© Wuhan University 2023

0 Introduction

The occupant restraint system is the sum of the devices that restrain the relative displacement of the occupant in the vehicle. Its core role is to effectively buffer the impact load of the occupant in the event of a collision, to avoid the secondary collision between the occupant and the components in the vehicle as much as possible. The occupant restraint system mainly consists of a seat belt, airbag, seat, headrest, etc. The restraint protection effect of a single component on the occupant is determined by its parameters, such as the elongation of the seat belt and the force limit value of the force limiter. There are many parameters affecting the performance of the occupant restraint system, and each parameter has a coupling effect, and there is a hidden nonlinear function relationship between each parameter and the occupant injury index.

Currently, researches on occupant restraint system are mostly carried out under certain working conditions. However, there are many uncertainties in the actual vehicle collisions, such as the random excitation of the road surface, boundary conditions, and initial conditions. In addition, there will be many considerations in the priority of protection for a passenger, for example, the acceleration of the head should be small, the chest's compression should not be too large, and the force of the leg should have a tolerance limit. These considerations are not independent or linearly related^[1]. Therefore, this is a complex optimization problem.

To improve the optimization efficiency, Qing et al^[2] proposed a method for identifying the uncertain parameters of the occupant constraint system based on Bayesian inference, which could effectively give the probability distribution of the parameters of the occupant constraint system, and ensure the global convergence of parameter optimization. Ma et al ^[3] used a multi-island genetic algorithm to optimize the multi-parameters of vehicle occupant restraint systems based on global sensitivity analysis and the Kriging agent model, and provided a reference for practical engineering design. Yuan et al ^[4] carried out multi-objective optimization of the constrained system based on the non-dominated sorting genetic algorithm-II(NSGA-II) genetic algorithm. The limitation of these optimization algorithms is that they need to spend a lot of energy to perform many different super parameter optimization tasks, and sometimes they need to use the previous information to carry out the next super parameter optimization. To solve the multi-parameter and multi-objective optimization problem of occupant the constraint system well, it is necessary to find an approximate model, in which the global sensitivity analysis is used to reduce the dimension of the multi-parameter; then the global optimization algorithm is used for the initial optimization, and finally the local fine optimization is carried out according to the initial optimal solution.

1 The Design of the Optimization Method

This research mainly studies the impact of different parameters in the occupant restraint system on occupant injury in frontal collisions. By optimizing multiple parameters selected in the occupant restraint system, we seek a comprehensive evaluation method to minimize the damage to the key parts of the occupant, such as the chest, head, and pelvis. This is a typical multi-parameter and multi-objective optimization design.

1.1 Design Variable

There are four main categories of parameters affecting the performance of the occupant restraint system: the structural parameters of the vehicle body, the seat, the seat belt, and the airbag. Under normal circumstances, the structure parameters of the car body are fixed, then mostly the relevant parameters of the other three categories should be adjusted. The adjustable parameters of the safety belt system include the height, the force limit, the extension rate of the safety belt, etc. In the process of collision, the airbag mainly affects the head injury value of the occupant, and the main parameters include the weaving material of the airbag, the shape of the airbag, the size of the exhaust hole, the ignition time, and the type of the generator. In this paper, eight parameters are selected for injury simulation analysis as shown in Table 1.

1.2 Optimization Objectives and Constraints

The head, chest, and legs are the main sites of injuries to occupants in a head-on collision. To better evaluate the injuries of the crew, the Weighted Injury Criterion (WIC ) is given as follows^[5]:

$\begin{array}{l}\mathrm{W}\mathrm{I}\mathrm{C}=\mathrm{0.6}\left(\frac{\mathrm{H}\mathrm{I}{\mathrm{C}}_{\mathrm{36}\mathrm{m}\mathrm{s}}}{\mathrm{1}\mathrm{000}}\right)+\frac{\mathrm{0.35}\left(\frac{C_{\mathrm{3}\mathrm{m}\mathrm{s}}}{\mathrm{60}}+\frac{C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}}{\mathrm{75}}\right)}{\mathrm{2}}\\ +\frac{\mathrm{0.05}\left(F_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}+F_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}\right)}{\mathrm{20}}\end{array}$(1)

where $\mathrm{H}\mathrm{I}{\mathrm{C}}_{\mathrm{36}\mathrm{m}\mathrm{s}}$ is the head performance index of 36 ms, $C_{\mathrm{3}\mathrm{m}\mathrm{s}}$ (unit: g) is chest injury performance index of 3 ms, $C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ (unit: mm) is chest compression volume, $F_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}$ and $F_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}$ (unit: kN) are the left and right thigh forces, respectively.

