Aerodynamic multi-objective optimization on train nose shape using feedforward neural network and sample expansion strategy

Abstract

Feedforward neural network (FNN) models with strong learning ability and prediction accuracy are crucial for optimization. This paper investigates the effects of the number of training samples and the hidden layers on the accuracy of the FNN model. Meanwhile, under the premise of a high space-fillingness degree, a sample expansion strategy based on the max–min distance criterion is proposed, which ensures that the expanded sample set completely contains the pre-expanded. The strategy can eliminate the interference of sample differences. Furthermore, the multi-objective optimization on the train nose shape is accomplished by minimizing the aerodynamic lift force of the tail car (LT), as well as the aerodynamic drag force of the head (DH) and tail car (DT) using the FNN model. The results indicate that the number of training samples has a greater impact on the prediction error of the FNN model than the number of hidden layers does. Prediction errors decrease as the number of training samples increases and then stabilise, the most accurate one is chosen for nose shape optimization. The DH, DT, and LT all have prediction errors of less than 2%. Compared with the original high-speed train, the DH, DT, and LT of the optimal model are reduced by 5.24%, 3.74%, and 2.61%, respectively. Meanwhile, the correlation analysis reveals that the height of the cab window and the horizontal profile have a significant impact on the aerodynamic characteristics of the high-speed train.

KEYWORDS:

• Feedforward neural network
• sample expansion strategy
• multi-objective optimization
• high-speed train
• train nose shape
• aerodynamic characteristic

1. Introduction

The aerodynamic resistance of the high-speed train (HST) plays a dominant role in the running resistance with the increase of operating speed, accounting for more than 85% as the speed reaches 300 ∼ 350 km/h (Guo et al., 2020; Tian, 2019). The increased aerodynamic resistance severely affects the economy and speed improvement. Furthermore, the aerodynamic lift force of most models is upward and increases sharply as the running speed increases. Excessive aerodynamic lift force reduces the wheel-rail contact force, decreases the traction efficiency, and can even cause derailment under certain circumstances (Liang et al., 2022; Tian, 2019; Zhang et al., 2023). Therefore, improving the comprehensive aerodynamic performance of the HSTs has emerged as an important research focus (Dai et al., 2023; Krajnović, 2009; Liu et al., 2023; Zhang et al., 2021; Wang et al., 2022). Scholars have turned their attention to nose shape as one of the important factors influencing the aerodynamic characteristics of HST (Muñoz-Paniagua & García, 2019; Xiang et al., 2019; Yao et al., 2015).

The traditional optimization of the HST nose shape is an experience-based approach. Many models are established for comparative analysis using numerical simulations or wind tunnel tests, and then the shape with the best aerodynamic characteristic is chosen. This method is expensive, the design cycle is lengthy, and the indicators will conflict during the multi-objective optimization design (Li et al., 2016). As computer technology advances, multi-objective aerodynamic optimizations based on surrogate models have become increasingly popular in the HST nose shape design. Lee and Kim (2007) optimized the amplitude of the micro-pressure wave generated by trains travelling into a tunnel using the Kriging meta-model, and the working method and efficiency of the model are analysed in detail. Li et al. (2016) adopted the free-form deformation method for parametric modelling, performing mesh deformation without re-meshing. Then, the multi-objective optimization was conducted using the Kriging model. Moreover, aiming at reducing the resistance of the HST, Sun et al. (2010) optimized the shape of the China Railways High-speed 3 (CRH3) and found that the original nose shape of CRH3 owns an excellent aerodynamic performance. Yao et al. (2013; 2015) conducted three-dimensional parametric modelling of China Railways High-speed 380A (CRH380A) and established the Kriging surrogate model. The multi-objective optimizations were carried out through the genetic algorithm and ant colony algorithm, respectively, to reduce the aerodynamic resistance of the total train and the aerodynamic lift force of the tail car. The optimization of the HST aerodynamic performance using the Kriging surrogate model has achieved a lot. Furthermore, numerous academics have also carried out HST aerodynamic multi-objective optimization using the neural network surrogate model. Krajnović (2009) took the operational safety and aerodynamic resistance of the train in a crosswind environment as the optimization objectives and studied the influence of the polynomial model, radial basis functions neural network model. The results showed that the optimization result of the neural network model is slightly better. Yao et al. (2012) established a neural network model to carry out aerodynamic multi-objective optimization of HSTs, achieving a drag reduction of 8.7%. Muñoz-Paniagua et al. (2014) optimized the pressure wave and the aerodynamic resistance when the train travelled through the tunnel. 15 samples were drawn to build a neural network model, and the prediction error of the surrogate model was less than 4%. The genetic algorithm was employed to perform multi-objective optimization, and an HST model with better all-around aerodynamic performance was obtained. The same method was also adopted to optimize running stability in the crosswind and pressure pulse from trains passing by each other (Muñoz-Paniagua & García, 2019). The neural networks are composed of more complex and deep-level structures, which can learn more detailed characteristics and have stronger inductive learning abilities (Krajnović, 2009; Muñoz-Paniagua & García, 2019; Yao et al., 2012). Therefore, the feedforward neural network (FNN) surrogate model is used to perform multi-objective optimization of HST aerodynamic characteristics with the goals of reducing the aerodynamic drag force of the head and tail car and the aerodynamic lift force of the tail car.

