Structural optimization design of BIW using NSGA-III and entropy weighted TOPSIS methods

Multi-objective optimization is always a difficult and hot problem for the static and dynamic performance and lightweight optimization design of a body-in-white (BIW). In this regard, a hybrid method coupled with NSGA-III and entropy weighted TOPSIS methods was put forward here. Effects of each design variable on the BIW performance was studied by means of the contribution method so as to optimize various design variables. Further, an experimental design was carried out by means of a radial basis function model to approximate each performance index of the automotive body. On the base of that, a comprehensive method where the Pareto frontier is searched by means of NSGA-III while the optimal solution is selected by means of the entropy weighted TOPSIS method was proposed based on lightweight, static and dynamic performances to analyze BIW performances to determine the contradictory optimal solution among various BIW factors. Afterward, our numerical verifications demonstrate that the BIW performances are improved significantly so that it can be feasible to optimize static and dynamic BIW performances and the lightweight performance of the BIW can be guaranteed accordingly.


Introduction
Automotive lightweight technologies that can not only decrease energy consumption and reduce automobile exhaust emissions but also greatly improve the environmental quality. 1 Thus, automobile lightweight has always been a research hotspot.The BIW weight accounts for about 30% of an entire automobile so that the BIW lightweight shall be of great importance to lightweight of the entire automobile. 2 The BIW bending stiffness, torsional stiffness, and modal performance are the basic factors that must be taken into account for the BIW design because they are closely related to the vehicle safety, comfort, and reliability.As a result, effects of the BIW lightweight on the rigidity and modal performance should be quite concerned.
Higher vibration and noise levels significantly impact the psychological and physical health of occupants. 3arious vibration excitations and noises acts on the BIW where various automobile assemblies are equipped and then are passed on to the occupant's body whose stiffness is greatly burdened with attenuation of external excitations.Generally, the greater the BIW stiffness is, the less external excitations can be transferred into the automobile and the better the noise, vibration, and harshness (NVH) performances of the automobile can come off.Also, the BIW modal frequency distribution can greatly affect automotive NVH performances. 4It is necessary to prevent resonances between BIW and engine excitations.In this respect, the BIW design shall necessarily ensure the reasonable modal frequency matching, namely, its first order bending and torsional frequencies should be far from those excitation frequencies.Additionally, many static and dynamic performance indexes must be taken into account for the BIW design to achieve a multi-object optimization.
Each BIW contains a large number of thin-walled panels.If all panels are regarded as design variables, the optimization will be significantly more complicated than with a conventional computer due to such excessive design variables.Thus, it is necessary to select reasonable and critical panels as the design variables.The traditional variable screening is mainly based on direct selection of the sensitivity 5 or contributions. 6nluckily, the both methods cannot be divorced from subjective selection.Various parameters have their own effects on performances and the same parameter will affect various performances on different levels. 7For example, the thickness of a certain panel weakly affects the BIW stiffness while it radically impacts on the modal performance.Consequently, it is necessary to screen variables according to effects of various parameters on different performances and finally determine the multi-objective design space.Evidently, this variable screening process is a multi-criteria decision.A hybrid multi-decision method is presented here to more effectively and comprehensively select design variables and provides a reference for subsequent optimization.
There are a lot of grids in a BIW finite element model so that the calculation cost may be high while the optimization efficiency may be low.Optimization methods based on surrogate models, such as the response surface method, 8,9 radial basis function (RBF), 10,11 kriging method, 12,13 support vector machine, 14,15 and artificial neural network method, 16,17 are relatively more efficient.On the other hand, it remains challenging to determine the most suitable surrogate modeling method for approximation and optimization of the responses.So far, those existing findings show that the RBF method is at a relatively comprehensive advantage compared with other surrogate models for dealing with several design variables and relatively few experimental samples. 18Thus, the RBF model is applied here to determine multi-objective static and dynamic performances and achieve the BIW lightweight optimization.
Several different criteria of the system, such as the static and dynamic stiffness, and modal and total mass, must be taken into account for optimization of the BIW static and dynamic performances and lightweight so that it shall be a typical multi-objective optimization problem. 19The non-dominated sorting genetic algorithm (NSGA-II) is an evolutionary algorithm based on the concept of Pareto optimization to solve multiobjective optimization problems.Unfortunately, the NSGA-II relies on crowding distances to evaluate solutions as the objective functions increase; consequently, reduced convergence and diversity could occur.Thus, Deb proposed the reference-point-based nondominated sorting genetic algorithm (NSGA-III) is applied to solve multi-objective optimization problems with three or more objectives. 20NSGA-III is an algorithm based on reference points and non-dominated sorting that normalizes the target vector during evolution to ensure that the scale ratio of each target can be consistent.As a result, it is becoming more popular to specially solve optimization problems with three or more objective functions.NSGA-III is applied here optimize the BIW multi-objective static and dynamic performances and lightweight.
Multi-objective optimization is usually solved to output a set of non-dominated solutions, one of which is better than the other by at least one criterion.Thus, determination of the optimal trade-off scheme for a comprehensive optimization is no doubt a subject worthy of study, which is typically classified as a typical multi-criteria decision-making (MCDM) problem.Many methods such as technique for order preference by similarity to ideal solution (TOPSIS), 21,22 gray relational analysis (GRA), 23,24 and rank sum ratio (RSR) 25 have been introduced in the past few decades.Wang and Wang 26 applied TOPSIS to screen the initial design variables and successfully improve the crashworthiness and lightweight of b-pillars.TOPSIS can not only efficiently generate the screening variable results but also outperform the other methods in terms of computation time, simplicity, and stability.Thus, TOPSIS is applied here to sort the Pareto solutions and determine the optimal solution.
Another important step for obtaining the optimal optimization scheme is determining the weight of each performance index. 27Relative findings present a variety of weight methods such as principal component analysis (PCA), 28 analytic hierarchy process (AHP), 29 entropy method (EM), 30 coefficient of variation (CV), 31 and other entropy weighted TOPSIS methods, which are popular because they require only the monotonically increasing rather than decreasing characteristics of different utility functions and avoid the subjectivity of weight selection. 32The entropy weighted TOPSIS method is applied here to more comprehensively and reasonably present the weights.
Our study is organized structurally as follows: Section 2: description of our optimization method; Section 3: presentation of our BIW finite element modeling and experimental verification; Section 4: presentation of our optimization processes and verification of our optimization results; and Section 5: summary of our major contributions.

