Optimized Design of Large-Body Structure of Pile Driver Based on Particle Swarm Optimization Improved BP Neural Network

Abstract

Optimization of the pile driver’s large-body structure is important to achieve the driver’s overall light weight. This paper studies the large-body structure of a hydraulic static pile driver. We used the APDL parametric design language provided by ANSYS to construct a geometric model of the large-body structure and performed a static analysis of the finite element model. Under the assumption that the strength and stiffness meet the design requirements, the optimization model was constructed with the thickness of each plate of the large-body structure as the design variable, the structural strength and stiffness as the constraints, and the minimum mass as the objective function. Finally, two optimization algorithms were used to solve the model, and the comparison of the two sets of solutions shows that the improved BP neural network algorithm based on the particle swarm optimization algorithm performs better. The optimized mass of the large-body structure was reduced from 82,556.1 kg to 65,046.15 kg, a mass reduction of 21%. The lightweight design of the pile driver was achieved.

1. Introduction

Manufacturing is the mainstay of the national economy and the nation’s foundation. To promote the country as a whole to achieve a high degree of industrial modernization and become a strong manufacturing country, the “Made in China 2025” plan was proposed. It was mentioned that far-reaching industrial changes are taking place with the deep integration of the current generation of information technology and manufacturing [1]. Pile-driving machinery has played an increasingly prominent role in the construction of the Chinese economy. There are various types of pile-drive devices, mainly the cylinder diesel pile drive, hydraulic pile drive, hydraulic static pile drive, hydraulic vibratory pile drive, etc. Hydraulic static pile drivers are a type of heavy industrial machinery and equipment used in civil engineering pile foundation projects, the construction of which is illustrated in Figure 1.

Hydraulic static pile drivers have numerous advantages over pile-driving and vibratory hammers [2]. Their vibration-free, noise-free, and pollution-free working conditions make them more suitable for construction in urban areas; when the pile is evenly and uniformly pressed into the soil by static pressure, the pile is not subjected to tensile stresses caused by significant vibrations, thus avoiding damage to the body of the pile; extreme levels of mechanization can reduce the strength of the workforce; the rapid construction speed of hydraulic static pile drivers can effectively shorten the work cycle. Meanwhile, the hydraulic static pile driver is also an environmentally friendly piling machine with Chinese characteristics [3]. As a result, hydraulic static pile drivers are commonly used from civilian buildings to industrial facilities, and from high-rise buildings to municipal road and bridge construction sites. Different working conditions and other factors can affect pile presses, especially large-tonnage pile presses. During operation, the main load-bearing section of the pile driver chassis platform can crack, which can seriously impact the pile driver’s lifetime. Therefore, the hydraulic static pile press’s main load-bearing parts need great strength and stiffness [4]. To achieve superior performance, stability, and long service life, the conservative design of domestic hydraulic static pile drivers has resulted in excessive design redundancy, excessive material waste, and low economic value. To increase the economic value of the pile driver while ensuring its high strength and stiffness, it is particularly critical to design it as lightweight.

With the continuous development of computer technology, computer design tools have become powerful tools for mechanical design, and the use of computer software in mechanical optimization design has become widely popular. With the advent of richer theoretical models of optimization algorithms, the use of computer-aided design combined with optimization algorithms has further enhanced design research in mechanical engineering [5]. Experimental design methods are commonly used in optimization design to process data to build an optimization model. Ju et al. [6] designed a multi-objective optimized machine structure design based on an orthogonal test of the machine structure dimensions. Ren et al. [7] performed a Latin hypercube experimental design of the main design parameters of the bed and used a multi-objective optimization algorithm to solve the bed optimally. Optimization of multiple complex optimization objectives with correlations must be implemented with the aid of computers. Therefore, applying optimization algorithms is also the main direction of this study. In the study of optimal design combined with an optimization algorithm, Liu et al. [8] proposed an improved particle swarm optimization algorithm for the multi-objective optimal design of a gantry machine slide. Jiang et al. [9] proposed a tiny UAV low-speed airfoil optimization design method based on an improved PSO algorithm, and the results improved the search performance of the UAV. Song et al. [10] combined a BP neural network and genetic algorithm to construct a framework for the multi-objective optimization problem of the blade mixer. Wang et al. [11] used an optimization algorithm combining the BP neural network and genetic algorithm to obtain an optimal solution for the profile parameters of a twin-screw compressor. Xiong et al. [12] conducted a sensitivity analysis of the finite element model of the pile-driver press box. Based on the analysis results, the dimensions of the structure were optimized, eventually leading to a reduction in the total mass of the structure. Qiu et al. [13] used neural-network-based response surface and multi-objective genetic optimization algorithms for the sizing optimization of the pile-driver lifting turntables. By integrating the scholars’ research on the optimal design of mechanical products, it was found that there is little research on pile presses. Hydraulic static pile drivers have one of the highest market shares among pile-driving devices and are optimally designed to help promote economic construction.

