Advertisement

Automotive Innovation

, Volume 1, Issue 4, pp 342–351 | Cite as

Uncertainty Optimization Design of a Vehicle Body Structure Considering Random Deviations

  • Jianhua ZhouEmail author
  • Fengchong Lan
  • Jiqing Chen
  • Fanjie Lai
Article
  • 175 Downloads

Abstract

In vehicle body manufacturing, there are small differences between the actual value and design value of, for example, plate thickness and material characteristics. This is caused by the processing technology, environment and other uncertain factors. Therefore, the performance of the vehicle body processed according to the deterministic optimization solution fluctuates. The fluctuations may make structural performance fail to meet the design requirements. Thus, in this study, an optimization design is executed with 6\( \sigma \) robustness criteria and a Monte Carlo simulation single-loop optimization strategy based on the radial basis function neural network approximate model considering deviations in plate thickness, elastic modulus, and welding spot diameter, which is called the uncertainty optimization design method. As an example, considering the bending stiffness, torsion stiffness, and first-order frequency as constraints, the method is applied to the lightweight design of a car body structure, and the reliability of deterministic optimization design and uncertainty optimization design is compared. The results demonstrate that the uncertainty optimization design solution is effective and feasible without lowering the static stiffness and modal performance, and the weight is reduced.

Keywords

Body structure Lightweight Approximate models 6\( \sigma \) robustness criteria MCS single-loop optimization strategy 

1 Introduction

The optimization design of a vehicle body structure has resulted from the comprehensive optimization of multiobjective performance; however, in vehicle body manufacturing, because of the processing technology, environment, and other uncertain factors, there are small differences between the actual value and the design value of, for example, plate thickness, and material characteristics. Thus, deviations of design parameters randomly caused by the manufacturing process may make structural performance (e.g., strength, stiffness, and NVH) fail to meet the design requirements. During the optimization design, if these random deviations of parameters are neglected, this may lead to a serious deterioration in the vehicle body’s structural performance, that is, the optimization solution lacks structural reliability and engineering feasibility. For example, deviations in the plate thickness and welding spot diameter will lead to an obvious change in NVH performance. Therefore, during structural optimization design, considering the influence of random deviations in the structural physical parameters and applying the method of uncertainty design will guarantee that manufactured body structures meet the requirements for overall structural performance.

The method of uncertainty design considering the uncertainty of random factors mainly includes two aspects: robust design [1] and reliability design [2]. Robust design focuses on the insensitivity of design objective performance to variations of design variables, whereas reliability design focuses on the degree required to satisfy the uncertainty constraint. For example, during lightweight design, the structural weight should be reduced under the guarantee that its static and dynamic performance meets the requirements. Thus, static and dynamic performances are the reliability constraint conditions, and the probability of failure needs to be limited to an acceptable range. A great deal of research on this type of optimization design method has been conducted. Liu et al. [3] considered the spatial variability of connection parameters and developed a robustness design for a steel moment resisting frame. Cui et al. [4] applied the reliability design method to the design of front longeron based on the dual response surface model, and finally improved collision performance robustness while 3.32% of the total weight was reduced. Zhang [5] developed an optimization design for a thin-walled beam by applying the robustness design method, design of experiments (DOE), response surface model, and genetic algorithm, and finally reduced the structural weight under the guarantee of reliability. Considering uncertainties from manufacturing and the environment, Qiu et al. [6] applied the robustness design method executed with 6\( \sigma \) robustness criteria to the design of a B-pillar, and finally reduced 3.32% of the structural weight while improving the robustness of collision performance. Considering the requirements of several performance indices, such as mode, stiffness, and collision performance, Lan et al. [7] developed a robustness design of a vehicle body by applying the nondominant sort genetic algorithm and 6\( \sigma \) robustness criteria.

However, the uncertainty optimization design method considering both the requirements of robustness and reliability is rarely applied in practical engineering design. As there are a large number of uncertainties in the car body lightweight design problem, it is important to combine the optimization algorithm with the uncertainty optimization design method to improve its engineering feasibility. Thus, this article is based on a product R&D project and structured as follows: the uncertainty optimization design method is presented in Sect. 2. In Sect. 3, the optimization method and optimization algorithm are combined to solve the car body lightweight design problem. The conclusion is presented in Sect. 4. It shows that the lightweight design solution of the uncertainty optimization design method is effective and feasible and is achieved without decreasing the static stiffness and modal performance, while the weight is reduced. The main contribution of this article is the application of the uncertainty optimization design method in vehicle body lightweight design considering both the requirements of robustness and reliability. It provides a more effective and feasible optimization design method for lightweight design.

