Study on Aerodynamic Performance and Lightweight Multiobjective Optimization Design of Wheel With Entropy Weighted Grey Relational Analysis Methods

In order to improve the aerodynamic performance and optimization efficiency while wheel lightweight designing, a multi-objective optimization design method of wheel lightweight based on entropy weighted grey relational analysis (EGRA) was proposed in this article. The aerodynamic analysis finite element model of the assembled wheel was established, and the simulation accuracy was verified by experiments. Study the distribution law of performance parameters such as pressure and turbulent kinetic energy in flow field of the car, analyze the variation law of flow field velocity and turbulent intensity in front and rear wheel cavities of the assembled wheel, and analyze the cloud diagram distribution of temperature and surface convective heat transfer coefficient of the brake disc (<inline-formula> <tex-math notation="LaTeX">$h_{c}$ </tex-math></inline-formula>). Research on the influence of wheels with different disc structures on the aerodynamic drag coefficient of the car (<inline-formula> <tex-math notation="LaTeX">$C_{d}$ </tex-math></inline-formula>) and the <inline-formula> <tex-math notation="LaTeX">$h_{c}$ </tex-math></inline-formula>. Combined with grey relational analysis (GRA) and EGRA, the objective evaluation of the comprehensive aerodynamic performance of wheels with different disc structures was given. With the design of experiments (DOEs), 12 important design variables were screened out by contribution analysis method. Using the approximate model method, combined with the RBF surrogate model, a hybrid method combining EGRA and Non-Dominated Sorting Genetic Algorithm-II (NSGA- II) was proposed to lightweight and multi-objective optimize the assembled wheel. Comparing and analyzing the optimization platform recommending scheme, the technique for ordering preferences by similarity to ideal solution (TOPSIS) method preferring scheme and the EGRA method optimum scheme, it was found that the optimal compromise scheme was obtained by the EGRA method, the reduction of the <inline-formula> <tex-math notation="LaTeX">$C_{d}$ </tex-math></inline-formula> was more obvious, and the improvement rates of performance were also more balanced. After multi-objective optimization, the mass of the assembled wheel was reduced by 10.83%, the <inline-formula> <tex-math notation="LaTeX">$C_{d}$ </tex-math></inline-formula> was reduced by 5.02%, and the average convective heat transfer coefficient of brake disc (<inline-formula> <tex-math notation="LaTeX">$h_{a}$ </tex-math></inline-formula>) was reduced by 8.02%. The optimized assembled wheel has a weight reduction of 32.74% compared with the same type of cast aluminum alloy wheel, which has a remarkable lightweight effect and significant reduction on the <inline-formula> <tex-math notation="LaTeX">$C_{d}$ </tex-math></inline-formula>.