Contribution analysis
The response of the system can be approximated by the following variable parameter regression model.
where: Y is the system response; x p is the design variable; H p (x p ) represents the main effect of a variable; R pq (x p , x q ) represents the cross effect of any two variables; m is a constant; e is the error value; and k is the number of variables.
The main effect of a variable can be expressed by equation ( 2) and its corresponding contribution to the response can be defined by equation (3).
where: g p is the main effect coefficient of the variable obtained by least square estimation; and N x i is its contribution to the response.

Entropy weighted TOPSIS
A multi-response decision-making problem is transformed into a single-response decision-making problem by means of TOPSIS that is a multi-criteria decisionmaking technique, based on calculation of the relative Euclidean distance between the alternative, positive, and negative ideal solutions.Unluckily, TOPSIS brings about subjective selection while assigning weights to each response.Thus, the entropy weighted method is introduced here to objectively weigh each response based on TOPSIS and form an entropy weighted technique based on the ranking preference of ideal solution similarity.
The absolute value of the contribution of parameters to the response is described as a decision matrix to obtain the parameters of the initial variables that have a significant influence on the performance response.After regularization, the structural parameters are sorted according to the combined contribution value of the Euclidean distance to the performance to analyze their effects on the performance, and the favorable design variables for subsequent optimization can be selected.The specific steps are given as follows.
The contribution value of the initial variable to the performance is classified and defined as a decision matrix that is written as: where: , n) represents the contribution of the mth board parameter to the nth performance; m represents the number of panels; and n represents the number of performance indicators.For intuitive analysis, the above decision matrix is regularized as follows: where: r ij is the result of x ij normalization.Then, the entropy value of each index is calculated by means of the entropy weight method in accordance with the information entropy theory and the weights are obtained according to different entropies.The lower the entropy of the information is, the higher the weight is.The information entropy (E j ) is defined as: where: R ij is the result of r ij normalization and H is the adjustment factor.
where: v j denotes the jth weighting coefficient.The distance of the positive or negative ideal solutions is calculated from the best and worst alternatives.The positive and negative ideal solutions are described as follows.
where: A + and A 2 represents the positive and negative ideal solutions, respectively.Those parameters that have greater impacts on static and dynamic performances should be selected here.Thus, the positive and negative ideal solutions can be expressed as: where: v j + and v j 2 are the positive and negative ideal solutions under the jth performance, respectively.
Finally, the Euclidean distance is applied to calculate the separation of each alternative from positive and negative ideal solutions based on the following equations.
where: S i + and S i 2 represent the distance from the ith parameter to the positive and negative ideal solutions, respectively.
Thus, the relative closeness can be calculated correspondingly.
where: C i denotes the relative closeness of the ith parameter to performance

