Optimization Study of Driver Crash Injuries Considering the Body NVH Performance

Abstract

Optimal body structure design is a central focus in the field of passive automotive safety. A well-designed body structure enhances the lower threshold for crash safety, serving as a basis for the deployment of other safety systems. Frontal crashes, particularly those with an overlap rate below 25%, are the most frequent types of vehicular accidents and pose elevated risks to occupants due to variable energy absorption and force transmission mechanisms. This study aims to identify an optimized, cost-effective, and lightweight solution that minimizes occupant injuries. Using a micro-vehicle as a case study and accounting for noise, vibration, and harshness (NVH) performance, this paper employs Elman neural networks to predict key variables such as the first-order modes of the body, the body’s mass, and the head injury values for the driver. Guided by these predictions and constrained by the first-order modes and body mass, a genetic algorithm was applied to explore optimal solutions within the solution space defined by the body panel thickness. The optimized design yielded a reduction of approximately 173.43 in the driver’s head injury value while also enhancing the noise, vibration, and harshness performance of the vehicle body. This approach offers a methodological framework for future research into the multidisciplinary optimization of automotive body structures.

1. Introduction

The rapid advancement of the automotive industry has led to a substantial increase in the global automobile population. Simultaneously, this surge in automobiles has resulted in a heightened frequency of traffic accidents. Mitigating injuries resulting from these accidents hinges on both legislative measures and public safety awareness. Additionally, advancements in active and passive safety technologies play a crucial role in this regard [1]. Unlike active safety technologies, passive safety technologies are adept at minimizing personal injuries during accidents [2]. Notably, the development of an exceptional body structure design remains a focal point in passive safety research. A dependable body structure design can elevate the lower threshold of automobile crash safety, forming a strong basis for the subsequent integration of safety systems. Consequently, vehicular crash safety holds enduring significance within the domain of automotive research and development (R&D).

In the early stages of vehicle R&D, extensive reliance was placed on physical crash tests. This approach was characterized by its inherent limitations, including low precision, protracted R&D cycles, and exorbitant costs, largely attributed to its empirical and test-based nature. In a concerted effort to expedite R&D cycles and reduce costs, alternative methods such as crash analysis, multirigid body dynamics, and the finite element method were successively introduced [3,4,5,6,7,8]. Among these methodologies, the finite element method has gained widespread acceptance in automotive crash safety research due to its exceptional precision, ease of model adaptation, and reliable results. Through continuous research and international scholarly contributions, notable progress has been made in the field of automotive crash safety.

Abbasi et al. [9] employed the thicknesses of six internal vehicle components as design parameters. The optimization objectives centered on occupant chest acceleration during frontal crashes and body mass. Liu, HC et al. [10] integrated experimental design with the multiobjective particle swarm optimization method to enhance the crashworthiness of automobile structures. This approach effectively addressed the low accuracy issue associated with traditional response surface methods across the entire design space. Du, XP et al. [11] introduced a data-mining-based design method for multiobjective vehicle design. This approach combined experimental design with crash finite element analysis, efficiently achieving the design objectives. To enhance the light weight, crash safety, and optimization efficiency of the body-in-white, Zhang, S et al. [12] employed entropy-weighted gray correlation analysis along with an improved nondominated sorting genetic algorithm for multiobjective optimization. The results notably improved vehicle lightweighting and crash safety performance. Gao, DW et al. [13] proposed an adaptive radial basis function neural network with a machine learning point addition (ARBF-MLPA) method, reducing computational costs for solving multiobjective problems. This approach was applied alongside the non-dominated sorting genetic algorithm II (NSGA-II) for vehicle multiobjective optimization during side-effect crashes. Zhu, TJ et al. [14] introduced a structural lightweight design approximation model based on a sequential quadratic programming algorithm. The objective was to minimize the body weight without compromising bending stiffness performance through a comprehensive multiobjective optimization method. Wang, Q et al. [15] developed a multiobjective optimization model for occupant restraint system (ORS) design, considering interval uncertainty and its correlations. The interval multiobjective optimization model was transformed into a deterministic model, and the problem was solved using the multiobjective genetic algorithm (GA) coupled with the interval expansion method. Sanjay Patil et al. [16] proposed a novel square section thin-walled column design to enhance crashworthiness under axial loading. The evaluation was based on energy absorption (EA), statistical energy analysis (SEA), and crash force efficiency (CFE) metrics, with a focus on maximizing statistical energy analysis and crash force efficiency using nondominated sorting genetic algorithm II. Guangyong Sun et al. [17] presented a multiobjective optimization method for automotive bumper design under various effect load scenarios. This approach considered multiple effect load scenarios to optimize bumper system intrusion and energy absorption, improving crash safety in diverse vehicle conditions. Wang, DQ et al. [18] investigated the effect of front-end structures and restraint systems on occupant injuries during crashes. Their study explored coupling relationships and optimization methods based on vehicle crash dynamics to reduce occupant injuries, providing insights into occupant injury protection in traffic accidents. Kohar, CP et al. [19] integrated material and process development for the structural design of front longitudinal beams to maximize crash energy absorption characteristics. Structural size optimization was conducted using response surface methodology, artificial neural network metamodeling, and simulated annealing optimization techniques to maximize energy absorption while minimizing mass. Tanlak, N et al. [20] conducted a pertinent study on optimizing the bumper shape. The parameters of the beam cross-sectional shape were used as design variables and were optimized using a hybrid search algorithm, resulting in significant improvements in both crashworthiness and resistance to the low-speed effects of the bumper beam. Gandikota et al. [21] equipped a vehicle model with a dummy and an occupant restraint system. The crash finite element model of the body-in-white was employed for vibration analysis, facilitating a multidisciplinary vehicle optimization design for crashworthiness, occupant safety, noise, vibration, and harshness (NVH), and other relevant properties of the vehicle.

