RBF-Based Integrated Optimization Method of Structural and Turning Parameters for Low-Floor Axle Bridge

Abstract

The axle bridge plays a crucial role in the bogie of low-floor light rail vehicles, impacting operational efficiency and fuel economy. To minimize the total cost of the structure and turning of axle bridges, an optimization model of structural and turning parameters was built, with the fatigue life, maximum stress, maximum deformation, and maximum main cutting force as constraints. Through orthogonal experiments and multivariate variance analysis, the key design variables which have a significant impact on optimization objectives and constraints (performance responses) were identified. Then the optimal Latin hypercube design and finite element simulation was used to build a Radial Basis Function (RBF) model to approximate the implicit relationship between design variables and performance responses. Finally, a multi-island genetic algorithm was applied to solve the integrated optimization model, resulting in an 8.457% and 1.1% reduction in total cost compared with the original parameters and parameters of sequential optimization, proving the effectiveness of the proposed method.

1. Introduction

Using a concave “U”-shaped axle bridge structure to connect two wheels with interference, 100% low-floor light rail vehicles can reduce the height of their floor surface from the ground [1]. The axle bridge is an important component of the vehicle bogie, and its—design structure is of great importance to improving the operating performance and fuel economy of light rail vehicles. Moreover, reasonable selection of the turning parameters of the axle bridge is an important way to ensure machining accuracy, improve production efficiency, and reduce production costs. Therefore, it is necessary to conduct structural and process parameter optimization for axle bridges.

Qu et al. [2] developed a multi-objective optimization model to reduce cutting force and surface roughness, and to improve the material removal rate, which was successfully solved by using the NSGA-II algorithm. Tian et al. [3] established a multi-objective optimization model aiming to minimize carbon emissions and processing time. In this case, the NSGA-II algorithm was used to obtain the optimal solution. With the spindle speed, feed per tooth, and transverse cutting depth as design variables, Yang et al. [4] formulated an optimization model targeted at minimizing carbon emissions and costs in milling, and used the genetic algorithm to obtain the optimal solution. Pourmostaghimi et al. [5] constructed a turning parameter optimization model with the material removal rate and processing cost as objectives, and surface roughness as the constraint. The particle swarm optimization algorithm was utilized to solve the model, yielding the optimal turning parameters. Qu et al. [6] investigated the effects of dry grinding, wet grinding, and nano-carbon fluid minimum lubrication on the grinding performance of ceramic matrix composites. They determined the optimal machining method through a controlled variable approach and orthogonal experiments, ultimately enhancing machining quality. Li et al. [7] developed a theoretical model for predicting grinding force in the grinding process, and optimized force parameters through a genetic algorithm.

In the process of optimization design, conventional methods often necessitate multiple iterations of finite element simulation analysis [8,9]. However, these methods involve high computational costs, which cannot meet the swift and efficient design requirements for engineering products. Consequently, surrogate models have emerged as a promising technology in the design field [10,11], extensively applied in complex equipment such as aircraft, ships, and automobiles [12]. Commonly employed surrogate models in engineering encompass the Polynomial Response Surface (PRS) [13], Radial Basis Function (RBF) [14], and Support Vector Regression (SVR) [15], among others. Wang et al. [16] established a multi-objective optimization model with the objectives of maximizing the machining efficiency and minimizing the tooth surface roughness. They utilized a regression model to approximate the implicit relationship between grinding parameters and tooth surface roughness, and employed the particle swarm optimization algorithm for the optimization solution. Wang et al. [17] constructed the response surface model to optimize the front collector structure using different materials. The feasibility of the response surface model was validated through metrics R2 and RMSE, and the MIGA algorithm was employed for seeking the optimal solution. Gao et al. [18] employed the Kriging model and a genetic algorithm to optimize welding process parameters, resulting in the optimal weld geometry shape. Wang et al. [19] integrated sensitivity analysis with structural contribution analysis to identify design variables for the lightweight body frame of an electric bus. Subsequently, RBF and NSGA-II were utilized to fit and solve the collaborative optimization model. Li et al. [20] assessed the fitting accuracy of RBF, PRS, SVR, and Kriging models through cross-validation error. RBF, exhibiting the highest accuracy, was employed to approximate the implicit relationship between EDM parameters and the performance response of 304 steel. Finally, NSGA-II was used for multi-objective optimization. Gao et al. [21] focused on the welding frame of the bogie, establishing the lightweight design model with static strength and welding fatigue as constraints. The Kriging model and an adaptive sequential approximation optimization method were then applied for the solution.

In existing cost-driven optimization research focusing on both structural and process parameters, the majority of researchers employ a sequential optimization approach (as illustrated in Figure 1a), leading to a separation between structural and process design, and making it challenging to minimize the total cost of both. Moreover, during the optimization process, the results of a performance response analysis (finite element simulations or physical experiments) are often directly utilized, resulting in high optimization costs. Therefore, this paper proposes an integrated optimization method for the axle bridge structure and process of low-floor light rail vehicles. It establishes an optimization model with the axle bridge structural parameters and process parameters as variables, and cost as the objective. Additionally, it introduces an RBF model to replace the implicit relationship between the axle bridge structure, process parameters, and performance response. Finally, the optimized model is solved using a multi-island genetic algorithm to obtain the optimal combination of axle bridge structure and process parameters. The innovation lies in the integration of optimization models, which reduces overall costs compared to previous methods, where structural and parameter optimizations were conducted sequentially, as depicted in Figure 1b.