In this paper, we studied the large-body structure of the hydrostatic pile driver YZY400. ANSYS software was used to construct its geometric model. After static analysis, the large-body structure was optimized using a particle swarm optimization algorithm and a modified BP neural network algorithm based on the particle swarm optimization algorithm. The optimization results of the two algorithms were compared, and it was found that the modified BP neural network algorithm can better achieve the optimization objective. Applying BP neural networks to structural optimization design was studied, where a lightweight structure design was accomplished.

2. Modeling

The finite element method (FEM) is a numerical analysis method for solving complex engineering problems by converting complex calculations into simple calculations through a discrete approach [14]. This method was first used in the static and dynamic analysis of aircraft structures in the early 1950s to obtain deformations, stresses, self-oscillation frequencies, and modes of the structure. As computer technology evolves, finite element analysis is increasingly used for simulation in engineering. The most popular finite element analysis software programs are ANSYS, ADINA, ABAQUS, and MSC. With its rich functionality and wide range of applications, ANSYS software is the first choice of technical personnel and scientific researchers in various industries. The finite element analysis and optimization of the large-body structure of the pile driver must be based on a geometric model, and the accurate and economical construction of a geometric model of the large-body structure is the key to the problem. The APDL parametric design language provided by the finite element analysis software ANSYS was used to write the command flow to establish the geometric model of the large-body structure of the YZY400 hydraulic static pile driver. The constructed 3D geometric model is shown in Figure 2.

Experience shows that the accuracy of the FEA is less strong the closer it is to the true model. When modeling complex structures, excessive accuracy in constructing the exact model details may cause the FEA calculation to fail or make errors. Therefore, a reasonable simplification of the actual model when performing the FEA effectively improves the analysis’s efficiency and ensures the analytical results’ accuracy. To improve the efficiency and ensure the convergence and correctness of the finite element analysis results when performing finite element modeling, components that have a minor effect on the overall strength and stiffness of the large-body structure, such as bolts connecting the outrigger structure to the large-body structure, are not considered. Because the bolt grade determines the strength of the bolt connection, it has a minor effect on the overall performance of the large-body structure. Thus, the details of the connection between the legs and the large-body structure are not considered, and they are stiffened to the large-body structure.

3. Static Analysis

3.1. Grid Division

The pile driver’s large-body structure is a box-type structure of thin steel plates welded together. The steel plate structure is contained in a plate-and-shell unit-type structure subject to torsional or bending deformations when subjected to force. Therefore, Shell63 cells were chosen when selecting cell types. Shell63 cells are commonly used in finite element analysis to model structures under pressure or tension as well as bending and torsional deformations.

The model needs to be meshed after selecting the cell types for the large-body structures using ANSYS APDL. The MSHAPE command was used to divide the mesh, and the LESIZE command was used to set the edge length of the mesh to control the mesh size, hence the mesh’s number and quality. We employed free-mesh splitting for the complex large-body structures to save time and improve design efficiency. Figure 3 shows an overall model of the large-body structure of the press pile driver after dividing the mesh.

The quality of the meshing needs to be checked after meshing. We imported the generated Mesh model into ANSYS Workbench and entered the Mesh interface, where one can view the full information about the quality of meshing in the Detail of Mesh [15]. Various criteria for assessing the meshing quality for large-body structures led to the results in Figure 4.

According to the obtained results, the mesh quality of the large-body structure satisfies the criteria, which indicates that the meshing quality is excellent and validates the correctness of the adopted meshing method, so the subsequent finite element analysis boundary treatment and analysis can continue.

