Multiobjective Lightweight Optimization Design Method for a Dump Truck Carriage under Multiple Working Conditions

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Introduction
Te issues of fossil fuel emissions and global climate change have attached wide concerns in the world. Terefore, reducing fossil fuel consumption has posed a huge challenge for many industries [1][2][3].
With the continuous improvement of environmental protection policies, vehicle emission requirements are also further enhanced. How to meet the emission requirements under the environmental protection policy has become a challenge for all vehicle companies. Vehicle lightweight, a potentially efective solution to mitigate emissions, is widely employed by vehicle companies. Studies have revealed that when the mass of the vehicle is shrunk by 100 kg, the braking and handling stability of the vehicle will be improved, and the fuel consumption and emissions are reduced by 5-8% and 4.5%, respectively. In addition, the fuel utilization rate increased by 3.8% correspondingly [4][5][6].
Up to now, there are still a lot of studies focused on lightweight. In the past decades, various weight reduction methods have been intensively studied to achieve vehicle weight reduction. Lee combines the optimal design of lightweight materials and rigid materials to achieve a 10% reduction in the weight of the instrument panel [7]. Additionally, the GPA and AHP methods are employed to select the body lightweight materials and determine the weight of each attribute, respectively [8]. In [9], a cost-effective and reliable joining technique for multimaterial designs, such as the joining of aluminum to press-hardened boron steel and welding of resistance elements, is investigated. To solve the problem of unique surface smoothness and low-dimensional accuracy of Ti 49.4 Ni 50.6 (at. %) shape memory alloy (SMA), Takale and Chougule improved the surface smoothness and dimensional accuracy of SMA by studying a series of processing processes [10]. By comparing the accuracy of the approximate model method under multiple working conditions, a lightweight design of the cargo compartment is executed by Li and Ma, and a 7.47% mass reduction is achieved [11]. Te genetic algorithm is utilized to study concrete bridges in [12]. Te results display that efective and well-calibrated structure identifcation is successfully realized by the inversion technique of the genetic algorithm. Moreover, Ye et al. have improved vehicle acceleration, braking, and handling to a certain extent by optimizing the design of load-bearing frame components [13]. And Sookchanchai et al. developed a topology optimization method for automotive suspension control arms; it is worth noting that this approach has practical guidance for other automotive stamping lightweight designs [14]. Te optimal Latin equation design and response surface model are employed to optimize the thickness of each pipe frame by Wang et al., and the results show that 14.38% weight reduction on the premise of the brake pedal intrusion is reduced by 25.31% and realize the 20.02% reduction of the cockpit of peak acceleration [15]. Meschut et al. proposed a multiobjective particle swarm optimization algorithm to fnd the optimal layout of parallel P2 hybrid electric vehicles [16]. Long et al. proposed a multiresponse weighted adaptive sampling (MWAS) method based on a hybrid surrogate model, which can signifcantly improve the efciency of car seat model optimization [17]. Te detailed model and calculation process of GMM-SOMVSA-CO to solve the RBMDO problem is put forward by Meng et al. so that the uncertainties brought by other disciplines do not need to be considered in the process of single discipline analysis and optimization [18].
