# Novel Bionic Design Method for Skeleton Structures Based on Load Path Analysis
## 3 Case Study
### Multi-Objective Size Optimization  
Based on the above design, the parametric model of control arm is established and the multi-objective size optimization is carried out. This case takes the maximum structural stiffness and strength and the minimum mass as optimization objective, and several important dimensions as design variables are shown in Figure 14, including the diameter of lightening holes \(\phi\), the angle of bionic stiffeners \(\alpha\), the thickness of the control arm \(h\), the thickness of upper, middle and lower plate \(h_{1}\), \(h_{2}\), \(h_{3}\).  
In order to improve the efficiency of size optimization, the sensitivity analysis is applied to obtain the influence degree of these dimensions to the optimization objective by the Equation (8) in Section 2.5, where, the \(x_{i}\) is the vector of design variables \(x_{i}=[h,\alpha,\phi,h_{1},h_{2},h_{3}]\).  
Figure 14: Parametric model of the control arm.  
Figure 13: The design idea of bionic sub-branch: (**a**) the main-branch of control arm; (**b**) the bionic model with fish bone; (**c**) the bionic model with skin; (**d**) the final bionic model.  
The parametric model in Figure 14 is input in the commercial FEA software ANSYS 18.0, and the sensitivity analysis is carried out. The result is shown in Figure 15. From the sensitivity analysis results, the bionic structure is mainly sensitive to the plate thickness and the angle of the secondary bone. Therefore, five parameters of \(\alpha\), \(h\), \(h_{1}\), \(h_{2}\), \(h_{3}\) are selected as design variables.  
According to the design and installation requirements of the control arm, if the size is too large, it will interfere with the control arm and other components and can not work normally. However, if the size is too small, it will not guarantee enough stiffness, and it will also make it difficult to apply biological characteristics to structural sub-branch. Therefore, the maximum and minimum values of the five design variables are set near the initial values, and the specific ranges are shown in Table 2.  
And the mathematical model of size optimization is established, as shown in Equation (10).  
\[\left\{\begin{array}{ll}\text{find}&h,\ h_{1},h_{2},h_{3},\alpha\\ \text{min}&f(x)=\min[M,d_{1},\sigma_{1},d_{2},\sigma_{2}]\\ \text{s.t.}&h\in[25,35];\ h_{1}\in[3,5];\ h_{2}\in[3,5]\\ &h_{3}\in[3,5];\ \alpha\in[40,50]\end{array}\right. \tag{10}\]  
The FEM of the bionic model of the control arm is established and the size optimization is carried out. A set of optimum sizes are obtained and rounded, as shown in Table 3, the final bionic model is established.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design Variables** & **Initial Value** & **Minimum Value** & **Maximum Value** \\ \hline \(h\) (mm) & 32.0 & 25 & 35 \\ \(h_{1}\) (mm) & 3.5 & 3 & 5 \\ \(h_{2}\) (mm) & 4.5 & 3 & 5 \\ \(h_{3}\) (mm) & 3.5 & 3 & 5 \\ \(\alpha^{(\circ)}\) & 45.0 & 40 & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variation range and initial value of design variables.  
Figure 15: The sensitivity analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Initial Value** & **Optimal Value** & **Round Value** \\ \hline \(h\) (mm) & 32.0 & 29.91 & 30.0 \\ \(h_{1}\) (mm) & 3.5 & 3.86 & 4.0 \\ \(h_{2}\) (mm) & 4.5 & 4.11 & 4.0 \\ \(h_{3}\) (mm) & 3.5 & 3.93 & 4.0 \\ \(\alpha(^{\circ})\) & 45.0 & 42.80 & 43.0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The results of size optimization.  
In order to verify the manufacturability, the control arms are machined by numerical control machine tool, as shown in Figure 16, the total weight of the control arm is reduced from 1.01 kg to 0.76 kg.