3.2. Work Condition Selection

During the operation of the hydraulic static pile driver, shown in Figure 1, the crane is responsible for lifting the pile. Hydraulic cylinders in the chassis platform then provide power to perform the pile-pressing and pile-pulling actions, which are attended to by the pile-clamping and pressure mechanisms on top of them. The pile press frame, crane, clamping pile press mechanism, and hydraulic system are all mounted on the large-body skeleton of the pile press. When the pile driver is in pileup mode, the pileup mechanism on the bulk structure interacts with it. The gravitational force of the pileup mechanism and the reaction force of the pileup form a combined force of size F₁ in the opposite direction of the pileup. This force acts upwards on the bulk structure via the longitudinal transfer mechanism, and the transverse transfer rotational mechanism acts with its gravity on the large-body structure and the contact surface of the cylinder via the uniform pressure Q₁. The maximum pileup force is 400 T, and the combined force F₁ value is at its maximum. So, the large-body skeleton is under maximum force during this working condition, i.e., the pileup operation, and will produce the largest deformation. Therefore, this study performed a static analysis of the bulk structure during the pileup operation; that is, under the most dangerous working conditions.

3.3. Statics Analysis of Large-Body Structures

Static analysis of large-body structures is primarily concerned with assessing the ability of large bodies to resist deformation under a static load based on an analysis of the most dangerous working conditions for large-body structures. At this time, the maximum pileup force is 400 T. The combined force of the gravity of the pileup mechanism and the pileup reaction force is 3,898,719 N, acting on the bulk structure in the reverse direction of the pileup force; the gravity of the traverse slewing mechanism acts as a uniform load on the large-body structure and the contact surface of the cylinder, which has a value of 3.061 MPa for a uniform load. In setting the constraints, in addition to the translational degrees of freedom in the Z-direction, a total of five degrees of freedom of motion and rotation in all other directions were constrained. The aim was to simulate the normal action of the cylinder legs of a hydraulic static pile driver under realistic working conditions. The finite element model of the large body after setting the constraints is shown in Figure 5. The above loads were applied to the large-body structure using the SFA command in ANSYS APDL. The results of the deformation and stress analysis of the large-body structure under the most dangerous working conditions are shown in Figure 6.

According to the analysis results, the maximum stress appears at the junction between the frame and the legs, with a maximum stress value of 110 MPa. The large-body structure material is Q235 with an allowable stress of 156.67 MPa, which satisfies the strength requirements but has more room for optimization. The largest deformation also occurred at the junction between the frame and the outrigger, with a maximum deformation of 2.016 mm, as required by design.

4. Structural Optimization Design

In order to achieve a lightweight design of the large-body structure of the hydraulic static pile driver that satisfies the requirements of strength and stiffness, its structure needs to be optimized. The aim is to find an optimal set of design variables such that the large-body structure satisfies the strength and stiffness requirements and enables its mass reduction. Considering that the members of the large-body structure are all thin steel plates, the thickness of the bulk plate was chosen as the structural design variable. The large-body structure of the pile press is mainly composed of a side sealing plate, pile opening sealing plate, column connecting plate, inner longitudinal plate, crane support plate, outrigger connecting plate, middle longitudinal plate, middle sealing cover plate, and side sealing cover plate [16]. The main structural dimensions are shown in Figure 7.

The above six design variables indicate the thicknesses of the crane support plate, the side cap plate, the inner longitudinal plate, the side seal plate, the pile opening seal plate, and the intermediate longitudinal plate. The dimensions of each plate thickness and the range of values are shown in

Table 1. The thickness of each plate and the range of values.

Variables	Initial Value/m	Range of Values/m
$R_1$	0.04	[0.02, 0.06]
$R_2$	0.02	[0.01, 0.03]
$R_3$	0.03	[0.015, 0.045]
$R_4$	0.035	[0.02, 0.05]
$R_5$	0.05	[0.025, 0.075]
$R_6$	0.06	[0.03, 0.09]

, based on the actual dimensions of the large-body structure of the YZY400 hydraulic static pile driver.

