Multi-parameter Sensitivity Analysis and Application Research in the Robust Optimization Design for Complex Nonlinear System

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CHINESE JOURNAL OF MECHANICAL ENGINEERING

Vol. 28,aNo. 1,a2015 ·55·

DOI: 10.3901/CJME.2014.1107.164, available online at www.springerlink.com; www.cjmenet.com; www.cjmenet.com.cn

Multi-parameter Sensitivity Analysis and Application Research in the Robust

Optimization Design for Complex Nonlinear System

MA Tao

*

, ZHANG Weigang, ZHANG Yang, and TANG Ting

State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, China

Received January 24, 2014; revised July 8, 2014; accepted November 7, 2014

Abstract: The current research of complex nonlinear system robust optimization mainly focuses on the features of design parameters,

such as probability density functions, boundary conditions, etc. After parameters study, high-dimensional curve or robust control design is used to find an accurate robust solution. However, there may exist complex interaction between parameters and practical engineering system. With the increase of the number of parameters, it is getting hard to determine high-dimensional curves and robust control methods, thus it’s difficult to get the robust design solutions. In this paper, a method of global sensitivity analysis based on divided variables in groups is proposed. By making relevant variables in one group and keeping each other independent among sets of variables, global sensitivity analysis is conducted in grouped variables and the importance of parameters is evaluated by calculating the contribution value of each parameter to the total variance of system response. By ranking the importance of input parameters, relatively important parameters are chosen to conduct robust design analysis of the system. By applying this method to the robust optimization design of a real complex nonlinear system-a vehicle occupant restraint system with multi-parameter, good solution is gained and the response variance of the objective function is reduced to 0.01, which indicates that the robustness of the occupant restraint system is improved in a great degree and the method is effective and valuable for the robust design of complex nonlinear system. This research proposes a new method which can be used to obtain solutions for complex nonlinear system robust design.

Keywords: complex nonlinear system, global sensitivity analysis, robust optimization design, grouped variables

1 Introduction



Each product or system is designed for the use of certain requirements, product in its life cycle should keep its function steadily[1]_{. In practical engineering, the}

performance of a product or system is not only influenced by the design parameters, but also affected probably by boundary conditions. If a product is a dynamic nonlinear complex system, then system performance usually varies to a great extent. Therefore, robust design is necessary to be considered for complex nonlinear system[2]_.

In the past several decades, a lot of research works have been done about the robust analysis of high dimensional nonlinear systems, and significant results have been achieved. For example, PIET, et al[3]_{and YAN, et al}[4]_,_first

studied the probability distribution of all design parameters, and tested point by point through the design space using Monte Carlo method and finally got the robust solution of a nonlinear system. LOCKE, et al[5]_{, proposed a useful}

strategy using Sobol sampling and simulated annealing

* Corresponding author. E-mail: [email protected]

Supported by National Natural Science Foundation of China (Grant No. 51275164)

method. ALVES, et al[6]_,_{presented a random search for}

kinetic parameters to find all plausible models. STELLING, et al[7]_,_{explored the two-dimensional spaces to all kinetic}

parameters. Although their research works made a great step for approaching to global analysis, there were still limitations due to the calculation complexity. WAGNER[8]

performed the Monte Carlo simulation for dynamic models to simulate a target function by randomly perturbing all parameters, but the huge computational resource with an increase in the numbers of all parameters is hard to accept. OKUYAMA, et al[9]_,_{presented a method of the robust}

stability condition for sampled-data control systems, but it is applicable only to the sampled-data control system. KIM, et al[10]_, _{applied the complex-step method in nonlinear}

robust performance analysis, the accuracy of the solutions seemed to be significantly better than those achieved using gradient approximation methods. However, it needs cost function of dynamical systems, and it’s not suitable for all nonlinear systems. KÖKSOY[11] _{proposed a nonlinear}

programming solution to solve robust quality problems, but sometimes the accuracy of the estimated models is not satisfied. YANG, et al[12]_,_{first used T-S fuzzy models to}

describe complex nonlinear systems, and then investigated the global robust stability of T-S fuzzy models. ZHANG, et al[13]_,_{put forward moderate sensitivity matrix of a system}

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MA Tao, et al: Multi-parameter Sensitivity Analysis and Application Research in the Robust Optimization Design for Complex Nonlinear System

