Mathematical Problems in Engineering
Volume 2018 (2018), Article ID 3159637, 14 pages
https://doi.org/10.1155/2018/3159637

Research Article

A Systematic Optimization Design Method for Complex Mechatronic Products Design and Development

Jie Jiang,¹ Guofu Ding,¹ Jian Zhang,¹ Yisheng Zou,¹ and Shengfeng Qin²

¹School of Mechanical Engineering, Southwest Jiaotong University, Chengdu 610031, China
²Department of Design, Northumbria University, Newcastle upon Tyne NE1 8ST, UK

Correspondence should be addressed to Jian Zhang; jerrysmail@263.net and Shengfeng Qin; sheng-feng.qin@northumbria.ac.uk

Received 11 October 2017; Accepted 24 December 2017; Published 11 February 2018

Academic Editor: Qian Zhang

Copyright © 2018 Jie Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Designing a complex mechatronic product involves multiple design variables, objectives, constraints, and evaluation criteria as well as their nonlinearly coupled relationships. The design space can be very big consisting of many functional design parameters, structural design parameters, and behavioral design (or running performances) parameters. Given a big design space and inexplicit relations among them, how to design a product optimally in an optimization design process is a challenging research problem. In this paper, we propose a systematic optimization design method based on design space reduction and surrogate modelling techniques. This method firstly identifies key design parameters from a very big design space to reduce the design space, secondly uses the identified key design parameters to establish a system surrogate model based on data-driven modelling principles for optimization design, and thirdly utilizes the multiobjective optimization techniques to achieve an optimal design of a product in the reduced design space. This method has been tested with a high-speed train design. With comparison to others, the research results show that this method is practical and useful for optimally designing complex mechatronic products.

1. Introduction

It is very difficult to optimally design a complex mechatronic product for many reasons. First, there are a big number of design parameters, usually greater than 100. Second, there are many key performance indicators as either goals or constraints. Third, these parameters are multiple-disciplines related and their determination needs multidisciplinary collaborative efforts. Furthermore, the coupled relations among these parameters and performance indicators are highly nonlinear and vague. In a word, optimally designing a complex mechatronic product is very challenging in a big design space. Therefore, the optimization efficiency is low and it is difficult to obtain a satisfactory solution. In addition, complex mechatronic products are usually composed of many subsystems having different parameters, and their performances are correlated. It is difficult to have an effective system model to describe the relationships between parameters, subsystem performances, and the whole system performances. Thus, Data-Driven Modelling techniques such as artificial neural networks- (ANNs-) based surrogate modelling provide alternative solutions to this problem.

Designing a complex mechatronic product optimally requires considering numerous design parameters and ways of identifying a set of best design variables and obtaining a best design solution effectively under major constraints such as safety/security and stability. The challenges are threefold: the number of design variables or design space is very big; these design parameters have complex system coupling relationships, and it is difficult to take all design parameters in optimization, needing a design space reduction; and a practical model for predicting the coupled vast system performances involves several subsystem performances. Taking a high-speed train as an example, its dynamics performances are related to high-speed train dynamics, high-speed pantograph-catenary dynamics, high-speed train aerodynamics, and high-speed train-track coupling dynamics. Therefore, even with changes in a small set of design variables in practice, the evaluation of the system performances is not straightforward because of their coupled relationships. If each subsystem is modelled as a component, it can be seen that some variables may affect several subsystem models and others may strongly influence only one subsystem. These design variables are coupled and affect different subsystem models. There is a need to construct an overall design performance evaluation model or a goal function for optimal design because so far there is no one established. The ultimate challenge is to develop a systematic optimization design method to solve the above challenges properly and support optimal design of complex mechatronic products.

This paper presents a systematic optimization design method for designing complex mechatronic products. At the beginning, it uses each subsystem dynamics model to conduct design parameter sensitivity analysis and identify key design parameters for design space reduction. Then, a neural network-based surrogate model is established between the identified key design parameters and key performance indicators to describe the whole system performance. Upon the neural network surrogate model, an optimization design model is developed, and finally, an optimal design computing is realized with an improved optimization algorithm for better quality and efficiency. In the current design practice, designing a typical complex mechatronic product such as a high-speed train is mainly by the trial and error method. It lacks a systematic design optimization method for its design. Therefore, we take the optimal design of a high-speed train as a case study to verify the effectiveness of the proposed method.