4.1. Experimental Design

In the structural optimization design of a large body, it is expected that there are six design variables to be examined, namely the above six dimensions. However, in practice, not every variable significantly influences the optimization objective, and too many variables with small degrees of influence are examined, which takes longer and requires more effort during optimization. Therefore, it is necessary to select from several design variables, i.e., test factors, through the experimental design to obtain the test factors that have a greater influence on the large body’s structural performance (mass, maximum stress, and maximum deformation). The experimental design methods commonly used for structural optimization include orthogonal test design, central composite design, Box–Behnken design, Latin square design, Latin hypercube design, etc. [17]. Orthogonal experimental design is a space-based method commonly used for design problems with multiple factors. Its most striking feature is that the sample points are uniformly and more neatly distributed in the sample space. The results obtained from the experimental design can be compared with each other. This approach typically uses orthogonal test tables to schedule the tests and allows statistical treatment of the test results using extreme difference analysis. Compared with alternative experimental methods, the orthogonal experimental design method uses fewer trials to obtain more comprehensive spatial information about the sample, and the experimental results obtained through the experimental orthogonal design table can be analyzed to determine the effect of each factor on the output response. There are six factors affecting the mass, maximum stress, and deformation of the bulk structure, and the orthogonal test tables with six factors and five levels are chosen according to the range of values of the variables in Table 1, as shown in

Table 2. Orthogonal test table.

Test Number	${\mathit{R}}_1$	${\mathit{R}}_2$	${\mathit{R}}_3$	${\mathit{R}}_4$	${\mathit{R}}_5$	${\mathit{R}}_6$
1	1	1	1	1	1	1
2	1	2	2	2	2	2
3	1	3	3	3	3	3
4	1	4	4	4	4	4
5	1	5	5	5	5	5
6	2	1	2	3	4	5
7	2	2	3	4	5	1
8	2	3	4	5	1	2
9	2	4	5	1	2	3
10	2	5	1	2	3	4
⁞	⁞	⁞	⁞	⁞	⁞	⁞
25	5	5	4	3	2	1

.

Twenty-five trials were performed according to the testing protocol in Table 2, and the results were analyzed for polar differences using the EXCEL tool. The main order of the factors affecting the quality M is $R_1>R_6>R_4>R_3>R_5>R_2$, the main order of the factors affecting the maximum stress S is $R_1>R_6>R_5>R_4>R_2>R_3$, and the main order of the factors affecting the maximum deformation U is $R_1>R_6>R_5>R_4>R_3>R_2$. The analysis results show that $R_1$ (crane support plate thickness) and $R_6$ (thickness of the center longitudinal plate) have the maximum impact on the mass, maximum stress, and maximum deformation; $R_2$ has the least impact on all three and reduces the effort of optimizing the design. A total of five parameters, $R_1$, $R_3\sim R_6$, were chosen as design variables for the optimization of the large-body structure.

4.2. Constructing the Optimal Model

The essence of optimal design is the problem of finding extrema, establishing mathematical relationships between the design variables and optimization objectives, and generating mathematical models [18]. The three elements of the optimization model are the design variables, the constraints, and the objective function. Combined with the practical situation of optimization of the large-body structure, three elements can be defined: a total of five parameters, $R_1$, $R_3\sim R_6$, were chosen as the design variables based on the results of the orthogonal tests in Section 4.1; the constraints were the stiffness and strength of the large-body structure; and the quality of the bulk structure was an objective function. Having identified the three elements, we constructed the optimal design model, the objective function relation equation.

This study used the optimal Latin square method to conduct experiments to obtain the data for constructing the relational equation of the objective function [19]. The optimized Latin square method has a better and more uniform filling of the sampling space, which can reflect the characteristics of the sample space while retaining the advantages of the original Latin square method in terms of efficiency and effectiveness. The ANSYS APDL software was used to obtain the mass, maximum stress, and deformation values for large structures under the most dangerous conditions for different design variables, and the resulting test results data were organized into test tables. Some of these are shown in

Table 3. Table of partial test results.

${\mathit{R}}_1$ (m)	${\mathit{R}}_3$ (m)	${\mathit{R}}_4$ (m)	${\mathit{R}}_5$ (m)	${\mathit{R}}_6$ (m)	M (kg)	S (Mpa)	U (m)
0.036	0.027	0.0315	0.045	0.054	74,760.1	114	0.002402471
0.0206	0.0245	0.03447	0.0446	0.0511	55,629.1	338	0.006521
0.04231	0.03595	0.03296	0.02701	0.04538	81,664	87.9	0.001984
0.02683	0.03912	0.02347	0.03656	0.08548	70,820	194	0.003683
0.03387	0.03746	0.03854	0.05113	0.05503	76,087	27	0.002569
⁞	⁞	⁞	⁞	⁞	⁞	⁞	⁞
0.038668	0.030999	0.0315	0.051665	0.061998	80,786.5	96.6	0.002107169

.