show that the flow field around the wheel is mainly vortex 85 in nature, and the number and strength of developed vortex 86 structures is strongly dependent on the applied yaw angle 87 level. Su et al. [12] discussed the influence of the spoke 88 characteristics on the C d , studied the improved MIRA model 89 on Fluent, researched the relationship between the C d under 90 different spoke offset distances and curvatures, and analyzed 91 the reasons for aerodynamic drag changes in different situa-92 tions. The results show that a smaller spoke offset distance 93 is beneficial to the reduction of drag coefficient. When the 94 wheel spoke offset distance is 10 mm, the minimum value of 95 C d is 0.2514. Malizia and Blocken [13] accurately modeled 96 two bicycle wheels, investigated the effect of the presence of 97 the ground and type of wheel/ground contact on the wheel's 98 aerodynamic drag, and provided flow field visualizations 99 to elucidate around spoked wheels in crosswind conditions 100 flow behavior. The results show that the gap between wheel 101 and ground should be a maximum of 10 mm, lower than 102 without crosswind, and the step height should be less than 103 10 mm. Jia et al. [14] used the stable Reynolds-averaged 104 Navier-Stokes calculation in the simulation, combined with a 105 wind tunnel experiment, to study local flow, surface pressure 106 coefficient, aerodynamic drag coefficient and lift coefficient 107 of wheel under different conditions. The results show that 108 the rotating wheel has a significant effect on the flow around 109 the isolated wheel, and rotation reduces the differential pres-110 sure, drag coefficient and lift coefficient, thereby improv-111 ing aerodynamic performance. Martins et al. [15] studied the 112 aerodynamic interaction of three-element wing and wheels 113 in ground effect by performing a 3D computational fluid 114 dynamics analysis on a simplified quarter model of a Formula 115 One racing car using a detached-eddy simulation approach. 116 The results show that the wheel wake is influenced by flap 117 configuration, and different flap configurations produce dif-118 ferent up wash flow fields, resulting in a change in the sep-119 aration point at the top of the tire. As the separation point 120 moves back, the downwash generated in the central region of 121 the wheel wake gradually increases, resulting in a shorter and 122 longer combined wake. 123 Wang et al. [16] conducted an experimental study on a 2/5 124 scale vehicle equipped with 2/5 scale rotating wheels. The 125 results show that the near wake of the wheel has a more 126 local effect on the aerodynamic lift and drag of the car and 127 the low-pressure region of the underbody has an effect on 128 aerodynamic pressure. The rear wheel wake interacts with 129 the car wake, exerting pressure conditions on the bottom 130 of the body and affecting the drag of car. Zhang et al. [17] 131 analyzed the influence of the tire profile on the aerody-132 namic characteristics of the vehicle, established a paramet-133 ric model based on the tire size parameters, optimized the 134 design parameters of the tire profile, and reduced the aero-135 dynamic resistance. Zhou et al. [18] analyzed the influence 136 of tire profile and tread pattern structure on tire aerodynamic 137 performance, studied the load characteristics of different 138 tire pattern structures, and revealed the difference between 139 the flow characteristics and the flow field around the tire. 140 of the outside surface of brake disc through experiments. 167 The results show that the ratio of the diameter of the brake 168 disc to the wheel determines the convective heat transfer 169 characteristics on the outside surface of the brake disc. conditions. The results show that the heat dissipation perfor-174 mance of disc brakes is related to the shape, material and 175 working conditions of the brake disc. 176 The above studies mostly focus on the simulation and test 177 of the aerodynamic performance of wheels and brake discs, the performance above the wheel, gives suggestions for wheel 197 design, and conducts a multi-objective optimization design 198 for the assembled wheel. Finally, the optimal design scheme 199 is given based on the EGRA. The contribution analysis method mainly uses the regression 203 of DOEs to calculate the contribution, which is used for rank-204 ing the contribution of design variables to the performance 205 target and screening the design variables in high dispersion 206 or high nonlinearity analysis to reduce the computational 207 cost and improve the efficiency of optimal design, and its 208 analytical calculation steps are as follows.

209
Step 1: Normalize processing 210 A DOEs approach was used to obtain a sample of experi-211 ments between design variables and response characteristics. 212 The design variables have different design spaces, contribu-213 tion values also change in the design space, and the sample 214 data input needs to be normalized using formula (1): wherex is the average value of the sample data; σ is the 218 standard deviation; N is the total number of sample data; 219 x i is the original input and x * i is the normalized input.

221
If there are k design variables (x 1 , x 2 , . . . , x k ), then any 222 response characteristic can be expressed by a multiple regres-223 sion model as: R ij x i , x j is the crossover effect of any two design 229 variables; µ is a constant term and ε is the deviation.

230
The main effect of the design variable can be expressed by 231 formula (3): Therefore, the contribution of the design variables can be 234 defined by formula (4): If the response characteristic has the characteristic of 261 ''lower is better'', the normalization method can be expressed 262 as: If the response characteristic is ideal with respect to a 266 specific value, the normalization method can be expressed as: 269 where x * i (k) is the k-th response characteristic value of the and T is the specific value.

277
The GRC of the k-th response characteristic of the 278 i-th experiment is expressed as: is the initial experimental sequence; 0i (k) = |x * 0 (k)x * i (k)| 282 is the absolute difference between x * 0 (k) and x * i (k); 283 max = max i max k 0i (k) and min = min i min k 0i (k) are the 284 maximum and minimum values of 0i (k), respectively; ξ is 285 the distinguishing coefficient, ξ ∈[0, 1], which is generally 286 defined as 0.5.

287
Step 3: Calculate the GRG 288 The GRG is calculated by averaging the GRC and 289 expressed as: where is the GRG, and n is the number of response 292 characteristic.