The application of neural networks in addressing nonlinear functions has gained prominence in recent years within the domain of automobile design and development. Douglas W. Kononen et al. [22] developed a logistic multivariate regression model to predict occupant injuries in crashes, utilizing various crash scenarios as input parameters. Chinmoy Pal et al. [23] expanded upon traditional crash variables by introducing two additional variables: intrusion and maximum deformation position. They applied Kononen’s algorithm for logistic regression analysis, resulting in a substantial enhancement in the accuracy of predicting driver injuries during crashes. Chen, QJ et al. [24] pioneered the development and implementation of a machine learning computational algorithm, constructing a deep neural network capable of determining and identifying the initial effect conditions of car crashes based on their final material damage states and permanently deformed structural configurations (wreckage). This approach holds broader significance in addressing the inverse problem for engineering failure analysis and vehicle crashworthiness evaluation. Xie, YX et al. [25] introduced a variational Bayesian-learning-based computational algorithm aimed at “inversely” identifying the deformation field of crashed cars and their residual strain fields, leveraging the final damaged structural configurations (wreckage). This research exhibited the unique capability of the developed machine learning algorithm in practically identifying the deformation field of real crashed cars, demonstrating significant potential for forensic analysis of car crashes and vehicle crashworthiness evaluation. Teoh, ER et al. [26] employed logistic regression modeling to predict the risk of driver fatalities in small-overlap frontal crash tests with the intention of rating vehicle safety levels.

The utilization of neural networks serves to complement the limitations of the traditional response surface method, which heavily relies on test points. Consequently, this paper selects the Elman neural network model as the foundational model for predicting occupant injuries. While neural networks have made notable strides in the field of vehicle R&D, there remains ample room for improvement in terms of the prediction accuracy. Thus, selecting the appropriate neural network for specific optimization problems is of primary importance. Liang, Y et al. [27] applied an Elman neural network to accurately simulate and predict temperature changes during the digester digestion process. Li, XY et al. [28] employed the Elman neural network for real-time battery capacity prediction. Abdelhafez, E et al. [29] correlated human-related weather variables with active corona virus disease 2019 (COVID-19) cases using an artificial neural network (ANN) approach, with the Elman model exhibiting the best performance and the most accurate correlation coefficients, particularly in relation to the weather variables and active COVID-19 cases in Jordan. Li, XM et al. [30] proposed a modeling and prediction method for indoor temperature hysteresis response characteristics, utilizing the time-lag neural network and an Elman neural network. The results indicated that the Elman network, with its simpler structure, reduced storage requirements and delivered a higher prediction accuracy, showing that this model serves as a superior modeling method for indoor temperature prediction. Guo, ZQ et al. [31] established a dynamics model for an electrohydraulic composite steering system, optimizing the steering assist torque distribution through genetic algorithms. They designed a fuzzy-compensated Elman neural network predictive controller to address the nonlinear challenges of the electrohydraulic composite system. The simulation results demonstrated that their coordinated control strategy substantially improved the electrohydraulic composite steering system in terms of the steering feel and energy consumption when compared to traditional electrohydraulic power steering systems.

Therefore, this paper opts for the Elman neural network to predict driver and passenger injury values in crash accidents, subsequently comparing the prediction accuracy with other neural network approaches. A 25% overlap frontal crash is more prone to causing injuries to personnel compared to a standard frontal crash due to distinct energy absorption mechanisms. This presents one of the most challenging conditions to address in terms of vehicle crash safety. As research in this domain has deepened, several scholars have made significant strides in tackling this issue. By analyzing energy transfer pathways during crashes and optimizing vehicle structures, material thickness, and related factors, researchers have achieved notable improvements in terms of crash safety [32,33,34,35,36,37]. However, previous optimization studies have focused primarily on enhancing crash safety without adequately considering other critical vehicle properties. This one-sided approach has often led to a trade-off, where improving crash safety results in the compromise of other vehicle attributes. In this paper, drawing from both domestic and international research on vehicle crashes, a specific compact car serves as an illustrative case study. This research aims to optimize crash safety while simultaneously considering the NVH performance of the vehicle. This optimization approach introduces constraints related to the body’s first-order modal properties and its mass while prioritizing the minimization of driver head injury as the primary optimization objective [38]. The objective is to explore an optimization design method employing neural network algorithms to comprehensively address both crash safety and NVH performance in vehicle design.