There are two primary ways to create surrogate models in general for the HSTs aerodynamic optimization. On the one hand, sampling directly selects a sufficient quantity of samples. The majority of the samples are chosen to construct models, and a small portion of the samples are used as the validation set for the model accuracy test (Li et al., 2016; Sun et al., 2010). The surrogate model, on the other hand, is built using the initial samples obtained through a sampling technique. Other samples are then added until the convergence requirements are satisfied by combining the infill criterion (Yao et al., 2015). The precision of the surrogate model is closely correlated with the training sample size. A lack of samples is a common source of low precision, and providing more samples does not always improve the situation. The surrogate model established by too many samples may have an overfitting phenomenon, poor generalisation ability, and even lead to low accuracy (Li et al., 2021a). There has been little study into the relationship between sample size and model accuracy. This study proposes a sample expansion strategy based on the max–min distance method (Johnson et al., 1990) to guarantee that the expanded sample set has a higher space-fillingness degree. Meanwhile, the expanded sample set obtained using the sample expansion strategy completely contains the pre-expansion, eliminating the differences caused by different samples. It’s different from the sampling method to extract the different numbers of samples directly. Therefore, the impact of the sample size on the accuracy of the FNN surrogate model will be more intuitively reflected. Furthermore, the sample expansion strategy proposed in this paper is different from the traditional sample infill criteria. This strategy can select the most suitable sample in case the sample database exists, while the samples obtained using the sample infill criteria do not necessarily exist in the database. Therefore, this strategy can make full use of the samples in the existing database.

This paper establishes sample sets using the sample expansion strategy to study the relationship between the accuracy of the FNN surrogate model and the number of samples. In the meantime, the impact of the number of hidden layers on the accuracy of the FNN surrogate model is being investigated. Following that, the multi-objective aerodynamic optimization on the HST nose shape is performed using the FNN surrogate model with the highest accuracy. Finally, a train nose shape with better overall performance is obtained, providing guidance and suggestions for the HST and other aspects of performance optimization.

2. Computational models

2.1. Mathematic model

The FNN model contains input, hidden, and output layers. Meanwhile, each layer is composed of multiple neurons. The sample is defined as (x, y), and the input vector is x = (x₁, x₂,  … , x_m)^T, where m is the dimension of the input layer. The output vector is f = (f₁, f₂,  … , f_c)^T, where c is the dimension of the output layer. Assuming that the k^th (k = 1, 2,  … , L) hidden layer contains n_k neurons, the corresponding hidden layer vector is h = (h₁, h₂, … , h_k)^T. ${\mathit{W}}^k=(w_{ij}^k{)}_{n_k\times n_{k-1}}$ and ${\mathit{W}}^{L+1}=(w_{ij}^{L+1}{)}_{c\times n_L}$ are weight matrixes between the (k-1)^th hidden and k^th hidden layer, the L^th hidden and output layer, respectively. After then, the output of each layer of the FNN is. (1) $\begin{array}{r}\{\begin{array}{l}{\mathit{h}}_1={\sigma }_{{\mathit{h}}_1}({\mathit{W}}^1\mathit{x}+{\mathit{a}}^1)\\ {\mathit{h}}_k={\sigma }_{{\mathit{h}}_k}({\mathit{W}}^k{\mathit{h}}_{k-1}+{\mathit{a}}^k)\\ \mathit{f}={\sigma }_c({\mathit{W}}^{L+1}{\mathit{h}}_R+{\mathit{a}}^{L+1})\end{array}\end{array}$(1) where a is the offsets, $\sigma$ is the activation functions (Rumelhart et al., 1986).

