## 8 Results
Significant decreases of displacement of bone segment D\({}_{\text{P}}\) (68.31%) (Figure 16) and interfragmentary displacement D\({}_{\text{d}}\) (R) (Figure 18) are achieved by increasing the diameter of the external fixation device tree (56.37%), which results in a significant increase of the resistance moment (108.52%), and a reduction of the deformations and von Mises stresses (48.21%) at the fixator tree (Table 7). Figure 16: Reduction of displacement of proximal bone segment (D\({}_{\text{P}}\)) during optimization process. Figure 17: Difference between calculated and allowed von Mises stresses (\(\sigma_{\text{vm}-}\)) at fixator tree during the optimization process. Figure 18: Comparison diagram of displacements at the place of fracture R and at the place of load for initial and optimized design. \begin{table}
\begin{tabular}{c c} \hline \hline
**Initial Design** & **Optimized Design** \\ \hline \multicolumn{2}{c}{Fixator tree with its cross section in the middle} \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Distribution of von Mises stresses on fixator components for initial and optimized design. The initial and optimized design of external fixation device shown with displacement field are given in Figure 18. The maximal displacement of the design is marked. This displacement represents the maximal resulting displacement of proximal bone segment \(\mathrm{D_{P}}\), and at the same time, it is optimization constrain \(\mathrm{g_{1}(x)}\). The distribution of von Mises and principal stresses on initial and optimized design of external fixation device is given in Figure 20. Control places are marked: von Mises stress at control place \(\mathrm{MP}-\), (\(\mathrm{\sigma_{vm-}}\)) and von Mises stress at clamping plate (\(\mathrm{\sigma_{vm,P}}\)). These stresses are also optimization constrains at the same time--\(\mathrm{g_{2}(x)}\) and \(\mathrm{g_{3}(x)}\) (Table 7). The distribution of von Mises stress on all non-standard parts of initial and optimized design of external fixation device is given in Table 8. Using the structural optimization of the external fixation device, better distribution of von Mises stress on fixator tree is achieved, with a significant reduction of maximal value from 354.3 MPa to 183.5 MPa at control place \(\mathrm{MM}\)-, because the diameter of the fixator tree is increased. At the same time, tree wall thickness is reduced (Table 8). The optimization process of the coupling plate resulted with the decrease of coupling plate thickness from 3 mm to 1.965 mm, without significant changes in distributions and maximal values of von Mises stress (Tables 7 and 8). Moreover, the same thing happened with the coupling body; thickness on the central part is decreased (Table 8). The coupling carrier body had significant geometrical changes; the thickness of the long part of the coupling carrier body decreased from 1.5 mm to 0.974 mm; this resulted in von Mises stress increasing up to the value of allowed stress. Because of functional purposes, dimensions of diameters on this part must be adjusted to fixator tree (Tables 1 and 8). For coupling ring, the dimensions are also changed, mostly because it needs to be adjusted to new dimensions of the fixator tree and coupling carrier body. In addition, the thickness of the outer wall of coupling ring is reduced, which resulted in better von Mises stress distribution and the better use of materials (Tables 1 and 8). the loads of external fixation device in real applied conditions, and design parameters. It is necessary to choose the proper dimensions, shape, and place for all external fixation device components. Structural size optimization of external fixation device is carried out, with a goal to reduce interfragmentary displacements at the place of bone fracture, to reduce the mass of the fixator, and to achieve better distribution of von Mises stress at critical places on the design. Interfragmentary displacements and stiffnesses of external fixation device have significant influence on the speed and quality of the healing process.