The paper is structured as follows. Section 2 reviews related work and Section 3 introduces the proposed systematic optimization design method for developing complex mechatronic products. Section 4 shows the case study results, followed by conclusions in Section 5.

2. Related Work

Multiobjective and multidisciplinary optimization in engineering is closely related to our research problems. Many scholars or engineers have conducted a lot of research on optimization frameworks and algorithms. Fabio et al. [1] proposed the use of design optimization techniques to find the ideal truncated full-scale design considering the dynamic effects. Wei et al. [2] proposed a comprehensive framework including a multiobjective interval optimization model and evidential reasoning approach to solve the unit-sizing problem of small-scale integrated energy systems. Lei et al. [3] built a constrained multiphysics model of a motor wheel for an electric vehicle and then optimized it. Liu et al. [4] established a multihierarchical integrated product design data model supporting the multidisciplinary design optimization (MDO) in the Web environment and a Web services-based framework considering uncertainties was proposed. Zheng and Liao [5] improved particle swarm algorithms, which could be applied to many other parameter identification and optimization problems. Zhang et al. [6] presented a modified multiobjective evolutionary algorithm based on the decomposition approach to solve an optimal power flow problem with multiple and competing objectives. Lee et al. [7] proposed a Web services-based MDO framework that enabled the synthesis of available disciplinary and cross-disciplinary resources for MDO via the Globus Toolkit. Gong et al. [8] demonstrated a design sensitivity analysis (DSA) method. Some scholars also applied multidisciplinary optimization to deal with engineering problems in mechanical systems and multibody systems. Kuzmanovic et al. [9] considered damping optimization in a mechanical system excited by an external force. He and McPhee [10] presented a methodology for the design optimization of multibody systems by using genetic algorithms. Hosder et al. [11] considered surrogate functions as an important tool in multidisciplinary design optimization to deal with noisy functions, high computational cost, and the practical difficulty of integrating legacy disciplinary computer codes.

The above methods usually need to establish a large number of equations and formulas for derivation. Thus, their efficiency is low and the solution is not guaranteed. Some scholars proposed using surrogate models to solve optimization problems. Wang and Shan [12] described the meta-modelling techniques in support of engineering design optimization. Golovidov and Kodiyalam [13] described the ideas and methods of how to use an approximate model to do multidiscipline optimizations. Jiang et al. [14] used a neural network model to realize a multiobjective optimization involving process parameters. Kim et al. [15] combined differential and genetic methods for a suspension system design. Yuan et al. [16] promised a methodology for the optimal design of complex mechatronic systems. James and Azad [17] presented two case studies on the use of agent-based modelling in the design of complex systems. Xu et al. [18] proposed two improved strategies for supporting system design optimization. Adaptive meta-model approaches were also proposed. Yi and Malkawi [19] utilized a neural network model for energy simulation. Cheng and Lee [20] explored an efficient back-propagation neural network-based meta-model for approximating optimal solutions. Cheng and Yang studied the optimal design of suspension system parameters for high-speed trains with Kriging model [21].

However, the optimization design of complex mechatronic products is a systematic problem, which involves parameter identification, design space reduction, and optimization strategies. Some scholars studied some of the related problems. Wang [22] considered optimization strategies including sensitivity analysis, surrogate models, and searing algorithms to enable global engineering optimization. Cai et al. [23] presented a general multiagent control methodology for an energy system optimization in a “plug-and-play” manner. Park et al. [24] described the properties of sensitivity analysis between some of the suspension characteristics of the Korean high-speed train as the design variables and the dynamic performance as the response variables. Li et al. [25] presented a new meta-model-based global optimization method using fuzzy clustering for design space reduction. Forrester and Keane [26] reviewed the work on constructing surrogate models and their use in optimization strategies, while Shyy et al. [27] reviewed the fundamental issues arising in surrogate-based analysis and optimization. Zhang et al. [28] presented a new method to identify the key design variables based on the sensitivity analysis for high-speed train design. Kim et al. [29] used a separate meta-model for each performance indicator, requiring multiple meta-models for a multiple-objective optimization. Ma et al. [30] proposed a global sensitivity analysis method by dividing variables in groups. Queipo et al. [31] applied a robust optimization design method to a real complex nonlinear system design. Coello [32] provided a comprehensive discussion on the fundamental issues arising from the use of surrogate-based analysis and optimization.