In order to obtain optimal values for each design variable, the bulk structure was then optimized to reduce the redundancy of the design. It was necessary to construct expressions for the five design variables as a function of the bulk structure’s mass, maximum stress, and maximum deformation. The experimental data obtained from the optimized Latin square design was first processed using regression analysis. The EXCEL regression analysis tool is more convenient, simpler, and more efficient than other methods. Therefore, a regression analysis of the experimental data using the data processing tool EXCEL was selected from a variety of regression analysis methods. First, the specific influence coefficients of the five design variables on the objective function were analyzed, and a table of the regression analysis results for the expression of the large-body structure mass function is shown in

Table 4. Table of regression analysis results on mass M.

	Coefficients	Standard Error	t Stat	p-Value
Intercept	4595.752	0.051667	88,949.98	0.001
X Variable 1	1,216,390	0.577331	2,106,921	0.001
X Variable 2	169,125.3	0.769774	219,707.7	0.001
X Variable 3	241,171.9	0.659807	365,519.1	0.001
X Variable 4	34,127.85	0.461865	73,891.46	0.001
X Variable 5	234,725.8	0.384887	609,856.2	0.001

.

Collating the data in the table yields the following expression for the regression function of the mass M on the five design variables:

$$M=1216390X_1+169125.3X_2+241171.9X_3+34127.85X_4+234725.8X_5+4595.752$$

(1)

The results of the regression analysis of the expression for the maximum stress function of the large-body structure subjected to the analysis can be seen in

Table 5. Table of regression analysis results for maximum stress S.

	Coefficients	Standard Error	t Stat	p-Value
Intercept	389,540,494.2	18,298,354.5	21.28828001	1.20376 × 10−52
X Variable 1	−6,239,216,457	197,832,147	−31.53793027	2.70762 × 10−78
X Variable 2	−38,555,965.68	263,724,057.2	−0.146198136	0.883916779
X Variable 3	−44,669,648.47	263,764,808.3	−0.169354088	0.865694592
X Variable 4	66,759,184.62	158,298,029.4	0.42173099	0.67368832
X Variable 5	−294,959,688.1	131,903,938.1	−2.236170447	0.026479707

.

Collating the data in the table yields the following expression for the regression function of the maximum stress S on the five design variables:

$$\begin{array}{l}S=-6239216457X_1-38555965.68X_2-44669648.47X_3+66759184.62X_4\\ -294959688.1X_5+389540494.2\end{array}$$

(2)

Finally, a regression analysis was performed on the expression of the maximum deformation function of the large-body structure to obtain a table of the analytical results shown in

Table 6. Table of regression analysis results for maximum deformation U.

	Coefficients	Standard Error	t Stat	p-Value
Intercept	0.005364	3.64 × 10−5	147.3953	5.1 × 10−245
X Variable 1	−0.06936	0.000407	−170.565	1 × 10−260
X Variable 2	−0.00191	0.000542	−3.52736	0.000499
X Variable 3	−0.00289	0.000465	−6.21437	2.14 × 10−9
X Variable 4	−0.00055	0.000325	−1.68518	0.093201
X Variable 5	−0.00603	0.000271	−22.2326	3.67 × 10−61

.

Collating the data in Table 6 yields the following expression for the regression function for the largest deformation U:

$$U=-0.06936X_1-0.00191X_2-0.00289X_3-0.00055X_4-0.00603X_5+0.005364$$

(3)