2 Uncertainty Optimization Design Method for a Vehicle Body Structure

2.1 Robust and Reliable Design Method

In the optimization design of a body structure, physical uncertainty refers to the distribution of physical design parameters, such as material characteristics, geometric size, and boundary conditions. These physical parameters are defined as design variables \( X = \{ X_{1} ,X_{2} , \ldots X_{n} \} \). Assuming that the design variables obey the normal distribution, this can be expressed as:
$$ X = \bar{X} + \varepsilon $$
(1)
where \( \bar{X} \) is the mean value of X and \( \varepsilon \) is a random distribution item subject to \( \varepsilon \sim N(0,\sigma^{2} ) \); so X is subject to \( X \sim N(0,\sigma^{2} ) \).
Robustness design focuses on the robustness of design objective performance. This means that objective performance is less sensitive to the deterioration of the design variables. For example, the design point × 1 is the final optimization solution when the uncertainties and performance fluctuations are not considered, as shown in Fig. 1. However, when the design parameter changes from x1 − Δx to x1 + Δx, the value of the objective function changes over a larger range from f1 to f1 + Δf1. Point x2 is the robust solution when the uncertainties and performance fluctuations are considered. When the design parameter changes from x2 − Δx to x2 + Δx, the value of the objective function changes within a smaller range of f2 − 0.5Δf2 to f2 + 0.5Δf2. The change of the objective function value at point x2 is much smaller than that at point x1 when the design parameter changes. This means that the objective function near point x2 is less sensitive to design variable changes than point x1. Thus, the purpose of robust optimization design is to determine the acceptable design points with a relatively stable objective function value when the uncertainties and performance fluctuations are considered.
Fig. 1

Robust optimization design

Reliability design focuses on the degree required to satisfy the uncertainty constraint. Considering a part design as an example, the stress distribution and strength requirement are shown in Fig. 2. The stress and strength are random quantities, and the three vertical dashed lines refer to their nominal values. Without considering the uncertainties and performance fluctuations, the nominal value of stress 1 is less than that of the strength, and the part is considered to be free from damage. However, in an actual scenario, there is uncertainty in both stress and strength. In this case, the shaded area of the two probability density curves shown in Fig. 2 represents the probability of failure.
Fig. 2

Reliability optimization design

Thus, the purpose of reliability optimization design is to make the failure probability of the design point within an acceptable range without lowering the target performance. Stress 2 in Fig. 2 is the stress distribution of the part using reliability optimization design. There is almost no overlapping area between the probability density curves of stress 2 and the strength requirement.

Considering the requirements of both robustness and reliability, the uncertainty design problem of a vehicle body structure can be presented as follows:
$$ \begin{aligned} & {\text{minimize}} \\ {\kern 1pt} & \varphi \left( {x_{i} } \right) = E\left( {f\left( {x_{i} } \right)} \right) + \beta \cdot \sqrt {Var\left( {f\left( {x_{i} } \right)} \right)} \\ & {\text{subject}}\;{\text{to}}\;P_{\text{r}} \{ G\left( {d,\mu_{x} } \right) \ge 0\} \ge P_{ 0} \\ & X_{\text{L}} \le X_{i} \le X_{\text{U}} \\ & X_{i} = (\bar{X}_{ 1} ,\bar{X}_{ 2} \ldots \ldots \bar{X}_{n} )^{\text{T}} \\ \end{aligned} $$
(2)
where \( f\left( {x_{i} } \right) \) is the objective function of the optimization problem, \( E\left( {f\left( {x_{i} } \right)} \right) \) is the expectation function of the objective function, \( Var\left( {f\left( {x_{i} } \right)} \right) \) is the variance function,\( \beta \) is the weighting coefficient used to control the distance between product quality and the expected value, and \( G\left( {d,\mu_{x} } \right) \) is the limit state function. The optimization problem constraint is defined in terms of the probability of not failing, which is the so-called probability of reliability \( P_{\text{r}} \), whereas \( P_{ 0} \) represents the minimum allowed probability of reliability of the structure.

2.2 Approximate Model of Vehicle Body Structural Performance

The approximate model is a mathematical model for describing complex systems by building a relationship between structural performance and the design variables. The process to build the approximate model is shown in Fig. 3.
Fig. 3

Approximate model building process

The approximate model to describe vehicle body structural performance can be presented as:
$$ y(x) = \tilde{y}(x) + \varepsilon $$
(3)
where \( y(x) \) is the actual value of structural performance and is equal to the structural performance analyzed by the finite element model (FEM) method, \( \tilde{y}(x) \) is the approximate value of the performance index and can be presented as a polynomial hypothesis, and \( \varepsilon \) is the random error between \( y(x) \) and \( \tilde{y}(x) \).