2. Optimization of Structural Parameters

As an important component of the bogies of low-floor light rail vehicles, the optimization design of the axle bridge is of great importance for improving the operating performance and fuel economy of light rail vehicles [22]. Low-floor light rail vehicles adopt concave “U”-shaped axle bridges to reduce the height from the floor to the ground [1]. The axle bridge structure used in this paper, which consists of the axle journal, axle head, axle body, and primary spring seat, is shown in Figure 2. In the actual working process of an axle bridge, stress is a critical factor affecting its strength, because excessive stress can lead to axle bridge failure. Deformation is a key factor affecting the stiffness of the axle bridge. Excessive deformation not only affects passenger comfort but also poses safety concerns. Fatigue life is crucial for the long-term safe working of an axle bridge. Therefore, the stress, deformation, and fatigue life are used as the constraints in the structural optimization of an axle bridge.

2.1. Finite Element Simulation

Based on the TB/T 3549.1-2019 [23] standard, the analysis and calculation of the magnitude of loads on an axle bridge were determined according to Formulas (1)–(3). The technical parameters utilized for these calculations are detailed in

Table 1. Main technical parameters of low-floor light rail vehicles.

	Unit	Name	Value
Mv	kg	Vehicle mass	10,200
m+	kg	Bogie mass	5400
p	kg	Normal passenger capacity	6240
g	m/s2	Gravitational acceleration	9.81

.

To ensure safe operation under extreme working conditions, the applied load of an axle bridge under the combined action of vertical, lateral, and torsional loads was comprehensively analyzed and calculated.

The vertical load is

$$F_{Z1}=F_{Z2}=\frac{g}{2n_b}(m_v+1.2p-n_b{m}^{+})$$

(1)

where $F_{Z1}$ (N) is the vertical load applied on the left side of the axle bridge; $F_{Z2}$ (N) is the vertical load applied on the right side of the axle bridge; and n_b is the number of bogies (take $n_b$ as 2 for two bogies per carriage).

The lateral load is

$$F_y=0.5(F_Z+0.5m^{+}g)$$

(2)

where, Fy (N) is the magnitude of the lateral load applying to each bogie.

Moreover, to simulate the distortion of the bogie under curved track conditions, a torque was applied at the spring seat of the axle bridge. The formulation of torque is

$$M=m_{v}g/8\times 0.315, N·m$$

(3)

Based on the Formulations (1)–(3), the vertical, lateral, and torque loads were 16,892.82 N, 21,689.91 N, and 3939.94 N·m, respectively. The axle bridge material used was 25CrMo low-carbon alloy steel (Trading Company, Liaocheng, China), and its main mechanical parameters [24] are shown in

Table 2. Mechanical parameters of 25CrMo.

Tensile Strength (MPa)	Yield Strength (MPa)	Elastic Modulus (MPa)	Poisson’s Ratio	Density (kg/m3)
678	≥420	2.05 × 105	0.28	7850

.

The 3D model of the axle bridge was drawn and simplified using SolidWorks, and meshed using ANSYS in a tetrahedral manner. The results are shown in Figure 3. The number of cells in the model was 112,977, and the number of nodes was 200,053.

The vertical load and torsional load were applied to the primary spring seat, and the lateral load was applied to the axle journal. The constraints were applied to the axle journal at both ends. The maximum stress and deformation nephogram were obtained through static analysis, as shown in Figure 4. The maximum stress was 286.233 MPa, and the maximum deformation was 5.23501 × 10⁻³ mm.

Based on the statics analysis results, the fatigue life analysis of the axle bridge was carried out. The fatigue damage and life nephograms were obtained, as shown in Figure 5. From the damage nephogram, the damage degree of each part of the axle bridge can be seen. The red is the weak area, which is located at the left end of the axle journal and near the arc of the connection between the axle journal and the axle head. This indicates that these areas are prone to fatigue failure, which is basically consistent with the results of the static strength analysis.

According to the analysis results, the minimum life of the axle bridge was $2.16\times {10}^{12}$ occurrences at the 10,479 node, which is far greater than the requirements of the enterprise. From the results of the statics and fatigue analysis of the axle bridge, the stress, deformation, and life of the axle bridge are surplus, thus there is still space for lightweight design.

2.2. Parametric Modeling

To reduce the finite element simulation time, it was necessary to conduct parametric modelling by means of APDL. Parametric modelling can achieve rapid adjustment of size parameters, automatic imposing of load and constraints, automatic generation of finite element analysis models, and rapid acquisition of response values such as stress, deformation, and fatigue life. The parametric modelling process of the axle bridge is shown in Figure 6.

