Metamodel-Based Multi-Objective Reliable Optimization for Front Structure of Electric Vehicle
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Abstract
In this paper, a multi-objective reliable optimization (MORO) procedure for the front body of an electric vehicle is proposed and compared with determinate multi-objective optimization (DMOO). The energy absorption and peak crash force of the simplified vehicle model under the full-lap frontal impact condition are used as the design objectives, with the weighted sum of the basic frequency, the first-order torsional and bending frequencies of the full-size vehicle model, and the weight of the front body taken as the constraints. The thicknesses of nine components on the front body are defined as design variables, and their geometric tolerances determine the uncertainty factor. The most accurate metamodel using the polynomial response surface, kriging, and a radial basis function is selected to model four design criteria during optimization, allowing the efficiency improvement to be computed. Monte Carlo simulations are adopted to handle the probability constraints, and multi-objective particle swarm optimization is employed as the solver. The MORO results indicate reliability levels of \(R = 100\)%, demonstrating the significant enhancement in reliability of the front body over that given by DMOO, and reliable design schemes and proposals are provided for further study.
Keywords
Multi-objective reliable optimization Electric vehicle body Metamodel technique Monte Carlo1 Introduction
Similar to conventional-fuel automobiles, poor body designs of electric vehicles will lead to many severe problems [1]. Consequently, structural design optimization at the body-in-white (BIW) stage is a major concern in the electric automotive industry. Of all the structural properties of the vehicle body, frontal crashworthiness has attracted the most research interest in terms of the safety consideration [2]. There are many practical metrics for evaluating a vehicle’s crashworthiness, such as the energy absorption, maximum acceleration, and maximum intrusion, although many of these conflict with one another. Therefore, to acquire the optimal crashworthiness, a multi-objective optimization (MOO) problem must be explored.
Numerous finite element method (FEM) simulations are required to evaluate the objectives and constraints in MOO problems, and these can be extremely time-consuming when using either gradient-based or evolutionary-based optimization techniques. To enhance the optimization efficiency, the widely used metamodel technique uses a set of samples determined by the design of experiment (DOE) during the FEM optimization [3]. The polynomial response surface (PRS), kriging (KRG), and radial basis function (RBF) are popular models in MOO for vehicle bodies. For instance, Liao et al. employed the PRS-based non-dominated sorting genetic algorithm II (NSGA-II) algorithm to perform MOO for vehicle body, using the weight, full-lap crashworthiness, and 40% offset frontal impact as design criteria [4]. By combining a sequential RBF-based metamodeling technique with a micro-multi-objective genetic algorithm, Chen et al. used MOO to simultaneously reduce the peak impact force in the event of a roof crash and decrease the weight of the car [5]. To maximize the absorbed energy of a bus frame under the rollover condition while ensuring as light a weight as possible, Fan et al. conducted MOO using NSGA-II [6]. Note that the types of metamodels used for structural mechanics depend greatly on the specific research objectives. Consequently, it is necessary to study the most accurate and appropriate metamodel for predicting multi-discipline response in vehicle BIW optimization.
The above-mentioned studies did not consider uncertainty factors such as the manufacturing tolerance, materials, loading conditions, or environment, resulting in considerable limitations in practical applications. In deterministic optimization, designs are often pushed to the limit, and small variations in design variables or parameters could cause the design to violate some crucial constraints. To take various uncertainties into consideration, reliable design optimization (RDO) has received increasing attention in automotive structure optimization. For example, a comparative study of multi-objective, deterministic, reliable, and robust design optimization for the crashworthiness improvement of a car’s B-pillar assembly was presented by Shetty et al. [7]. Fang et al. [8] explored multi-objective reliable optimization (MORO) for a vehicle door using Monte Carlo simulations (MCS) based on the probabilistic sufficiency factor method, metamodel technique, and multi-objective particle swarm optimization (MOPSO) algorithm. Song et al. [9] implemented RDO for an automotive knuckle component under different working conditions, where a constraint-feasible moving least-square method was adopted to model the functional inequality constraint. Shi et al. [10] developed a stochastic sensitivity analysis method to compute the sensitivities of the probabilistic response using a metamodel with MCS, where the metamodel is determined by a Bayesian metric with data uncertainty. Rais-Rohani et al. [11] optimized the shape and size of vehicle structures by examining the effects of different constraints and their associated uncertainties on the reliability and efficiency of the optimum designs for 100% or offset frontal crashes. Nevertheless, RDO for the front body structure of an electric vehicle considering multiple performance requirements and uncertainty factors has received limited attention in the literature.