This is an optimization problem aiming at minimizing occupant damage. The mathematical model can be expressed as follows:

Min WIC

$\mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\{\begin{array}{l}\mathrm{H}\mathrm{I}{\mathrm{C}}_{\mathrm{36}\mathrm{m}\mathrm{s}}\le \mathrm{800}\\ C_{\mathrm{3}\mathrm{m}\mathrm{s}}\le \mathrm{60}\mathrm{g}\\ C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}\le \mathrm{50}\mathrm{m}\mathrm{m}\\ F_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}\le \mathrm{10}\mathrm{k}\mathrm{N}\\ F_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}\le \mathrm{10}\mathrm{k}\mathrm{N}\end{array}$(2)

1.3 The Solution to the Problem

For the multi-parameter and multi-objective optimization of the occupant restraint system, a large number of occupant injury tests under frontal collision are needed, and taking simulation tests is the most efficient and economical way. Therefore, we establish a simulation model with high approximate accuracy and a real vehicle crash test. The number of trials also requires the use of orthogonal trials or optimal Latin squares to improve efficiency. Moreover, clarifying the functional relationship between design parameters and occupant injury indicators is difficult, so an approximate model should be selected, in which the high space dimension caused by too many parameters should be reduced. Therefore, the combination of the global optimization algorithm and local optimization algorithm can better approximate the real optimal solution. The process of problem solving is shown in Fig. 1.

Fig. 1 Mindset of solution seeking

2 Simulation Model of Occupant Restraint System

A reliable simulation model is constructed to reduce research costs and improve research efficiency. In mathematical dynamic model (MADYMO) engineering software, a real vehicle's occupant restraint system simulation model is established according to its layout and size, as shown in Fig. 2. The model consists of a vehicle body, dummy, airbag, and seat belt model. The dummy model invokes a Hybrid III 50th percentile male dummy from the MADYMO engineering software dummy library. The safety belt model is a combination of multiple rigid bodies and finite element models^[6].

Fig. 2 Simulation model of the frontal collision driver's side restraint system

In the simulation experiment, the acceleration in Y direction and Z direction below the B-pillar is very small, and the damage to all parts of the occupant is relatively small, so it can be ignored. Therefore, only the acceleration in the X direction below the B-pillar is taken as the input of the simulation model, which approximately replaces the acceleration in the X direction of the test vehicle. The acceleration waveform is shown in Fig. 3.

Fig. 3 X-direction acceleration curve of vehicle

We adjust the parameters of the dummy, body, seat belt, and airbag according to the data of the real car. From the continuous adjustment of the model and real test, the model and the real test have a better coincidence, and the effect of error control is achieved. The trend, shape, and peak time of each injury index curve of the simulation and the real test are compared.

Figure 4 shows the comparison of the damage response of various parts of the simulation test and real test on dummy.As seen from Fig. 4, the shape and trend of each part of the Hybrid III 50th percentile male dummy in this model are consistent with the injury curve of the dummy obtained in the real trolley test, and the peak time is also close. Therefore, the reliability of this model is relatively high.