The mean square error $E=\frac{1}{2}\sum {||f-y||}^2$ is adopted as the cost function in this study. The weight W and the offset a are continuously updated until the E no longer changes (Rumelhart et al., 1986). (2) $\begin{array}{rl}{\mathit{W}}^k& ={\mathit{W}}^k-\eta \frac{\mathrm{\partial }E}{\mathrm{\partial }{\mathit{W}}^k}\end{array}$(2) (3) $\begin{array}{rl}{\mathit{a}}^k& ={\mathit{a}}^k-\eta \frac{\mathrm{\partial }E}{\mathrm{\partial }{\mathit{a}}^k}\end{array}$(3) When the velocity is much less than the local speed of sound, further to that, when the running speed is less than 360 km/h, air can be treated as incompressible steady flow in HST aerodynamics (Li et al., 2021b). The governing equation is. (4) $\begin{array}{r}\frac{\mathrm{\partial }\rho \mathit{\Phi }}{\mathrm{\partial }t}+\text{div(}\rho \mathit{u}\mathit{\Phi })=\text{div(}\mathbf{\Gamma }\text{grad}\mathit{\Phi }\text{)}+\mathit{S}\end{array}$(4) where u represents the HST’s running speed, $\rho$ represents the air density. $\mathit{\Phi }$, $\mathbf{\Gamma }$ and S represents the flow field flux, diffusion coefficient, and source term.

Moreover, Shear Stress Transfer (SST) k-ω turbulence model and the incompressible Reynolds Average Navier-Stokes (RANS) are adopted (Li et al., 2023; Yao et al., 2020). The Semi-Implicit Method for Pressure-Linked Equations Consistent (SIMPLEC) is chosen to couple the velocity and pressure (Fluent Inc, 2021), and the pressure-based solver is chosen in the numerical method.

The aerodynamic force F is defined as (5) $\begin{array}{r}F=\sum _{i=1}^n\left(\sum _{j=1}^m\left(p_{i,j}\cdot \left({\mathit{d}}_{i,j}\cdot \mathit{e}\right)\cdot a_{i,j}\right)\right)\end{array}$(5) where the numbers of computational faces in the circumferential and longitudinal directions, respectively, are m and n. p_{i, j} and a_{i, j} are the mean pressure and area of the face. d_{i, j} is the unit normal vector, and e is the unit vector (Fluent Inc, 2021). When e is directed along x, F is the aerodynamic drag force, and when directed along z, F is the aerodynamic lift force.

2.2. Geometry model

A three-car model that includes the head, middle and tail car is established before the numerical simulation, bogies and simplified windshields are reserved. Moreover, rail and ballast are installed to simulate ground configurations and the distance between the ballast and the roof of the train is taken as the characteristic height (H). The characteristic height is 4.2 m, and the head and middle car lengths are 6.2 and 6.0 H, respectively. Figure 1 depicts further information on the HST and ground configurations.

Figure 1. High-speed train model and ground configurations.

3. Numerical information and optimization process

3.1. Numerical method and verification

The distance between the boundary and the nose tip of the head car is 12 H, and the nose tip of the tail car is 24 H away from the boundary. The height of the computational domain is 10 H, and the width is 24 H, as shown in Figure 2. Moreover, the inlet boundary is specified as the velocity-inlet condition with an incoming flow of 83.33 m/s. A pressure outlet boundary is applied for the outlet boundary of the computational domain, and the outlet pressure is 0. Meanwhile, the symmetry boundary condition is adopted on the top and sides of the computational domain.