2. Finite Element Modeling

2.1. Finite Element Model of a Whole-Vehicle Crash

Road traffic accidents predominantly involve frontal crashes, with frontal crashes featuring an overlap rate of less than 25% being particularly prevalent. Such crashes exhibit a higher incidence rate and pose a greater risk to the lives and well-being of drivers and passengers. This increased risk arises from the distinct energy absorption and force transmission mechanisms at play, which differ from those observed in other frontal crashes. In 2016, the evaluation protocol of the China insurance automobile safety index (C-IASI) incorporated 25% overlap frontal crashes. However, analysis of evaluation results over the past three years has revealed that fewer than one-third of the vehicle models have received an excellent rating. Hence, there is an urgent need to optimize certain models that are currently available in the market.

Figure 1 shows the 25% overlap frontal crash model. Different components are distinguished by colours. The model has a total mass of 962.914 kg, consisting of 1,950,979 nodes and 1,931,233 cells. Among these cells, 61,366 are triangular, accounting for approximately 4.12% of the total. To meet the precision requirements of crash simulations, a grid size of 8 mm was selected for the target cells, while the entire vehicle includes a grid range spanning 4–10 mm. The vehicle model underwent preprocessing using the Hypermesh (2019) software and was subsequently analyzed and solved using the LS-DYNA solver.

2.2. Finite Element Model of the Body

There are generally two models available for conducting a modal analysis of the body-in-white (BIW). The BIW comprises the crash-absorbing structure bolted onto the welded body parts, excluding elements such as glass, doors, hood panels, sunroofs, luggage compartment lids, and wing panels. The interior body (BIP) extends beyond the BIW by adding front and rear windshields and triangular windows. Typically, BIP modal analysis focuses on identifying overall torsion and overall bending modes.

The body’s NVH analysis model is simpler than the crash model. It retains the structures of the white body panels, subframes, body weld joints, weld seams, and adhesives. In NVH performance analysis, the deformation of the body parts is considered linear elastic deformation. Consequently, the material types assigned to the body panels, closures, and welded joints are specified as MAT1. To enhance accuracy, modifications were made to the model. Specifically, the constrained nodal rigid body (CNRB) rigid unit from the LS-DYNA crash model was adapted to the RBE2 unit within OptiStruct. Additionally, spot welds in the OptiStruct model are represented using area contact method elements with identical elastic properties as the welded parts. Figure 2 Shows the BIP body model. The model consists of a total of 550,602 shell cells, 2976 area contact method spot weld cells, and 1496 RBE2 cells.

3. Vehicle Crash Simulation Analysis

3.1. Simulation Model of 25% Overlap Frontal Crash of Whole Vehicle

Since the inception of the first crash dummy, vehicle crash test dummies have evolved significantly, progressing from a single dummy model to a diverse array of different types and groups. These various dummy models serve distinct purposes and allow researchers to gather test data pertaining to different groups of crash scenarios. Among these, the most commonly utilized models are the Hybrid III dummy and the test device for human occupant restraint (THOR) dummy. The THOR dummy, in particular, serves as an improved successor to the Hybrid III dummy, addressing the limitations of the latter by incorporating sensors for the face, neck, chest, and other body parts.

In the context of simulating driver injuries during a real crash scenario, the 50th percentile THOR dummy model plays a pivotal role in the crash process (Figure 3). This model closely replicates the material properties, physical structure, and mechanical response characteristics of a human occupant. The restraint system includes a two-dimensional seatbelt model, steering-wheel-mounted airbags, and side air curtains.

For the frontal 25% overlap crash simulation, the vehicle model collides with a rigid barrier at a velocity of 64 km/h. The solution employs the LS-DYNA solver with a designated solution time of 150 ms.

3.2. Analysis of Crash Results

Frontal crash accidents in traffic incidents often exhibit a high incidence rate and result in significant injuries. This is primarily due to the higher driving speeds associated with frontal crashes, leading to increased kinetic energy. As a consequence, two key factors come into play. First, the engine and other vehicle components endure the strong effects and compression from the frontal panel, reducing the survival space for the front passengers. Second, the occupants experience severe head and chest injuries, often resulting in fatalities, due to the substantial inertia forces generated by the significant effects.