Multi-objective optimization method
The RBF model coupled with the experimental design method, the NSGA-III and TOPSIS approach are integrated into our hybrid optimization method for multiobjective NVH and lightweight optimization design of vehicle panels, whose processes are schematically shown in Figure 1.In detail, the effectiveness of our finite element model is verified by measurements.On this basis, the final design variables can be selected.Next, the experimental design integrated with RBF model is applied to fit the initial model indicators including objectives and constraints, which are solved by the NSGA-III considering mass, bending stiffness, first-order torsional modal frequency as three objectives.Thereby, a set of optimal solutions is determined and the optimal solution is obtained by means of the TOPSIS method.In the end, the optimal model is compared with the initial model to demonstrate the superiority and feasibility of our hybrid method.

Numerical BIW model and validation
Our BIW which includes 6131 welding point elements and whose mass is 384.1 kg is meshed with thin-walled panels units (QUAD4 and TRIA3) and the grid size is 8 mm.Solder joints are modeled using the RBE2 and ACM units.The elastic modulus, Poisson's ratio, and  A high-precision finite element model is necessary for optimization of BIW multi-objective lightweight, static, and dynamic performances.To this end, our finite element simulations are verified with the corresponding measurements.
BIW static structural parameters include the bending and torsional stiffness; the former is primarily applied to evaluate the ability to resist deformation when bearing the mass of passengers or cargo while the latter is mainly utilized to evaluate the ability to resist torsional deformation on an uneven road surface.

Bending stiffness analysis
As for our analysis of static bending stiffness, those analysis points are arranged as follows: (1) the left mounting point of the BIW's front suspension is constrained for the second and third degrees of freedom; (2) the right mounting point is constrained for the third degree of freedom; (3) the left mounting point of the rear suspension is constrained for the first, second, and third degrees of freedom; (4) the right mounting point is constrained for the first and third degrees of freedom.
A vertical downward force of 1000 N is loaded on the left and right sides near the midpoint of the front and rear suspension connection lines (Figure 2).The displacement measuring point is the intersection of the extension lines of the load action line and frame rail.The bending stiffness is calculated by means of the following equation.
where: F (F = 1000 N) is the loading force; and |Z 1 | and |Z 2 | are the absolute values of the z-direction displacement of the left and right measuring points, respectively.

Torsional stiffness analysis
As for our analysis of the static torsional stiffness, the left and right mounting points of the BIW rear suspension are fixed fully.A torque of 2000 N m is applied to the mounting points on the left and right sides of the front suspension (Figure 3).On the other hand, the displacement measurement points are set as the installation points on the left and right sides of the front suspension for our post analysis.The torsional stiffness is calculated by means of the following equation.

Free modal analysis
In an actual driving process, an automobile produces vibrations under various excitations.While the exciting frequency is close to the natural frequency of the automotive body, the automotive body will approximately resonate so as to significantly influence the car comfort and reliability.Most of excitations is primarily in the lower frequency range.Thus, it is necessary to evaluate the low-order natural frequencies of the BIW, especially the first order bending and torsion frequencies.
The BIW modes and shapes are calculated here by means of the Lanczos method whose computational efficiency is high.Unconstrained analysis is carried out in the range of 1-100 Hz without calculation of the foremost six order rigid body modes.As a result, simulation can be completed in a shorter time.Then, the first order torsion and bending mode frequencies of BIW are 31.1 and 49.7 Hz, respectively (Figure 4).

Experiment validation
Static stiffness and modal experiments are carried out to verify the accuracy of our BIW finite element model.The actual modal and stiffness measurements are shown in Figure 5.
A comparison between static and dynamic stiffness simulations and measurements of the BIW is shown in Table 1.The results show that our simulations are very close to the corresponding measurements.Their relative errors are 2.18%, 2.64%, 2.39%, and 22.26%, respectively.Overall, our static and dynamic stiffness finite element model is accurate enough for our subsequent multi-objective lightweight and stiffness optimization.