Figure 2. Computational domain.

The HST that has carried out wind tunnel tests (Han & Yao, 2017) is numerically simulated using the above numerical method to verify its accuracy. The test model is 1:8 scaled, as shown in Figure 3(a) (Sun et al., 2020), and Figure 3(b) shows the numerical simulation model. Three meshes named Coarse, Medium, and Fine are generated, with 23.5, 28.21, and 35.34 million cells, respectively. Meshes are mainly composed of hexahedral elements, together with a small number of tetrahedral elements. The three meshes have the same boundary layer parameters. The first layer is 0.01 mm thick, the growth ratio is 1.2, and the total number of layers is 18, ensuring that the y+ is close to 1.

Figure 3. Models of high-speed train. (a) Wind tunnel test model (b) Numerical simulation model.

The aerodynamic drag coefficient is. (6) $\begin{array}{r}{C}_{\text{d}}=\frac{F}{0.5\rho u\text{A}}\end{array}$(6) where F represents the aerodynamic forces. A means the windward area of the train, which is 10.8 m².

Figure 4(a) shows the aerodynamic drag coefficients of each carriage and the whole vehicle obtained by wind tunnel tests and numerical simulations. Exp represents the data obtained from the wind tunnel test (Han & Yao, 2017). The coefficients of the three meshes are extremely close, with a relative error of less than 1% in the aerodynamic resistance coefficients of the head car. The maximum error in the wind tunnel test results, as well as the tail car, is about 2.2%. The value of the aerodynamic drag force coefficient of the middle car is small, and the difference in the result is only 0.001. The pressure distribution of a certain line behind the tail car on the longitudinal centre section is depicted in Figure 4(b), and the solid black line in the figure shows its position in the wake. According to the data, there is essentially no change at any of the other sites, with just a small error of about 3% in the region of maximum positive pressure for three meshes. When the number of cells reaches 23.5 million, the aerodynamic force and the pressure distribution reveal that the mesh has little effect on the numerical simulation. Meanwhile, the agreement between numerical simulation and wind tunnel test results indicates the accuracy of the numerical simulation method used in this study. In theory, there is a positive association between calculation accuracy and cell count. The numerical simulation should choose a mesh with as many cells as possible. However, calculation speed and efficiency are critical considerations. As a result, the Medium mesh is chosen for the following research to consider calculation accuracy and efficiency. Figure 5 shows the cells of the boundary layer, the refinement regions, and the cells surrounding the train body.

Figure 4. Results obtained from the numerical simulation and wind tunnel test. (a) Drag force coefficient (b) Pressure distribution behind the tail car.

Figure 5. Computational grids.

3.2. Sample expansion strategy

The max–min distance criterion proposed by Johnson et al. (1990) is one of the most widely used methods to evaluate the uniformity (space-fillingness) of a sampling plan. This paper proposes a sample expansion strategy based on the idea. The crowding degree between samples is measured using Euclidean Distance, and the distance between samples is defined as. (7) $\begin{array}{r}d({\mathit{x}}^{i_1},{\mathit{x}}^{i_2})=\sqrt{\sum _{j=1}^m{({\mathit{x}}_j^{i_1}-{\mathit{x}}_j^{i_2})}^2}\end{array}$(7) Where ${\mathit{x}}^{i_1}$ and ${\mathit{x}}^{i_2}$ are any two samples in the sample set.

The max–min distance is defined as. (8) $\begin{array}{r}l=max(min(d({\mathit{x}}^{new},\mathit{X}(i\left)\right)\left)\right),i=1,2,3\dots n.\end{array}$(8) Where x^new represents the newly added samples. X is the original sample set, and n is the number of samples in the original sample set.

With the max–min distance as the fitness function, the particle swarm optimization algorithm is used to find x^new to realise the expansion of the sample set.

3.3. Optimization process

The multi-objective optimization process is shown in Figure 6, and the detailed procedure is as follows:

1. Step1. Parameterise the HST nose, select design variables (also known as independent variables), and determine the change interval of each variable.

2. Step2. The Optimal Latin Hypercube Design (OLHD) method is used to generate the initial training and validation sets of independent variables (Park, 1994).