In summary, the surrogate model technology is useful for the optimization of complex electromechanical systems, and it has a successful application in the optimization of some products. There are a body of work on surrogate modelling and multiobjective optimization. However, for optimally designing complex mechatronic products, it still lacks a systematic approach to guide and guarantee the optimization design of such complex systems. Therefore, this paper proposes a systematic optimization method based on integral design space reduction and system surrogate modelling techniques for designing complex mechatronic products optimally.

3. Systematic Optimization Design Method

In order to improve the design efficiency and reduce the difficulty of performance evaluation (simulation) calculation, this paper puts forward a new systematic optimization design method based on the surrogate model and intelligent multiobjective optimization techniques for designing complex mechatronic products. It includes five stages. In Stage 1, according to the topology structure, design parameters, and boundary conditions of a complex system, the design parameters of the complex system are extracted. In Stage 2, the design parameter space is reduced by expert knowledge or by the parameter sensitivity analysis with each subsystem evaluation (simulation) model. In Stage 3, the key design parameters affecting the whole system design objectives (or running performances) are obtained, forming a reduced design space. In Stage 4, a surrogate model describing the relationship between the whole system performances and the key design parameters is established, and then a corresponding optimization model is developed based on the surrogate model. In Stage 5, intelligent optimization algorithms are used for the optimization design of a complex mechatronic product.

The system optimization design flow is shown in Figure 1. It includes specifying design parameters and objectives, conducting design space reduction, setting up a system surrogate model, setting up an optimization model, and conducting optimization computing.

Figure 1: Optimization design flow.

3.1. Specifying Design Parameters and Objectives

Figure 2 shows the optimal design problem space of a complex mechatronic product related to product structure, design parameters, and performance indicators. For a complex mechatronic product, there are many design parameters associated with a number of subsystems. The design parameters are divided into structural design parameters and performance design parameters. The number of parameters usually is very big and the data range is very wide. At the same time, the running performances of the complex mechatronic product are synthesized in different aspects, so there are a lot of performance indicators (or design objectives).

Figure 2: Optimal design problem space of a complex mechatronic product.

In order to solve the optimization design problem for a complex mechatronic product, it is necessary to define what the design parameters (variables) are and what the objective indicators are. The objectives of the optimization are defined by considering the comprehensive requirements of the performances. Usually setting up a set of the design objectives is very important, while the design goal (comprehensive objective) function is usually formed by summing up all of the weighted objectives. Weights for each individual objective are determined based on their relative importance and previous design knowledge. So the system optimization is transformed into solving the maximum or minimum value of the comprehensive objective function.

3.2. Design Space Reduction

Within a huge design space, there are many parameters and the coupled relationships between these design parameters and objectives are very complex and usually nonlinear. If is used to represent the number of design parameters and for objectives, the design space is a multidimensional problem. Because the design space is large and high dimensional, it is necessary but difficult to identify the key design variables in the optimal design.

To solve this problem, a method for design space reduction is proposed with two rounds. In round 1, important parameters are chosen by experts with related domain knowledge. In round 2, the important parameters will be used to establish surrogate models against each performance indicator, and then sensitivity analysis is conducted based on the established surrogate models. Finally, according to the sensitivity analysis results, the key design parameters are determined based on a predefined sensitivity threshold. The specific process is shown in Figure 3 and the details can be found in [28].

Figure 3: The flow of design space reduction.

3.3. Setting Up a System Surrogate Model

For a complex mechatronic product, usually, there is no practical model for describing whole system performances, and instead, there are several subsystem performance (simulation) models within multidisciplinary fields. The whole system performances are nonlinearly coupled with the subsystem performances. Thus, the direct use of subsystem performance simulation models in the optimal design process is difficult because it requires a coupled system simulation method with spatiotemporal synchronization process control over subsystem simulation computing [33]. One big problem with this kind of coupled system simulation is being time consuming and requiring huge computing resources. Especially for the whole system optimal design, this requires persistently iterative simulation and optimization. The cost of computing becomes much higher. For this reason, a whole system surrogate model is proposed as a very good alternative model, to reduce the difficulty of optimization and improve the efficiency of optimization.