The optimization design problem for a vehicle body structure is complicated, and the optimal Latin method [8] in which a few simple points can fulfill the entire design space is selected to conduct the DOE.

During numerical modeling, because the optimization design of a vehicle body structure is a complicated problem with multiple variables and high-order functions, a neural network model is used to build the approximate model.

There are two types of neural network model [9]: the radial basis function (RBF) neural network model and the elliptical basis function neural network model. The RBF neural network model is typically built using a linear combination of the radial symmetric function, considering the Euclidean distance function or other distance functions as the independent variables. Using this mathematical model, the problem of fitting and prediction in multidimensional space can be simplified to a one-dimensional problem. It can be written as:
$$ f(x) = p(x) + \sum\limits_{i = 1}^{n} {\omega_{i} \cdot \phi_{i} \left( {\left\| {x - x^{i} } \right\|} \right)} $$
(4)
where \( p(x) \) is a fitting polynomial based on a Gaussian function, \( \omega_{i} \) is the weighting coefficient, \( \left\| {x - x^{i} } \right\| \) represents the Euclidean distance between the sample point and the design reference point, n is the number of input variables, and \( \phi_{i} \) is a radial function that can be written as:
$$ \phi_{i} (y) = \left[ { - \frac{{\left\| {y - c_{i} } \right\|^{2} }}{{2\sigma_{i}^{2} }}} \right] $$
(5)
where \( c_{i} \) is the central vector of the basis function and \( \sigma_{i} \) is the perceptible variable.
To assess the precision of the approximate model of a vehicle body structure, the square error (\( R^{2} \)) [10] is used and can be written as:
$$ R^{2} = 1 - \frac{{\sum\limits_{i = 1}^{n} {\left( {y_{i} - \tilde{y}_{i} } \right)^{2} } }}{{\sum\limits_{i = 1}^{n} {\left( {y_{i} - \bar{y}_{i} } \right)^{2} } }} $$
(6)
where \( y_{i} \) is the actual value of the structural performance, \( \bar{y}_{i} \) is the mean value of the structural performance of the sample points, and \( \tilde{y}_{i} \) is the value estimated based on approximate mathematical models. If the value of \( R^{2} \) is closer to one, the global precision of the model is higher.

2.3 MCS Single-Loop Optimization Strategy

To solve the design problem effectively, the uncertainty optimization should be decoupled into two parts: deterministic optimization and reliability analysis. A single-loop optimization strategy based on a Monte Carlo simulation (MCS) method [11] is used. It first decouples the deterministic optimization and reliability analysis and then adjusts the constraint of the next deterministic optimization function according to the reliability analysis results of the previous deterministic design solution based on the MCS method. The strategy is shown in Fig. 4.
Fig. 4

MCS single-loop optimization strategy

3 Lightweight Design of a Vehicle Body Structure

3.1 Structural Performance

The FEM of the body in white is shown in Fig. 5. The total weight is 364.89 kg.
Fig. 5

FEM of the vehicle body in white

The simulated modal parameters of the body structure were obtained using the Lanczos method, and the first four simulated modes are shown in Fig. 6.
Fig. 6

First four simulated modes. a First mode: 39.2 Hz, b second mode: 42.7 Hz, c third mode: 44.6 Hz and d fourth mode: 48.7 Hz

The experimental modal parameters of the body structure are obtained through the experiments shown in Fig. 7, and the first four experimental modes are shown in Fig. 8.
Fig. 7

Experimental modal analysis of the body in white

Fig. 8

First four experimental modes. a First mode: 37.6 Hz, b second mode: 40.5 Hz, c third mode: 42.6 Hz and d fourth mode: 47.4 Hz

The simulated modal frequency and experimental modal frequency were compared and are shown in Table 1.
Table 1

Mode analysis of the body in white

 

Simulated frequency (Hz)

Tested frequency (Hz)

Relative error (%)

First mode

39.2

37.6

4.3

Second mode

42.7

40.5

5.4

Third mode

44.6

42.6

4.7

Fourth mode

48.7

47.4

2.7

By comparison, the modal shapes in the simulated result are consistent with the experimental result, and the maximum error of the inherent frequency is 5.4%. This means that the accuracy of the FEM can meet the requirements of an engineering application. Based on the model, the bending and torsion stiffness analysis results are shown in Figs. 9 and 10.
Fig. 9

Bending deformation

Fig. 10

Torsion deformation

3.2 Relative Sensitivity Analysis

Relative sensitivity analysis [12] of the plates was conducted to determine the plates that have a great influence on structural performance.