3. Step3. The sample expansion strategy is adopted to create the training set with varying numbers of samples. There are no identical samples in the training and validation sets.

4. Step4. Establish geometric models of the HST based on the design variables, and generate meshes.

5. Step5. The corresponding dependent variables are obtained using ANSYS.FLUENT. The optimization goal is to minimize the corresponding dependent variables, which are the aerodynamic drag force of the head car (DH), the aerodynamic drag force of the tail car (DT), and the aerodynamic lift force of the tail car (LT).

6. Step6. The FNN models are built using a variety of training sets and hidden layers, then optimization is applied with the highest accuracy FNN model. Finally, the Pareto front solution set and the correlation between the objectives and design variables are obtained.

Figure 6. optimization process.

3.4. Design of experiment

v₁, v₂, v₃, v_4, and v₅ five design variables, corresponding to 5 control lines of the train nose, are defined. It should be noted that this train is not the same model as the wind tunnel test. The height control line C₁, width control line C₃, cab window height control line C₂, first auxiliary control line C₄, and second auxiliary control line C₅ make up the five control lines. v₁, v_2, and v₃ stand for the distance of curve deformation. v₄ and v₅ represent the curve deformation ratio. Further to that, the point on the curve C₄ moves to (1 + v₄)P_y along the y direction, where P_y is the original y coordinate of the point. The definitions for v₄ and v₅ are the same. The values of the five design variables of the original model are all 0. Figure 7 depicts the design variables and their ranges.

Figure 7. Variables and their ranges.

The OLHD method, which ensures that the extracted subset has a high degree of space-fillingness in the design space, is adopted to create a sample set with 75 samples. The starting point of the validation set is then chosen at random from the sample set, and 55 samples are selected from the 75 samples using the sample expansion strategy to establish the validation set. The remaining 20 samples as the initial training set. Since 75 samples are gathered using the OLHD method, and the verification set is selected based on the max–min distance, both the validation set and the initial training set exhibit a high space-fillingness degree. Figure 8(a) shows the spatial distribution of the validation set and its projection in three directions using v₁, v₂, and v₃ as examples.

Figure 8. Spatial distribution of samples. (a) Validation set (b) Training sets.

The sample expansion strategy is used to generate the training sets with different numbers of samples based on the initial training set. There are 12 training sets in total, each with 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, and 240 samples. The sample size is made larger in order to investigate the effect of the number of samples on the accuracy of the FNN model. The training set with 40 samples is known as TS-40, and other training sets are the same. The validation and training sets contain no identical samples. The projections of points (only variables v₁, v₂, and v₃) of the initial training set, TS-40 and TS-60 in three directions are shown in Figure 8(b). The distribution of samples in each sample set is proved to be quite uniform. The 240 samples in the training set are numbered 1–240, and the 55 samples in the validation set are numbered 241–295, for a total of 296 samples including the original HST model. Parameterised modelling is performed according to the variables to generate 296 models. Subsequently, numerical simulations are carried out to determine the aerodynamic forces of 296 HST models using the commercial software ANSYS FLUENT, as shown in Table 1. Sample No. 0 in Table 1 is the original model.

Table 1. Samples of training and validation sets.

Model	v1/mm	v2/mm	v3/mm	v4	v5	DH/N	DT/N	LT/N
0	0	0	0	0	0	6473.2	5298.7	4834.8
1	28.0	82.0	−5.7	−0.085	0.072	6184.5	5154.4	4598.4
2	165.0	372.0	25.5	0.015	0.068	6267.5	5296.6	4756.9
3	185.0	−182.0	2.5	0.028	0.175	6236.7	5166.7	4694.3
…	…	…	…	…	…	…	…	…
293	17.0	−186.0	35.3	0.071	0.088	6144.5	5133.8	4842.8
294	159.0	−108.0	−22.3	0.022	0.297	6160.5	5149.7	4857.3
295	199.0	−18.0	11.3	0.100	0.224	6286.3	5140.4	4716.7

