## 6 Optimization Model
### Optimization Model of External Fixation Device  
For mobile design such as external fixation device, fast and reliably healing (decreasing the displacement at fracture gap) with light design is most important. Because of that, for the cost function of the optimization model, the minimum volume of design is selected:  
\[\mathrm{minf}(\mathbf{x})=\mathrm{minf}(\mathrm{x}_{1,}\mathrm{x}_{2,}\mathrm{ x}_{3})=\mathrm{minf}\sum_{i=1}^{61}\mathrm{A}_{i}\mathrm{L}_{i} \tag{15}\]  
where: \(\mathbf{x}\) is vector of reference construction parameters which are used for fixator parametrization (Table 6), A\({}_{1}\)L\({}_{i}\) is the volume of fixator elements, and i is the element of design. External fixation device has 61 elements in total. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name of the Parameter** & **Index** & **Range, mm** & **Step, mm** \\ \hline Outer tree diameter & x\({}_{1}\) & d\({}_{8}\) & 12 \(\div\) 20 & 0.5 \\ \hline Tree wall thickness & x\({}_{2}\) & \(\delta\) & 0.5 \(\div\) 1.5 & 0.1 \\ \hline Basic thickness of clamping plates & x\({}_{3}\) & \(\delta_{\mathrm{po}}\) & 1.5 \(\div\) 3 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Optimization parameters (referent design parameters) of external fixation device. For optimization parameters, referent design parameters are chosen. Parameters have the most important impact on cost function. Using numerical and experimental analysis, the most important parameters are determined. These parameters have a big influence on the mechanical stability of external fixation device, and on its volume. These parameters are optimization parameters (free parameters). Range and step can be defined for these parameters (Table 6). Using these optimization parameters and its range and steps, areas of possible design solution are reduced. Constrains of optimization parameters are defined as:  
\[ \begin{array}{l}\mathrm{x_{1d}}\leq\mathrm{x_{1}}\leq\mathrm{x_{1g}};\;12 \;\mathrm{mm}\leq\mathrm{x_{1}}\leq 20\;\mathrm{mm}\\ \mathrm{x_{2d}}\leq\mathrm{x_{2}}\leq\mathrm{x_{2g}};\;0.5\;\mathrm{mm}\leq \mathrm{x_{2}}\leq 1.5\;\mathrm{mm}\\ \mathrm{x_{3d}}\leq\mathrm{x_{3}}\leq\mathrm{x_{3g}};1.5\;\mathrm{mm}\leq \mathrm{x_{3}}\leq 3\;\mathrm{mm}\end{array}  \tag{16}\]  
The initial values of optimization parameters (Table 6) are defined at the beginning of the optimization process. The optimization algorithm uses these values, and tries to reduce its values to achieve convergences to its optimal values. If the search in one direction is successful and the local optimum is not found, the step is increased with a goal to increase the speed of the algorithm. Optimization design constrains in the form of stress and deformation constrains are defined in FEM model. The constrains of optimization model are defined in a form of interfragmentary displacements at fracture gap and as dimensional and stress constrains of external fixator device and its components. Dimensional constrains are defined using already mentioned ranges for optimization parameters. Allowed displacements at fracture gap are significantly reduced in comparison to initial design. This is done because small displacements increase the speed of bone healing. The constrains of interfragmentary displacement are treated as main constrain (Constraint 1). The equation of constrain of proximal segment displacement at fracture gap (Figure 12) is:  
\[\mathrm{g_{1}}(\mathbf{x})=\delta_{\mathrm{p1}}-\delta_{\mathrm{d}}\leq 0 \tag{17}\]  
where: \(\delta_{\mathrm{p1}}\) is maximal resulted displacement of proximal bone segment at fracture gap, and \(\delta_{\mathrm{d}}=1-2\;\mathrm{mm}\) is recommended for the maximal displacements of proximal bone segment at fracture gap for given fracture gap size [27]. Figure 12: Optimization constrains on FEM model. In the optimization model of the external fixation device, two stress constrains are given:  
* At control place at the middle of fixator, tree local sensor for von Mises stress (local sensor-von Mises stress) is defined.