The surrogate model is a mathematical model for fitting discrete data using an approximation approach, which can determine key parameters and their value ranges with less training samples and advanced trial design methods. [27, 32] Back-propagation neural network (BPN) is one of these effective surrogate models, and its structure is shown in Figure 4.

Figure 4: The framework of the BPN.

The design performances of a complex product are considered at the same time with different performance indexes. When performing an optimal design, it is necessary to meet all these indexes at the same time and thus synthesize these indexes into a comprehensive goal function. Lastly, a multiobjective optimization model can be established based on the surrogate model.

Another aspect is to establish a radial basis function network for the parameters. The establishment of the back-propagation neural network (BPN) parameters includes the accuracy of the model and diffusion factor. The mean square error of the same model can also affect the adjustment of the BPN surrogate model.

For having a surrogate model with high accuracy, RRMSE (Relative Root Mean Square Error) error criterion is used in training. RRMSE of the model is defined as follows:

Among them, is the actual values with response to the test points ; is the actual values, with response to the test points ; is the number of the test points; represents test samples set. refers to the mean value of the actual responses.

3.4. Setting Up the Optimization Model

The optimization model can be described with the surrogate model as follows:

(1)Design variables (key design parameters): (2)Optimization goal function with subperformance functions (indicators):

When constructing the optimization goal function , we need to obtain the best overall performance indicator with a set of design parameters within their constraints. In the optimal design, the target is to obtain the minimum of . The can be represented in (2) with subfunctions. For some subfunctions such as , they are transformed into the minimum value of with their reciprocal substitutions because they are expected to achieve the biggest values in real term

where represent objective 1, objective , and objective , and are the corresponding weight coefficients. The multiobjective optimization is to find the minimum solution. Thus, the goal is to obtain

3.5. Conducting Optimization Computing

The optimization of complex mechatronic products is a very complex problem, involving many parameters, indicators, and boundary conditions. Meanwhile, it also requires considering the efficiency and accuracy of optimization. Intelligent optimization algorithm is a good method for speeding up the optimization. Intelligent optimization algorithms include genetic algorithm, differential evolution algorithm, and particle swarm algorithm. Cai and Aref [34] developed a genetic algorithm- (GA-) based optimization procedure. Zheng and Liao [5] realized parameter identification of nonlinear dynamic systems using an improved particle swarm optimization. Qin et al. [35] utilized the differential evolution algorithm (DE) for global numerical optimization.

Each algorithm has its own characteristics. DE (differential evolution) is a parallel optimization algorithm evolved from GA (genetic algorithm), and it has excellent characteristics for global optimization. We select DE in our application because DE reportedly has a more effective evolutionary strategy [35] to generate new individuals, which makes it an efficient and powerful population–based stochastic search technique for solving optimization problems over continuous space and the implementation of DE is relatively easy.

The novelty of this systematic optimization design method has twofold. Firstly, it can effectively couple design space reduction and the system surrogate modelling techniques into the system optimization modelling. In the existing literatures, these two techniques are discussed separately; thus, when applying the surrogate modelling technique to develop a surrogate model for describing complex system relationships, there is a general difficulty in determining what parameters in both inputs and outputs should be chosen to establish an effective surrogate model for use in the optimization. Secondly, it demonstrates that the identified key variables from the space reduction are well qualified as the input variables for establishing the corresponding system surrogate model and the optimization model, thus, providing a general form of the system optimization modelling and solving techniques for complex mechatronic product optimal design.

4. Case Study

Here, we demonstrate the proposed optimal design method with a high-speed train design. We validate the system optimization from the following three aspects. The first is to approve that the proposed systematic method is necessary and solution-guaranteed by the comparison of a direct surrogate model without design space reduction and the one with this operation. The second is to compare the convergence speed and accuracy of the surrogate model under different design parameters. This indicates that, without a design space reduction operation, the resultant surrogate model will perform quite differently. The last is to prove the effectiveness of the system method in terms of its efficiency and accuracy. We overall demonstrate the usefulness of the method by comparing three different tests.