In the context of a 25% overlap frontal crash test, the front part of the vehicle overlaps with the barrier by 25%. In such collisions, the relatively small contact surface between the vehicle’s front and the barrier results in a concentrated effect at the crash site. This scenario presents a rigorous test for vehicle body safety under extreme conditions. Notably, the China new car assessment program (C-NCAP) test employs 40% offset deformable barriers for offset small-overlap crash tests, whereas rigid barriers, as used in this analysis based on C-IASI, tend to simulate crashes involving walls and utility poles. This distinction underscores the varied testing methodologies applied.

According to the C-IASI, the crash setup involves a nondeformable barrier, and the entire vehicle’s initial velocity during the crash is set at 64 km/h. The computation time is controlled at 150 ms, and additional control parameters, such as energy control, the time step, and file generation, are configured to export the solution file. Subsequently, this file is submitted for solving using the LS-DYNA solver.

Figure 4 illustrates the deformation and posture of the vehicle’s front section during the collision within the 150 ms timeframe. Between approximately 0 ms and 45 ms, the primary energy-absorbing components include the front engine cover, engine hood, front cross member, front longitudinal beam, energy-absorbing box, and radiator. However, due to the limited overlap area, the energy absorption is insufficient, and the primary energy-absorbing deformation area becomes the front passenger compartment after 45 ms. In this phase, the front left passenger compartment, left front longitudinal beam, and steering system experience deformation, with partial deformation occurring at the front of the left A-pillar. Approximately 100 ms after the collision, the vehicle’s rear section undergoes significant lateral movement as a result of a large moment deflection around the Z-axis in the Y-direction. This analysis also examines the model’s validity in terms of energy and mass changes throughout the collision.

Upon reviewing the simulation results of the entire vehicle model, the energy change and mass increase curves are depicted in Figure 5. The energy curve revealed that the system energy within the model is largely conserved throughout the collision process. The total energy remains relatively constant, with energy conversion primarily involving the transformation of kinetic energy into internal energy. Initially, the predominant form of energy is kinetic, but as the collision progresses, the kinetic energy steadily decreases, reaching its lowest point and stabilizing at approximately 70 ms. Concurrently, the internal energy increases with the vehicle’s motion, reaching its peak and stabilizing at approximately 70 ms. The overall mass increase during the collision process amounts to approximately 12.5 kg, equivalent to approximately 0.82% of the total mass. The energy and mass variations observed in the 25% offset collision model fall within the expected range. These results suggest that the energy changes in this computational model are reasonable, and the hourglass energy ratio and mass increase align with typical values, thereby validating its suitability for subsequent research analysis.

Head injury constitutes a primary form of harm in traffic accidents, and injuries of varying severity can lead to serious harm or fatality for the vehicle occupants. This study specifically focuses on head injuries sustained by individuals involved in traffic accidents. Such injuries can be categorized into two scenarios: the direct effects of the head colliding with the vehicle’s interior and brain tissue injuries resulting from the inertial forces acting on the head.

The head injury criterion ($HIC$) currently serves as the predominant metric for assessing human head injuries in trials [38]. The calculation is presented as follows:

$$HIC={\max }_{t_1,t_2}\left. [\left(t_2-t_1\right){\left(\frac{1}{\left(t_2-t_1\right)}{\displaystyle {\int }_{t_1}^{t_2}a}(\mathrm{t})\mathrm{d}t\right)}^{2.5}\right]$$

(1)

In Equation (1), $t_1$ and $t_2$ represent the starting and ending moments of the selected period within the entire simulation, respectively, while $a(t)$ signifies the synthesized acceleration of the human head’s center of gravity. The most commonly used head injury measures in vehicle crash testing are $HI{C}_{15}$ and $HI{C}_{36}$. $HI{C}_{15}$ reflects the maximum head injury for crash durations up to 15 ms, whereas $HI{C}_{36}$ includes the maximum head injury for crash durations up to 36 ms. $HI{C}_{36}$ provides a more comprehensive evaluation of the effects on the head in accidents due to the longer time frame considered. This extended duration allows for the detection of sustained effects and scenarios that could lead to severe consequences.

Furthermore, $HI{C}_{36}$ better captures the biomechanical characteristics of the human head, as it is more sensitive to longer-duration effects. Consequently, $HI{C}_{36}$ provides a more precise quantification of head injuries sustained in real accidents.

Table 1. AIS levels corresponding to the human head injury index $HI{C}_{36}$.

AIS Rating	HIC36	Damage	Degree of Damage
1	130–519	Have a headache and vertigo	Mild impairment
2	520–899	Temporary loss of consciousness; linear fracture	Moderate impairment
3	900–1254	1–6 h loss of consciousness; depressed fracture	Severity of an injury
4	1255–1574	6–24 h loss of consciousness; open fracture	Severe impairment
5	1574–1859	Prolonged loss of consciousness; cerebral hematoma	Critically injured Life-threatening injuries
6	>1860	Dead or unsalvageable	Deadly

below outlines the relationship between abbreviated injury scale (AIS) levels and the human head injury index $HI{C}_{36}$.