In this paper, a MORO procedure for an electric vehicle’s front body structure is presented, and the effects of uncertainties in the geometric parameters of components are examined. After constructing and implementing FEM for the baseline design of an electric vehicle body structure, there is still a relatively large optimization design space in its frontal part. The crashworthiness with full-lap frontal impact of this front structure, and the effect on the basic modal frequency and lightweight property of the whole body, is investigated using deterministic multi-objective optimization (DMOO) and the corresponding MORO, with the thicknesses of nine key components taken as the design variables. MCS is employed to address the probabilistic constraints in MORO, and different metamodels are screened to select the most suitable substitute for the costly FEM. The MOPSO algorithm is adopted to generate well-distributed Pareto solutions in both DMOO and MORO.
2 Theory and Methodology
2.1 Deterministic Multi-objective Optimization
2.2 Multi-objective Reliable Optimization
2.3 Monte Carlo Simulation
As stated in Eq. (2), probabilistic constraints should be repeatedly evaluated in MORO. MCS has been widely applied in approximating the probability of a series of random process output events by randomly sampling for uncertain variables [9, 11, 12, 13]. The MCS procedure is composed of the following steps: (1) generating the sample set of random variables based on the probability density function; (2) constructing the mathematical model of the limit state function to ensure the failure probability of known sample points for random variables; (3) calculating the probabilistic characteristics of the structural system response after simulating for the sampling points of the random variables.
2.4 Metamodel Technique
In general, direct structural optimization based on a simulation model might be inefficient or even infeasible, as iterative nonlinear FEM runs for objectives and constraints evaluation usually have an extremely high computational burden. As a model of models, a metamodel can be constructed from the relationship between the inputs, i.e., the set of design variables generated by DOE, and the outputs, i.e., the corresponding system responses, and this metamodel can be conveniently used to predict the response at other points within the design space. The metamodel technique has been widely adopted in DMOO and MORO [13]. In this paper, three types of metamodels are studied, namely PRS, KRG, and RBF.
2.4.1 PRS
2.4.2 KRG
2.4.3 RBF
2.4.4 Accuracy Evaluation Metrics for Metamodels
2.5 Multi-objective Particle Swarm Optimization Algorithm
- Step 1.
Set the number of iterations as t = 0, initialize the position (\(\mathbf{x}_i )\) and velocity (\(\mathbf{v}_i )\) of each particle, compute the objective vector corresponding to each particle, extract the non-dominated solutions, and place them in an external archive (\(\hbox {A}_i )\);
- Step 2.
Determine the optimal positions of each particle (\(\mathbf{p}_i )\) and the whole particle swarm (\(\mathbf{p}_g )\);
- Step 3.Under the condition of ensuring the particles move about the search space, update \(\mathbf{x}_i \) and \(\mathbf{v}_i \) according to$$\begin{aligned} \left\{ \begin{array}{l} \mathbf{v}_i (t+1)=\mathbf{v}_i (t)+r_1 c_1 \left[ {\mathbf{p}_i (t)-\mathbf{x}_i (t)} \right] \\ \qquad \qquad \qquad +\,r_2 c_2 \left[ {\mathbf{p}_g (t)-\mathbf{x}_i (t)} \right] \\ \mathbf{x}_i (t+1)=\mathbf{x}_i (t)+\mathbf{v}_i (t+1) \\ \end{array}\right. \end{aligned}$$(11)
- Step 4.
Update \(\hbox {A}_i \) and \(\mathbf{p}_g \) based on the newly derived non-dominated solutions;
- Step 5.
Terminate the algorithm if the convergence condition is satisfied; if not, return to step 3.
