1. Introduction
As an important part of the space camera’s optical machine structure, the optical machine structure greatly affects the imaging performance of the space camera’s mirror. To ensure the healthy working condition of the space camera ground assembly, ground testing, launch, and in-orbit operating conditions, multiple design criteria need to be considered to optimize and design complex mechanical structures such as optical machine supporting structure to meet the system performance indexes [
1,
2]. The system performance indicators mainly include light weight, stiffness, and strong resistance to external environmental interference [
3,
4]. Additionally, there may be interactions between different design criteria. During the launch process, the space camera will be greatly disturbed by random external vibrations. To avoid the space camera’s resonance during the launch process, the space camera’s optical machine system’s stiffness must be increased as much as possible [
5]. Simultaneously, to reduce launch costs, the space camera’s mass needs to be reduced as much as possible while maintaining the imaging performance and lifetime of the camera [
6].
With the continuous development of computer technology, finite element analysis (FEA) technology has been rapidly developed and widely used in many fields, including mechanical optimization and design [
7,
8,
9,
10,
11]. In the optimized design of complex mechanical structures for finite element analysis, it is necessary to discretize the mechanical continuous structure into many units and a large number of calculations. Each finite element analysis will take a long time.
The Monte Carlo method (MCM), a numerical analysis of mathematical models [
12], is a traditional method for optimizing mechanical structures by iterative finite element calculations with random sampling to continuously modify the model parameters [
13,
14,
15,
16]. The whole process requires a large number of samples and calculations, with high time and cost, and sometimes the optimization results have difficulty meeting the design accuracy requirements [
17]. Therefore, it is necessary to introduce new optimization design methods for complex mechanical structures to improve optimization design efficiency and design accuracy. Since the function relationship between input random variables and output response in the optical machine structure optimization design has strong nonlinearity, it is difficult to solve the implied function relationship, so it is necessary to establish a numerical surrogate model to solve the above problems. In the numerical surrogate model, the functional relationship between the input random variables and the output random variables is represented by fitting a small batch of data samples [
18].
Neural networks have been used to fit the nonlinear relationship between input variables and output responses in recent years due to their strong nonlinear fitting capabilities [
19,
20,
21,
22,
23,
24,
25]. Neural networks have been applied in many fields, such as space-based large mirror structure, turbine disks, automotive bushings, etc. As a machine learning method, the backpropagation neural network (BPNN) has great advantages in fitting nonlinear functions [
26,
27]. BPNNs have a strong function approximation ability, which can be approximated to continuous functions with arbitrary accuracy. However, in training samples, BPNNs suffer from over-fitting problems, local optimum issues, slow late convergence, and poor generalization ability. These problems will affect their training efficiency and prediction accuracy [
28].
The particle swarm optimization (PSO) algorithm proposed by Kennedy and Everhart is inspired by the social behavior of animals such as school of fish, insect swarms, and bird populations, ensuring the PSO algorithm has the abilities for global optimization [
29,
30,
31]. The Bayesian regularization (BR) algorithm enables BPNN to avoid overfitting and improves the generalization of BPNN [
32].
The purpose of this paper is to propose an efficient alternative model method named the backpropagation neural network method based on particle swarm optimization and Bayesian regularization algorithms (BMPB) to improve the computational efficiency and accuracy of fitting the implicit functions of nonlinear systems. In this method, the initial weight and initial threshold of ANNs are searched by the improved PSO algorithm; then, to obtain the trained neural network model, the BP algorithm training sample data determines the optimal weight and threshold. Finally, BMPB takes the optical machine system’s supporting structure as an example of the structural optimization design.
In the next parts of the paper, the basic theory of neural network, PSO algorithm, BR algorithm, and BMPB are introduced in
Section 2; the specific use of BMPB is discussed in
Section 3; BMPB is used in
Section 4 to optimize the design of the supporting structure of the optical machine system;
Section 5 is the conclusion of the paper.