The results of accelerometer measurements in the X, Y, and Z directions at the dummy head measurement point were obtained. The acceleration variations in these three head movement directions during the crash are presented in Figure 6. X-direction acceleration represents the forward and backward movement of the head. The X-direction acceleration curve reaches its peak value between 70 ms and 80 ms and then gradually decreases. Y-direction acceleration corresponds to the left and right movement of the head. The curve fluctuates within the range of approximately ±30 m/s². However, there is a notable peak in the Y-direction acceleration, reaching 69.3 m/s² at approximately 148 ms. This peak is attributed to the substantial lateral movement of the body during the crash. Z-direction acceleration illustrates the head’s pitching motion. The acceleration curve reaches a negative peak of −62.42 m/s² at 61.5 ms and then gradually declines due to the reaction forces. It subsequently reaches a positive peak of 64.56 m/s² at 85.1 ms.

The three-axis accelerations were synthesized to create a composite acceleration curve by taking the square root of the sum of squares of the three-axis acceleration curves. This composite curve is displayed in Figure 7.

Using Equation (1), the composite acceleration curve was utilized to calculate the $HI{C}_{36}$ value for the driver during the crash process. The $HI{C}_{36}$ value peaks at 845.96 between 59.30 ms and 95.30 ms. Referring to the relationship table between the abbreviated injury scale levels and the human head injury index $HI{C}_{36}$, it was concluded that the corresponding level of head injury sustained is moderate. Specifically, the injury can be described as a temporary loss of consciousness and a linear fracture.

4. Body NVH Performance Simulation Analysis

4.1. Modal Analysis

During the analysis of the NVH performance of a vehicle body, particular attention should be given to the consideration of the first-order overall modes of the body-in-white. These low-frequency modes are especially sensitive to low-frequency excitations such as those originating from the engine. Such excitations can induce vibrations and noise within the vehicle, ultimately influencing ride comfort. To mitigate these effects, it is advisable to address the first-order modal frequency of the body-in-white during the modal planning phase. The modal solution file is then submitted for calculation using Optistruct.

Table 2. Frequencies and shapes of the first eight non-zero modes of the body structure.

Inherent Frequency Order	Intrinsic Frequency/Hz	Vibration Pattern
First stage	28.99	Tank bracket transverse mode
Second stage	29.63	Longitudinal modes of the rear tailgate of the tank support
Third stage	30.87	Longitudinal modes of the rear tailgate of the tank support
Fourth order	37.24	Tank bracket transverse mode
Fifth order	39.75	First-order torsional modes
Sixth order	43.34	First-order bending modes
Seventh order	45.84	Tank bracket local modal
Eighth order	48.76	Torsional mode

presents some of the mode values and shapes, while Figure 8 and Figure 9 depict the first-order mode shapes. It is worth noting that the mass of the body-in-white in this experiment was 306.86 kg.

4.2. Plate Acoustic Sensitivity Analysis

Noise generation in a vehicle is often attributed to the vibration modes of the body panels. In these modal vibration patterns, there are specific points called modal nodes where the displacement is minimal or zero. Optimizing the vehicle structure to concentrate its mass near these modal nodes can significantly enhance the vehicle’s vibration characteristics. However, it is crucial to balance these improvements with other considerations, as altering the thickness of body panels can affect both the NVH performance and crash safety of the vehicle. To address these concerns, in this experiment, the thickness of body panels was utilized as a design variable in a manner that aimed to find an optimal combination of the panel thicknesses. This optimization process sought to enhance crash safety while also considering the NVH performance of the body.

Modal analysis was employed to identify the first-order torsional and first-order bending modes of the body, which serve as constraints for interior noise optimization. Furthermore, lightweight vehicle design can improve fuel efficiency and reduce costs. Therefore, the optimization process must not compromise the NVH performance while striving for lightweight design. To ensure this balance, the body mass was also included as a constraint in the optimization process.

The sensitivity analysis involved defining the input variables according to

Table 3. Variables for sensitivity analysis.

