Uncertainty Optimization Design of a Vehicle Body Structure Considering Random Deviations
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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 strategy1 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
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.
2.2 Approximate Model of Vehicle Body Structural Performance
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.
2.3 MCS Single-Loop Optimization Strategy
3 Lightweight Design of a Vehicle Body Structure
3.1 Structural Performance
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 |
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.
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
Boundary conditions of the design variables
Lower bound | Initial value \( x_{i0} \) | Upper bound | Interval | X _{ i} |
---|---|---|---|---|
0.6 | 0.5 < x_{i0} < 1.0 | x_{i0 } + 0.4 | 0.1 | i = 1–28 |
x_{i0} − 0.5 | 1.0 < x_{i0} < 2.0 | x_{i0 } + 0.5 | 0.1 | |
x_{i0 }− 0.6 | 2.0 < x_{i0} | x_{i0 } + 0.6 | 0.1 | |
5.7 | 6.0 | 6.3 | 0.1 | i = 29 |
199.5 | 210 | 220.5 | 10 | i = 30 |
Approximation models’ global precision
Iterations | Global precision R^{2} | ||
---|---|---|---|
\( K_{\text{B}} \) | \( K_{\text{T}} \) | 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 |
3.4 Deterministic Optimization and Reliability Analysis
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 | K_{B}/N/mm | K_{T}/Nm/° | f_{1}/Hz | m/kg | |
Results | 17483.2 | 20916.2 | 40.95 | 344.0 |
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
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.
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 |
Changes of the objective function and constraints
Iterations | Objective \( \varphi (x_{i} ) \)/kg | Increment of constraints | ||
---|---|---|---|---|
∆K_{B} | ∆K_{T} | ∆f_{1} | ||
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 |
Changes of structural performance reliability
Iterations | Structural performance reliability | ||
---|---|---|---|
P(K_{B}) | P(K_{T}) | P(f_{1}) | |
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 |
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
Accuracy of the uncertainty optimization solution
Responses | \( m\left( {x_{i} } \right) \)/kg | K_{B}/N/mm | K_{T}/Nm/° | F_{1}/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).
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