2.3. Structural Optimization

To reduce the blank costs, the optimization model of the axle bridge was built in Equation (4), which takes the diameter of the axle journal $d_1,d_2,d_3$, and the thickness of the primary spring seat $t$ as design variables, and takes the stress, deformation, and fatigue life as constraints.

$$\begin{array}{l}\mathrm{find}:\mathit{d}=\left[d_1,d_2,d_3,t\right]\\ \min :f(\mathit{d})=U_D\\ \mathrm{s}.\mathrm{t}.:L_{\min }(\mathit{d})\ge {10}^7\\ S_{\max }(\mathit{d})<320Mpa\\ D_{\max }(\mathit{d})<6.4\times {10}^{-3}mm\\ d_3-d_2\ge 10, d_2-d_1\ge 5\end{array}$$

(4)

where $\mathit{d}$ is the design variable; $U_B$ is the cost of the axle bridge blank, which can be calculated using the product of weight and unit price; L is the fatigue life; $D(\mathit{d})$ is the maximum deformation; and $S(\mathit{d})$ is the maximum stress.

The value range and initial value of the structural parameter of the axle bridge are shown in

Table 3. Design variables and value range in structural optimization.

Design Variable	Value Range	Initial Value
Diameter of the first section journal ×1 (mm)	[110, 130]	120
Diameter of the second section journal ×2 (mm)	[120, 140]	133
Diameter of the third section journal ×3 (mm)	[140, 160]	153
Thickness of spring seat ×4 (mm)	[50, 70]	60

.

A full factorial design (FFD) with 4-factor and 3-level was conducted, and 64 groups of structural parameters were randomly selected from the FFD samples. APDL was then performed for parametric modeling to obtain the stress, deformation, and fatigue life values at the 64 groups of structural parameters. Partial sample points and performance response values are shown in

Table 4. Partial sample points for structural optimization of axle bridge.

No.	Design Variable				Performance Response
	x1	x2	x3	x4	Blank Cost	Maximum Stress	Maximum Deformation	Minimum Life
	(mm)				($)	(MPa)	(10−3 mm)	(Times)
1	110	130	160	50	546.94	331.365	9.106	5.22 × 108
2	120	140	160	50	560.93	328.764	6.010	2.24 × 100
3	120	130	140	60	570.96	286.491	3.225	9.13 × 1012
4	110	140	160	50	550.08	344.752	11.200	8.14 × 107
5	120	130	150	60	575.59	281.54	4.161	2.16 × 1012
6	110	130	150	60	564.73	288.221	9.259	7.11 × 108
7	120	130	160	70	603.28	250.875	5.018	3.23 × 1012
8	110	130	150	70	587.47	260.510	9.291	7.08 × 108
…	…	…	…	…	…	…	…	…
64	110	120	160	50	544.03	325.990	4.466	9.04 × 109

.

Based on the 64 groups of structural parameters and corresponding simulation results, an RBF model was built to fit the implicit relationship between design variables and responses. Then multi-island genetic algorithm was used to solve the structural optimization model. The parameters of multi-island genetic algorithm were iterations of 2000, islands number and population size of 10, and probability of crossover and variation of 0.01. The comparison of the axle bridge structural parameters before and after optimization is shown in

Table 5. Comparison of axle bridge structural parameters before and after optimization.

Design Variable	d1 (mm)	d2 (mm)	d3 (mm)	t (mm)
Before optimization	120	133	153	60
After optimization	110.04	123.29	151.71	50.921

.

The structural parameters after optimization were imported into the parametric model for statics analysis and fatigue life analysis. The maximum stress was 313.408 MPa, the maximum deformation was 6.395 × 10⁻³ mm, and the minimum life was 1.029 × 10⁹ times. Therefore, under the premise that the stress, deformation, and service life meet the constraints, the weight of the optimized axle bridge was reduced by 37.836 kg and the weight reduction rate was 8.05%. Multiplied by the unit price, the total cost of the axle bridge blank was reduced by $47.56 after structural optimization.

3. Optimization of Turning Parameters

In turning, the reasonable selection of cutting parameters is an important way to ensure machining accuracy, improve production efficiency, and reduce production costs. Cutting force is an important factor influencing machining quality; excessive cutting force can cause vibration in the machining system, thereby affecting machining accuracy. Therefore, using the optimal structural parameter in Section 2, the optimization model for the axle bridge turning parameters was built with turning cost as the objective and cutting force as the constraint.

3.1. Optimization Model of Turning Parameters

In practice, multi-pass turning is used to machine the axle journal. The turning parameter optimization model was built with turning cost as the objective, and the cutting speed, feed rate, and depth of cut in the rough machining, semi precision machining, and precision machining stages as variables, as shown in Equation (5). The design variables and value range are shown in

Table 6. Design variables and value range in optimization of turning parameters.