Variable Number	Variable Name	Initial Value/mm	Initial Interval/mm
N1	Front apron	0.8	[0.6, 1.0]
N2	Front pillar upper inner plate	1	[0.8, 1.2]
N3	Front pillar lower inner plate	0.8	[0.6, 1.0]
N4	Trench floor	1	[0.8, 1.2]
N5	Front plate	0.7	[0.5, 0.9]
N6	Front lower edge beam	2	[1.8, 2.2]
N7	Center column inner plate	1	[0.8, 1.2]
N8	Awning (under ceiling)	1.2	[1.0, 1.4]
N9	Midplane	0.7	[0.5, 0.9]
N10	Backplane	0.7	[0.5, 0.9]
N11	Tailgate	0.6	[0.4, 0.8]
N12	Rear wheel cover plate	0.9	[0.7, 1.1]
N13	Rear pillar lower plate	0.7	[0.5, 0.9]
N14	Upper front panel	0.7	[0.5, 0.9]
N15	Sidebar	0.8	[0.6, 1.0]
N16	Rear pillar upper plate	0.7	[0.5, 0.9]
N17	Rear pillar reinforcement plate	0.8	[0.6, 1.0]
N18	Center column	0.7	[0.5, 0.9]
N19	Hinge reinforcement plate	2	[1.8, 2.2]
N20	Lower plate of center column	2.5	[2.3, 2.7]
N21	Lower center column reinforcement plate	1.2	[1.0, 1.4]
N22	Topside beam	2.1	[1.9, 2.3]
N23	Sill plate	1	[0.8, 1.2]
N24	Rear lower side girder	1.4	[1.2, 1.6]
N25	Threshold reinforcement plate	1.8	[1.6, 2.0]
N26	Taillight panel	1.4	[1.2, 1.6]
N27	Front roof beam	0.8	[0.6, 1.0]
N28	Backplane	1	[0.8, 1.2]
N29	Rear shock absorber mounting plate	0.7	[0.5, 0.9]
N30	Front seat support beam	2.5	[2.3, 2.7]
N31	Rear seat support beam	1.5	[1.3, 1.7]

, while the first-order torsional modes, first-order bending modes, and body mass were defined as the output responses. The Plackett–Burman sampling method was utilized for the sensitivity analysis, allowing for the calculation of the main effects of the factors with a minimal number of trials, particularly when all factors are binary-level.

Figure 10 shows the sensitivity coefficients for each response. The sensitivity analysis focused on examining the effect of variations in design variables within their upper and lower limits on each response. It should be noted that the interaction effects between the design variables were disregarded. The results indicate that an increase in the body panel thickness leads to an increase in the body modes, resulting in positive modal sensitivities. Additionally, the plate thickness is directly proportional to the plate mass, leading to a positive mass sensitivity. In selecting the design variables, consideration was given to the first-order torsional and bending modes as well as to the first two plates with a higher mass sensitivity. Notably, the side circumference has a more significant effect on both the modal and mass sensitivity.

5. Prediction of Driver Injury Values

5.1. Experimental Design

The objective, constraints, and design variables for predicting occupant injuries were established following a comprehensive analysis that included the 25% overlap frontal crash simulation of the entire vehicle and the NVH performance analysis of the body. The primary objective function in this context was the $HI{C}_{36}$ injury value associated with the driver’s head during a crash. Additionally, several constraints were considered, including the first-order torsional mode of the body-in-white, the first-order bending mode, and the overall body mass, which includes the body-in-white and enclosed parts. Twelve panels, comprising elements such as the front longitudinal beam outer panel, front longitudinal beam inner panel, and upper longitudinal beam, were designated as variables for the subsequent experimental design, and their respective values are outlined in

Table 4. Experimental design variables.

Variable Number	Variable Name	Initial Value/mm	Initial Interval/mm
X1	Front longitudinal beam outer plate	1.7	[1.5, 1.9]
X2	Front longitudinal beam inner plate	1.9	[1.7, 2.1]
X3	Upper longitudinal beam	1.2	[1.0, 1.4]
X4	Longitudinal girder connection beam outer plate	1.3	[1.1, 1.5]
X5	Longitudinal girder connection beam inner plate	1.7	[1.5, 1.9]
X6	Front apron	0.8	[0.6, 1.0]
X7	Sidebar	0.8	[0.6, 1.0]
X8	Front pillar upper inner plate	1.4	[1.2, 1.6]
X9	Front pillar lower inner plate	0.8	[0.6, 1.0]
X10	Awning (under ceiling)	0.8	[0.6, 1.0]
X11	Center column inner plate	1	[0.8, 1.2]
X12	Tailgate	0.6	[0.4, 0.8]

.

Various sampling methods are available for experimental design, including full factorial design, Latin hypercube sampling, the Taguchi method, and Hammersley sampling. The Hammersley sampling method offers distinct advantages, as it can yield reliable estimates of output statistics with fewer samples. Moreover, it achieves a uniform distribution across a k-dimensional hypercube and involves an efficient identification process. Importantly, the Hammersley sampling method only requires sampling on the input and output signals, obviating the need to identify the internal state of the system. Consequently, the Hammersley sampling method was selected for the design of the experiments to acquire the subsequent training and test sets.