4.1. Specifying Design Parameters and Objectives of High-Speed Train

High-speed train is a typical complex mechatronic product. As a means of rapid transportation, it has been widely appreciated and vigorously developed. However, design and development of a high-speed train need to evaluate its dynamics behaviors and performances under various running environments to meet its safety and running performance requirements. It has several subsystems (shown in Figure 5), and its total number of design parameters are very big (more than 100), involving parameters related to structure design, mechanical design, bogie design, dynamics performance design, and so on.

Figure 5: High-speed train and its running environment.

Therefore, here, we only take dynamics performance-related design as an example because it is the top level of design for high-speed trains. The running performances of a high-speed train include 7 performance indicators as our design objectives; the number of initial design parameters possibly affecting the performance indicators is more than 100 (shown in Table 1). In addition, some performance indicators might be affected and coupled with other indicators. For instance, its safety indicator is mainly influenced and affected by coupled system dynamics among the train, the track, the catenary, and the airflow. The 7 performance indicators are lateral stability , vertical stability , derailment coefficient , the ratio of wheel load reduction , lateral wheelset force , overturning coefficient , and critical speed .

Table 1: The all design parameters of high-speed train.

Parameter
Name
Unit

Parameter 1
Nominal wheel radius

Parameter 2
Distance between backs of wheel flanges

Parameter 3
Wheelset roll moment of inertia
Kg⋅

Parameter 4
Wheelset yaw moment of inertia
Kg⋅

Parameter 5
Longitudinal stiffness of primary suspension per axle side

Parameter 6
Vertical damping of primary suspension per axle side
KN⋅s/m

Parameter 7
Longitudinal stiffness of axle box tumbler joint per axle side

Parameter 8
Yaw damper lateral span

Parameter 9
Lateral stiffness of secondary suspension per bogie side

Parameter 10
Vertical stiffness of secondary suspension per bogie side

Parameter 11
Secondary vertical damper
KN⋅s/m

Parameter 12
Secondary lateral damper
KN⋅s/m

Parameter 13
Wheelset mass

Parameter 14
Lateral stiffness of axle box tumbler joint per axle side

Parameter 15
Longitudinal stiffness of secondary suspension per bogie side

Parameter 16
Longitudinal stiffness of Yaw damper joint per bogie side

Parameter 17
Carbody roll moment of inertia
Kg⋅

Parameter 18
Lateral distance between the secondary suspension of the two sides of the bogie

Parameter 19
Carbody mass

Parameter 20
Lateral stiffness of primary suspension per axle side

Parameter 21
Longitudinal distance between bogie centers

Parameter 18
Carbody yaw moment of inertia
Kg⋅

Parameter 19
Vertical distance from the rail surface to the center of gravity

Parameter 20
Wheelbase

Parameter 21
Vertical stiffness of primary suspension per axle side

Parameter 22
Carbody pitch moment of inertia
Kg⋅

Parameter 23
Framework mass

Parameter 24
Wheelset pitch moment of inertia
Kg⋅

Parameter 25
Vertical damping joint stiffness per axle side

Parameter 26
Nominal wheel radius

Parameter 27
Distance between backs of wheel flanges

Parameter 28
Wheelset roll moment of inertia
Kg⋅

Parameter 29
Air spring vertical stiffness (per spring)

Parameter 100
Lateral damper joint stiffness of secondary suspension per bogie side

Parameter 101
Lateral stop clearance of secondary suspension

Parameter 102
Vertical damping transverse span of secondary suspension

Parameter 103
Yaw damper lateral span

Parameter 104
Traction drawbar mass

Parameter 105
Swing stiffness of traction joint of secondary suspension

Due to the complexity of mechatronic products, the optimal design of a mechatronic product is a very complex problem. Some papers suggest that the surrogate modelling technique can be applied to this problem [12–15]. When focusing on high-speed train, we choose the typical neural network surrogate model based on design space reduction and intelligent optimization algorithm to optimize its design.

The number of initial design parameters of high-speed train is more than 100. The relationship between design parameters is very complicated and highly nonlinearly coupled in the wheel/rail contact model. The construction of a surrogate model of a complex electromechanical product requires big enough samples to train the model. Therefore, in order to get enough good samples, a Railway System Dynamics simulation Package SIMPACK Rail is utilized to generate a set of corresponding data between a set of inputs of design variables and a set of performance indicators. When we prepare the design variable values for simulation, there is a difficulty in knowing the right value range of each design parameter. Thus, we use experts’ guessed values as our references.

