## 4 Structural Design Optimization  
In order to optimize the key energy-absorbing structure, the parametric model of the seat was developed. The length of the projection of the AB in the horizontal direction is \(l_{1}\). The length of the projection of the BD in the horizontal direction is \(l_{2}\). The length of the CD is \(l_{3}\). The angle between the AB and the horizontal plane is \(\alpha\). The angle between the BD and the horizontal plane is \(\beta\). See Figure 9 for details.  
Figure 8: Spine load–time curve during the crash. The arrow indicates the change of the spine load during a short time when the peaks of the test and the simulation occur, respectively.  
Figure 7: Comparison in terms of acceleration and velocity between the testing and the simulation. (a) the comparison of acceleration (b) the comparison of velocity.  
As mentioned in Section 2 of this paper, the value of \(\mathbf{F}_{p}\) varies with time and peaks at one point when the above four structural parameters are determined. The change in each of the four structural parameters may affect the energy absorption capacity of the key energy-absorbing structure, which can thereby affect its peak. When the combination of the four parameters minimizes the peak, the seat has the best protection for the human body during the crash. The mathematical model for the optimization of the key energy-absorbing structure of the seat is shown in Equation (19).  
\[\left\{\begin{array}{l}\min F_{p-\max}(\mathbf{x})\;\mathrm{i}=1,2,\cdots, \mathrm{k}\\ s.t.\mathbf{x}^{L}\leq\mathbf{x}\leq\mathbf{x}^{U}(\mathbf{x}\in\mathbf{R}^{n} )\\ g_{u}(\mathbf{x})\geq 0,\;u=1,2,\cdots,p\\ h_{v}(\mathbf{x})=0,\;\mathrm{v}=1,2,\cdots,\mathrm{q}\\ \end{array}\right. \tag{19}\]  
\[\mathbf{x}=\{\alpha,\beta,l_{1},l_{3}\} \tag{20}\]  
\(F_{p-\max}(\mathbf{x})\) is the objective function which is expected to be minimized in the optimization process. It is a function based on the connection between the parameters and the maximum value of \(\mathbf{F}_{p}\). \(\alpha\), \(\beta\), \(l_{1}\) and \(l_{3}\) are the four variables of the objective function, respectively. See Equation (20) for details. \(\mathbf{x}^{L}\) and \(\mathbf{x}^{U}\), which are two column vectors, are the upper and lower limits of the variables, respectively. \(g_{u}(\mathbf{x})\geq 0\) are inequality constraints and \(\mathrm{h}_{v}(\mathbf{x})=0\) are equation constraints.  
According to the H135 flight manual [36], the minimum distance inside the cabin from the floor to the top is 1150 mm and the distance between the two seat cushions placed in the opposite direction is 1342 mm. The seated height of the Hybrid III 50 th Percentile ATD is 884 mm. The distance from the hip to the outside of the knee is 592 mm and the thickness of the knee is about 119 mm. The thickness of the cushion is 39 mm. On the basis of the above, the scope of variables as well as equality or inequality constraints can be determined by considering the comfort level of the occupant. See Equation (21) for details.  
\[\left\{\begin{array}{l}0^{\circ}<\alpha\leq 90^{\circ}\\ 0^{\circ}<\beta\leq 90^{\circ}\\ 0\leq l_{1}\leq l=l_{1}+l_{2}=434\;\mathrm{mm}\\ l_{3}\leq l_{2}\cdot\sec\beta\\ l_{1}\cdot\tan\alpha\leq 202\;\mathrm{mm}\\ l_{1}\cdot\tan\alpha<l_{2}\cdot\tan\beta\leq 227\;\mathrm{mm}\\ \end{array}\right. \tag{21}\]  
Additionally, in this way, the optimal solution which minimizes the peak spine load during the crash can be obtained and the structure can be optimized.