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and use Taylor expansion to solve the optimal problem for the robust design of a nonlinear system with analytical formulations. However, for many complex nonlinear mechanical systems, it cannot be found out exact mathematical expressions to reflect the relation between the output responses and design variables. Furthermore, with the increase of the number of design parameters, the costs of the calculation rise sharply while the computational efficiency declines heavily. With the development of sensitivity analysis methods in recent years, a lot of researches have been done to evaluate the importance of parameters by calculating the sensitivity and uncertainty of parameters[14–16]_{. DOODMAN, et al}[17]_{, achieved further}

progress, they investigated the use of global sensitivity analysis(GSA) and harmony search(HS) algorithm for design optimization of air cooled heat exchangers(ACHEs). GSA was performed to identify the non-influential parameters, then HS algorithm was applied to the optimization of relatively influential parameters. Finally, the robust stochastic approach was achieved.

In this paper, general global sensitivity analysis is replaced by sensitivity analysis based on grouped variables which needs less computation time. By analyzing the sensitivity and uncertainty of design parameters based on grouped variables, the importance of parameters is evaluated by calculating the contribution of each parameter to the total variance of the response. The surrogate model of the system is established by using important variables and the accuracy is verified. Finally, robust optimization design is conducted and an example is given out to show the effect of the new method.

2 Multidimensional Sensitivity Analysis

Using Group Sensitivity

2.1 Global sensitivity analysis

There are two kinds of sensitivity analysis methods, one is local sensitivity analysis and the other is global sensitivity analysis. Local sensitivity analysis is simple and easy to realize, but not suit for high dimensional and nonlinear system[18]_{. Global sensitivity analysis studies the}

system response influenced simultaneously by all design parameters in the whole design space. The analytical methods mainly include response surface method(RSM), Fourier amplitude sensitivity test(FAST), regression analysis method(RAM) and variance-based sensitivity analysis(VBSA), among which the variance-based Sobol’ method[19]_{is very popular and widely used in many fields.}

However, it is difficult to calculate the sensitivity indices by using Sobol’ method directly. SALTELLI, et al[20]_and

TARANTOLA, et al[21]_{, proposed a new method for global}

sensitivity analysis based on Monte Carlo sampling and found that the Sobol’ method had better performance than Latin Hypercube Sampling(LHS), particularly in the case where there is a larger subset of inputs interacting with each other strongly. The theory of the method is as follows:

First, two input matrices A and B are generated using Monte Carlo sampling, each row of the two matrices is a set of design variables. n and N stands for the variable number and sampling number respectively. Input matrices A and B are written as

11 12 1 21 22 2 1 2 , n n N N Nn x x x x x x x x x       é ù ê ú ê ú ê ú = ê ú ê ú ê ú ê ú ë û Α 11 12 1 21 22 2 1 2 . n n N N Nn x x x x x x x x x       é ¢ ¢ ¢ ù ê ú ê _¢ _¢ _¢ ú ê ú = ê ú ê ú ê _¢ _¢ _¢ ú ê ú ë û Β

Matrix Ci is got through matrix B by replacing the ith column with the ith column of matrix A. Matrix C–i is got

through matrix A by replacing the ith column with the ith column of matrix B. Matrices Ci and C–i are as follows:

11 12 1 1 21 22 2 2 1 2 , i n i n i N N Ni Nn x x x x x x x x x x x x           é ¢ ¢ ¢ ù ê ú ê ¢ ¢ ¢ ú ê ú = ê ú ê ú ê _¢ _¢ _¢ ú ê ú ë û C 11 12 1 1 21 22 2 2 1 2 . i n i n i N N Ni Nn x x x x x x x x x x x x           -é ¢ ù ê ú ê _¢ ú ê ú = ê ú ê ú ê _¢ ú ê ú ë û C

The input matrices are submitted into the model to calculate the result, and the corresponding output vectors are obtained. Vectors yA, yB, yC are the output column vectors corresponding to the input matrices. Estimations are obtained by using sensitivity analysis based on Monte Carlo method.