Design Variables	Value Range	Initial Value
Cutting speed for rough machining ×1 (m/min)	[80, 150]	115
Feeding speed for rough machining ×2 (mm/r)	[0.5, 0.8]	0.65
Cutting speed during semiprecision machining ×3 (m/min)	[180, 200]	190
Feeding speed during semiprecision machining ×4 (mm/r)	[0.3, 0.5]	0.4
Depth of cut during semiprecision machining ×5 (mm)	[0.8, 2]	1.4
Cutting speed during precision machining ×6 (m/min)	[220, 235]	227.5
Feeding speed during precision machining ×7 (mm/r)	[0.1, 0.3]	0.2
Depth of cut during precision machining ×8 (mm)	[0.2, 1]	0.6

.

$$\begin{array}{l}\mathrm{find}: \mathit{X}=\left[x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8\right]\\ \min : f\left(\mathit{X}\right)=U_M\\ \mathrm{s}.\mathrm{t}.: F_C\left(\mathit{X}\right)\le F_{C\max }\end{array}$$

(5)

where $U_M$ [25] is the turning cost, $F_C$ is the main cutting force, and $F_{C\max }$ is the maximum allowable value of $F_C$. According to the literature [26], $U_M$ is calculated as follows:

$$\begin{array}{l}{U}_M=C_M+C_I+C_R+C_T\\ =K_0\left[t_m+\left(t_{\mathrm{c}}+t_l\right)+t_e\frac{t_m}{T_p}\right]+K_t\frac{t_m}{T_p}\end{array}$$

(6)

where $C_M$ is the actual processing cost; $C_I$ is the cost of workpiece installation and disassembly, as well as the cost of machine idle; $C_R$ is the tool replacement cost; and $C_T$ is the tool cost. $K_0$ is the unit time cost; $t_m$ is the cutting time; $t_{\mathrm{c}}$ is the workpiece installation and disassembly time; $t_l$ is the machine idle time; $t_e$ is the tool replacement time; $K_t$ is the cost of per blade of the tool; and $T_p$ is the tool life. Formula (4) is used to calculate $t_m$.

$$t_m=\frac{\pi \left(d_2+d_t\right)L}{1000x_4{x}_5\left(\frac{d_t-2x_7}{2}\right)}+\frac{\pi \left(d_2+2x_3\right)L_2}{1000x_1{x}_2{x}_3}+2\frac{\pi d_2{L}_1}{1000x_4{x}_5\left[\frac{\left(d_2-d_1\right)-2x_8}{4}\right]}+\frac{\pi \left(d_1+2x_8\right)L_1}{1000x_6{x}_7{x}_8}$$

(7)

where $L$, $L_1$, and $L_2$ are the length of different segments of the axle journal; $d_1$ and $d_2$ are the diameters of axle journal (in Figure 2); and $d_t$ is the blank allowance.

The parameters and values involved in Equations (6) and (7) are shown in

Table 7. Parameter value of turning parameter optimization model of axle bridge.

Parameters	Values	Parameters	Values (mm)
$t_{\mathrm{c}}$	35 min	$L$	380
$t_e$	15 min	$L_1$	305
$K_t$	5 $/blade	$L_2$	75
$K_0$	2.5 $/min	$d_t$	10
$T_p$	25 min

.

3.2. AdvantEdge Turning Simulation

To obtain the cutting force at different turning parameters, AdvantEdge turning simulation was conducted in this paper. The AdvantEdge turning simulation process includes preprocessing the simulation parameter setting; solving; and result postprocessing; as shown in Figure 7. The simulation parameters for AdvantEdge turning are shown in

Table 8. Parameter value of AdvantEdge turning.

Parameters	Values	Parameters	Values
Cutting Method	3D External Turning	Workpiece Diameter (D)	8 mm
Workpiece Length (L)	4 mm	Workpiece material	25 CrMo4
Tool shape	Rhombus	Back rake angle	0°
Side rake angle	−5°	Front clearance angle	−25°
Major cutting-edge angle	90°	Tool tip radius	0.4 mm
Tool material	Carbide-General	Tool coating material	TiAlN
thickness	0.005 mm

. According to the cutting theory, the ratio of main cutting force, back force, and feed force in cutting is about 1:0.4:0.25 [27]. Among the cutting forces, the main cutting force is the largest, followed by the back force, and the feed force is the smallest. Therefore, the main cutting force is generally used as the response index during the cutting process.

In this paper, YT15 cemented carbide was used as the cutting tool material. The mechanical properties of YT15 is as follows: strong hardness (91 HRA), bending strength (1180 MPa), and high density (12 g/cm³). Moreover, YT15 exhibits a high bonding temperature and robust oxidation resistance. Therefore, YT15 was used in this paper to turn the axle bridge.

To reduce the optimization cost of the turning parameters, RBF was used to fit the implicit relationship between the turning parameters and the main cutting force. In this paper, an 8-factor and 4-level orthogonal experiment was designed to perform an AdvantEdge turning simulation to obtain the main cutting force. The main cutting force $F_{C1}$ (N) for rough machining; $F_{C2}$ (N) for semi precision machining; $F_{C3}$ (N) for precision machining; and the turning cost f(x) ($), are shown in

Table 9. Sample points of orthogonal experiment in optimization of turning parameters.