The relative sensitivity was analyzed based on the sensitivity of the bending stiffness (\( S_{\text{b}} \)), the sensitivity of the torsional stiffness (\( S_{\text{t}} \)), the sensitivity of the first-order frequency (\( S_{\text{f}} \)), and the sensitivity of the weight (\( S_{\text{m}} \)). After the relative sensitivity analysis, 28 plates that had a greater or lesser impact on performance were chosen as optimization design variables, as shown in Fig. 11. The relative sensitivity analysis results of 28 chosen plates are shown in Table 2.
Fig. 11

Chosen plates

Table 2

Relative sensitivity analysis result of plates

Part

Weight (kg)

Relative sensitivity

Bending stiffness

Torsional stiffness

First-order frequency

1

2.911

2.15e5

2.75e2

− 5.08e1

2

5.502

2.68e5

1.66e6

− 6.07e1

3

2.457

1.15e5

2.23e5

− 8.07e1

4

1.078

1.08e5

4.41e3

− 7.90e1

5

3.121

3.34e5

1.78e2

2.73e2

6

2.537

8.15e4

4.06e3

− 7.90e1

7

2.520

4.37e5

4.28e5

2.53e2

8

0.813

2.44e5

1.60e3

− 7.13e1

9

0.780

3.40e5

2.47e5

− 9.74e1

10

3.600

5.41e4

4.17e3

− 6.07e1

11

1.528

2.21e2

1.83e5

− 6.07e1

12

0.890

1.25e1

1.23e4

− 7.13e1

13

4.346

3.08e5

3.94e3

− 6.07e1

14

4.687

2.48e5

5.41e3

− 9.74e1

15

1.528

1.24e3

4.83e4

− 6.07e1

16

0.735

− 1.19e4

2.47e5

− 9.74e1

17

3.611

2.45e4

3.79e5

− 7.13e1

18

2.604

3.78e5

1.28e2

2.39e2

19

0.890

2.15e3

1.43e5

− 7.13e1

20

0.626

− 5.95e3

6.36e5

− 6.46e1

21

1.192

− 7.76e3

1.84e4

− 8.59e1

22

4.611

− 1.77e4

1.78e4

− 0.49e1

23

2.573

− 1.06e5

2.23e4

2.53e2

24

2.673

2.58e5

3.86e3

− 5.58e1

25

3.124

2.55e4

5.32e3

− 9.74e1

26

0.337

1.60e3

2.36e5

9.49e1

27

1.216

1.25e1

5.32e3

1.11e2

28

1.173

1.06e3

2.03e3

− 1.38e2

3.3 Approximate Models of Bending Stiffness, Torsional Stiffness, and First-Order Frequency

According to the process requirements of the plates, the design boundary conditions of the 28 chosen plates’ thicknesses are presented in Table 3. Taking the practical process of manufacturing the body in white into account, the welding spot diameter and material elastic modulus are considered as noise factors in the design of the uncertainty optimization.
Table 3

Boundary conditions of the design variables

Lower bound

Initial value \( x_{i0} \)

Upper bound

Interval

X i

0.6

0.5  <  xi0 <  1.0

xi0  + 0.4

0.1

i = 1–28

xi0 − 0.5

1.0   <  xi0 <  2.0

xi0  + 0.5

0.1

xi0  −  0.6

2.0   <  xi0

xi0   + 0.6

0.1

5.7

6.0

6.3

0.1

i = 29

199.5

210

220.5

10

i = 30

Based on the integration platform that combines Nastran with Isight [13], the global approximation models of the bending stiffness, torsional stiffness, and first-order frequency were constructed using an RBF neural network modeling strategy and considering the 28 plates’ thicknesses as the optimization design variables, and the welding spot diameter and material elastic modulus as noise factors. The global precision convergence index of the model was set to 0.95 and the changes during iterations are shown in Table 4.
Table 4

Approximation models’ global precision

Iterations

Global precision R2

\( K_{\text{B}} \)

\( K_{\text{T}} \)