Multi-objective lightweight design
First of all, variables are screened by means of the contribution method to not only efficiently perform our  multi-objective integrated design but also save calculation time.Then, the sample points with design parameter boundaries are determined by means of the optimal Latin hypercube experimental design method and an optimization model where the BIW performance indicators are taken into account is established.The BIW is also optimized based on application of NSGA-III.Finally, our optimal solution is selected from the Pareto front calculations by means of the entropy weighted TOPSIS method.

Component screening
The relationship between variables and responses is analyzed by means of the above method (Section ''Contribution analysis'').The measured design variables and responses are calculated by means of the regression method.Figure 6 shows the contributions of 30 design variables to various performance index parts, where M, K B , K T , f B , and f T represent the BIW mass, bending and torsional stiffness, First order bending modal frequency, and torsional modal frequencies, respectively, where positive bar values indicate positive correlations while negative bar values represent negative correlations.Based on our contribution analysis, 30 panel groups are screened.However, the number of panels remains large.Thus, 13 groups of panels are further screened to save the calculation time and improve the optimization efficiency (Figure 7).

Design of experiments
Our experiments is designed to obtain sample points and further establish an approximate model.Common experimental design methods include full factor design, partial factor design, orthogonal array design, central combination design, Hamersley design, Latin hypercube design, and optimal Latin hypercube design.Compared with other experimental design methods, the optimal Latin hypercube method has many advantages, such as better uniformity of the design space, fewer experiments, and higher precision of design space exploration.Thus, the optimal Latin hypercube method is applied here to sample 13 groups of design variables for our optimization.A total of 120 samples are generated, of which 100 samples act as training sets and 20 samples are used as test sets.Their ranges are listed in Table 2.

RBF surrogate model
Three surrogate modeling methods (namely the RSM, RBF, and Kriging methods) are firstly applied here to approximate the functional relationship between the input thickness variables and output indexes while the  RBF method is selected finally because it has a higher prediction accuracy.The RBF model approximates the following function. 33 where: N is the number of sampling points; x À x i k k is the Euclidean norm between design variable vector (x) and the ith sampling point (x i ); f represents any function that is named as a RBF in case of f(x) = f( x k k); v i is the unknown weighting factor positioned at x i of the basis function; p j (x) is a polynomial term; C j (j = 1, 2, Á Á Á , M) is the coefficient for p j (x); and M is the number of polynomial terms.
Additionally, 20 sample points are selected to verify the prediction accuracy of our surrogate model, where the R-square (R 2 ) and maximum absolute relative error (MARE) are two important evaluation criteria.They are expressed as: where: y i is the simulation value of the ith response; ŷi is the surrogate-based prediction of the ith response; and y i is the mean value of y i .The fact that the MARE value decreases along with growth of R 2 indicates that our surrogate model has a higher prediction accuracy.Comparison of various models are shown in Table 3.The mass and the NVH have larger R 2 values and smaller MAREs by means of the RBF method rather than RSM and Kriging methods.In addition, 20 sample points are used to verify the accuracy, whose error scatters are shown in Figure 8, where error points are distributed along a 45°diagonal line.As a result, our fitting accuracy is good and the RBF surrogate model has a higher prediction accuracy.It can be used for the subsequent BIW multi-objective lightweight and NVH performance optimization.