No.	x1	x2	x3	x4	x5	x6	x7	x8	FC1	FC2	FC3	f(x)
1	80	0.5	180	0.3	0.8	220	0.1	0.2	3487.08	1154.42	151.011	1416.113
2	100	0.8	180	0.3	1.2	235	0.25	0.5	5194.55	1516.73	532.188	788.662
3	150	0.5	180	0.35	1.6	225	0.25	1	3568.04	2268.77	1059.06	738.875
4	125	0.8	180	0.35	2	230	0.1	0.3	4592.23	2507.41	191.173	1113.071
5	125	0.6	180	0.4	0.8	235	0.15	1	3827.83	1497.79	689.17	777.732
6	150	0.7	180	0.4	1.2	220	0.3	0.3	4561.46	1962.6	318.087	826.584
7	100	0.6	180	0.5	1.6	230	0.3	0.2	3705.94	3065.46	330.981	908.317
8	80	0.7	180	0.5	2	225	0.15	0.5	4166.67	3308.97	381.564	872.050
9	150	0.7	187	0.3	1.6	235	0.15	0.2	4247.22	1968.68	198.979	1104.153
10	125	0.6	187	0.3	2	220	0.3	0.5	4133.8	2063.03	683.916	774.064
…	…	…	…	…	…	…	…	…	…	…	…	…
32	150	0.8	200	0.5	2	235	0.3	1	5203.67	3541.95	1287.64	708.977

.

3.3. Optimization Solution

The multi-island genetic algorithm was used to solve the optimization model of axle bridge turning parameters, and optimal turning parameters were obtained. The comparison of turning parameters before and after optimization is shown in

Table 10. Comparison of turning parameters before and after optimization.

Variable	x1 (m/min)	x2 (mm/r)	x3 (mm)	x4 (m/min)	x5 (mm/r)	x6 (m/min)	x7 (mm/r)	x8 (mm)
Before optimization	115	0.65	190	0.4	1.4	227.5	0.2	0.6
After optimization	118	0.7458	184.9	0.3451	1.4687	227.76	0.2443	0.5207

.

Based on the spindle speed, feed rate, and depth of cut after optimization, an AdvantEdge turning simulation and cost calculation were conducted. The main cutting forces for precision machining, semi precision machining, and rough machining were 539.56 N, 2340.98 N, and 4460.17 N, respectively, all of which met the constraint conditions. After optimization, the turning cost was $110.63, which is $4.99 less than that before optimization, with a decrease of 4.316%.

4. Integrated Optimization of Structural and Turning Parameters

In order to further reduce the cost of the structural and turning parameters, the integrated optimization method was developed in this paper. Orthogonal experiments and multivariate variance analysis were used to reduce the dimension of optimization problems and improve the optimization efficiency. Through orthogonal experimental design, and utilizing ANSYS and AdvantEdge for finite element and turning simulations, samples and corresponding performance response values were obtained. Subsequently, multivariate variance analysis was employed to screen design variables, selecting those significantly impacting the objective and performance responses as optimization variables. Utilizing the selected design variables, optimal Latin hypercube sampling was employed to acquire sample points, and response values were obtained through simulation. Based on the OLHS samples and corresponding performance responses, the RBF model was constructed to fit the implicit relationship between design variables and responses. Finally, an integrated optimization model was established with total cost as the optimization objective, incorporating axle bridge structural parameters and turning process parameters as variables. The specific process is illustrated in Figure 8.

4.1. Radial Basis Function

Radial Basis Function (RBF) is a typical three-layer (input layer, hidden layer, output layer) feedforward neural network [28], which is characterized by strong nonlinear approximation capability, efficiency, simplicity, and adaptability [29]. Therefore, RBF was chosen in this study to approximate the implicit relationship between variables and responses. During the optimization process, using the RBF model instead of the finite element simulation and turning simulation can significantly reduce the optimization cost. Furthermore, utilizing the accurate RBF approximation of the variable-response implicit relationship enables the selection of more precise optimization algorithms, consequently improving optimization accuracy.

To verify the effectiveness of RBF, the nonlinear numerical example used was as follows [30]:

$$\begin{array}{l}g(x)=1-{(x_1+x_2-5)}^2/30+{(x_1-x_2-12)}^2/120\\ 0\le x_1\le 10,0\le x_2\le 3\end{array}$$

(8)

Twenty samples obtained via Latin hypercube sampling were selected to construct the RBF model. The samples (red cross), true curve of $g(x)$ (red solid line), and predicted curve using RBF (blue dashed line) are shown in Figure 9. As can be seen, the RBF model has good fitting ability to the nonlinear example with fewer samples. Therefore, RBF was used in this paper to approximate the implicit relationship between optimization variables and performance responses.

4.2. Multivariate Variance Analysis

The design parameters for integrated optimization include structural parameters ($d_1,d_2,d_3,t$); rough turning process parameters ($x_1,x_2$); semi precision turning process parameters ($x_3,x_4,x_5$); and precision turning process parameters ($x_6,x_7,x_8$). To reduce the cost of integrated optimization, orthogonal experiments and multivariate variance analysis were used to screen design variables that have a great influence on the performance responses. In this section, orthogonal experiments with 12-factor and 4-level were used for experimental design (as shown in

Table 11. Table of factor level for orthogonal experiments.