F 1

0

0.83251

0.85267

0.73356

1

0.92346

0.93492

0.78234

2

0.96594

0.98214

0.85573

3

0.92168

4

0.95734

Because the relationship between the structural weight, plate thickness, welding spot diameter, and material elastic modulus is linear, the approximation model of the structural weight was built using a linear polynomial model. Its global precision was 0.99038 and it is given by:
$$ \begin{aligned} m\left( {x_{i} } \right) & = \, 277 + \, 1.50x_{1} + \, 5.76x_{2} + 1.93x_{3} + \, 4.02x_{4} \\ & \quad + \, 2.21x_{5} + \, 1.32x_{6} + \, 1.28x_{7} + \, 1.33x_{8} + 4.78x_{9} \\ & \quad + \, 2.84x_{10} + \, 1.38x_{11} + \, 1.33x_{12} + \, 3.20x_{13} \\ & \quad + \, 1.74x_{14} + \, 3.19x_{15} + \, 3.48x_{16} + \, 1.21x_{17} \\ & \quad + 0.74x_{18} + \, 1.27x_{19} + \, 2.06x_{20} + \, 20.50x_{21} \\ & \quad + \, 2.78x_{22} + \, 1.08x_{23} + \, 2.97x_{24} + \, 2.17x_{25} \\ & \quad + \, 0.53x_{26} + \, 1.45x_{27} + \, 1.76x_{28} + \, 6.54*10^{ - 2} x_{29} \\ & \quad {-} \, 5.86*10^{ - 7} x_{30} \\ \end{aligned} $$
(7)

3.4 Deterministic Optimization and Reliability Analysis

The deterministic vehicle body structure lightweight design problem is written as follows:
$$ \begin{aligned} & {\text{minimize}}\;m(x_{i} ) \\ & {\text{subject}}\;{\text{to}}\;K_{\text{B}} (x_{i} ) \ge 17218.2\,{\text{N/mm}} \\ & K_{\text{T}} (x_{i} ) \ge 20743.8\,{\text{Nm/}}^\circ \\ & f_{1} (x_{i} ) \ge 39.2\,{\text{Hz}} \\ & x_{\text{L}} \le x_{i} \le x_{\text{U}} \\ {\kern 1pt} & i = 1,2, \ldots 27,28 \\ & x_{29} = 6\,{\text{mm}}\quad x_{30} = 2 1 0\,{\text{GPa}} \\ \end{aligned} $$
(8)
where \( m(x_{i} ) \) is the structural weight, \( K_{\text{B}} (x_{i} ) \) is the bending stiffness, \( K_{\text{T}} (x_{i} ) \) is the torsional stiffness, and \( f_{ 1} (x_{i} ) \) is the first-order frequency.
Based on the above approximation models, the adaptive simulated annealing (ASA) algorithm [14] was adopted to solve the deterministic lightweight design problem, and its solution is shown in Table 5.
Table 5

ASA optimization solution

Variables

X 1

X 2

X 3

X 4

X 5

Results (mm)

1.0

0.8

0.9

0.8

1.5

Variables

x 6

x 7

x 8

x 9

x 10

Results (mm)

0.6

0.7

0.7

0.7

0.9

Variables

x 11

x 12

x 13

x 14

x 15

Results (mm)

1.0

0.7

0.75

0.8

0.8

Variables

x 16

x 17

x 18

x 19

x 20

Results (mm)

0.7

0.7

0.9

0.9

1.0

Variables

x 21

x 22

x 23

x 24

x 25

Results (mm)

0.6

0.7

0.7

1.0

2.8

Variables

x 26

x 27

x 28

  

Results (mm)

1.1

1.0

1.0

  

Responses

KB/N/mm

KT/Nm/°

f1/Hz

m/kg

Results

17483.2

20916.2

40.95

344.0

Assuming that the vibrations of the 30 design variables obey the normal distribution with the variance coefficient of 0.05 and the mean value as the optimized result obtained in the deterministic optimization, the reliability analysis of the deterministic optimization scheme was performed using an MCS method. The reliability analysis results are shown in Fig. 12.
Fig. 12

Reliability of the deterministic optimization solution. a Reliability analysis of bending stiffness, b Reliability analysis of torsional stiffness, c Reliability analysis of 1st-order frequency

According to the reliability analysis results, the structural weight fluctuated within the range of 343.39 kg to 344.54 kg. During 1000 random sampling experiments, the probability of the first-order frequency greater than 39.2 Hz was 100%, whereas the probability of bending stiffness greater than 17,218 N/mm was 55.4% and the probability of torsional stiffness greater than 20,743 Nm/° was 70.8%. This means that if the deterministic optimization design solution is applied to vehicle body manufacturing, although the target of weight loss is achieved, the possibility of failure still exists.

