# Shape Optimization of Structures by Biological Growth Method
## 3 The 3D Structural Element Optimization
### Shape Optimization of 3D Structural Element (CAO Method)  
The objective of the shape optimization for the structural element under study was to minimize the maximum stress variations in the design elements. No optimization or geometric constraints were defined in the above example. A model was used for the shape optimization, the mesh of which is shown in Figure 12b. The result of the shape optimization of the structural element in the computational cycle \(n=20\) is shown in Figure 14a. In order to perform a static analysis of the structural element after shape optimization, it was necessary to extract its mesh from the results. Also in this case, the directly extracted mesh with predefined dimensions and arrangement of elements was used, with subsequent modification in the form of manually moving some nodes to such a position to obtain smoother design lines. The 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 shape optimization is shown in Figure 14b. It consisted of 14,385 elements and 23,939 nodes. The last step of the optimization by the biological growth method was to perform a static analysis under the given loading conditions but this time for a model that had undergone both topological and shape optimization. To compare the results with those of previous model designs (Figures 11 and 13b), we kept the same outputs: the field of equivalent stresses according to [PERSON] theory (Figure 15b). Figure 14: (**a**) Motion of 3D structural element nodes during shape optimization. (**b**) Finite element mesh of 3D structural element model after shape optimization. In Figure 15b it can be seen that after the shape optimization, the maximum stress dropped to \(\sigma_{max}=127\) MPa. The comparison of the stress distribution before and after the shape optimization is shown in Figure 15. For a good visualization, the upper limit of the color contours was set to 120 Mpa. From Table 2 it can be deduced that the main objective of shape optimization was to reduce the maximum stress or to distribute the equivalent stress more homogeneously along the contour lines of the design. The maximum stress in the investigated design element was reduced by the shape optimization by 67.44%. In Figure 15b it can be seen that most of the optimized region (the gray part in Figure 10a) was loaded with a stress in the range \(\sigma=50\div 70\) MPa. After the shape optimization, the maximum stress (\(\sigma_{max}=127\) MPa) was significantly closer to this interval, and thus, the stress in the structure was distributed more homogeneously than is the case in Figure 13b. In this case, the shape optimization had a minimal effect on the change in mass (or volume). The mass of the component increased by 0.84%, confirming the claim that the shape optimization had no direct effect on it and its change was random in nature. From Figure 15, it is clear that the optimized shape still needs to be modified by the designer for production purposes in order to smooth out surface irregularities that create local indentations that are unsuitable from aesthetic, production, and user points of view. In this process it is necessary to smooth the surface so that material is not removed from the part and any sharp edges are not created causing new indentations with consequent stress concentration. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Parameters of the Design Element**} & \multicolumn{1}{c}{**The Difference**} & \multicolumn{1}{c}{**The Difference in \%**} \\ \hline
**Before Shape Optimization** & **After Shape Optimization** & **The Difference in \%** \\ \hline \(\sigma_{max}=390\) MPa & \(\sigma_{max}=127\) MPa & \(\Delta\sigma_{max}=-263\) MPa & \(-67.44\) \\ \hline \(m=0.119\) kg & \(m=0.120\) kg & \(\Delta m=+0.001\) kg & \(+0.84\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of 3D structural element results before and after shape optimization.