	${\mathit{d}}_1$	${\mathit{d}}_2$	${\mathit{d}}_3$	$\mathit{t}$	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$	${\mathit{x}}_6$	${\mathit{x}}_7$	${\mathit{x}}_8$
Level 1	110	120	140	50	80	0.5	180	0.3	0.8	220	0.1	0.2
Level 2	117	127	147	57	100	0.6	187	0.3	0.8	225	0.15	0.3
Level 3	124	134	154	64	125	0.7	194	0.4	1.6	230	0.25	0.5
Level 4	130	140	160	70	150	0.8	200	0.5	2	235	0.3	1

). Finite element analysis and turning simulation experiments were then carried out for each group of sample points. The performance response values such as weight; stress; deformation; life; rough machining main cutting force $F_{C1}$; semi precision machining main cutting force $F_{C2}$; and precision machining main cutting force $F_{C3}$ were obtained, as shown in

Table 12. Sample points of orthogonal experiment in integrated optimization.

No.	Design Variable		Response
	Structure Parameters $({\mathit{d}}_1,{\mathit{d}}_2,{\mathit{d}}_3,\mathit{t}$)	Turning Parameters $({\mathit{x}}_1,\sim {\mathit{x}}_8$)	Blank Cost ($), Stress (MPa) Deformation (10−3 mm), Life (Times)	FC1 (N)	FC2 (N)	FC3 (N)
1	124, 140 154, 50	125, 0.7, 180, 0.3, 1.2, 225, 0.25, 1	562.53, 309.363, 6.237 1.48 × 107	1516.73	3723.56	1059.06
2	130, 134 147, 70	150, 0.6, 200, 0.4, 1.6, 235, 0.25, 0.2	609.93, 328.198, 6.297 5.8 × 107	2286.1	3886.51	349.649
3	124, 140 154, 57	100, 0.8, 194, 0.35, 1.6, 230, 0.25, 0.2	578.45, 311.924, 6.219 1.48 × 107	880.815	5194.55	423.531
4	124, 127 160, 70	100, 0.5, 180, 0.4, 1.2, 220, 0.15, 1	606.99, 315.039, 6.208 1.59 × 107	1962.6	3767.96	658.867
5	124, 134 160, 64	80, 0.8, 187, 0.35, 2, 225, 0.1, 0.3	595.47, 318.499, 6.179 1.58 × 107	2342.86	4018.12	196.11
6	130, 140 140, 57	100, 0.5, 194, 0.5, 0.8, 235, 0.1, 0.5	588.65, 323.065, 6.426 5.46 × 107	1733.97	3767.96	374.606
7	124, 127 160, 64	125, 0.6, 194, 0.5, 1.6, 235, 0.15, 0.2	593.34, 313.538, 6.2116 1.58 × 107	3041.92	3827.83	198.979
8	124, 134 160, 70	150, 0.7, 200, 0.3, 0.8, 230, 0.1, 0.5	609.11, 315.716, 6.17 1.59 × 107	1066.37	4247.22	292.187
…	…	…	…	…	…	…
49	130, 140 160, 70	150, 0.8, 200, 0.5, 2, 235, 0.3, 1	618.21, 326.137, 6.162 4.13 × 107	3541.95	4210.54	1287.64

.

Based on the sample points of orthogonal experiments, multivariate variance analysis was conducted on the design variables and performance responses. The detailed process of multivariate variance analysis can be found in [31]. The analysis results are shown in

Table 13. Results of Multivariate Variance Analysis.

Optimization Object	p (Blank Cost)	p (Stress)	p (Deformation)	p (FC2)	p (FC3)	p (Turning Cost)
$d_1$	0.000 **	0.000 **	0.006 **	—	—	—
$d_2$	0.000 **	0.557	0.003 **	—	—	—
$d_3$	0.000 **	0.047 *	0.001 **	—	—	—
$t$	0.000 **	0.675	0.008 **	—	—	—
$x_1$	0.000 **	—	—	0.284	—	0.752
$x_2$	—	—	—	0.000 **	—	0.279
$x_3$	—	—	—	0.000 **	—	0.906
$x_4$	—	—	—	—	—	0.825
$x_5$	—	—	—	—	—	0.767
$x_6$	—	—	—	—	0.997	0.945
$x_7$	—	—	—	—	0.004 **	0.000 **
$x_8$	—	—	—	—	0.000 **	0.000 **

Note: “* means p < 0.05, ** means p < 0.01”.

, where p < 0.05 indicates that the design variable has great influence, and p < 0.01 indicates that the design variable has even greater influence.

According to the results of the multivariate variance analysis (as shown in Table 11), $d_1,d_2,d_3,t$ were regarded as having affected the weight and deformation significantly, $d_1$ and $d_3$ were regarded as having affected the maximum stress significantly, $x_7$ and $x_8$ were regarded as having affected the main cutting force and turning cost significantly. Therefore, six optimization variables were selected based on their significant influence on the objective and response values: structural parameters $d_1,d_2,d_3,t$ of the axle bridge, as well as feed rate $x_7$ and cutting depth $x_8$ in the turning process.