3.5 Uncertainty Optimization

The uncertainty lightweight design of a vehicle body structure considering the fluctuations of design variables is written as follows:
$$ \begin{aligned} & {\text{minimize}} \\ & \hbox{min} \;{\kern 1pt} \varphi \left( {x_{i} } \right) = E\left( {m\left( {x_{i} } \right)} \right) + \beta \cdot \sqrt {Var\left( {m\left( {x_{i} } \right)} \right)} \\ & {\text{subject}}\;{\text{to}}\;P(K_{\text{B}} (x_{i} ) \ge 17218\,{\text{N/mm}}) \ge P_{0} \\ & P(K_{\text{T}} (x_{i} ) \ge 20743\,{\text{Nm/}}^\circ ) \ge P_{0} \\ & P(f_{1} (x_{i} ) \ge 39.2{\text{Hz}}) \ge P_{0} \\ & x_{\text{L}} \le x_{i} \le x_{\text{U}} (i = 1,2, \ldots 27,28) \\ & \bar{x}_{29} = 6{\text{mm}}\quad \bar{x}_{30} = 2 1 0\,{\text{GPa}} \\ \end{aligned} $$
(9)
where \( E(m(x_{i} )) \) is the expectation function, \( Var(m(x_{i} )) \) is the variance function of the structural weight of the body in white, \( \beta \) is used to control the robustness of the structural performance based on the 6\( \sigma \) robustness criteria and its value is 3, and \( P_{0} \) is the minimum allowed probability of reliability and its value is set to 0.99865 to ensure a feasible solution of the uncertainty optimization design problem.
The relationship between the structural weight and design variables is linear and can be written as \( m(x_{i} ) = a + b_{i} x_{i} \), where \( a \) is a constant and \( b_{i} \) is the coefficient of each variable. Thus, \( E\left( {m\left( {x_{i} } \right)} \right) \) and \( Var\left( {m\left( {x_{i} } \right)} \right) \) can be converted to:
$$ E\left( {m\left( {x_{i} } \right)} \right) = E\left( {a + b_{i} x_{i} } \right) = a + b_{i} \cdot E(x_{i} ) = m\left( {x_{i} } \right) $$
(10)
$$ \begin{aligned} Var \left( {m\left( {x_{i} } \right)} \right) & = Var\left( {a + b_{i} x_{i} } \right) = b_{i}^{2} Var\left( {x_{i} } \right) \\ & = b_{i}^{2} \cdot \left( {\xi \cdot Ex_{i} } \right)^{2} = \gamma \cdot X \end{aligned} $$
(11)
$$ \begin{aligned} \gamma & = 1.0 \times 10^{ - 9} \times [2.250\quad 33.178\quad {\kern 1pt} 3.725\quad 16.160 \\ & \quad 4.884\quad 1.742\quad 1.638\quad {\kern 1pt} 1.769\quad 22.848\quad 8.066 \\ &\quad 1.904\quad 1.769{\kern 1pt} \quad 10.240\quad {\kern 1pt} 3.028{\kern 1pt} \quad 10.176{\kern 1pt} \quad 12.110 \\ &\quad 1.464\quad 0.546\quad 1.613\quad 4.244\quad {\kern 1pt} 420.250\quad 7.728{\kern 1pt} \\ & \quad 1.166\quad 8.821\quad 4.709\quad {\kern 1pt} 0.285\quad 2.103\quad 3.098{\kern 1pt} \\ & \quad {\kern 1pt} 4.264 \times 10^{ - 3} \quad 3.439 \times 10^{ - 13} ] \\ \end{aligned} $$
where \( \xi \) is the variance coefficient.
According to the single-loop optimization strategy, the uncertainty optimization problem should be transformed into the initial deterministic optimization problem first. The initial deterministic optimization problem is written as follows:
$$ \begin{aligned} & {\text{minimize}} \\ & \varphi \left( {x_{i} } \right) = E\left( {m\left( {x_{i} } \right)} \right) + \beta \cdot \sqrt {Var\left( {m\left( {x_{i} } \right)} \right)} \\ & {\text{subject}}\;{\text{to}}\;K_{\text{B}} (x_{i} ) \ge 17218.2\,{\text{N/mm}} \\ & K_{\text{T}} (x_{i} ) \ge 20743.8\,{\text{Nm/}}^\circ \\ & f_{1} (x_{i} ) \ge 39.2\,{\text{Hz}} \\ & x_{\text{L}} \le x_{i} \le x_{\text{U}} \\ {\kern 1pt} & i = 1,2, \cdot \cdot \cdot 27,28 \\ & \bar{x}_{29} = 6\,{\text{mm}}\quad \bar{x}_{30} = 2 1 0\, {\text{GPa}} \\ \end{aligned} $$
(12)

3.6 Constraint Correction and Reliability Analysis

First, the initial deterministic optimization problem was solved using the ASA algorithm and then reliability was analyzed using the MCS method. According to the analysis results, the constraints of the initial deterministic optimization problem were finally adjusted toward the requirement.