4. Multi-objective optimizations

4.1. The factors affecting the prediction accuracy

The number of training samples and hidden layers are important parameters that affect the prediction accuracy of the FNN surrogate model. As a result, FNN models with two, three, and four hidden layers are constructed, containing 20, 30, and 20 neurons, respectively, as illustrated in Figure 9. The first hidden layer consists of 5 neurons. It should be emphasised that the 2-hidden-layer FNN model has the same number of neurons as the first two layers of the 3-hidden-layer FNN model, and this is also true for the 4-hidden-layer FNN model. Moreover, the activation functions of the hidden and output layers are all tanh. The loss function is MSE, the optimizer is ADAM, and the learning rate is 0.01. Beta1 and beta2 are 0.9 and 0.999, respectively. The number of iterations is 5000 steps. Figure 10(a) illustrates the link between the mean error of optimization objectives and the number of hidden layers and training samples.

Figure 9. Structure of the FNN model.

Figure 10. Relationship between prediction errors and the hidden layers and training samples. (a) Mean prediction error of objectives (b) Prediction errors of 3-hidden-layers FNN model.

The prediction errors of FNN models with three different hidden layers are all large when the training samples are fewer and gradually decrease as the training samples increase, as shown in Figure 10(a). When the number of training samples reaches 140, the prediction errors remain stable. The mean error is denoted as e_mean,140 when the number of the training sample is greater than 140, and the minimum error is e_min,140. The e_mean,140 of the 2-, 3-, and 4-hidden-layer FNN models are 2.0%, 1.7%, and 2.2%, respectively, and e_min,140 is 1.9%, 1.5%, and 2.1%. The number of training samples is 180, 180, and 160, correspondingly. It can be drawn that the FNN model prediction error is less affected by the number of hidden layers, while it is relatively more affected by the number of training samples. The FNN model with three hidden layers is chosen for further study since it has the lower e_mean,140 and e_min,140. The prediction errors of the DH, DT, and LT of the 3-hidden-layer FNN model are shown in Figure 10(b). The convergence law of prediction error is consistent with Figure 10(a). The e_mean,140 of DH, DT, and LT are 1.4%, 1.6%, and 2.2%, respectively. The e_min,140 is 1.2%, 1.5%, and 1.9%, respectively, corresponding to 180 training samples. Meanwhile, the prediction error is basically stable after the training samples reach 140, while too many samples will reduce the construction efficiency of the surrogate model. Therefore, considering the factors of prediction accuracy and efficiency, the FNN model that is ultimately chosen has three hidden layers and is trained using 180 samples. Due to the different sample spatial distributions from DH, DT, and LT, the prediction error of LT is slightly larger than that of DH and DT. The standard deviation of LT in TS-240 is 180.3, while the standard deviation of DH and DT is 118.8 and 126.6. Different prediction errors result from varying data dispersion. However, the DH, DT, and LT prediction errors are all less than 2%, indicating that the FNN model can accurately anticipate the aerodynamic forces acting on the HST.

4.2. Multi-objective optimization of HST nose shape

The Non-dominated Sorting Genetic Algorithm III (NSGA III) (Deb & Jain, 2013) is employed to optimize the aerodynamic characteristic of the HST based on the FNN surrogate model. The optimization goal is to minimize the DH, DT, and LT, which is a three-objective optimization. The population size is 300, the crossover probability is 0.9, and the evolutionary algebra is 300. A 16-core parallel Xeon(R) Gold 5218 CPU @2.3 GHz is used for surrogate model training and optimization calculations. The training process takes 106.2 s, the prediction of each sample takes 20 ms, and the optimization calculation takes 217.8 s. Figure 11 depicts the Pareto front solution set obtained through multi-objective optimization. The data from the original model is represented by the black cube, and the majority of the points in the Pareto front solution set outperform the original model.

Figure 11. Pareto solution set of three-objective optimization.

According to previous findings, an FNN model with three hidden layers is obtained by training with 180 samples, and two-objective optimization is carried out. The goal of optimization is to reduce the aerodynamic resistance of the HST (DTotal) and the aerodynamic lift force of the tail car (LT). Same validation set consisting of 55 samples is used to verify the accuracy of the FNN model. The average prediction error of the DTotal is 1.1%, and the prediction error of the LT is 2.0%, which is consistent with the prediction accuracy of the three-objective FNN model. The Pareto front solution set is obtained by performing optimization computations using the NSGA III and the two-objective FNN surrogate model.