4.3. Integrated Optimization Model

To reduce the total cost of the blank and the turning process of an axle bridge, a mathematical model for integrated optimization was built with total cost as the objective, and the dimensions and turning process parameters of the axle bridge structure as design variables, as shown in Equation (9).

$$\begin{array}{l}\mathrm{find}: \mathit{x}=\left[x_1,x_2,x_3,x_4,x_{11},x_{12}\right]\\ \min : f\left(\mathit{x}\right)=U_D+U_M\\ \mathrm{s}.\mathrm{t}.: L_{\min }\left(\mathit{x}\right)\ge {10}^7\\ S_{\max }\left(\mathit{x}\right)<D_s\\ D_{\max }\left(\mathit{x}\right)<D_G\\ F_{\max }\left(\mathit{x}\right)<F_C\end{array}$$

(9)

where $U_D$ and $U_M$, respectively, represent the cost of the blank and the turning process of an axle bridge; $L_{\min }$ represents the minimum allowable life of the axle bridge; $S_{\max }$, $D_{\max }$ and $F_{\max }$, respectively, represent the maximum stress, maximum deformation, and maximum main cutting force of the axle bridge during FEM and turning simulation; and $D_s$, $D_G$, and $F_C$ are the maximum allowable values for the corresponding performance responses.

4.4. Optimal Latin Hypercube Sampling

Based on the screened design variables, the design of the experiments was reconducted. Compared with traditional sampling methods, Latin Hypercube Sampling (LHS) has a more uniform filling performance in the design domain. However, some sample points in LHS cannot be evenly selected in the design domain, so the fitting accuracy of the surrogate model may be biased. Optimal Latin Hypercube Sampling (OLHS) improves LHS by increasing the distance between sample points. It distributes all sample points as evenly as possible within the design domain, which can show better uniformity and space filling. It can thus fit the relationship between variables and responses more accurately and truly.

As can be seen from Figure 10a, sample points in some areas of LHS are too concentrated, and samples in other areas are missed. By contrast, sample points in OLHS are more even and there is no missing area, as shown in Figure 10b.

Based on the results of multivariate variance analysis, 60 samples were selected via the OLHS method. Statics and turning simulation analysis were conducted to obtain the weight, stress, deformation, life, and main cutting force at these sample points. Partial sample points are shown in

Table 14. Partial sample points of optimal Latin hypercube sampling.

No.	${\mathit{d}}_1,{\mathit{d}}_2,{\mathit{d}}_3,\mathit{t}$$, {\mathit{x}}_7$$, {\mathit{x}}_8$	Blank Cost ($)	Stress (MPa)	Deformation (10−3 mm)	Life (Times)	Cutting Force (N)	Total Cost ($)
1	121.35, 127.57, 154.59, 58.92, 0.1, 0.589	576.17	308.461	6.274	9.82 × 106	330.438	676.355
2	128.11, 138.92, 154.05, 64.05, 0.132, 0.73	599.05	320.797	6.222	6.41 × 107	465.454	701.416
3	121.35, 130.27, 148.39, 58.11, 0.297, 0.232	572.14	309.701	6.317	9.74 × 106	381.504	674.949
4	124.86, 135.68, 156.22, 54.32, 0.268, 0.914	573.08	313.908	6.219	3.74 × 107	1035.92	675.163
5	115.14, 132.16, 153.51, 65.95, 0.222, 0.2	586.08	338.994	6.276	5.63 × 106	323.932	686.018
6	114.86, 128.92, 143.24, 67.03, 0.287, 0.449	582.39	344.813	6.414	3.71 × 106	667.398	682.821
7	117.03, 1274.3, 158.11, 52.43, 0.154, 0.924	558.23	329.541	6.275	8.01 × 106	651.253	660.062
8	112.43, 129.19, 157.84, 69.46, 0.251, 0.654	592.40	307.256	6.235	2.07 × 107	792.161	692.223
9	115.41, 140, 145.14, 60.27, 0.260, 0.33	571.95	345.105	6.334	3.97 × 106	396.776	671.198
10	114.32, 125.95, 146.76, 58.65, 0.116, 0.265	563.50	294.359	6.369	2.35 × 107	188.412	664.645
……	……	……	……	……	……	……	……
60	115.68, 131.35, 140.27, 67.84, 0.214, 0.87	584.51	343.689	6.42129	4.47 × 106	783.644	684.693

.

4.5. Integrated Optimization Process

In order to keep the axle bridge strength and cutting force within the allowable range, and minimize the total cost of structure and turning processes, the RBF model and multi-island genetic algorithm were introduced in the integrated optimization of the axle bridge. The integrated optimization process is shown in Figure 11.

Step 1: Determine the initial design variables and value range. In this paper, the existing structural and turning parameters were used as the initial design variables, whose ranges were determined according to the requirements of the enterprise.