3 FEM of Electric Vehicle Body-In-White
3.1 FEM Simulation for Free Modal Analysis
Three natural frequencies of the BIW
Natural frequency | Basic frequency | First-order torsional frequency | First-order bending frequency |
---|---|---|---|
Value (Hz) | 26.22 | 28.49 | 37.52 |
3.2 FEM Simulation for Crashworthiness with Full-Lap Frontal Impact
As one of the most dangerous extreme working conditions of passenger vehicles, this paper examines the 100%-overlap frontal crashworthiness. In this case, structural deformations mainly occur at the front end of the vehicle to absorb kinetic energy and reduce the force passing into the passenger compartment. A single simulation of the crash process using a detailed FEM model of the entire vehicle requires enormous computational effort and has a high risk of divergence. In view of this, only the FEM model of the front body structure is used, and other parts of the vehicle are substituted and represented by a board with a uniformly distributed mass rigidly attached to the front body [18, 19]. Thus, Blytskho–Tsay shell elements and piecewise linear plasticity material models are arranged here. The FEM model of this vehicle for 100%-overlap frontal crashworthiness analysis involves 90267 elements (see Fig. 3). An initial velocity of 50 km/h is considered, and the crash process occurs within 25 ms.
4 Optimization Design for the Front BIW
Detailed information on nine design variables
Design variables | Baseline design (mm) | Range (mm) | CoV (%) |
---|---|---|---|
\(x_{1}\) | 0.7 | 0.5–2.0 | 1 |
\(x_{2}\) | 0.9 | 0.5–2.0 | 1 |
\(x_{3}\) | 1.4 | 0.5–2.0 | 1 |
\(x_{4}\) | 1.6 | 0.5–2.0 | 1 |
\(x_{5}\) | 1.0 | 0.5–2.0 | 1 |
\(x_{6}\) | 1.4 | 0.5–2.0 | 1 |
\(x_{7}\) | 0.8 | 0.5–2.0 | 1 |
\(x_{8}\) | 0.9 | 0.5–2.0 | 1 |
\(x_{9}\) | 0.7 | 0.5–2.0 | 1 |
Accuracy assessment for different metamodels
Design criteria | First-order PRS | Second-order PRS | Third-order PRS | Fourth-order PRS | KRG | RBF | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(R^{2}\) | RAAE | \(R^{2}\) | RAAE | \(R^{2}\) | RAAE | \(R^{2}\) | RAAE | \(R^{2}\) | RAAE | \(R^{2}\) | RAAE | |
E | 0.9072 | 0.2431 | 0.9259 | 0.1762 | 0.9210 | 0.1955 | 0.9091 | 0.2074 | 0.9188 | 0.1849 | 0.8754 | 0.2297 |
\(F_{\mathrm{max}}\) | 0.9978 | 0.0431 | 0.9990 | 0.0245 | 0.9991 | 0.0234 | 0.9989 | 0.0294 | 0.9986 | 0.0293 | 0.9978 | 0.0386 |
\(f_{\mathrm{w}}\) | 0.8012 | 0.3024 | 0.9840 | 0.0875 | 0.9865 | 0.0741 | 0.9887 | 0.0661 | 0.9535 | 0.1258 | 0.9846 | 0.0759 |
Comparison between the baseline and the knee-point plan in DMOO
Items | Baseline design | Knee-point plan in DMOO | ||||
---|---|---|---|---|---|---|
Knee point | FEM | Error | Reliability | |||
Objectives | E | 16.18 kJ | 12.41 kJ (− 23.3%) | 11.92 kJ | + 4.1% | – |
\(F_{\mathrm{max}}\) | 421.93 kN | 263.58 kN (− 37.5%) | 255.17 kN | + 3.3% | – | |
Constraints | M | 18.03 kg | 15.37 kg (− 14.8%) | 15.65kg | − 1.8% | 100% |
\(f_{\mathrm{w}}\) | 30.74 Hz | 30.92 Hz (+ 0.6%) | 30.82 Hz | − 0.3% | 55% |
5 Results and Analysis
5.1 Selection and Analysis of Metamodels
As the weight of the front structure of this BIW has a linear relationship with the thicknesses of its components, the first-order PRS is selected to model M. To evaluate the accuracy of the different metamodels, 10 extra validation points are generated at random over the whole design space by OLHS. The computational results are compared in Table 3.
From Table 3, it is clear that the third-order PRS, second-order PRS, and fourth-order PRS are the most accurate and appropriate metamodels for predicting \(F_{\mathrm{max}} \), E, and \(f_\mathrm{w} \), respectively, as they give the highest \(R^{2}\) values in conjunction with the lowest RAAE values for each design criterion.