Three objective functions were identified for the optimal design of the large-body structure based on real-world situations: mass, maximum deformation, and maximum stress, which are multi-objective optimization problems. These objectives differ significantly in importance, with the quality objective being the most valuable and having the highest weight, and the maximum deformation and maximum stress objectives being the next most important and being weighted approximately equally. This paper transformed the multi-objective problem into a single-objective optimization problem with optimal solutions. Having identified the three elements of the optimization model in this subsection, the mathematical model for the optimization of large-body structures can be obtained as follows:

$$\left\{\begin{array}{c}\genfrac{}{}{0pt}{}{Minimize M\left(x\right)}{x = [x_1,x_2,x_3,x_4,x_5]}\\ \genfrac{}{}{0pt}{}{\begin{array}{c}0 < \mathrm{S} \le 156.67\\ 0 < \mathrm{U} \le 0.006\\ 0.02 \le x_1 \le 0.06\end{array}}{\begin{array}{c}0.015 \le x_2 \le 0.045\\ 0.02 \le x_3 \le 0.05\end{array}}\\ \genfrac{}{}{0pt}{}{0.025 \le x_4 \le 0.075}{0.03 \le x_5 \le 0.09}\end{array}\right.$$

(4)

After constructing the mathematical model of the optimized large-body structure, the algorithm needs to be applied to solve the model to find the optimal solution of the model.

5. Application of the Optimization Algorithms

5.1. Particle Swarm Optimization Algorithm

The particle swarm optimization algorithm is an intelligent optimization algorithm, also known as the bird flocking foraging algorithm, based on the fundamental principle of exploiting cooperation and information sharing among individuals in a population to find the best solution. Compared with other affine optimization algorithms, the particle swarm optimization algorithm has simpler rules and converges faster, demonstrating its superiority in practical engineering optimization problems. The particle swarm optimization algorithm is also stochastic, and its flowchart of operations is shown in Figure 8 [20,21].

Two essential factors determine whether a particle swarm optimization algorithm can find an optimal solution: the number of particle swarms and the speed of progress of the individual swarm optimization process. If the number of particle swarms is not large enough, the mass of the particles is not large enough, and the initial value of the optimization will be high, which is not conducive to finding a globally optimal solution. Too large a velocity for each particle in the swarm during the optimization process is detrimental to finding the optimal local solution of the model. At the same time, too small a velocity for the advancing particle increases the running time, but the search quality is higher. Therefore, the number of particle swarms should be large, and the velocity of particle precession should be appropriate. The initial number of particle swarms is 200, and the maximum number of forward steps is 1000. Once the program is written, the search for large-body structure masses begins.

5.2. Improved BP Neural Network Optimization Algorithm Based on Particle Swarm Optimization

The BP network is a widely used multilayer feed-forward network that uses the error between the actual and predicted values to propagate in the opposite direction in the network. The BP neural network continuously learns the mapping relationship between input and output through many calculations and error feedback, calculates the error between the new data and the fitted data, and adjusts the coefficients such as weights ω and configuration b. The BP neural network ends the computation until the error meets the set accuracy requirement or the training times reach the maximum. Thus, the BP neural network is, in fact, a prediction algorithm that is widely used in engineering by simulating input–output mapping functions closer to the actual ones. This feature can help to address the shortcomings of particle swarm optimization algorithms with high-fitness function requirements. The expressions obtained from the regression analysis were replaced with functional expressions for the three objective values fitted by the trained BP neural network. These were then applied to the particle swarm optimization algorithm to construct a BP neural network optimization algorithm based on particle swarm optimization. The number of nodes in the hidden layer of a BP neural network is not fixed, and this number can affect the performance of the constructed neural network architecture. The number of neuron nodes in the hidden layer should be between the numbers of neuron nodes in the input and output layers. In this study, the number of neuron nodes in the hidden layer was set to 50, and no additional hidden layer was added, to avoid the overfitting problem caused by increasing the number of hidden layers. The constructed neural network algorithm model is shown in Figure 9.

The number of samples used to train the neural network was set to 200. The dataset was partitioned randomly, with the partitioning unit being each datum. The neural network’s training, validation, and test sets were 80%, 10%, and 10%, respectively. The training learning rate was 0.1, the maximum number of training sessions was 200,000, and the training objective error was 0.0001. Training stopped when the maximum number of training sessions was reached or when the error between the training prediction value and the actual target value was smaller than the set error value [22]. The predicted versus actual values for 200 samples for the three objective functions are plotted in Figure 10, Figure 11 and Figure 12.

As can be seen from the above figures, the predicted values of the trained neural network nearly coincide with the actual values, and the high prediction accuracy is in line with expectations. It provides excellent conditions for using neural networks to train models for particle swarm optimization solutions.