3. Optimal Design Process of an Optical Machine Structure
During the ground manufacturing phase and the ground assembly phase, the mirror surface shape of the space camera and the relative positions between the different mirrors are affected by gravity, which ultimately affects the imaging performance. In addition, space cameras are subjected to random vibrations during the launch phase, which may result in resonance. In order to ensure the excellent optical performance of the space camera, it is crucial to optimize the optical machine structure. The diagram of the optical system is shown in
Figure 2. Maintaining a high stiffness of the optical structure can reduce the influence of gravity on the imaging performance of the space camera and avoid the resonance of the space camera during the rocket launch process. In order to reduce the launch cost, the space camera needs to minimize the space camera’s mass on the basis of ensuring the excellent imaging performance and lifetime of the space camera.
3.1. Finite Element Calculation and Integrated Analysis
As an important component of a space camera, the supporting structure of the secondary mirror consists of a supporting ring and four supporting legs. In order to ensure the imaging performance and light weight of the space camera, the four supporting legs of the supporting structure need to be optimized. The simplified diagram of the secondary mirror supporting structure is shown in
Figure 3. The outer diameter and wall thickness of each supporting leg are
and
, respectively.
Firstly, the dimension parameters of the supporting structure of the secondary mirror are extracted for parametric modeling, and the 3D model is obtained. The finite element method is then used to divide the 3D model into grids, assign materials, and impose boundary constraints. Finally, the first-order modal frequency (H1) of the supporting structure of the secondary mirror is obtained by modal analysis.
The Latin Hypercube Sampling method was used to sample the secondary mirror’s supporting structure’s dimension parameters to obtain 400 data samples. The integrated analysis and design method were used to carry out the data samples’ finite element calculation to obtain training samples.
3.2. Training Neural Network Model
The training samples were randomly divided into a training set and a test set. BMPB was used to train the training set. After obtaining the optimal weight and threshold, the test set was tested to verify the prediction accuracy.
3.3. Optimization of Optical Machine Structure Performance Index
Firstly, the optimization objectives are confirmed and constraints are set. Then the improved PSO algorithm is used to optimize the trained neural network model to obtain the best optimization results.
3.4. Optimization Flow Chart
The optimization flow chart is shown in
Figure 4. The optimization flow chart is mainly divided into two major parts. The first part is to build the BMPB prediction model; firstly, the integrated analysis system is used to obtain the training samples and test samples of BPNN, then the improved PSO algorithm is used to optimize the weight and threshold of BPNN, and finally, the BR algorithm is used to train the BPNN to obtain the BMPB prediction model. The second part is objective optimization, which firstly determines the objective function, then sets the constraints and input variable ranges, and finally uses the improved PSO algorithm to optimize the objective to obtain the best optimization results.
5. Conclusions
BMPB is developed to fit the nonlinear function between the input variable and the output response in the optical machine structure’s optimal design. Furthermore, the applicability of the proposed method is verified using the example of a supporting structure. Some conclusions of this paper are summarized as follows.
(1) The BMPB introduces the improved PSO algorithm into BPNN, which effectively improves the convergence efficiency of BPNN and avoids falling into the local optimum.
(2) The BMPB integrates the BR algorithm into BPNN, which enables BPNN to avoid overfitting and improves the generalization of BPNN;
(3) The supporting structure’s final optimization result is verified by finite element calculation, and the prediction accuracy is up to 99.4%.
(4) The BMPB can fit the nonlinear function between the input variable and the output response in the optimal design of the optical machine structure, with high prediction accuracy and the reduction of a lot of finite element calculation time.
We provided a way to optimize the design of optical and mechanical structures. This paper proposes the application of BMPB in the field of optimal design of optical machine structures, and the BMPB prediction model has the advantages of high prediction accuracy and avoidance of overfitting. It is expected that BMPB will play a great role in improving the process of performing complex multi-physics field analysis (such as gravity field analysis and thermal field analysis, etc.) for optical machine structures.