5.2. Elman Neural Network

A neural network is a complex system composed of interconnected neurons. Based on how information flows within the network during its operation, neural networks can be classified into two fundamental types: feedforward neural networks and feedback neural networks. Feedforward networks utilize hidden layers and nonlinear transfer functions to create complex nonlinear mapping functions. The output of a feedforward network depends solely on the current input and the weight matrix, and it is independent of previous network outputs. In contrast, feedback neural networks incorporate inputs with time delays or feedback from output data, rendering them dynamic systems. Consequently, feedback neural networks are also referred to as recurrent neural networks or regression networks.

The architecture of the Elman neural network is depicted in Figure 11. The Elman neural network introduces a context layer into the feed-forward network as a one-step delay mechanism to facilitate memory retention. Typically, an Elman-type neural network comprises four layers: the input layer, the hidden layer, the context layer, and the output layer. As illustrated in the figure, the connections between the input, hidden, and output layers resemble those of a feed-forward network, with the units in the input layer serving as signal transmitters and the units in the output layer acting as weightings. The units within the hidden layer incorporate both linear and nonlinear functions, while the context layer is responsible for storing the output of the hidden layer units from the previous time step and feeding it back into the network as input. The Elman neural network possesses the capability of approximating any nonlinear mapping with a high precision due to its self-association, which makes it sensitive to historical data. Additionally, the internal feedback network enhances the network’s capacity to process dynamic information.

The structure of the Elman neural network is presented in Figure 11, and its nonlinear spatial expression can be mathematically defined as follows:

$$y(k)=g(w^{3}x(k))$$

(2)

$$x(k)=f(w^1{x}_c(k)+w^2(u(k-1)))$$

(3)

$$x_c(k)=x(k-1)$$

(4)

In Equation (2), $y$ is the dimensional output vector, $x$ is the $n$ dimensional node unit vector of the middle layer, $u$ is the $r$ dimensional input vector, $x_c$ is the $n$ dimensional input vector, $w^3$ is the weight of the middle layer to the output layer, $w^2$ is the weight from the input layer to the middle layer, $w^1$ is the weight from the takeover layer to the middle layer, $g(*)$ is the transfer function of the neurons of the output layer, which is a linear combination of the outputs from the middle layer, and $f(*)$ is the transfer function of the neurons of the middle layer.

6. Analysis of Elman Forecast Results

The data used for model fitting were collected from numerous simulation experiments. To ensure data quality, a data cleaning process was applied to remove individual outliers and anomalies. After cleaning, the data underwent normalization to prevent issues arising from excessively large or small eigenvalues. Min–max normalization was employed to linearly transform the original data, mapping the results to the range [0, −1]. Data normalization accelerates the convergence of weight parameters. The processed data were split into a training set and a test set, with cross-validation selected as the verification method. The neural network was tasked with predicting the $HI{C}_{36}$ value of the driver’s head, the first-order modes of the body-in-white, and the body mass during a crash. Figure 11 presents a comparison between the prediction results of the driver’s head $HI{C}_{36}$ value and the simulation results. To assess the prediction accuracy in comparison with other neural network algorithms, the radial basis function (RBF) neural network, the generalized regression neural network (GRNN), and the back propagation (BP) neural network were also used for the prediction. Evaluation metrics such as the mean squared error, root mean squared error, mean absolute error, and mean absolute percentage error were employed to assess the prediction accuracy, as displayed in

Table 5. Comparison of errors of algorithms in predicting head injury values.

Neural Network	MSE	RMSE	MAE	MAPE
Elman	1847.729	42.985	35.131	3.835
RBF	10,995.161	104.858	91.378	11.345
GRNN	9030.242	95.028	90.139	10.936
BP	24,737.367	157.21	135.874	15.975

.

From the table, it is evident that the predictions made by the Elman neural network exhibit a high degree of agreement with the actual values and have lower prediction errors compared to the other neural networks. The ordering of the prediction accuracy, from highest to lowest, is as follows: Elman, GRNN, REF, BP.

The mean squared error ($MSE$) quantifies the sum of the squares of the differences between each predicted value and the actual value. Smaller values indicate lower prediction errors and a higher accuracy.

$$MSE=\frac{1}{n}{\displaystyle \sum _{i=1}^n{(a_o-y_o)}^2}$$

(5)

The root mean squared error ($RMSE$) is the square root of the MSE and measures the deviation of observations from the true values.

$$RMSE=\sqrt{MSE}$$

(6)

The mean absolute error ($MAE$) is not affected by the sign of the deviation due to its use of absolute values, providing a better reflection of the actual prediction error.

$$MAE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|a_o-y_o\right|}$$

(7)

The mean absolute percentage error ($MAPE$) offers a more intuitive representation of forecast accuracy, resembling the concept of the relative error. It compares the difference between the predicted value and the actual value to the actual value itself to determine the percentage error.

$$MAPE=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|\frac{{\widehat{y}}_i-y_i}{y_i}\right|}$$

(8)

The Elman neural network algorithm also predicts the first-order modes of the body-in-white, and the associated prediction errors for the body mass are detailed in

Table 6. Elman neural network algorithm prediction error.