As the important component of the dump truck and main load bearing, the traditional carriages are designed as thicker and heavier as possible to ensure safety and service life. However, it will lead to the waste of materials and a dramatic increase in the cost of transportation and energy consumption. Fortunately, the lightweight design of the carriage will improve the economy, power, braking, and safety of the dump truck [19]. In the initial study, the lightweight of dump trucks is achieved by the replacement of raw materials with higher strength steel with various optimization algorithms. With the continuous maturity of automobile manufacturing materials, how to more efectively improve the utilization rate of materials has become more prominent in the role of theoretical algorithms. Tus, the utilization of diferent algorithms or the improvement of the original algorithm is the focus of lightweight.
Nevertheless, there are still many problems in the lightweight design of dump truck compartments. It can be found that most of the research studies on dump truck carriages only take the carriage quality as a single objective to optimize the design, which leads to many problems in the manufactured carriages in the past ten years. For example, problems such as bumps, cracks, and fractures that may occur in the carriage owing to insufcient stifness will cause the carriage's service life to plummet. Furthermore, the shrunk of the carriage service life contributed by the reduction in the vehicle mass will increase the vehicle manufacturing cost to a certain extent. Tus, the carriage stifness is a signifcant factor for optimization objective design. Additionally, the variation of vibration frequency is afected by the change in the quality of the carriage. To prevent the resonance of the carriage from afecting the life, safety, and cost of the carriage, the frequency is also studied and analyzed in this study.
Te methods to achieve lightweight can be classifed into four categories: (1) the materials with higher stifness, better energy absorption, and lower density are utilized to replace traditional materials individually or as a whole [20], (2) a more mature processing technology is adopted to guarantee the material properties do not change while operating the plates connected [21], (3) the objective optimization is carried out in the aspects of topological, structural, and size via leveraging fnite element techniques [22,23], and (4) a novel approximate model construction method is adopted to improve the accuracy of the approximate model. In addition, the lightweight design of the vehicle is realized by integrating it into a new or improved algorithm [24,25]. Currently, the study on lightweight is implemented via adopting a variety of methods combined and selecting high-strength steel to replace traditional materials, as well as executing the optimization for the size of the objective. Note that the processing techniques, employed to connect a variety of materials, will be reasonably selected based on the nature of the connection between diferent materials. Additionally, for larger components, topology optimization is generally adopted frst, then size optimization is carried out via optimal design. Ultimately, structural optimization is executed according to actual requirements to improve the utilization rate of energy and materials [26].
Te criteria for the lightweight design of the dump truck compartment is whether the economic efciency of the vehicle has been improved after taking the lightweight, which includes the terms of bending stifness, service life, and vehicle cost. By comparing the accuracy of the dualcondition carriage models constructed by diferent approximation methods, a method combining linear and specifc numerical moving least squares (MLSM) is selected to ft the approximate model of the carriage in this study. Additionally, the mass, stifness, and frequency of the carriage are analyzed utilizing a multiobjective genetic algorithm and lightweight optimization design.