293
According to formula (9), the GRG of each design scheme 294 can be obtained. According to the size of GRG, each design 295 scheme can be sorted to obtain the optimal scheme with 296 comprehensive performance. Through calculation, the design 297 scheme with the highest GRG value represents the scheme 298 with the best comprehensive performance. The entropy weight method is used to determine the weight 302 of objective function. The higher entropy weight, the greater 303 the weight of objective function in optimization process.

304
The mapping function f i :[0,1]→[0,1] applied in the 305 entropy must satisfy the following three conditions, , and f i (x) must be monotonically increasing in 307 interval x ⊂(0,0.5); Therefore, the function w e (x) is defined 308 as the mapping function of the entropy weight method: where w e (x) takes its maximum value at x = 0.5, that is, 311 w e (0.5) = 0.6487. At the same time, in order to ensure that 312 the mapping function can take values within the range [0,1], 313 the following new entropy is defined: According to the above definition and the GRC, the steps 316 for determining the weight of each objective function are as 317 follows:

318
(1) In all design schemes, the sum of the GRC correspond-319 ing to the k-th response characteristic: (2) Normalize the coefficients: (3) The entropy of each response characteristic can be 324 written as:

331
(6) The weight of the k-th response characteristics can be 332 written as: Since the relative significance of each response charac-335 teristic may be different, the simple averaging method of 336 formula (9) may lead to inaccurate evaluation of the GRG.

337
Therefore, according to formula (17), the weight of each 338 response characteristic can be obtained, and different weights 339 are assigned to the response characteristic to carry out EGRA: where ω k is the weight of the k-th response characteristic.

342
According to formula (18), EGRA sorting can be per-343 formed to obtain optimal solution for comprehensive perfor- components are provided for researchers to choose, which 375 can be freely combined to form different configuration mod-376 els according to needs: the chassis are divided into Detailed 377 (D) and Smooth (S); the rearview mirror are divided into 378 with Mirrors (wM) and without Mirrors (woM); wheels are 379 divided into reserved with Wheels (wW) and without Wheels 380 (woW); ground conditions are divided into Ground Simu-381 lation (with GS) and without Ground Simulation (woGS). 382 And all of them provide corresponding wind tunnel test 383 data for researchers to use in CFD numerical simulation for 384 validation.

385
In order to achieve the accuracy of CFD simulation anal-386 ysis, it is necessary to adjust the calculation scheme of 387 numerical simulation based on wind tunnel test data to deter-388 mine the appropriate meshing strategy, boundary condition 389 parameters and solution method [26], [27]. Thus, this article 390 uses the DrivAer model and its wind tunnel test data to 391 verify the correctness of the simulation model. Considering 392 the research focus of wheel aerodynamic performance, the 393 combination configuration of a smooth chassis, removing the 394 rear-view mirror, keeping the wheels and mobile ground were 395 selected [28]. As shown in Figure 2.   (19): where y + is the wall distance, µ is the dynamic viscosity of

449
In the sensitive area of the car, the gradient of the flow 450 field parameters is large, and mesh refinement is required. 451 Therefore, the grid near the boundary of the computational 452 domain can be divided sparsely, and the grid near the car 453 needs to be refined, and three gradually encrypted density 454 boxes were set up. The research focus of this article is the 455 aerodynamic performance of wheel, so it is necessary to 456 refine the meshes around the front, rear and wheels, and the 457 mesh refinement strategy is shown in Figure 4. The flow 458 field near the boundary of computational domain is divided 459 by a hexahedral structured mesh, and the flow field near 460 car outside the boundary layer is divided by a tetrahedral 461 unstructured mesh by the center interpolation method [31]. 462 The half-car model flow field has a total of 5.324 million 463 calculation grids, as shown in Figure 5.

IV. BOUNDARY CONDITIONS AND EXPERIMENTAL
where k is turbulent kinetic energy, ρ is air density, ε is  Table 1.  Table 2.

499
In Table 2, I is the turbulence intensity; D is the hydraulic 500 diameter; u is the inlet wind speed and the speed of the car; 501 ω is the wheel rotation angular velocity; and p is the outlet 502 pressure of the computational domain.