The total variance ˆ( )V Y is estimated as

T 1 ( ) _A( _A _B), V Y N = y y -y (1) Meanwhile, fˆ₀2 1 T_{A B}, N = y y (2) T 1 ˆ _, i i A C U N = y y (3) T 1 ˆ _. i i A C U N -- = y y (4)

The estimation of the main effect index S of the input x_i parameter xi is 2 0 ˆ ˆ . ˆ( ) i i X U f S V Y -= (5)

The estimation of the total effect index T

i

x

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CHINESE JOURNAL OF MECHANICAL ENGINEERING ·57· parameter xi is 2 T ˆ( ) (ˆ ˆ0)_. ˆ( ) i i X V Y U f S V Y -- -= (6)

The main effect index S is a measure for the variance _x_i

contribution of the individual parameter xi to the total variance, which indicates the influence of a single parameter to the system responses. T

i

x

S is the result of the main effect of xi and all its interactions with other parameters. Input matrices A, B, Ci, C–i are required to

calculate the main effect and total effect indices of all input parameters when using this algorithm, and sampling number is N(n+2)[22]_.

2.2 Sensitivity analysis based on grouped variables As to high dimensional and nonlinear system, the sensitivity indices calculation work is too huge to accept by using global sensitivity analysis method based on Monte Carlo algorithm. An effective analytical method is to decompose the system into several low-dimensional sub systems[23–24]_{, then analysis is conducted on the}

low-dimensional systems. Group sensitivities strategy is an analytical algorithm based on this method, the main idea is that the related variables are divided into one group and all the groups are uncorrelated. The detailed methodology is presented below.

Consider that a model M can be divided into two sub models M1 and M2, and sensitivity analysis is made for each

model, sensitivity indices Sj1_{for M}

1 and Sj2 for M2 are

computed, then sensitivity indices SjM_{can be obtained as} follows.

(1) Assume that all input variables are different in M1

and M2, i.e.,

1 1( , ,1 p),

M = f x  x (7)

2 2( p 1, , p q).

M = f x +  x + (8)

If 1≤j≤p, for xj, the conditional expectation of M can be calculated by Eq. (9):

[

]

1 2

| _j | _j .

E M x_êé_ë _û_úù=E M xé_ê_ë ù_ú_û+E M (9)

If p+1≤j≤p+q, for xj, the conditional expectation of M can be calculated by Eq. (10):

[ ]

2 1

| j | j .

E M xé_ë_ê ù_ú_û=E M_êé_ë x _ûù_ú+E M (10)

The terms E[M1] and E[M2] are constant, and thus the

variance of the conditional expectation V(E[M1|xj]) and

V(E[M2|xj]) for given xj can be got by Eqs. (11), (12):

(

1| j

)

(

1| j

[

2

]

)

,

V E M x_ë_êé ù_ú_û =V E M xé_ê_ë ù_ú_û+E M (11)

(

2| j

)

(

2| j

[ ]

1

)

.

V E M_ë_êé x _ú_ûù =V E Mé_ë_ê x _û_úù+E M (12)

Thus, sensitivity indices of M can be obtained by indices of M1 multiplying V(M1)/(V(M1)+V(M2)) and M2

multiplying V(M2)/(V(M1)+V(M2)). It is easy to verify that

all sensitivity indices relative to interaction between variables of M1 and M2 are equal to zero.

(2) Assume that models M1 and M2 have the same input

variables, then the conditional expectation of M for given a variable xj is equal to the sum of the conditional expectation of M1 and the conditional expectation of M2.

Thus, the first-order sensitivity indices of M are obtained by Eq. (13): 1 ( 1) 2 ( 2) ( ) ( ) M j j j V M V M S S S V M V M = ´ + ´ + 1 2 2 ( , ) . ( ) j j Cov E M X E M X V M         ₍₁₃₎

In practice, the cost of the estimation of the covariance is very expensive as the direct estimation of sensitivity indices of M.

(2) Assume that M1 and M2 have part common variables,

i.e.,

1 1( , ,1 p),

M = f x  x M₂= f x₂( , ,₁ x_{p q}₊ ).

For the common variables, the sensitivity indices are calculated as above; and for the different variables (xp+1,,

xp+q), indices are calculated by multiplying sensitivity

indices of M2 with V(M2)/V(M).

3 6σ Robust Optimization Design

Robust optimization aims at developing a solution that is insensitive to variations of the nominal design, and realizes the target of optimal mean and minimum variation. In order to quantify the performance of the robust design, this paper uses 6σ as the evaluation index. 6σ is a quality philosophy at the highest level relating to all processes[25]_{. The signal σ}

refers to standard deviation.