Step 2: Screening of design variables. Firstly, 49 groups of sample points were obtained via 12-factor and 4-level orthogonal experiments. The performance responses of each group of sample points were obtained using ANSYS statics and AdvantEdge turning simulation analysis. Then, multivariate variance analysis was conducted to select the design variables which have great impact on weight, stress, deformation, and the main cutting force.

Step 3: Optimal Latin hypercube sampling and RBF modeling. Based on the screened design variables in Step 2, 60 groups of sample points were selected using the OLHS method, and simulation analysis was carried out. On this basis, the RBF surrogate model was built to fit the implicit relationship between the design variables and performance responses.

Step 4: Optimization solution and result validation. The multi-island genetic algorithm was used to optimize the surrogate model, and the results were verified.

4.6. Optimization Solution

In the integrated optimization process of the axle bridge structure and process, the initial design variables consisted of the structural parameters of the axle bridge, and machining parameters including roughing, semi-finishing, and finishing cutting, denoted as $x_1$ to $x_8$. Through multivariate variance analysis on these initial design variables, key variables significantly impacting the performance responses were identified, including the axle bridge dimension parameters $d_1,d_2,d_3,t$, the feed rate in the finishing process $x_7$, and the turning depth $x_8$. The reduction in dimension parameters after optimization achieved a structural weight reduction, thereby lowering the raw material cost of the axle bridge. Additionally, the reduction in turning parameters led to a decrease in machining costs, resulting in reduced overall cost.

The integrated optimization model of the axle bridge was solved via multi-island genetic algorithm. The comparison of design variables before and after optimization is shown in

Table 15. Comparison of design variables before and after optimization.

Design Variables	${\mathit{d}}_1$ (mm)	${\mathit{d}}_2$ (mm)	${\mathit{d}}_3$ (mm)	$\mathit{t}$ (mm)	${\mathit{x}}_7$ (mm/r)	${\mathit{x}}_8$
Before optimization	120	133	153	60	0.2	0.6
After optimization	111.98	128.66	156.93	50.043	0.14175	0.69155

and

Table 16. Performance before and after optimization, and comparison between predicted and true values.

	Blank Cost ($)	Stress (MPa)	Deformation (10−3 mm)	Main Cutting Force (N)	Total Cost ($)
Before optimization	590.77	281.540	5.761	539.919	706.392
After sequential optimization	543.75	313.408	6.295	580.458	653.843
After integrated optimization	547.16	306.415	6.29572	469.704	646.65

.

As can be seen from Table 16, compared with the initial value, the total cost after sequential optimization was reduced by 7.439%, from $706.392 to $653.843. Furthermore, the total cost of integrated optimization is $646.65, which is 1.1% lower than that of sequential optimization.

Based on the design variables after optimization, the 3D model and turning model of the axle bridge were rebuilt. After simulation, the stress, deformation, and precision machining cutting force were obtained, as shown in Figure 12 and Figure 13, which are all within the allowable range.

5. Conclusions

In this paper, an integrated optimization approach for both axle bridge structure and turning processes based on RBF was proposed.

(1)

The optimization of the structural parameters of the axle bridge was conducted. A full factorial experiment with 4-factor and 3-level was designed, from which 64 groups of sample points were randomly selected, and the parametric model was used for simulation. Based on the simulation data, the RBF model was built to fit the implicit relationship between the design variables and the performance responses. In addition, a multi-island genetic algorithm was used for the optimization solution. The total cost was reduced by $47.56 after structural optimization, with a reduction ratio of 8.05%.

(2)

Using the optimal structural parameters of the axle bridge, the optimization of the turning parameters of the axle bridge was conducted. An orthogonal experiment with 8-factor and 4-level was designed, whose performance responses were simulated using the 3D turning simulation model built by AdvantEdge software 7.3. Based on simulation data, the RBF model was built to fit the implicit relationship between the turning parameters and performance responses. Finally, a multi-island genetic algorithm was used to solve the optimization model of the process parameter. The total turning cost after optimization was $110.63, which was $4.99 lower than the cost before optimization, with a decrease of 4.316%.

(3)

The integrated optimization of the structural and turning parameters was conducted. By combining orthogonal experiments and multivariate variance analysis, design variables that have great impacts on the performance responses were selected. The optimal Latin hypercube experiment and simulation were then used to select 60 groups of sample points, which were used to build the RBF model to fit the implicit relationship between design variables and performance responses. Finally, a multi-island genetic algorithm was used to solve the integrated optimization model. Through verifying tests, the total cost after integrated optimization was $646.65, which was 8.457% lower than the total cost before optimization, and 1.1% lower than the total cost of sequential optimization, proving the effectiveness of the proposed method.

(4)

Through the integrated optimization, the total cost of the axle bridge structure and turning is reduced. The proposed method can also be used in structural and machining process optimization for other products, which provides theoretical insights for enterprises to reduce manufacturing costs and enhance product competitiveness. However, in practical applications, there may be deviations in manufacturing costs among different companies for the same product (e.g., due to variations in labor hours), and raw material prices may fluctuate over time. Therefore, in future research, consideration will be given to these uncertainties during the optimization process, further enhancing the practical effectiveness of the integrated optimization method.