503
The roughness height and the roughness constant settings 504 of each boundary are shown in Table 3.

D. SIMULATION AND EXPERIMENTAL VERIFICATION
The test data of DrivAer in different combinations were pro-508 vided by the Technical University of Munich. Figure 6 shows 509   Table 4.  The error between the simulation value and the test value is 517 0.40%, which shows that the set calculation scheme is correct 518 and feasible. The boundary conditions of the brake disc are set as shown 550 in Table 6. In the DrivAer model, the original wheel was replaced with 554 the assembled wheel, and the above ventilated brake disc 555 model was added, the 3D model of DrivAer after recon-556 figuration is shown in Figure 9(a). The FEM is shown in 557 Figure 9(b). The same meshing strategy as in Section III is used to 559 divide the reconfigured DrivAer model and flow field, half-560 car flow field calculation grids is 5.669 million. The refined 561 part and the finite element model of the flow field are shown 562 in Figure 10 and Figure 11, respectively. The finite element 563 model is set up using the same boundary conditions and solu-564 tion method as in Section IV. Since heat transfer needs to be 565 calculated, the energy equation is opened and the calculation 566 is performed in ANSYS/FLUENT. The y + value of the first layer grid in the near-wall region 569 of body satisfies 30 < y + < 300, in order to indicate 570 that the numerical calculation can obtain good simulation 571 results. Figure 12 shows the y + value distribution cloud map 572 of reconfigured DrivAer boundary layer grid when the car 573 speed is 30 m/s.

574
The y + value of the boundary layer grid on the surface of 575 the car is distributed around 34.50, and this article estimates 576   As shown in Figure 13(b), the turbulent kinetic energy is 595 large when the airflow passes through the front of the body, 596 the windshield, the front of the bottom of the front wheels 597 and the rear wheels, resulting in a lot of energy consumption. 598 When the airflow passes through the front of the body, due 599 to the pressure difference, the airflow enters the wheel cav-600 ity, resulting in turbulent kinetic energy increas and energy 601 consumption.

603
Due to the rotation of the wheel and the negative pressure, 604 the airflow flowing into the wheel cavity will form strong 605 turbulence or vortex core inside the wheel, which will affect 606 the flow field near the car, dissipate energy and increase 607 aerodynamic drag. The velocity vector of the flow field and 608 the distribution of turbulence intensity at the wheel cavity are 609 shown in Figure 14.

610
As shown in Figures 14(a) and (b), the airflow hits the 611 front of the lower part of the wheel to form positive pressure 612 and wheel cavity vortex, which increases the C d . The airflow 613 velocity at the front wheel is the largest, and it also has the 614 greatest impact on the C d and the h c .

615
As shown in Figures 14(c) and (d), the airflow forms a 616 vortex core at front and rear wheels off the ground, and 617 the vortex core expands into a vortex, which is separated in 618 the direction of the car leaving. The maximum turbulence 619 intensity at the front wheel is 23.62%, and the rear is 17.83%. 620 Because the front wheel is directly impacted by the incom-621 ing flow, the turbulent flow intensity is greater; while the 622 turbulent flow intensity of the rear wheel is smaller, but 623 it has a great impact on the flow field at the rear of the 624 body; the tail of body forms two wake vortices that revolve 625 around its own vortex core, and a back flow phenomenon 626 occurs. Therefore, the flow field near the wheel cavity can 627 be changed by optimizing the design of the disc structure to 628 reduce the C d .

630
The high-speed rotating wheel causes the change of the flow 631 field near the wheel cavity, and the change of the flow field 632 determines the h c . The cloud map of temperature and h c of 633 front and rear brake discs are shown in Figure 15; the aver-634 age temperature and average surface convective heat transfer 635 coefficient of brake disc (h a ) are shown in Table 7.  As shown in Figure 15, the h c at the bottom and the surface 637 near the wheel is better than other parts, which is related to 638 the turbulence intensity of the flow field near the disc.

639
As shown in Table 7, the average temperature of the front 640 disc is 20.82% lower than that of the rear disc, and the h a of 641 the front disc is 23.18% higher than that of the rear disc.