Predictors	MSE	RMSE	MAE	MAPE
Body-in-white first-order Torsional modes/Hz	1.883 × 10−2	1.372 × 10−1	1.183 × 10−1	2.985 × 10−1
Body-in-white first-order Bending modes/Hz	2.845 × 10−1	5.334 × 10−1	3.941 × 10−1	9.306 × 10−1
Body mass/kg	2.424 × 10−5	4.923 × 10−3	3.603 × 10−5	1.167

.

Table 5 and Table 6 clearly demonstrate that the Elman neural network outperforms the other three neural network algorithms, obtaining the smallest prediction error for the driver’s head injury value across the four metrics of $MSE$, $RMSE$, $MAE$, and $MAPE$. The Elman neural network also exhibits a low prediction error when predicting body mass and the first-order modal state of the body-in-white, thus meeting the prediction requirements. To ensure processing accuracy, the step size for each input variable in the solution space was set to 0.1, resulting in 244,140,625 feasible solutions. The Elman neural network was employed to predict the outputs corresponding to these feasible solutions, including the driver injury value, body mass, and first-order modes. The optimization problem aimed to minimize the driver’s head injury value while avoiding an increase in body mass and preserving the first-order modal state of the body-in-white as constraints. The mathematical expression is as follows:

$$\left\{\begin{array}{l}\min f(X_i)\\ h_1(X_i)\ge 39.75\\ h_2(X_i)\ge 43.34\\ m(X_i)\le 306.86\end{array}\right.$$

(9)

This problem involved a large solution set, which is typically addressed through iterative operations. However, conventional iterative methods are prone to becoming trapped in local minima or “dead loops”, hindering the optimization process. Genetic algorithms, which are global optimization algorithms inspired by biological evolution, excel in overcoming this drawback. Compared to traditional optimization methods, genetic algorithms offer strong convergence, reduced computation time, and high robustness.

The optimization process using genetic algorithms involved generating multiple solution groups in the solution set space. The predicted values corresponding to each group of solutions were obtained through the trained Elman neural network, and only those that satisfied the constraints were retained until the population size met the requirements (Figure 12). The fitness of individuals in the population was calculated based on the difference between the initial solution for the driver’s head injury value and the predicted injury value of each solution, with larger differences indicating a better fitness. This process continued iteratively as the Elman neural network performed genetic, crossover, and mutation operations on individuals for the prediction. The initial solution served as the global optimal solution at the start, and if a solution better than the current optimal solution was found, it was replaced until the iteration terminated.

The optimization of the solution set space yielded optimal solutions that adhered to the constraints, resulting in values as follows: 1.7, 2, 1, 1.3, 1.5, 0.7, 0.8, 1.3, 0.6, 0.8, 1.2, 0.8. To assess the final prediction solution of the Elman neural network algorithm, these optimal plate thickness values were applied to the finite element model attributes, which were then submitted to the Optistruct and LS-DYNA solvers for simulation calculations. The optimization results are detailed in the

Table 7. Comparison before and after optimization of each constraint response.

Responsive	Pre-Optimization	Projected Value	Actual Value	Forecast Error (%)
Driver’s head $HI{C}_{36}$ value	845.96	627.1895	672.53	6.74
Body mass /kg	306.86	304.921	306.62	0.55
Body-in-white first-order Torsional modes/Hz	39.749	40.209	39.972	−0.59
Body-in-white first-order Bending modes/Hz	43.338	44.504	44.800	0.66

, showcasing a reduction of 173.43 in the $HI{C}_{36}$ value of the driver’s head, minimal changes in the body mass, and an improved optimization of the first-order modes compared to the original data.

7. Conclusions

Using the Hypermesh software, simulation models for body and vehicle crashes were established, allowing for the analysis of the NVH and frontal offset crash performance. Additionally, an optimization method leveraging an Elman neural network was introduced to comprehensively consider both the NVH performance and frontal offset crash performance of the vehicle. The results clearly demonstrate significant improvements in terms of the frontal offset crash performance of the entire vehicle, all while maintaining the NVH performance of the body. This approach successfully avoids the limitation of single-objective optimization and affirms the possibility of enhancing safety while simultaneously improving passenger comfort during body design. Furthermore, the prediction errors of the Elman neural network algorithm in predicting each response remained within 7%, aligning with the needed prediction accuracy. This case serves as a compelling example of the effective utilization of advanced intelligent algorithms for predicting the NVH performance and crash performance of vehicle bodies. It provides valuable insights for engineers and technicians to apply in practical engineering projects.