Finite Element Establishment and Rigid Strength Analysis of Dump Truck Carriage
Te fnite element establishment and rigid strength analysis of dump truck are explained in the following sections.

Establishment of the Finite Element Model of the Carriage.
In this study, UG software is employed to model the carriage, and it is seamlessly imported into HyperMesh for fnite element meshing. First, the fnite element model preprocessing is applied to simplify the parts with small mass and complex structure. Ten, a fnite element model with a grid size of 10 mm for the treated carriage is performed, and the welding between the panels and the quality of the grid is checked. Ultimately, the actual thickness, material properties, and loading force of the model are added to the fnite element model to prepare for the working condition analysis.

Static and Dynamic Working Conditions and Modal
Analysis for Carriage. Dump trucks have the characteristics of long single transportation time, heavy load, and severe working environment during operation. Tus, as the main bearing component of the dump truck, it is essential to analyze the working conditions of the carriage in practical.
In this study, three wording conditions of the carriage are studied, i.e., relatively static full-load and uniform transportation, static and dynamic 0°lift, and dynamic 30°lift.

Stifness and Strength Analysis of the Carriage under the
Full-Load Working Condition. Generally, the dump truck carriage works under full-load conditions, and the service life of the carriage is most tested during long-distance transportation. Since the dump truck is mainly leveraged to load small goods such as sand and gravel, the sand is employed to carry out the force analysis on the full-load condition of the carriage in this study. In fully loaded conditions, the dump truck is mainly in a state of uniform motion. Tus, it can be assumed that the carriage remains relatively stationary under specifc conditions. By constraining the translational degrees of freedom in the X, Y, and Z directions of the main longitudinal beam, the force diagram under the full-load condition of the carriage is shown in Figure 1: Te force analysis of the fnite element model of the carriage is done as shown in Figures 2 and 3.
In Figures 2 and 3, it can be observed that the maximum strain of the carriage under the loaded condition is 10.84 mm and the maximum stress is 287.5 MPa. Te strain is mainly concentrated at the two ends of the carriage foor: one end is at the contact surface between the carriage and the lifting mechanism and the other end is at the contact surface between the carriage and the subframe. Te results derived from Table 1 show that the carriage meets the vehicle manufacturer's design requirements, in which the whole carriage is made of A610L material, the maximum deformation is not more than 15 mm, the safety factor is 1.2, and the maximum yield stress is 425 MPa.

Analysis of 0°Lifting Condition of Carriage.
In this study, the 0°lifting condition is analyzed when the carriage is fully loaded. When 0°is lifted, the carriage is subjected to the vertical upward lifting force of the lifting mechanism, which includes a full load of goods and the dead weight of the carriage. Te main longitudinal beam of the carriage constrains the X-direction translation and Y-and Z-direction rotational degrees of freedom; the side plate constrains the Y-direction translation; the front plate is all restrained except for releasing the Y-direction rotational degrees of freedom. Te forces applied are shown in Figure 4.
In Figure 5, Q is the total length of the bottom plate, which is valued at 3.5 m; Q1 : Q2 � 2 : 3; the rated load of the carriage is 5 t, and the dead weight of the carriage is 3 t. Te lifting force is loaded into the fnite element model and the working conditions are analyzed, and the strain stresses in Figures 5 and 6 are derived.
It can be seen from Figures 6 and 7 that when the carriage is lifted at 0°, the maximum strain occurs on the contact surface between the carriage and the subframe, and the maximum stress occurs on the contact surface between the lifting mechanism and the carriage. Since the contact area between the lifting mechanism and the subframe and   Mathematical Problems in Engineering the carriage is the stressed part, strain and stress concentration are generated on these two sides, where the maximum strain is 13.59 mm and the maximum stress is 444.31 MPa. Compared under the full-load condition, the strain and stress in the 0°condition have an increased to a certain extent. Te 0°working condition data table is shown in Table 2.

Analysis of Working Conditions When the Carriage Is
Lifting at 30°. When the lift mechanism lifts the carriage to 30°, nearly half of the cargo will be unloaded, and the carriage is mainly subjected to the lift force of the lift mechanism and the support force of the subframe. As shown in Figure 4, the loading force is added to the carriage. Additionally, the results of the 30°lift working condition analysis of the carriage are presented in Figures 7 and 8. Figures 8 and 9 describe the stress-strain diagram of the carriage under the working condition of 30°, in which the strain concentration and stress concentration, respectively, exist at the contact surface between the lifting mechanism and the subframe, and at the welding point between the subframe and the carriage. Note that the maximum strain is 10.27 mm and the maximum stress is 362.20 MPa. Te 30°w orking condition data are shown in Table 3.