Fig. 4 Comparison of damage response in real test on dummy injury and simulation test in the frontal collision model (a) Acceleration of head; (b) Acceleration of chest; (c) Acceleration of pelvis; (d) Compression deformation of chest

3 The Process of Multi-Parameter and Multi-Objective Optimization

3.1 Building of an Approximate Model

3.1.1 Kriging approximate model

The Kriging method has the characteristics of local estimation, and the continuity and derivability of the correlation function are good, so it can often achieve a better fitting effect when solving the problem with a high degree of nonlinearity. It can be expressed as^{[7, 8]}:

$f\left(x\right)=g\left(x\right)+z\left(x\right)$(3)

where $f\left(x\right)$ represents the response of the system; $g\left(x\right)$ is a deterministic part, representing a polynomial function with independent variable x, which is called deterministic drift; $z\left(x\right)$ is called fluctuation, and it is a nonzero stochastic process. $z\left(x\right)$ has the following statistical properties:

$E\left(z\left(x\right)\right)=\mathrm{0}$(4)

$\mathrm{v}\mathrm{a}\mathrm{r}\left(z\left(x\right)\right)={\sigma }^{\mathrm{2}}$(5)

$\mathrm{c}\mathrm{o}\mathrm{v}\left(z\left(x_i\right),z\left(x_j\right)\right)={\sigma }^{\mathrm{2}}\left(R\left(\theta ,x_i,x_j\right)\right)$(6)

where ${\sigma }^{\mathrm{2}}$ represents variance; $x_i$, $x_j$ are any two points in the sample; $R\left(\theta ,x_i,x_j\right)$ is a correlation function with parameter $\theta$, which reflects the correlation of sample point space. The common correlation function is Gaussian function:

$R\left(d\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-d^{\mathrm{2}}/{\theta }^{\mathrm{2}}\right)$(7)

and exponential function:

$R\left(d\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(-d/\theta \right)$(8)

Compared with other approximate model techniques, Kriging is suitable for this study for it has the following advantages: 1) the Kriging model only uses a certain point to estimate the relevant information around the point, that is, it is based on the known information of structural dynamics; 2) Kriging model has both local and global statistical characteristics, which can analyze the trend and dynamics of known information.

3.1.2 Building of the Kriging approximation model

The optimal design problem in this study has eight design variables. Firstly, the optimal Latin square test method was used to extract 60 sample points in Isight software, and then MADYMO software was used to simulate and calculate the 60 sample points respectively to obtain the damage values of each part and calculate the WIC value of the objective function. Finally, Gaussian function was selected as the correlation function to establish the Kriging approximation model. In order to verify the accuracy of Kriging's approximate model, 9 sample points were randomly selected for comparison, and the results were shown in Table 2. The calculation error of Kriging approximation model and simulation test model is less than 10%. It can be seen that the Kriging approximate model has high reliability and can replace the simulation test to optimize the multi-objective parameters of the occupant restraint system.

3.2 Global Sensitivity Analysis

3.2.1 Principle of Sobol sensitivity analysis

Sensitivity analysis refers to the uncertainty study on the change degree of a model output responding to the input parameters. A large degree of change indicates a high sensitivity of the parameters.

To determine the main factors in the model and reduce the dimension of the design space, the Sobol method is used to judge the global sensitivity. The Sobol method is the most representative global sensitivity analysis method based on model decomposition and it can obtain the sensitivity of parameters of the first order, second order, or higher order, respectively, which can reflect the main effect of the parameter^{[9, 10]}.

Sobol method decomposes the model into a single parameter and the function of parameter combination. The hypothesis model is

$Y=f\left(x\right)\left(x=x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_m\right)$(9)

$x_i$ obeys [0,1] uniform distribution and $f^{\mathrm{2}}\left(x\right)$ is integrable, so the model can be decomposed into

$f\left(x\right)=f\left(\mathrm{0}\right)+{\displaystyle \sum _{i=\mathrm{1}}^n}{f}_i\left(x_i\right)+{\displaystyle \sum _{i<j}^n}{f}_{ij}\left(x_{}\right)+\cdots +f_{\mathrm{1,2},\cdots ,n}\left(x_{\mathrm{1}},x_{\mathrm{2}},\cdots ,x_k\right)$(10)

Then the total variance of the model can also be decomposed into the influence of a single parameter and each parameter portfolio:

$D={\displaystyle \sum _{i=\mathrm{1}}^n}{D}_i+{\displaystyle \sum _{i=\mathrm{1}}^n}{\displaystyle \sum _{\begin{array}{c}i=\mathrm{1}\\ i=j\end{array}}^n}\left(D_{ij}+\cdots +D_{\mathrm{1,2},\cdots ,n}\right)$(11)

To normalize this formula, we set

$S_{i_{\mathrm{1}},i_{\mathrm{2}},\cdots ,i_n}=D_{i_{\mathrm{1}},i_{\mathrm{2}},\cdots ,i_n}/D$(12)

The sensitivity S of a single parameter of the model and the interaction between parameters can be obtained from Equation (11):

$\mathrm{1}={\displaystyle \sum _{i=\mathrm{1}}^n}{S}_i+{\displaystyle \sum _{i=\mathrm{1}}^n}{\displaystyle \sum _{\begin{array}{c}i=\mathrm{1}\\ i=j\end{array}}^n}\left(S_{ij}+\cdots +S_{\mathrm{1,2},\cdots ,n}\right)$(13)

where $S_i$ is called the first sensitivity, $S_{ij}$ is the second sensitivity, and so on. $S_{\mathrm{1,2},\cdots ,n}$ is the sensitivity of n times, and there are ${\mathrm{2}}^n-\mathrm{1}$ terms in total. The total sensitivity $S_{\mathrm{T}j}$ is defined as:

$S_{\mathrm{T}j}=\sum S_{\left(i\right)}$(14)

where $S_{(i)}$ represents the sensitivity of all parameters including the i-th one.

3.2.2 Global sensitivity analysis based on the Sobol method

The Monte Carlo method was used to extract 6 000 sets of sample points in the design space, and then the calculation was carried out in the established Kriging approximation model. The Sobol method was applied to calculate the first-order and total sensitivity of each parameter.

Table 3 shows the sensitivity index values and the ranking order. Due to the estimation error of the Monte Carlo method, the sensitivity may be negative^[4].

As can be seen from Table 3, for the first-order sensitivity, the influence order is FLFL > TRAD > BER > BMFSF > TRPP > TRRL > PHPSB > DAV. For total order sensitivity, the order is FLFL > TRAD > BER > DAV > BMFSF > TRRL > PHPSB > TRPP. The top four design parameters listed in the total sensitivity of Table 3 are FLFL, TRAD, BER, and DAV, which are selected as parameters to be optimized.

3.3 Optimization of Parameters

Most optimization algorithms have a strong dependence on the initial point design, which is not suitable for applying in multi-parameter and nonlinear system such as the occupant restraint system. Multi-island genetic algorithm (MIGA) is a parallel genetic algorithm based on population grouping. With a better global solving ability and higher computational efficiency.

Non-Linear Programming by Quadratic Lagrangian (NLPQL) is a method for solving sequential quadratic programming (SQP) with smooth continuously differentiable objective functions and constraints. The algorithm uses quadratic approximation of Lagrange functions and linearization of constraints. One of the features of this approach is to stop searching once a local minimum is found. Therefore, the obtained result largely depends on the initial value given by the algorithm, which is a suitable algorithm for local optimization. After using the global optimal solution of MIGA, it is a good choice to use NLPQL for further precise optimization.

According to the sensitivity analysis results in Section 3.2, FLFL, TRAD, BER and DAV are selected as optimization targets, while the other four parameters remain unchanged, and the initial values remain unchanged. The value range of design variables remains unchanged because it is related to the vehicle model. 10 sample points are initially selected in the design space through the optimal Latin square test method, and simulation calculation is carried out by MADYMO software. The damage values of each part were obtained, and the WIC value of the objective function was calculated. The results are shown in Table 4, as shown in the results of the first round of Latin square test. From the data, the Kriging approximate model constructed by these 10 sample points is not very accurate. The same method was used again to extract 10 sample points for simulation calculation. The results are shown in Table 4. The Kriging approximation model was re-established according to a total of 20 sample points in the two rounds, and then 5 sample points were randomly selected to verify the accuracy of the proxy model. The verification results are shown in Table 5. The calculation of the established Kriging approximation model is no more than 5%, which is highly accurate and can be optimized in the next step.