Figure 6 illustrates a high-speed train topological structure and its dynamics model. The software Package SIMPACK Rail is an “add-on” module for use with SIMPACK to simulate rail system dynamics (http://www.simpack.com/uploads/media/datasheet_wheel-rail.pdf). In the SIMPACK software, the design parameters are the inputs to the system and then SIMPACK uses embedded wheel/rail contact model to calculate and output performance indicators such as Ride Comfort indicator and Derail coefficient. It is worth noting that using such system dynamics analysis software is quite time consuming, but it is still doable and cheaper comparing with real tests; thus, we use SIMPACK Rail to generate our sampling data.

Figure 6: Topological structure and dynamics model in SIMPACK software of high-speed train.

Next, we take a unified approach to determine the range of each parameter. Based on the initial range value of the parameter, the upper and lower 10% are used as the initial range value of each parameter. We use this range to carry out the experimental design to get hundreds of test data sets. We use the experimental data as input and use the SIMPACK software to simulate performances of each sample design. Due to the mismatch of parameters, only 58 sets of 100 can produce simulation results. For the remaining 42 sets, we cannot get simulation results; thus, as a result, there are no enough samples to train the surrogate model with only 58% of the calculation results.

Therefore, it is not feasible to use the full parameters in the original design space to establish a surrogate model. In order to solve this problem, we need a system method to have a guaranteed solution. Thus, in this paper we propose a method with design space reduction as a key step.

4.2. Design Space Reduction

The proposed design space reduction method has two rounds. In round 1, important parameters are selected by experts with related domain knowledge. In round 2, the important parameters will be used to establish surrogate models against each performance indicator, and then sensitivity analysis is conducted based on established surrogate models.

The details of the design space reduction technique have been reported in [28] with some domain expert involvement. As a result, in round 1 reduction, the 29 important design parameters (shown in Table 2) are selected by experts from more than 100 parameters and in round 2 reduction, the 16 key design variables are finally identified from the sensitivity analysis based on individual performance indicator models [28]. According to the sensitivity analysis, the parameters are sorted according to their importance (shown in Table 2).

Table 2: The design parameters sorted as the importance.

Parameter
Name
Unit
Section

Nominal wheel radius

395–430

Distance between backs of wheel flanges

1,351–1,355

Wheelset roll moment of inertia
Kg⋅
500–750

Wheelset yaw moment of inertia
Kg⋅
500–800

Longitudinal stiffness of primary suspension per axle side

800–1,150

Vertical damping of primary suspension per axle side
KN⋅s/m
10–30

Longitudinal stiffness of axle box tumbler joint per axle side

5–10

Yaw damper lateral span

2,400–2,800

Lateral stiffness of secondary suspension per bogie side

100–200

Vertical stiffness of secondary suspension per bogie side

120–300

Secondary vertical damper
KN⋅s/m
20–60

Secondary lateral damper
KN⋅s/m
30–50

Wheelset mass

1,800–2,200

Lateral stiffness of axle box tumbler joint per axle side

4–10

Longitudinal stiffness of secondary suspension per bogie side

100–200

Longitudinal stiffness of Yaw damper joint per bogie side

5–13

Carbody roll moment of inertia
Kg⋅
70,000–120,000

Lateral distance between the secondary suspension of the two sides of the bogie

2,400–2,500

Carbody mass

28,000–40,000

Lateral stiffness of primary suspension per axle side

800–1,200

Longitudinal distance between bogie centers

17,000–18,000

Carbody yaw moment of inertia
Kg⋅
1,100,000–1,500,000

Vertical distance from the rail surface to the center of gravity

1,400–1,600

Wheelbase

2,400–2,600

Vertical stiffness of primary suspension per axle side

1,000–1,500

Carbody pitch moment of inertia
Kg⋅
1,200,000–1,700,000

Framework mass

2,100–3,100

Wheelset pitch moment of inertia
Kg⋅
65–100

Vertical damping joint stiffness per axle side

3–6

4.3. Setting Up the System Surrogate Model

Based on the results obtained from the design space reduction, the system surrogate model with the 16 key design parameters and performance indicators was established. In order to prove the system approach and justify the reason for choosing the 16 key design parameters in the system surrogate model (named as BPN2), we construct other 5 surrogate models for comparison. The result comparisons with the 6 surrogate models are detailed in the next section.