Three-objective optimization can only obtain the DH and DT, while the aerodynamic drag force of the middle car (DM) is absent. It cannot be compared with the DTotal of the two-objective optimization. The data analysis, however, discovered that the difference between the maximum and minimum DM in the training set was only 156.6 N. The shape of the middle car has not changed between samples, making a smaller difference in the DM. Therefore, the average value (DM_mean) of the DM in the training set is taken, and the DTotal of three-objective optimization is obtained by adding DM_mean to DH and DT. The value of the DM_mean is 4082.2N. The Pareto front comparison between the three-objective optimization and the two-objective one is shown in Figure 12. As can be observed, there is barely any change in the shape of the Pareto front derived by the two- and three-objective optimization, with only a small difference in value. The DTotal calculated by the two-objective optimization is about 10 N larger than the three-objective one, and the LT is about 20 N smaller. The difference in LT is approximately 0.4% of its value, while the difference in DTotal is only about 0.06%. It demonstrates that the optimization of the DH and DT is almost equivalent to the optimization of the DTotal, owing to the small change in the DM. Since the three-objective optimization can more intuitively illustrate the change law of DH and DT, the following analysis will primarily focus on the three-objective optimization.

Figure 12. Pareto front comparison between the two- and three-objective optimization.

To confirm the optimization results and the precision of the FNN surrogate model, three validation points based on the best performance of DH, DT, and LT are chosen, respectively, as shown in Figure 11. Table 2 shows the results of the three samples obtained from the FNN surrogate model and numerical simulations. The prediction errors of all optimization objectives are less than 2%, which is consistent with the conclusions during the construction of the FNN surrogate model. The findings also demonstrate the high prediction accuracy of the FNN surrogate model. Point 1 is chosen as the optimal solution of multi-objective optimization with a preference for smaller DH and DT. The DH of the optimal solution is reduced by 5.24% compared with the original model, and the DT and LT are reduced by 3.74% and 2.61%, respectively.

Table 2. Prediction results and numerical simulation results of the validation points.

Validation points	optimization objectives	FNN model (N)	Numerical simulation (N)	Error
1	DH	6088.41	6150.66	1.01%
	DT	5047.81	5107.48	1.17%
	LT	4606.81	4711.67	1.93%
2	DH	6091.77	6040.62	0.85%
	DT	5039.10	5090.56	1.01%
	LT	4584.35	4660.23	1.63%
3	DH	6121.42	6051.02	1.16%
	DT	5051.74	5084.54	0.65%
	LT	4459.61	4543.89	1.85%

The five design variables of the original model are all 0, and the five design variables of the optimal solution model are 58.67, −194.43, 0.18, −0.090, and −0.144 in sequence. Figure 13 depicts a comparison between the original model and the optimal solution. The variables v₂ and v₅ of the optimal solution vary greatly and both are smaller than the original model, which gives the nose a concave shape. v₄ also has a larger amount of change that concaves the first auxiliary control line, which results in a smoother transition from the nose tip to the second auxiliary control line. Meanwhile, the optimal solution has a lower cab window height than the original model. In conclusion, the shape of the optimal solution model is sharper than the original model from the comparison of the design variables and the shape.

Figure 13. Comparison of the original model and optimal solution.

The surface pressure distribution of the head and tail cars is shown in Figure 14. The positive pressure at the nose tip has no obvious difference between the two models, but there is a difference in the driver’s cab. The positive pressure area of the optimal solution model is significantly smaller than that of the original model. The pressure distribution at the y = 0 cut plane of the two models is extracted to further quantify the pressure difference, as shown in Figure 15. Pressure coefficient (9) $\begin{array}{r}{C}_{\text{p}}=\frac{P}{0.5\rho v^2}\end{array}$(9) where P is the pressure.

Figure 14. Surface pressure distribution of the original and the optimal cars. (a) Original head car (b) Original tail car (c) Optimal head car (d) Optimal tail car.

Figure 15. Pressure coefficient distribution at the y = 0 cut plane.