We have written a program based on a modified BP neural network optimization algorithm based on the particle swarm optimization algorithm, where the main program invokes the particle swarm optimization algorithm to optimize the solution of the fitted function expression. The initial population size is 40, and the number of evolutions is 1000 in the subroutine invoked by the particle swarm optimization algorithm [23]. Due to the use of the BP neural network to train the model, a small number of particle swarms can achieve a good solution.

6. Results

The mass-finding curve of the particle swarm optimization algorithm based on the large-body structure is shown in Figure 13.

As can be seen from Figure 13, after 1000 iterations, the mass of the large-body structure is optimized, and the parameters before and after optimization are shown in

Table 7. Parameters before and after optimization with the particle swarm optimization algorithm.

Parameter Types	Before Optimization	After Optimization
Crane support plate thickness (m)	0.04	0.037
Inner longitudinal plate thickness (m)	0.03	0.022
Side seal sealing plate thickness (m)	0.035	0.034
Pile mouth sealing plate thickness (m)	0.05	0.036
Mid-longitudinal plate thickness (m)	0.06	0.039
Large-body structure quality (kg)	82,556.1	71,157.39
Maximum stress (MPa)	110	150.87
Maximum deformation (m)	0.00201	0.00242

.

The mass of the large-body structure optimized using the particle swarm optimization algorithm is 71,157.39 kg, the maximum stress is increased from 110 MPa to 150.87 MPa, and the maximum deformation is increased from 0.00201 m to 0.00242 m. The design redundancy in the strength and stiffness of the large-body structure is mostly reduced. The plots of the mass ergodic, deformation ergodic, and stress ergodic obtained by running the BP neural-network-based optimization of the particle swarm optimization algorithm are shown in Figure 14, Figure 15 and Figure 16, and the parameters before and after the optimization are shown in

Table 8. The BP neural network algorithm, improved based on the particle swarm optimization algorithm, optimizes various parameters before and after.

Parameter Types	Before Optimization	After Optimization
Crane support plate thickness (m)	0.04	0.036
Inner longitudinal plate thickness (m)	0.03	0.015
Side seal sealing plate thickness (m)	0.035	0.02
Pile mouth sealing plate thickness (m)	0.05	0.025
Mid-longitudinal plate thickness (m)	0.06	0.03
Large-body structure quality (kg)	82,556.1	65,046.15
Maximum stress (MPa)	110	156.66
Maximum deformation (m)	0.00201	0.00255

.

From Table 8, it can be seen that the maximum stress of the bulk structure obtained by solving with the BP neural network algorithm based on the modified particle swarm optimization algorithm increased from 110 MPa to 156.66 MPa and the maximum deformation increased from 0.00201 m to 0.00255 m. The design redundancy in the strength and stiffness of the large-body structure was largely reduced. The mass of the large-body structure was reduced from 82,556.1 kg to 65,046.15 kg, which is a significant optimization effect.

Compared with the two sets of optimization results, the modified BP neural network algorithm based on the particle swarm optimization algorithm performed better than the particle swarm optimization algorithm, which indicates that the accuracy of the objective function model obtained with BP neural network training is higher than that of regression analysis. The advantage of BP neural networks in fitting mathematical models with higher accuracy was fully exploited.

7. Conclusions

In this paper, two optimization algorithms were used to complete the lightweight design of the large-body structure of a hydrostatic pile driver. The APDL parametric design language provided by the finite element software ANSYS was used to construct parametric models of the large-body structure. Finite element analysis was used to perform a static analysis of the large-body structure under the most dangerous working conditions, checking for strength and stiffness. The optimization model was developed based on regression analysis, with the slab thickness of the large-body structure as the design variable and the mass as the objective function. Finally, we introduced the particle swarm optimization algorithm and the modified BP neural network algorithm based on the particle swarm to optimize the model, respectively, and showed that the optimization algorithm combined with the BP neural network gave better results than the individual algorithms. The optimized mass of the large-body structure is 65,046.15 kg, 21% lower than the pre-optimized mass of 82,556.1 kg, reducing the amount of material while meeting the design requirements for strength and stiffness and increasing its economic value to some extent. This study provides a solution for the optimal design of mechanical products such as pile drivers.