We use the same structure (see Figure 7) to construct the 6 surrogate models, namely, BPN1, BPN2, BPN3, BPN4, BPN5, and BPN6, with the numbers of design parameters: 12, 16, 18, 20, 25, and 29, respectively. When a parameter in Table 2 is not chosen as a design variable, its value is fixed to the middle of its range values. In this way, the sampling design of a design available is based on Latin hypercube sample design method and the sample data are produced with SIMPACK software. The details associated with the six surrogate models BPN1 to BPN6 are shown in Table 3.

Table 3: Six surrogate models.

BPN1
BPN2
BPN3
BPN4
BPN5
BPN6

Data set
U
V
W
X
Y
Z

The number of parameters
12
16
18
20
25
29

The number of iterations
68
93
126
195
256
291

Figure 7: The surrogate model structure of high-speed train based on different parameter groups.

Figure 7 shows the structure of the neural network surrogate model. For example, when equals 29, the 29 design variables are in the surrogate model. The 29 variables first generate 100 samples, and then these samples are inputted into SIMPACK simulation software to generate the corresponding performance indicator values. Because these parameters range differently, all parameters in the input layer and the output layer are then normalized for training a model with 95 samples and testing the model with the other 5 samples in its establishment process. In this way, we establish the six surrogate models.

The convergence rates of these surrogate models are shown in Figure 8. The numbers of iterations for each model are shown in Table 3 (last row). It is clear that the number of iterations increases as the sampling parameter increases. The maximum number of times is 5 times more than the minimum number. Therefore, the more the sampling parameters are, the slower the convergence of the model is, and the more resources and time it takes.

(a) BPN1

(b) BPN2

(c) BPN3

(d) BPN4

(e) BPN5

(f) BPN6

Figure 8: The convergence rate of BPN surrogate models.

4.4. Setting Up the Optimization Model

Design variables are corresponding to the 6 BPN models, and they are a subset of for each model (shown in Table 2), where . According to the design requirements of the high-speed train, the main design objectives are the 7 performance indicators: .

According to the design requirements and specifications of high-speed train in China, the range of the 7 indexes can be obtained. In this way, we regard the performance requirements as the boundary conditions of the optimization design. Constraints are

with these boundary conditions for to , the smaller the better regarding the performance of high-speed train. However, to critical speed , the higher the better regarding performance of high-speed train. With the reciprocal of into the optimization function, we get the goal function.

The function is

where is the weight coefficient. For high-speed trains, we think the 7 indicators are of the same importance. In practice, because the different units of the variables need to be normalized into dimensionless variables, the weight coefficient can be set to a specific value:

The optimization function is simplified as

4.5. Conducting Optimization Computing of High-Speed Train

In order to compare the optimization results, we use each of the 6 surrogate models as an optimization analyzer to get the corresponding results between the input variables and 7 key indicator outcomes; in this way, we establish 6 optimization models corresponding to the BPN1, BPN2, BPN3, BPN4, BPN5, and BPN6 surrogate models. We use the same optimization algorithm (differential evolution algorithm) to solve the 6-optimization models. The optimization algorithm is implemented within MATLAB.

The optimization times for each case are shown in Figure 9. For example, it takes 2.2 minutes to obtain the optimization result from BPN1 with 12 variable parameters. Figure 9 shows that the optimization time with BPN6 is 14 times of that with BPN1 and 10 times of that with BPN2. Therefore, the number of design variables involved in optimization affects the optimization time greatly and it is necessary to identify a suitable number of parameters in optimization with acceptable quality of the results.

Figure 9: The time histogram required for obtaining optimal results based on six surrogate models.

The optimization results are shown in Table 4 with varied performance parameters in the goal function. From Table 4, it can be seen that the optimization method is with good accuracy and effectiveness. In comparison with the initial performance parameters, the corresponding optimization results with the performance improvement percentages are shown in Table 5.