The stagnation point pressure coefficients of the noses of the original and optimal head cars are basically around 1, indicating that the numerical simulation results are accurate. The pressure distribution under the cab window of the head car is different, and the pressure amplitude of the optimal solution model is much smaller than that of the original model, which is an essential reason for the minor resistance of the optimal solution. Furthermore, the optimal tail car has a larger area of positive pressure in the streamlined zone than the original model. Meanwhile, the pressure amplitude in the window of the original model is smaller than that of the optimal solution, as shown in the enlarged figure of Figure 14. Therefore, the aerodynamic drag and lift forces of the tail car of the optimal solution are smaller.

The main effect is the changing law of the optimization objective in the design space of each variable, and the slope of the curve reflects the influence of the independent variable on the optimization objective. The main effects of the design variables on the DH, DT, and LT are shown in Figure 16(a ∼ c). The DH increases as the independent variables v₁, v₂, v₃, and v₅ increase, and decreases as v₄ increases. Meanwhile, data analysis is performed to obtain the correlation coefficients between the optimization objectives and independent variables, which quantifies the influence of the independent variables, as shown in Figure 16(d). The design variable v₂ has the greatest impact on the DH, while v₁ has the least. Besides, the DT is most affected by the independent variables v₂, v₃, and v₅ and increases with their rise, supporting the conclusion in Figure 13. The LT increases with v₃ and v₅, with v₃ having the largest effect, followed by v₅ and v₁. In conclusion, the independent variables v₂, v_3, and v₅ have a greater effect on the aerodynamic characteristic of the HST, while the influence of v₁ and v₄ is minor.

Figure 16. Main effect and correlations between the design variables and objectives., (a) DH (b) D (c) LT, (d) Correlation coefficients between the design variables and objectives.

5. Conclusions

The effects of the number of training samples and the number of hidden layers on FNN model accuracy are investigated, and an FNN surrogate model with relatively high accuracy is generated. The HST nose shape is then multi-objectively optimized with the goal of minimizing the aerodynamic lift force of the tail car (LT), the aerodynamic drag force of the head (DH) and the tail car (DT). The main conclusions can be summarised as follows.

1. The prediction error of optimization objectives using FNN is affected more by the number of training samples, whereas the number of hidden layers has a relatively smaller impact. As the number of training samples rises, prediction errors fall until they stabilize at 140 training samples. The final FNN model chosen has three hidden layers, is trained with 180 samples, and the prediction error of the DH, DT, and LT are all less than 2%.

2. There is barely any change in the shape of the Pareto front derived by the two- and three-objective optimization, with only a small difference in value. The difference in LT is approximately 0.4% of its value, while the difference in DTotal is only about 0.06%. Owing to the small change in the DM, the optimization of the DH and DT is almost equivalent to the optimization of the DTotal.

3. A Pareto solution front set is obtained via multi-objective optimization, from which three validation samples with different preferences are chosen. The prediction errors between the numerical simulation and the FNN model are less than 2%. Point 1 is then selected as the optimal solution of multi-objective optimization with a preference for smaller DH and DT. Compared with the original model, the DH, DT, and LT of the optimal solution are reduced by 5.24%, 3.74%, and 2.61%, respectively.

4. The independent variables v₂, v₃, and v₅ have a significant impact on the DH and DT, which increase with the value rises of the variables. The LT is highly influenced by the independent variables v₃ and v₅, and LT increases as v₃ and v₅ increase. In general, v₂, v_3, and v₅ have a greater impact on the aerodynamic characteristics of the HST, while the effect of v₁ and v₄ is relatively small.

Acknowledgement

This project was supported by the Sichuan Science and Technology Program (2023JDRC0062), National Natural Science Foundation of China (12172308), Project of State Key Laboratory of Rail Transit Vehicle System (2023TPL-T05) and New Interdisciplinary Cultivation Fund Program of Southwest Jiaotong University (YG2022006).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project was supported by the Sichuan Science and Technology Program (2023JDRC0062), National Natural Science Foundation of China (12172308), Project of State Key Laboratory of Rail Transit Vehicle System (2023TPL-T05) and New Interdisciplinary Cultivation Fund Program of Southwest Jiaotong University (YG2022006).