642
The above analysis shows that the greater the airflow  10-spokes wheels. The second group of wheel models has 667 the same disc spoke opening area, but different spoke styles: 668 keep the wheel disc opening area of 29927 mm 2 unchanged, 669 and design common 6 different spoke style wheels. The third 670 group of wheel models has the same number of spokes, but 671 different disc opening area: the number of spokes is designed 672 to be 5, and the spoke widths are 100 mm, 85 mm, 70 mm, 673 55 mm, 40 mm and 25 mm wheels. Three groups of wheel 674 models with different disc structures are shown in Figure 16. 675 Keep the brake disc unchanged in the reconfigured Dri-676 vAer model, replace wheels with different spoke structures, 677 adopt the same CFD analysis preprocessing method and 678 calculation scheme as in Section III and Section IV. Using 679 ANSYS/FLUENT for numerical calculation, the C d and the 680 h a are obtained (the average value of front and rear brake 681 discs is taken). Combined with GRA and EGRA in Section II, 682 the aerodynamic performance of wheels with different disc 683 structures is ranked, as shown in Table 8.

684
As shown in the first group of wheel models in Table 8, the 685 6-spokes wheel has the smallest C d , and the 5-spokes wheel 686 has the largest h a . The opening area of disc of each wheel 687 is the same, the air flow entering the wheel cavity, vortex 688 formation and turbulence intensity are basically close, so the 689 C d does not change significantly due to the change of the 690 number of spokes. However, wheels with fewer spokes have 691 larger holes between individual spokes, which is conducive 692 to the concentration of airflow into the wheel cavity and 693 enhances the heat convection effect of brake disc. As the 694 number of spokes increases, the h a decreases.

695
Based on the GRA ranking, the 5-spokes wheel has the 696 highest comprehensive performance index of 0.8571; based 697 on the EGRA ranking, the 5-spokes wheel has the highest 698 comprehensive performance index of 0.8795. Both methods 699  can give an optimal solution with a high comprehensive 700 performance index based on the conflicting the C d and 701 the h a .

702
After sorting and comparing the two methods, it is pointed 703 out that the 5-spokes wheel has high comprehensive perfor-704 mance, which avoids the blindness of subjective selection. 705 The results show that when the opening area of disc is the 706 same, reducing the number of spoke is beneficial to reduction 707 of the C d and the improvement of the comprehensive aerody-708 namic performance of wheel.

709
As shown in the second group of wheel models in Table 8,  Based on the GRA ranking, the comprehensive perfor-719 mance index of wheel with the f-style disc is the highest 720 of 0.6796; based on the EGRA ranking, the comprehensive 721 performance index of wheel with the f-style disc is the highest 722 of 0.7483. Both methods give an optimal solution with a high 723 comprehensive performance index, after sorting and com-724 parison, both methods point out that the wheel with f-style 725 disc has higher comprehensive performance, which avoids 726 the blindness of subjective selection. The results show that 727 when the opening area of the spokes is the same, increasing 728 the closed area of the top of the spokes connected to the rim 729 is beneficial to the reduction of the C d , but is not conducive 730 to the improvement of the h a . 731 VOLUME 10,2022 As shown in the third group of wheel models in Table 8  nodes, so as to construct the parameterized variables of the 785 wheel model. The assembled wheel parametric model in this 786 article defines 21 design variables, denoted as DV1, DV2,. . . , 787 DV21, as shown in Figure 18. Table 9 gives the description, 788 initial value and range of each design variable.

790
Due to the complex structure of the wheel and the large 791 number of initial design variables, in order to improve the 792 efficiency of multi-objective optimization of the assembled 793 wheel, the initial design variables should be screened in com-794 bination with the contribution analysis. Based on the initial 795 value and range of design variables in Table 9, the optimal 796 Latin hypercube design (OLHD) was adopted, and 100 sam-797 ples were selected to analyze the contribution of 21 initial 798 design variables of the assembled wheel. Figure 19 shows the 799 contribution values of 21 design variables to the performance 800 of the wheel. In the contribution analysis graph, a positive bar 801 value indicates a positive correlation, and a negative bar value 802 indicates a negative correlation.

