial design model of a 3D structural element. Figure 12: (a) Material density for the 3D structural element during topological optimization. (b) Finite element mesh of the 3D structural element model after topological optimization. sions and element arrangement was used. The only modification of the mesh occurred by manually shifting some nodes into position to obtain smoother design lines. The nodes that were displaced were not at the locations of the stress concentrators, and therefore, their new positions did not have a significant effect on the maximum stresses in the structure. As a rule, after the optimization phase using the programs, the designer still needs to smooth the whole structure and take into account possible technological requirements of the production. The finite element mesh consisted of tetrahedron elements. An element was made up of 20 nodes (a quadratic element), with each node having six degrees of freedom. The finite element mesh of the model after the topological optimization is shown in Figure 12b. It consisted of 14,279 elements and 23,838 nodes. Figure 13a shows the stress distribution in the structural element before the topological optimization. The next optimization step was to perform a static analysis under the given load conditions but this time for a model that had already undergone topological optimization. To compare the results with the initial design of the model (Figure 11), we kept the same outputs: the field of equivalent stresses according to [PERSON] theory (Figure 13b). In Figure 13b it can be seen that a maximum stress arose for the structural element after the topological optimization, \(\sigma_{max}=390\) MPa. For a good visualization of the comparison of the results before and after optimization (Figure 13), the upper limit of the colored contours was set to \(\sigma=120\) Mpa. A clearer display and comparison of the results of the design element before and after topological optimization is described in Table 1.  
From Table 1 it can be read that the main goal of topological optimization was to reduce the weight of the structural element, but at the same time there was an increase in \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters of the Design Element** & **The Difference** & **The Difference in \%** \\ \hline \(\sigma_{max}=54\) Mpa & \(\sigma_{max}=390\) Mpa & \(\Delta\sigma_{max}=+336\) Mpa & \(+622.22\) \\ \hline \(m=0.182\) kg & \(m=0.119\) kg & \(\Delta m=-0.063\) kg & \(-34.62\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of 3D structural element results before and after topological optimization. Figure 13: Comparison of stress distribution of 3D structural element (**a**) before topological optimization and (**b**) after topological optimization.