## 8 Results  
The goal of the developed KBE system is to achieve better design of the external fixation device without major design changes. The volume of the external fixation device is reduced from 1.143 \(\times\) 10\({}^{5}\) mm\({}^{3}\) to 1.032 \(\times\) 10\({}^{5}\) mm\({}^{3}\). The best solution for cost function is achieved in the sixteenth iteration of the SA algorithm (Figure 14). After that, the algorithm tries to find a better solution (from 16. to 33. iteration), but constrains go out of the allowed range. After finding the optimal solution using the global algorithm, CG algorithm is activated. CG algorithm did not improve the optimal solution significantly; the value of cost function decreases from 1.036 \(\times\) 10\({}^{5}\) mm\({}^{3}\) to 1.032 \(\times\) 10\({}^{5}\) mm\({}^{3}\) (Figure 14). Changes in optimization parameters (design parameters), during the optimization process, are shown in Figure 15. The diameter of fixator tree is increased ds (56.37%), and at the same time, tree wall thickness \(\delta\) (35.07%) and basic thickness of clamping plates \(\delta_{\text{op}}\) (34.5%) are decreased (Figure 15 and Table 7). In addition, the values of interfragment  
Figure 14: Optimization of volume of external fixation device. Figure 13: Algorithm of developed KBE system. tary displacement at fracture place R, which indicate the stability of fracture place, are decreased for 67.25% (Table 7). The SA algorithm, after 33 iterations, completely converges to the best solution of cost function (Figure 14), and achieves the optimal solution of analyzed design parameters, which minimizes the value of the external fixation device volume (Figure 15). In addition, optimization constrains in the form of displacements and von Mises stresses are fulfilled (Figures 16 and 17). The CG algorithm did not have significant influence on the optimization parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Properties/Parameter** & **Index** & **Units** & **Initial Design** & **Optimized Design** \\ \hline Volume & f(x), V & mm\({}^{3}\) & \(1.143\times 10^{5}\) & \(1.036\times 10^{5}\) \\ \hline Outer tree diameter & x\({}_{1}\), d\({}_{8}\) & mm & 12 & 18.764 \\ \hline Tree wall thickness & x\({}_{2}\), \(\delta\) & mm & 1.5 & 0.974 \\ \hline Basic thickness of clamping plates & x\({}_{3}\), \(\delta_{\text{op}}\) & mm & 3 & 1.965 \\ \hline Section modulus of the fixator connecting rod & \(Z\) & mm\({}^{3}\) & 115.97 & 241.82 \\ \hline [PERSON] stress at control place MP\(-\) (pressure) & g\({}_{2}\)(x), \(\sigma_{\text{vm}-}\) & MPa & 354.3 & 183.5 \\ \hline [PERSON] stress at control place MP+ (tensile) & \(\sigma_{\text{vm}+}\) & MPa & 330 & 162 \\ \hline [PERSON] stress at clamping plate & g\({}_{3}\)(x), \(\sigma_{\text{vm,p}}\) & MPa & 450 & 517.8 \\ \hline Maximal axial displacement at the place of load & d & mm & 4.18 & 1.44 \\ \hline Components of displacement vector of proximal & D\({}_{\text{p}(x)}\) & mm & 0.53 & 0.327 \\ bone segment at the place of fracture. & D\({}_{\text{p}(y)}\) & mm & 4.14 & 1.15 \\ \cline{2-5}  & D\({}_{\text{p}(\alpha)}\) & mm & \(-4.36\) & \(-1.5\) \\ \hline Maximal resulted displacement of proximal bone & g\({}_{1}\)(x), D\({}_{\text{p}}\) & mm & 6.28 & 1.99 \\ \hline Components of displacement vector of distal bone & D\({}_{\text{d}(y)}\) & mm & 0.53 & 0.327 \\ \cline{2-5} segment at the place of fracture. & D\({}_{\text{d}(y)}\) & mm & 4.29 & 1.16 \\ \cline{2-5}  & D\({}_{\text{d}(\alpha)}\) & mm & 0.22 & 0.061 \\ \hline Maximal interfragmentary displacement at the place & R & mm & 4.58 & 1.5 \\ \cline{2-5}   of fracture & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison between initial and optimized design of external fixation device. Figure 15: Changes of optimization parameters.