After six iterations, the uncertainty lightweight design problem of the vehicle body structure was solved. The changes of design variables, objective function \( \varphi \left( {x_{i} } \right) \), increment of constraints, and reliability of structural performance for each iteration are shown in Tables 6, 7, and 8. The reliability analysis result is shown in Fig. 13.
Table 6

Changes of design variables

Design variables/mm

Iterations

1

2

3

4

5

6

x 1

1.0

1.0

1.0

1.0

1.0

1.0

x 2

0.8

0.8

0.8

0.8

0.8

0.8

x 3

0.9

0.8

0.9

0.9

0.8

0.8

x 4

0.8

0.8

0.8

0.8

0.8

0.8

x 5

1.5

1.5

1.5

1.5

1.5

1.6

x 6

0.6

0.6

0.6

0.7

0.6

0.6

x 7

0.7

0.6

0.6

0.6

0.6

0.6

x 8

0.7

0.8

0.8

0.7

0.9

0.8

x 9

0.7

0.7

0.7

0.7

0.7

0.7

x 10

0.9

0.8

0.8

0.8

0.8

0.8

x 11

1.0

1.2

1.2

1.1

1.0

1.0

x 12

0.7

0.7

0.8

0.7

0.7

0.8

x 13

0.75

0.85

0.75

0.75

0.8

0.8

x 14

0.8

0.8

0.8

0.9

0.8

0.8

x 15

0.8

0.9

1.0

1.0

1.0

0.9

x 16

0.7

0.6

0.6

0.7

0.6

0.6

x 17

0.7

0.7

0.7

0.7

0.8

0.9

x 18

0.9

1.3

0.9

0.9

1.0

0.7

x 19

0.9

0.7

0.7

0.7

0.8

0.7

x 20

1.0

1.0

1.0

1.0

1.0

1.1

x 21

0.6

0.6

0.6

0.6

0.6

0.7

x 22

0.7

0.8

0.7

0.7

0.7

0.7

x 23

0.7

0.7

0.7

0.6

0.6

0.6

x 24

1.0

1.0

1.0

1.0

1.0

1.0

x 25

2.8

2.9

3.0

2.9

2.9

3.0

x 26

1.1

1.0

1.2

0.9

1.0

1.3

x 27

1.0

1.0

1.0

1.0

1.0

1.0

x 28

1.0

1.0

1.0

1.1

1.1

1.0

Table 7

Changes of the objective function and constraints

Iterations

Objective \( \varphi (x_{i} ) \)/kg

Increment of constraints

KB

KT

f1

1

345.92

0

0

0

2

346.44

345.93

437.92

0

3

346.52

40.45

69.97

0

4

346.51

13.05

10.03

0

5

346.32

0.78

3.57

0

6

346.5

0

2.12

0

Table 8

Changes of structural performance reliability

Iterations

Structural performance reliability

P(KB)

P(KT)

P(f1)

1

0.5540

0.7080

1.0000

2

0.9840

0.9690

1.0000

3

0.9950

0.9800

1.0000

4

0.9970

0.9900

1.0000

5

0.9990

0.9950

1.0000

6

0.9990

0.9987

1.0000

Fig. 13

Reliability of the uncertainty optimization solution. a Reliability analysis of bending stiffness, b Reliability analysis of torsional stiffness, c Reliability analysis of 1st-order frequency

Among the six iterations, the structural weight fluctuated within the range of 343.91 kg to 345.13 kg. The probability of a first-order frequency greater than 39.2 Hz was 100%, the probability of bending stiffness greater than 17,218 N/mm was 99.9%, and the probability of torsional stiffness greater than 20743 Nm/° was 99.87%.

Considering the fluctuations of design variables, the reliability of the bending stiffness and torsional stiffness of the vehicle body after uncertainty optimization increased by 44.6% and 29.2%, respectively, compared with the deterministic optimization solution.

3.7 Accuracy Analysis of the Uncertainty Optimization Solution

To verify the accuracy of the uncertainty optimization solution, the bending stiffness, torsional stiffness, and first-order frequency of the body in white were analyzed again based on the FEM and the plates’ thicknesses from above solution. The results are shown in Table 9.
Table 9

Accuracy of the uncertainty optimization solution

Responses

\( m\left( {x_{i} } \right) \)/kg

KB/N/mm

KT/Nm/°

F1/Hz

Solution

346.40

17620.41

21246.50

40.98

FEM

346.95

17121.29

20616.11

39.82

Error (%)

0.16

2.92

3.06

2.91

From Table 9, the relative errors between the results of the uncertainty optimization solution and those of FEM analysis are very small. Thus, the uncertainty optimization solution based on the RBF neural network model and MCS single-loop strategy is accurate and feasible.