Multi-objective optimization model
The BIW bending stiffness, and first order bending and torsional modes are taken as constraints whose values are no less than 95% of their initial values.The panel thickness is regarded as a design variable for 13 groups of parts to obtain our multi-objective optimization model that is described as follows: find : where: K B0 , K T 0 , f B0 , and f T 0 are the initial values of the bending stiffness and torsional stiffness, first order bending and torsional mode frequencies, respectively; x i is the value of the design variable; and x u i and x l i are the upper and lower limits of x i , respectively.
Since the BIW static and dynamic stiffness and lightweight can come down to a multi-objective optimization problem and there are mutual constraints between the objectives, NSGA-III is applied here to solve such problem, whose crossover and mutation operation ensures the solution diversity and whose reference point selection strategy improves the solution search speed.Its specific steps are listed as follows: Step 1: Initialization of the population of chromosomes, each representing a process parameter solution.
Step 2: Doing crossover and mutation operations and creation of a new population.
Step 3: Evaluation: the old and new populations are coupled into a temporary population; then, the fitness values of each chromosome are assessed for all target functions.
Step 4: A fast non-dominated sort.
Step 5: Application of our reference point strategy to guide the search.
Step 6: Repeating Step 2 to satisfy the stop condition.
The population size is set as 100.Simultaneously, a crossover probability of 0.85, mutation probability of 0.15, crossover parameter of 20, and mutation parameter of 20 are used in NSGA-III.For verification of our NSGA-III accuracy, NSGA-II is correspondingly selected as comparisons.The NSGA-II and III-based Pareto frontier solution sets are shown in Figure 9.The fact that the NSGA-II and III-based optimal solution regions partially overlap indicates that their optimization directions are basically the same.In our threeobjective optimization, NSGA-III rather than NSGA-II results in more evenly distributed points whose distribution range is wider so that NSGA-III is superior to NSGA-II.Considering the different distributions of the Pareto frontier points, the entropies and weights based on application of the entropy weight method are different for NSGA-II and NSGA-III; and the BIW panels based on application of the entropy weighted TOPSIS method are also different.
The TOPSIS method is introduced here to determine the optimal solution from the Pareto frontier.The weight of each target must be calculated prior to introduction of the relative closeness.A mixed method is applied to determine the weight of each response (Section ''Entropy weighted TOPSIS'').The weights of K B , f T , and M are determined as 0.34, 0.36, and 0.30, respectively.Figure 10 illustrates the relative closeness distribution of the Pareto frontier.The 188th design point peaks up to 0.6191 so that the optimal result shall be derived from the Pareto set.

Optimization results
After the multi-objective optimal design scheme of the BIW NVH and lightweight has been obtained, its effectiveness shall be verified.As shown in Table 4, the bending stiffness and the first order torsional and bending modal frequencies fall by 1.58%, 1.71%, and   0.02%, respectively, while the torsional stiffness rises by 6.31% through our NVH and lightweight multi-objective optimization.The BIW weight can be reduced by 12.5 kg and the corresponding weight reduction ratio can be up to 3.25%.Our simulations (Figure 11) indicate that the lightweight effect is well marked.

Conclusion
A new comprehensive optimization method is put forward here for the BIW and its finite element model is established.On this basis, a multi-objective optimization model is correspondingly established.The BIW design parameters are then optimized by means of NSGA-III to solve out the Pareto frontiers.Finally, the optimal structural parameters are selected by means of TOPSIS.The conclusions are as follows: (1) Our BIW finite element model has good accuracy.Simulations and measurements demonstrate that the corresponding multi-objective NVH and lightweight optimization is feasible and reliable.(2) The RBF method can be applied to approximate the mapping between the BIW input panel thickness and output performance indexes.In case of there being more design variables and fewer experimental samples, the approximation accuracy will be much better.Thus, our RBFbased multi-objective optimization shall be reliable and efficient.(3) The contribution analysis method can be applied to effectively screen lightweight optimized components.According to the contribution of each variable to the optimization goal, the variables with greater impacts can be identified and retained while the variables with lesser impacts can be eliminated.(4) The thickness of the optimized parts can be optimized by means of such method.Relative closeness that is based on the entropy weight method and verified by means of TOPSIS is more objective and efficient for determination of the optimal combination of the thicknesses of the optimized parts.(5) Based on our optimizations, the BIW NVH and lightweight performance are optimized to guarantee the overall automobile mechanical properties.The total body mass can be effectively reduced by 12.5 kg, or 3.25%.Simultaneously, the overall torsional stiffness and first order torsional mode frequency rises while the bending stiffness and first order torsional mode frequency fall.Thus, our multi-objective NVH and lightweight optimization methods shall be feasible and efficient.

Figure 1 .
Figure 1.Flowchart of our hybrid method.
where: T (T = 2000 N m) is the loading torque; |Z 3 | and |Z 4 | are the absolute values of the z-direction displacement of the left and right measuring points, respectively; and L is the distance between the measuring points.

Table 1 .
Result comparison of performance indicators.

Table 2 .
Ranges of initial design variables.

Table 3 .
R 2 and MARE for our surrogate model.

Table 4 .
Comparison of optimization objectives.