The difference of robust optimization and deterministic optimization is shown in Fig. 1. The x-axis represents the uncertain parameters, including design variables and noise factors, while the vertical axis represents the objective function to be minimized. If it is not considered the influence of the uncertain factors acting on the objective function, then point 1 is the deterministic optimization solution; If there is a variation of x, the response will have a bigger variationf1in the left and a relatively flat area in the right, which is more robust for design parameter

x. Point 2 is also within the design space when the design

variable varies in x, but the perturbation causes a much smaller change in objective function. That is to say, point 1 is highly sensitive to the parameter perturbation and

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usually cannot be recommended in practice, though it has a better mean value than point 2[26]_.

Fig. 1. difference of deterministic optimization and robust optimization

1. Deterministic optimization, 2. Robust optimization

Generally, robust design can be formulated as the following reliability-based design problem of optimization.

Find the set of design variables x that

L U min ( , ), s.t. ( , ) 0, , j F G x x x x x       ≤ ≤ ≤ ìïï ïï íï ïï + -ïî (14)

where x means the input parameters that may be design variables or random variables, j stands for the number of constraint function, xU, xL stand for the upper and lower

specification limit of the design variables.

Both input and output constraints are formulated to include mean performance and a desired “sigma level”.

The robust design objective with optimal mean and minimum variation can be formulated as follows:

2 2 1 2 1 2 1 ( ) , l i i i i i i i i D D F M S  S  = é ù ê ú = _ê - + _ú ë û

å

(15)

where D1i and D2i are weight factors, S1i and S2i are scale

factors for the “optimal mean” and “minimum variation” respectively for response i, Mi is the target for response i, and l is the number of response included in the objective. For those cases in which the mean performance is to be minimized or maximized, the objective formulation of Eq. (15) can be modified as shown in Eq. (16) ,where there is a “+”sign before the first term when the response mean is to be minimized, and the “-”sign when the response mean is to be maximized: 2 2 1 2 1 2 1 ( / ) ( ) . l i i i i i i i i D D F M S  S  = é ù ê ú = _ê+ - - + _ú ë û

å

(16)

In Eq. (16), constraints can be modified as quality constraints, which mainly contain the mean value of the performance and standard variation. The quality constraints are formulated as follows:

, n L - ≥ (17) , n U + ≤ (18)

where L, U stand for lower and upper specification limit of constraints.

In order to get quality of sigma level, the constraints should be changed to make the system performance defined on the boundary of sigma level. Six sigma robust optimization design based on Monte Carlo technology defines n as 6. To get robust design for six sigma is to both shift and shrink a performance distribution, as illustrated in Fig. 2. In Fig. 2(a), given the increased reliability, the potential worst-case performance associated with the left tail may be undesirable. With the shrunken distribution of Fig. 2(b), the level of reduced possible may not be sufficient to achieve an acceptable reliability level without shifting the mean performance. Only with both shift and shrink in Fig. 2(c), the reliability level is achieved, and the robustness is improved[27]_.

Fig. 2. Design for six sigma: shift and shrink

In this article, global sensitivity analysis and robust optimization method are combined effectively. Compared with the research works others had done, general global sensitivity analysis is replaced by global sensitivity analysis

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based on grouped variables which needs less computation time. Then, robust optimization design is conducted on important and influential parameters. Finally, the robust design solution of complex nonlinear system is achieved.

The flow chart of the robust optimization design for complex nonlinear system is shown in Fig. 3.

Fig. 3. Flow chart of robust optimization design

4 Case Study

4.1 Occupant restraint system model for vehicle crash Occupant system model is constructed for frontal impact analysis by using MADYMO software, which includes floor, steering wheel and column, seat, seat belt, airbag and Hybrid III dummy, as shown in Fig. 4.

The acceleration pulse curve of the frontal crash is defined in the model, which is measured by the accelerometer mounted on the B-pillar in the crash test. The occupant injury in frontal crash involves head, chest, leg and so on, and the Weighted Injury Criterion (WIC)[28] _is

used to evaluate the occupant injuries, as shown in Eq. (19): 36 3 0.6 0.35 2 60 75 1000 comp ms ms C HIC C WIC= æèççç ø÷÷÷÷ö+ èçççæç + ÷÷÷÷öø + L R 0.05 , 20 F F æ + ö_÷ ç _÷ ç _÷÷ çè ø (19)

where HIC36ms stands for head injury criterion, C3ms is the

value of chest 3ms acceleration, Ccomp is the chest depression, FL is the maximum axial force on left

thighbone, FR is the maximum axial force on right

thighbone.