Free Mode Analysis of the First 8th
Order of the Carriage. Dump trucks work on complex and harsh roads with low surface fatness leveling, and the requirements of the dynamic performance of the dump truck are higher. Terefore, the structural vibration obtained by the modal analysis can well refect the dynamic performance of the dump truck. Te frequency and vibration pattern are determined by a modal analysis of the carriage to prevent resonance. Generally, the frst eight orders are analyzed in the carriage because when the eighth orders are greater than 40 Hz, there is no need to analyze the following orders. Modal analysis is the analysis of the vibration characteristics of a system. For a linear system, when the linear conditions do not change, the dynamic characteristics of the structure can be expressed as where M, C, and K represent the mass matrix, damping matrix, and stifness matrix of the system structure, respectively, X is the displacement vector of the system, _ x and € x denote the velocity and acceleration vectors, and f (t) is the excitation force vector.
Te inherent frequency of the ith order can be denoted as   Figure 5: Force diagram at the moment of lifting 0°.  Te results of the frst 8th order modal analysis of the carriage in Hyperview are shown in Figure 10.
Te frst eight orders of modal frequencies are shown in Table 4.
Te 6-cylinder and 4-stroke diesel engine is adopted in this study, and the rotational speed of the engine at idle speed is 800 r/min. Te frequency at engine idle speed can be expressed as Te magnitude of the excitation frequency of the carriage road surface depends on the vehicle speed and road surface type. Generally, the driving speed of the dump truck in the sand mine area is 50 km/h, and the road adhesion coefcient is 0.4. Tus, the frequency can be calculated as It can be seen from Figure 10 that the value of the sixthorder frequency analysis of the carriage is 34.126 Hz, which is relatively close to the road excitation frequency. Terefore, as a constraint condition for carriage optimization, the sixthorder frequency needs to be not equal to 34.722 Hz.

. Multiobjective Optimal Design under the Construction of the Dual Working Condition Approximation Model
Trough the above analysis, it can be observed that the stifness and strength of the carriage are severe under the full-load condition and the 0°lift condition, and the sixthorder frequency need to be constrained. Terefore, optimal analysis is carried out for the dual operating conditions established comprehensively under the full-load condition and the 0°operating condition. Ten, the dual working condition model and modal model are imported in the Hyperstudy, in which 18 carriage thicknesses are defned as design variables for multiobjective optimization of car mass and stifness.

Latin Hypercube Sampling.
LHS is a space-flling method for arbitrary dimensions and has no number of times of sampling in space. In this study, LHS is leveraged to sample the dual working conditions and modes of the carriages to ensure that each sampling point has no other test points in the same row and in the same row. In addition, 260 optimization analysis on three working condition models are performed by LHS, and the results of working condition diagram, modal diagram, mass diagram, and stifness diagram are shown in Figures 11-14.
Due to a large number of design variables for the carriage, it is necessary to screen out the design variables that have a great infuence under the working conditions of the carriage and the optimization objective of the carriage. According to the sensitivity Figures 15-17, the design variables of 63 mm and 8 mm are selected as the predictions for the mode, mass, and stifness. Te relationship between the predicted design variables and the response is shown in Figures 18-20. Table 5 shows the table of design variables that have a high impact on the carriage optimization objective using Latin hypercube extraction screening.

Selection of Methods for Approximate Model Construction.
Te main methods for constructing approximate models at this stage are radial basis functions, HyperKriging, least square regression, and moving least square method. By comparing the accuracy obtained by the above four diferent approximation model methods, an approximation method more suitable for the carriage model is selected. Te results of precision are shown in Table 6.
It can be found in Table 6 that the ftting accuracy of the MLSM approximation method is higher and is adopted to construct the approximate model of the dual working conditions of the carriage. Te construction of the carriage      model utilizing the MLSM method with linear parameters will lead to the issue of lower accuracy for modal and fullload strain. Terefore, a customized numerical value is constructed for the response with low precision, and the precision results are shown in Table 7. As shown in Table 7, the accuracy of the approximation model for the dual working conditions of the carriage is improved by changing the MLSM method. In addition, the actual and simulated results of the fnite element model can be refected by the scatter plot, i.e., the closer the scatter plot to the y � x function, the higher the ftting accuracy of the model. Te accuracy graph of the response value is shown in Figure 21.

Mathematical Problems in Engineering
Te calculated formula for the coefcient of determination R 2 of the approximate model ftting function is expressed as where the random test sample n test is utilized to construct an approximation of the accuracy of the model, P and y p are the target predicted response value and the actual target value of the pth sample point, respectively, and y denotes the mean value of the actual target response of the sample point. It can be concluded that the closer R 2 is to 1, the higher the accuracy of the approximate model is. According to the numerical results of the approximate model accuracy of each target response value displayed in Table 7, the decisive coefcients of the 7 approximate models constructed are all greater than 0.90, which meets the requirements of engineering analysis.