5.2 Results and Analysis of DMOO
Comparison of results from MORO and DMOO
Items | Results of DMOO | Results of MORO | |||
---|---|---|---|---|---|
Knee point | Reliability | Knee point | Reliability | ||
Obj. | E | 12.41 kJ | – | 10.58 kJ | – |
\(F_{\mathrm{max}}\) | 263.58 kN | – | 241.86 kN | – | |
Cons. | M | 15.65 kg | 100% | 17.39 kg | 100% |
\(f_{\mathrm{w}}\) | 30.92 Hz | 55% | 31.61 Hz | 100% | |
D. V. | \(x_{1}\) | 0.55 mm | – | 0.91 mm | – |
\(x_{2}\) | 1.33 mm | – | 1.29 mm | – | |
\(x_{3}\) | 1.05 mm | – | 1.01 mm | – | |
\(x_{4}\) | 1.07 mm | – | 0.86 mm | – | |
\(x_{5}\) | 1.34 mm | – | 1.91 mm | – | |
\(x_{6}\) | 0.72 mm | – | 0.94 mm | – | |
\(x_{7}\) | 1.01 mm | – | 0.84 mm | – | |
\(x_{8}\) | 0.97 mm | – | 0.75 mm | – | |
\(x_{9}\) | 0.94 mm | – | 0.83 mm | – |
Table 4 compares the mechanical responses of the vehicle body regarding the design criteria for the baseline design and the knee- point plan in the DMOO. Note that the optimization effect is enormous, as the relevant \(F_{\mathrm{max}} \) is decreased by 37.5%, although E is degraded by \(\sim \)23.3%. Additionally, M and \(f_\mathrm{w} \) are reduced by 14.8% and improved by 0.6%, respectively, over the baseline design. This analysis validates the accuracy of the knee-point plan, because the differences between the four design criteria and the corresponding FEM results are all within 5%. However, the knee-point plan in DMOO does not give a suitable value of \(f_\mathrm{w} \): the reliability computed by MCS is only 55%, which means \(f_\mathrm{w} \) has a 45% chance of being less than its baseline value (30.74 Hz). In view of this, MORO is required for this front body.
5.3 Results and Analysis of MORO
In consideration of the uncertainty from the manufacturing tolerances of the components corresponding to the nine design variables, this section considers the application of MORO for the front body. The probability evaluation for the constraints in Eq. (14) is performed by metamodel-based MCS with a MOPSO optimizer. The final POF given by MORO is contrasted with that from DMOO in Fig. 9. The POF of MORO is located to the right of the DMOO POF, demonstrating that conservative optimum results are obtained by MORO. Furthermore, these two POFs are becoming closer toward the lower right corner, which indicates a relative sacrifice in E rather than \(F_{\mathrm{max}}\) to accommodate the randomness of the design variables in MORO. Therefore, Pareto solutions placed on the left half of the POF represent alternative design schemes for designers.
The optimally balanced knee-point plan is also extracted as the optimum design scheme for MORO. This is compared with that from DMOO in Table 5. To account for the uncertainty factor, E in the conservative MORO knee-point plan is further reduced to 10.58 kJ. \(F_{\mathrm{max}} \) is decreased to 241.86 kN, lowering the safety risk for this vehicle in a frontal crash. After MORO, M has increased to 17.39 kg, which still has a clear margin from the boundary value of 18.03 kg. The larger values of design variables \(x_{1}\) and \(x_{2}\) contribute most to the increased weight of the front body, as their corresponding components are the two largest in the front assembly. Compared with the results given by DMOO, \(f_\mathrm{w} \) is obviously ameliorated to 31.61 Hz, away from the design criterion of 30.74 Hz. Therefore, both M and \(f_\mathrm{w} \) have reliability values in excess of the 99% target, indicating the effectiveness of the knee-point design scheme in MORO.
6 Conclusions
- (1)
The second-order PRS, third-order PRS, and fourth-order PRS are the most appropriate metamodels for the energy absorption, maximum crash force, and weighted-sum frequency of the entire body, respectively.
- (2)
The reliability of the weighted-sum frequency in the knee-point design scheme from DMOO is 55%, which is much less than the 99% design requirement.
- (3)
The knee-point scheme from MORO is much more reliable and achieves an acceptable sacrifice of the overall performance of the vehicle body.
Notes
Acknowledgements
This work is supported by the Science and Technology Planning Project of Beijing City (Z161100001416007) and the National Key R&D Program of China (2017YFB0103801).
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