Fig. 4. Occupant restraint system model

4.2 Group variables and parameter sensitivity analysis The design variables include lock time of seat belt retractor, pre-load time, pre-load length, limited force, height of D-ring, the relative elongation of webbing, volume of airbag, coefficient of vent area, switch time and mass flow rate. Among these ten parameters, six parameters related with seat belt and are classified in one group, four parameters related with airbag and classified in another group. Global sensitivity analysis is conducted in groups of parameters. The initial value and the range of each design variable can be seen in Table 1 and Table 2.

Table 1. Design range of seat belt parameters

Design variable Initial value

Lower limit

Upper limit

Lock time of retractor T1/ms 1.0 0.5 2.0

Pre-load time T2/ms 15 10 20 Pre-load length L/mm 35 30 80 Limited force F/N 6000 2500 6500 Height of D_ring H/m 1.00 0.97 1.03 Relative elongation of webbing Q/% 16 7 18

Table 2. Design range of airbag parameters

Design variable Initial value Lower limit Upper limit Volume of airbag V/L 28.40 26.98 29.82

Coefficient of vent area S 1.0 0.9 1.1

Switch time T/ms 25 20 30

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Forty points are sampled respectively in the two groups of variables and are used to construct Kriging surrogate models. After the accuracy of the surrogate models is validated, 5000 samples are generated by using Monte Carlo method and sensitivity analysis is conducted to calculate the global sensitivity indices, which can be used to rank the design variables of each group on WIC. The ranking results can be seen in Tables 3 and 4. The variance of seat belt model M1 and airbag model M2 are calculated,

as shown in Eqs. (20), (21): 1 ( ) 0.000 373, V M = (20) 2 ( ) 0.001 011. V M = (21) And so: 1 1 2 ( ) 0.269 5, ( ) ( ) V M V M +V M = 2 1 2 ( ) 0.730 5. ( ) ( ) V M V M +V M =

Table 3. Global sensitivity indices and ranking for seat belt design variables on WIC

Input variable Sj Ranking SjT Ranking

Limited force 0.491 1 1 0.639 1 1

Height of D_ring 0.032 4 4 0.054 6 4

Pre-load length 0.170 4 2 0.332 2 2

Lock time of retractor -0.019 4 6 -0.000 2 5

Pre-load time 0.011 9 5 -0.000 7 6

Relative elongation

of webbing 0.084 2 3 0.094 9 3

Table 4. Global sensitivity indices and ranking for airbag design variables on WIC

Input variable Sj Ranking SjT Ranking

Mass flow rate 0.301 3 1 0.542 3 1

Coefficient of vent

area 0.168 8 2 0.361 2 2

Volume of airbag -0.047 6 4 0.021 4 4

Switch time 0.077 9 3 0.111 8 3

Total effect indices are selected to simplify the mathematical model using the method based on group variables. The parameters of total sensitivity indices ranking top four in the sensitivity analysis are chosen as design factors, which are mass flow rate, coefficient of vent area, limited force and pre-load length. The design range and associated distributions of variables can be seen in Table 5.

Table 5. Robust design factors and associated distributions

Design factors Initial _value Lower bound

Upper

bound Distribution

Limited force F/N 6000 2500 6500 Normal

Pre-load length

L/mm 40.0 30.0 80.0 Normal

Mass flow rate P 1.0 0.9 1.1 Normal

Coefficient of vent

area S 1.0 0.9 1.1 Normal

4.3 Robust optimization analysis

The parameters of total sensitivity indices ranking top four in the sensitivity analysis are chosen as design factors, optimal latin experimental method is used to generate 50 samples, and the results of calculation are used to build Kriging metamodel of WIC. Then six stochastic samples are chosen to check the relative absolute error, the check result indicates that the metamodel is very good and can replace the simulation model to solve the problem.

The deterministic optimization of occupant restraint system is formulated as follows:

min WIC, s.t. maxHIC_36ms≤700, C3ms≤55 g, Ccomp≤40 mm, 2500 N≤F≤6500 N, 30 mm≤L≤80 mm, 0.9≤P≤1.1, 0.9≤S≤1.1.