Multiobjective Optimization Design of Dump Truck
Compartment. In order to reduce the mass while ensuring the improvement of stifness, a multiobjective optimization design is carried out on the mass and stifness of the carriage. Under the condition of full load and 0°lift, the model is not equal to 34.722 Hz, the strain is not more than 15 mm, and the stress is not more than 425 MPa as constraints, the mathematical model of the carriage optimization algorithm can be expressed as where m represents the mass, EI is the bending stifness, disp is the strain under dual conditions, and stress and MT are the stress under dual conditions and sixth-order mode, respectively Based on the constructed MLSM approximation model with dual working conditions, the genetic algorithm is used to carry out the multiobjective optimization design of the carriage. Te multiobjective genetic algorithm transforms the multiobjective into a single objective for the optimal design according to a weighted approach with the following expression: where ω k is the weighting of the kth objective and f k is the normalized kth objective value. To obtain the solution set of the multiobjective genetic algorithm based on Pareto ranking, frstly, the distance between various individuals needs to be obtained and the formula is shown in (1): where f k max is the maximum target value and f k min is the minimum target value. For I, the individual niche count is obtained by calculating the adaptation of the niche count for an individual with the following equation: In the obtained 16448 iteration results, the optimal solution set with a large gap is screened as shown in Table 8. Furthermore, the Pareto frontier diagrams of the stifness and the total mass of the carriage obtained by multiobjective optimization are illustrated in Figures 22 and 23.
It can be found from Table 9 that if the ultimate goal is to ensure that the carriage is rigid enough to minimize the mass of the carriage, case 1 is the best choice. If the maximum stifness of the carriage is required and the mass of the carriage is reduced, scheme 2 can be selected; if the mass stifness of the carriage needs to be well uniform, schemes 3, 4, and 5 can be selected; moreover, to ensure the thickness of the carriage plate, schemes 6, 7, and 8 are a preferable choice. To achieve the goal of greatly reducing the quality of the carriage and improving the utilization rate of materials, scheme 2 with better comprehensive performance is selected and the whole circle inspection is carried out to verify whether the requirements of stifness and strength of the carriage are satisfed. Te weight of the carriage is 2.9 t after removing the complex and relatively negligible mass of the components. And the results are illustrated in Table 9.     Te comparison results of stifness and strength before and after full-circle optimization are shown in Table 9. It can be found in Figure 24 that the maximum deformation of the full-load uniform transport carriage before and after the optimization of the full circle changes from 10.84 mm to 12.14 mm, with a change rate of 11.99%. In Figure 25

Conclusion
(1) By synthesizing the stifness and strength analysis of various working conditions of the carriage, the optimal Latin hypercube sampling and approximate model are leveraged to construct a comprehensive model of the carriage with full-load uniform transport and 0°lifting dual working conditions. (2) Compared with diferent approximate model construction methods, it is found that the MLSM model has higher modeling accuracy. Furthermore, to ensure the accuracy of the subsequent multiobjective optimization design, the MLSM construction method is improved by changing the constructed single value to multivalue construction. (3) Te results of the dual working conditions and modal optimization design of the fnite element model of the carriage show that the optimization efect of the carriage is obvious, in which the fnal mass is reduced by 1.215 t and the stifness is increased by 16300 N·mm −1 . Additionally, the results verify the optimized full-circle carriage model meets the design requirement. In this study, the analysis under a variety of severe working conditions has been done and a preferable result is obtained. Nevertheless, to achieve a more realistic scenario, more severe working conditions will be integrated into the lightweight design analysis in our future work.

Data Availability
All data used to support the fndings of the study can be obtained from the corresponding author upon request.  Figure 27: Stress at 0°lift after rounding.

Conflicts of Interest
Te authors declare that they have no conficts of interest.