Insight software is used to integrate global optimization algorithms and numerical optimization algorithms to optimize the design space. Firstly, a multi-island genetic algorithm is used to locate the region where the target extreme value is in the design space, and then the NLPQL algorithm is applied to accurately optimize this region, which can give full play to the global optimization algorithm and have the high efficiency of the numerical algorithm^[9]. Through 1 001 iterations, the global optimal solution is obtained as follows: the force limiting value of the force limiter is 2 985.603 N, the airbag point explosion time is 27.585 ms, the belt extension rate is 12.684% and the airbag exhaust hole diameter is 27.338 mm. Figure 5 shows the 1 001 iteration curve.

Fig. 5 Iterative convergence curve

The obtained optimal solution was substituted into MADYMO engineering software for simulation calculation, and the results were compared with the optimal solution, as shown in Table 6.

As can be seen from Table 6, the error is controlled within 10%. The reason why the optimization solution value of the Kriging model is larger than the simulation value of MADYMO is caused by the error in the iterative operation process of Insight software.

The optimization results show that after optimization, the parameters of head injury $\mathrm{H}\mathrm{I}{\mathrm{C}}_{\mathrm{36}\mathrm{m}\mathrm{s}}$ , chest displacement $C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$, left thigh force $F_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}$ and weighted injury index WIC decrease by 22.60%, 7.29%, 0.84%, and 12.97%, respectively. The 3 ms synthetic acceleration of the chest $C_{\mathrm{3}\mathrm{m}\mathrm{s}}$ and the force of the right thigh $F_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^{\mathrm{f}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{r}}$ increased by 1.17% and 2.34%, which were controlled within 3%, and met the requirements of the high-performance index stipulated by C-NCAP2014. The protective effect of the occupant restraint system was further improved.

4 Conclusion

The Latin experimental design, Kriging's model-like technology, multi-island genetic algorithm, and NLPQL algorithm are combined to optimize the performance parameters of the occupant restraint protection system, and the optimal matching parameters obtained after optimization effectively reduce the occupant damage value compared with the initial design. The Kriging model of head $\mathrm{H}\mathrm{I}{\mathrm{C}}_{\mathrm{36}\mathrm{m}\mathrm{s}}$, chest 3 ms injury $C_{\mathrm{3}\mathrm{m}\mathrm{s}}$ and chest compression $C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ was analyzed for each design variable. The results show that: When the force limiting value of the force limiter is 2 985.603 N, the airbag point explosion time is 27.585 ms, the belt extension rate is 12.684%, and the airbag exhaust hole diameter is 27.338 mm, the optimal matching parameters of the occupant restraint system are obtained. At this time, the WIC of the human body weight damage index decreases by 12.97% compared with the initial value. Occupant head injury $\mathrm{H}\mathrm{I}{\mathrm{C}}_{\mathrm{36}\mathrm{m}\mathrm{s}}$ decreased by 22.60%, and chest displacement $C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}$ decreased by 7.29%.

Therefore, a set of perfect occupant restraint protection system can effectively reduce the damage to the occupant in the collision. The results also show that adjusting the performance parameters of the occupant restraint protection system can effectively reduce the risk of occupant injury in the collision. Therefore, according to the matching parameters of the car seat, the performance parameters of the occupant restraint protection system can be adjusted to the best value according to the different occupant figures in the collision, and the protective effect of the occupant restraint protection system can be changed.

References

All Tables

All Figures

	Fig. 1 Mindset of solution seeking
In the text

	Fig. 2 Simulation model of the frontal collision driver's side restraint system
In the text

	Fig. 3 X-direction acceleration curve of vehicle
In the text

	Fig. 4 Comparison of damage response in real test on dummy injury and simulation test in the frontal collision model (a) Acceleration of head; (b) Acceleration of chest; (c) Acceleration of pelvis; (d) Compression deformation of chest
In the text

	Fig. 5 Iterative convergence curve
In the text