Correspondingly, 6σ robust optimization model is formulated as follows:

max -(WIC) 6 (+  WIC), max 36 max 36

s.t.  (HIC ms) 6+  (HIC ms)≤700,

max(C3ms) 6+ max(C3ms)≤55 g, max(Ccomp) 6+ max(Ccomp)≤40 mm,

2500 N 6+ F≤F≤6500 N 6- F,

30 mm 6+ L≤ ≤L 80 mm 6- L,

0.9 6+ P≤P≤1.1 6- P,

0.9 6+ S≤ ≤S 1.1 6- S.

In order to solve the robust optimization problem, Sequential Quadratic Programming is used first to get the deterministic design solution: F=6000 N, L=40.1 mm,

P=0.93, S=1.09. In this study, it is assumed that the design

variables follow a Gaussian normal distribution. The mean value for each variable is set to be the same as the optimal design. The standard deviations of the variables are given as 5 percent of their mean values. Then, Monte Carlo descriptive sampling is used to generate 2000 samples, 6σ robust design values are obtained when the samples are calculated by using verified Kriging model.

In order to illustrate the efficiency and advantage of this method, a general dimension reduction method for reliability-based robust design is used in solving this problem. First, General global sensitivity analysis is applied to reduce the dimension of high nonlinear system. It comes out that the four parameters of coefficient of vent area, mass flow rate, limited force and pre-load length rank the top four, but the computational time is huger than the method of sensitivity analysis based on grouped variables. Then reliability-based robust design analysis is conducted on the results of general global sensitivity analysis. At last,

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CHINESE JOURNAL OF MECHANICAL ENGINEERING ·61·

a solution of reliability-based robust design analysis has been obtained.

The comparison of deterministic optimization, reliability-based robust design and 6σ robust optimization is shown in Table 6, from which it can be seen that, although the deterministic optimization value is better than 6σ robust optimization, the response fluctuation of objective function σ is 0.018, which means that the quality requirement is not satisfied. The WIC value of the robust optimization is relative higher than deterministic optimization, but the response fluctuation of objective function σ is reduced to 0.01, which means that the robustness of the occupant restraint system is improved obviously. Although the value WIC of reliability-based robust design is close to 6σ robust optimization, the response fluctuation of objective function σ is higher than the result of 6σ robust optimization. More computational time is used to compute the values of reliability-based robust design. All of these indicate that the new method raised in this paper is relatively effective and progressive.

Table 6. Result of optimization

Parameter Baseline Deterministic solution Reliability-based robust design 6σ robust optimization Limited force F/N 6000 6000 5468 5631 Pre-load length L/mm 40.0 40.1 40.8 43.2 Mass flow rate P 1.00 0.93 0.93 0.92 Coefficient of vent area S 1.00 1.09 0.96 0.97 Weighted injury criterion WIC 0.674 6 0.562 4 0.596 9 0.595 9 Response fluctuation of objective function  – 0.018 0.014 0.01

The contrast of dummy injury curves before and after 6σ robust optimization is shown in Fig. 5.

5 Conclusions

(1) Robustness is an important criterion to assess the design quality of a product or system. It is usually complicated and time-consuming to obtain robust optimization result for complex nonlinear system due to multi-parameter interaction, uncontrollable factors and dynamic system characteristics. Global sensitivity analysis based on grouped variables proposed in this paper is an effective method and can be used to obtain the robust optimization result properly.

Fig. 5. Contrast of dummy injury before and after 6σ robust optimization

(2) By applying this method to the robust optimization design of a real complex nonlinear system – a vehicle occupant restraint system, good results have been obtained with occupant injury reduced effectively, and the robustness of the restraint system is improved significantly. References

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Biographical notes

MA Tao, born in 1989, is a master student at State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, China. His research interests include parameter match design and robust optimization of occupant restraint system. Tel: +86-18796983836; E-mail: [email protected]

ZHANG Weigang, born in 1966, is currently a professor at Hunan University, China. He received his PhD degree from Hunan University, China, in 2002. His research interests include theory and methods of vehicle crash.

ZHANG Yang, born in 1987, is a master student at State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, China. Her research interests include parameter analysis and optimization of occupant restraint system. Tel: +86-18273130354; E-mail: [email protected]

TANG Ting, born in 1988, is a master student at State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, China. Her research interests include analysis and optimization of occupant restraint system.