D. APPROXIMATE MODEL AND ACCURACY VERIFICATION 810
The relationship between design variables and performance 811 index can be obtained using approximate model. The opti-812 mization based on the approximate model method, combined 813 with the multi-objective optimization algorithm, can real-814 ize multi-objective optimization. Kriging and RBF surrogate 815    Use the coefficient of determination (R 2 ), root mean square 828 error (RMSE) and Mean Absolute Percentage Error (MAPE) 829 to evaluate the accuracy of the approximate model. If the 830 R 2 value is closer to 1 and the RMSE value is closer to 0, 831 it indicates that the overall prediction accuracy of approxi-832 mate model is higher, and when the MAPE value is smaller, 833 it indicates that the local prediction accuracy of approximate 834 model is higher.    The design variables and range in Table 9

874
The optimization platform is shown in Figure 21. 269 non-dominated optimal solutions is obtained, as shown in 879 Figure 22.  Figure 23 shows the Pareto frontier GRG obtained by 894 using EGRA, and the non-dominated optimal solution with 895 the largest GRG is selected as scheme A. Therefore, the 896 232nd non-dominated optimal solution with the largest GRG 897 of 0.7181 is considered as the design solution with the best 898 comprehensive performance.    Table 12 shows the comparison between the optimization 931 results of design variables of scheme A and the initial values 932 before optimization.

933
According to Scheme A in Table 11, the mass of 934 the assembled wheel after multi-objective optimization is 935 5.524 kg, which is 10.83% lower than the initial value; the 936 C d is 0.2215, which is 5.02% lower than the initial value; 937 the h a is 64.00, which is 8.02% lower than the initial value. 938 The mass of a 16 × 61.2 J cast aluminum alloy wheel on the 939 market is 8.213 kg. Compared with this wheel, the weight 940 of the assembled wheel after multi-objective optimization is 941 reduced by 2.689 kg, which is 32.74%.

942
As shown in Table 12, the optimized value of width 943 of spoke grooves (x 7 ) is 4.54, which basically does not 944 interfere with air flow due to its smaller width, which can 945 reduce the C d , but also weakens the h a . The width and 946 thickness of the bottom spokes and the thickness of hub are 947 reduced, while the optimized assembled wheels use magne-948 sium alloy rims, which contribute significantly to the reduc-949 tion of wheel mass. Ultimately, the material and structural 950 changes resulted in a multi-objective optimized assembled 951 wheel with improved overall performance. The change of the disc structure of assembled wheel after 955 optimization affects the flow field of wheel cavity and car, 956 and then affects the C d and the h a . Therefore, it is necessary 957 to compare and analyze the aerodynamic performance of the 958 optimized front and rear assembled wheels. In the recon-959 figured DrivAer model in Section V.B, the pre-optimized 960 assembled wheel is replaced with an optimized assembled 961 wheel, and the same calculation scheme as in Section V is 962 used for simulation analysis.

963
As shown in Figure 25, the maximum turbulent intensities 964 of the X-direction section of front and rear wheel cavity of 965 optimized assembled wheel are 20.65% and 15.81%; Before 966 optimization, they were 23.62% and 17.83% respectively, 967 VOLUME 10, 2022    which were both lower than those before optimization.

968
As shown in Figure 26 and Table 13, after optimization, the 969 local and average temperatures of front and rear brake discs 970 both increased, and the h a decreased significantly.

971
After optimization, the width and thickness of the bottom 972 spokes of assembled wheel was decreased, which reduces the 973 impact of the rotating disc on the airflow and the interference 974 of the outer flow field of car; at the same time, the spoke slot 975 becomes smaller, which increases the closed area of the disc, 976 especially the closed area close to the rim, reduces the air flow 977 entering the wheel cavity, and weakens the interference of 978 multiple airflows, the generation of vortices and the intensity 979 of turbulence. These structural changes reduce aerodynamic 980 drag, but also result in a reduction in the convective heat 981 transfer performance of brake disc.

983
This article proposes a multi-objective optimization design 984 method for wheel lightweight based on EGRA. The aerody-985 namic analysis finite element model of the assembled wheel 986 was established, and the simulation accuracy was verified by 987 experiments. The distribution law of performance parameters 988 such as pressure and turbulent kinetic energy of car flow field 989 is studied, and the variation law of the flow field velocity 990 and turbulent flow intensity at front and rear wheel cavity 991 of assembled wheel was analyzed. The influence of wheels 992 with different disc structures on the C d and the h c was 993 researched, and an objective evaluation of the comprehen-994 sive aerodynamic performance of wheels with different disc 995 structures was given. Using the approximate model method, 996 the lightweight multi-objective optimization based on the 997 aerodynamic performance of assembled wheel was carried 998 out, and the lightweight effect and the C d are significantly 999 reduced.