Table 4: Comparison table of optimization results.

Lateral stability
Vertical stability
Derailment
The ratio of wheel load
Lateral wheelset force/10 (KN)
Overturning
Critical speed/200 (km/h)

Initial performances
2.3804
2.0245
1.497
1.978
1.2232
1.98
2.98

BPN1
2.2162
2.0235
1.568
2.104
1.12513
2.054
3.03

BPN2
2.1935
2.0021
1.272
1.828
1.02541
1.939
3.43

BPN3
2.1925
2.001
1.253
1.82
1.02465
1.915
3.40

BPN4
2.192
2.0003
1.248
1.806
1.01683
1.867
3.44

BPN5
2.178
1.9986
1.232
1.79
1.01485
1.819
3.35

BPN6
2.1715
1.9927
1.211
1.781
1.01312
1.804
3.39

Table 5: The table of performance improvement percentage (%).

Lateral stability
Vertical stability
Derailment coefficient
The ratio of wheel load reduction
Lateral wheel force
Overturning coefficient
Critical speed

BPN1
6.90%
0.05%
%
%
8.02%
%
1.68%

BPN2
7.85%
1.11%
15.03%
7.58%
16.17%
2.07%
15.22%

BPN3
7.89%
1.11%
16.30%
7.58%
16.23%
3.28%
14.28%

BPN4
7.91%
1.19%
16.63%
8.70%
16.87%
5.70%
15.69%

BPN5
8.50%
1.28%
17.70%
9.50%
16.87%
8.13%
12.62%

BPN6
8.78%
1.36%
19.11%
9.96%
17.17%
8.89%
14.30%

In order to show the optimization results more intuitively, the optimized performance results in Table 4 are shown in Figure 10, while the performance improvement percentages in Table 5 are illustrated in Figure 11.

Figure 10: Comparison chart of optimization results.

Figure 11: The chart of performance improvement percentage.

Figures 9, 10, and 11 together lead to three new understandings of the big system optimization. First, when the number of design variables is inadequate comparing to the number of the key variables, the optimization results are quite poor. Therefore, we need to avoid this kind of case. Second, when the number of the design variables is bigger than that of key variables, the optimization results are better. Third, when the number of the design variables is much bigger than the number of key variables, the optimization result is just slightly better but the optimization times are far worse. For example, the results from BPN5 with 25 variables are not significantly improved when comparing with BPN2, but the computing time and resources are greatly increased.

In summary, using just the key variables can give a system surrogate model with a better balance between the quality of optimization results and computational costs. This is a balanced system solution. In our case study, the surrogate model BPN 2 with 16 key variables is identified as the system surrogate model and supports the system optimization very well with a good balance between the optimization quality and time. It can be concluded that the system optimization method based on the system surrogate model with only key design parameters is more effective in terms of optimization accuracy and computational cost. The importance of the design space reduction and the effectiveness of the system optimization method for complex mechatronic products are proved.

5. Conclusion

Due to the complexity of complicated mechatronic products, the multiplicity of parameters, and the intricate relationship between design parameters and performance indicators, the optimal design of such a product is very sophisticated with hard problems. Therefore, it is almost impossible to ensure that all parameters are optimal. Because of the complex relationship among subsystems, the whole system simulation model is difficult to build, and the coupled simulation time is too long and the cost is huge, it is necessary to provide a data-driven modelling and optimization design method for complex mechatronic product design from a systemic perspective to balance the system executing time and effectiveness.

This paper has proposed a systematic design optimization method for complex mechatronic products from the identification of initial design parameters and objectives, to design space reduction for key variable identification, setting up a system surrogate model with just the key variables, establishing a system optimization model, and optimization computing.

The implementation of this method has been demonstrated through a case study of China high-speed train design with 6 different surrogate models. From the comparison study, it can be seen that the appropriate design space reducing is very important, which leads to not only the identification of key variables but also the establishment of the system surrogate and optimization models. The system optimization method based on the system surrogate model established with just key design parameters is more effective in terms of optimization accuracy and computational cost. The ability to produce a better design solution with the proposed method has been validated and demonstrated.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the National Natural Science Fund of Intelligent Mapping Research for Requirement Driven Fine Design Process of High-Speed Railway Vehicle (no. 51575461), China.

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