Ultimately, the structural weight was reduced by 17.94 kg without decreasing the bending stiffness, torsional stiffness, and modal performance. Although the weight of the vehicle body design using the uncertainty optimization solution was 2.4 kg heavier than that using the deterministic optimization solution, the reliability of the bending and torsion stiffness response was higher.

4 Conclusions

Considering the deviations of plate thickness, welding spot diameter, and material elastic modulus, in this article, the uncertainty optimization design method was applied to the lightweight design of a car body in white based on the requirement of reliability and robustness. The structural weight of the vehicle body design according to the uncertainty optimization solution was reduced by 17.94 kg, whereas the reliability of the bending stiffness and torsional stiffness increased by 44.6% and 29.2%, respectively, compared with the deterministic lightweight design solution.

Because the lightweight design of the car body structure is a comprehensive problem with multiple variables and multiple performance response indices, more uncertain factors that affect the structural performance and nonlinear performance response, such as collision performance and fatigue performance, should also be considered. In the future, further research needs to be conducted.

Notes

Acknowledgements

This study was supported by the National Natural Science Foundation of China (51775193) and the Science and Technology Planning Project of Guangdong Province, China (2016A050503021, 2015B0101137002, and 2017B010119001).

References

  1. 1.
    Lin, X.H., Feng, Y.X., Tan, J.R., et al.: Robust optimization design method for quality characteristics of mechanical products based on quantifier constraints. J. Mech. Eng. 49(15), 169–179 (2013)CrossRefGoogle Scholar
  2. 2.
    Du, X.P., Guo, J., Beeram, H.: Sequential optimization and reliability assessment for multidisciplinary systems design. Struct. Multidiscip. Optim. 35(2), 117–130 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Liu, Z.F., Atamturktur, S., Juang, H.C.: Reliability based multi-objective robust design optimization of steel moment resisting frame considering spatial variability of connection parameters. Eng. Struct. 76, 393–403 (2014)CrossRefGoogle Scholar
  4. 4.
    Cui, J., Zhang, W.G., Chang, W.B., et al.: Optimization design of collision safety robustness based on double response surface model. J. Mech. Eng. 47(24), 97–103 (2011)CrossRefGoogle Scholar
  5. 5.
    Zhang, Y., Li, G.Y., Zhong, Z.H.: Research on application of reliability-based multidisciplinary design optimization in lightweight design of thin-walled beams. China Mech. Eng. 20(15), 1885–1889 (2009)Google Scholar
  6. 6.
    Qiu, R.B., Chen, Y., Lei, F., et al.: Research on lightweight of b-pillar for car side impact body based on 6σ robust design. Mech. Strength 38(03), 502–508 (2016)Google Scholar
  7. 7.
    Xie, R., Lan, F.C., Chen, J.Q., et al.: Multi-objective optimization method of lightweight body structure to meet reliability requirements. Chin. J. Mech. Eng. 47(4), 117–124 (2011)CrossRefGoogle Scholar
  8. 8.
    Chen, G.D., Han, X.: Multi-objective optimization method based on agent model and its application in vehicle body design. J. Mech. Eng. 50(09), 70 (2014)Google Scholar
  9. 9.
    Chan, C.L., Low, B.K.: Probabilistic analysis of laterally loaded piles using response surface and neural network approaches. Comput. Geotech. 43, 101–110 (2012)CrossRefGoogle Scholar
  10. 10.
    Goel, T., Stander, N.: Comparing three error criteria for selecting radial basis function network topology. Comput. Methods Appl. Mech. Eng. 198(27–29), 2137–2150 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Biancardi, M., Villani, G.: Robust Monte Carlo method for R&D real options valuation. Comput. Econ. 49(3), 481–498 (2017)CrossRefGoogle Scholar
  12. 12.
    Chen, C., Yu, D.K.: Lightweight design of medium passenger car frame based on relative sensitivity analysis. Automot. Technol. 06, 27–30 (2014)Google Scholar
  13. 13.
    Lai, Y.Y.: Isight parametric optimization theory and detailed examples. Beijing University of Aeronautics and Astronautics Press, Beijing (2012)Google Scholar
  14. 14.
    Ekbal, A., Saha, S.: Combining feature selection and classifier ensemble using a multiobjective simulated annealing approach: application to named entity recognition. Soft. Comput. 17(1), 1–16 (2013)CrossRefGoogle Scholar

Copyright information

© China Society of Automotive Engineers (China SAE) 2018

Authors and Affiliations

  1. 1.School of Mechanical and Automotive EngineeringSouth China University of TechnologyGuangzhouChina
  2. 2.Guangdong Provincial Key Laboratory of Automotive EngineeringGuangzhouChina

Personalised recommendations