# Research on Optimization of Structural Parameters of Equipment Cabin Bottom Cover

Authors: Hua Zou; Qifeng Wu; Yonggui Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Since the railway vehicle structure has lots of parameters and several complex constraints, this study establishes a method for structural parameter optimization based on sensitivity analysis and surrogate models. Fatigue crack problem of the equipment cabin bottom cover of the EMU is taken as an example to optimize its structural parameters. First, establish the finite element (FE) model of the bottom cover and compare it with the bench test results to verify the accuracy of the load and restraint conditions. The sensitivity analysis method is used to determine the main parameters. The input samples are obtained by Latin hypercube sampling method, and the output samples are obtained by the method jointly developed by ABAQUS+Python and the surrogate model between the input and output samples is obtained by fitting, and its accuracy is verified. According to the design requirements, the optimization objective function and constraint conditions are established, and the optimization result is obtained by optimization algorithm. The results were substituted into the FE model for verification. The results show that the maximum equivalent stress of the bottom cover is reduced from 126.7 MPa to 78.9 MPa under a cyclic aerodynamic load of ±4 kPa, which is 37.7% optimized, and the effect is significant. This method avoids the iterative optimization of the FE model and improves the optimization efficiency.


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## Body

## 1. Introduction

In the application of large mechanical structural parts, fatigue cracks have always been the main factor leading to their failure [1–3]. Take railway vehicles as an example; fatigue cracks are found on the equipment cabin bottom cover. The reason is that the alternating aerodynamic load under the working conditions when trains enter and exit tunnels is the main factor leading to the initiation and propagation of cracks [4]. Therefore, it is necessary to study the structural optimization design method to reduce the stress level at the crack under alternating aerodynamic loads.Nowadays, there has been in-depth research on the optimization method of structural parts [5]. The authors in [6–11] separately optimized the structure of energy-absorbing structure, carbody structure, and the subway vehicle air-conditioning suspension. Their weights were reduced while meeting the requirements of strength and modal. But for the bottom cover, its protection function and manufacturability need to be considered, so the topology optimization results often do not have engineering application value. The authors in [12, 13] optimized the shape of the frame and wheel. But the shape optimization required modification of the floor structure. At the same time, it was also necessary to consider constraints such as installation, connection, and manufacturability, so the space for modification was limited. The size optimization does not require major changes to the floor structure, which is relatively easy to implement and has high engineering value. Therefore, this article uses the size optimization method to optimize the bottom cover.Now scholars have begun to use a variety of methods to optimize the parameters of the structure [14–16]. Among them, approximate methods have been widely used [17]. For the railway vehicle structure, the structure is complex and the boundary conditions are numerous, which often requires a lot of time in the simulation calculation. In order to improve the optimization efficiency, many scholars have conducted research on the optimization method of size parameters. Myzrglob and Zielinski [18, 19] studied the optimization method of structural parameters under multi-axial high-cycle fatigue. Sun et al. and Hudson et al. [20, 21] studied the optimization method of carbody parameters considering multiple factors. Sun et al. [22] optimized the modal frequency of the railway vehicle carbody based on SA. Miao et al. [23] used the ant colony algorithm to optimize the composite sandwich structure of the bottom cover. Zhi et al. [24] studied the fuzzy optimization method of bogie frame parameters based on response surface model. Baek et al. [25] optimized the wagon frame structure based on the Chebyshev polynomial model. The authors in [26–28] used surrogate models and multi-objective genetic algorithms to optimize the design of the carbody collision energy absorption structure. The authors in [29–31] optimized the fatigue strength of the welded frame weld by the surrogate model method.The above research used mathematical models to express the relationship between structural parameters of vehicle components such as carbody, bogie frame, and energy-absorbing structure with modal and strength indicators and optimized them with optimization algorithms to improve optimization efficiency. However, the optimization research on the equipment cabin bottom cover is still lacking at the present stage, and the connection and contact between the various components cannot be ignored for this type of assembly. There are many size parameters and installation parameters of components, and each parameter has a complex nonlinear relationship with the stress response of key points. Establishing a FE model that can reflect the actual components and meet the requirements of optimization accuracy is a prerequisite for optimization design. For this reason, this paper carried out the static strength test of the bottom cover and compared the FE calculation results with the test results to verify the accuracy of the FE model. Considering the complex preprocessing of the FE model and the long calculation time, this paper first extracts the key parameters of the bottom plate according to the sensitivity analysis (SA) method and uses the optimal Latin hypercube sampling to obtain the input sample set of the key parameters. Through the ABAQUS + Python co-simulation method, the FE model under different input parameters is obtained in batches, and the output parameter samples are obtained. Then, the surrogate model is fitted to replace the FE model, and optimization is performed based on the optimization algorithm. This method can effectively improve optimization efficiency.


## 2. Structural Parameter Optimization Design Method

### 2.1. Parameter Optimization Process Based on FE Model

In the FE model, due to the model scale, contact nonlinearity, material nonlinearity, and other issues, it often takes a long time to simulate. At the same time, optimization research needs to modify the parameters multiple times for iterative calculations, so there are many repeated calculations. If the FE model is directly called, it will consume a lot of time and be inefficient. Therefore, in order to improve the optimization efficiency under the premise of ensuring accuracy, the optimization method shown in Figure1 can be used for the complicated FE model.Figure 1
Bottom cover optimization method.According to Figure1, firstly, the SA method is used to screen the optimized parameters, and then the optimal Latin hypercube sampling or orthogonal test is used to sample in the range to obtain a uniform and representative parameter sample set. Use the Python program to read the samples in order, modify the parameters of the FE model in ABAQUS, submit calculations, and read the stresses at the key points. Summarize the stress values under different samples and use them as input and output parameters. Use the above input and output parameters as samples and import them into the Isight software to fit the surrogate model and evaluate the goodness of fit. Based on the surrogate model, optimization algorithms such as steepest descent method and genetic algorithm are used for optimization. Then, the optimal solution of the model is obtained and imported into ABAQUS for verification.


### 2.2. Parameter SA Method

When there are many optimization parameters, the amount of calculation in sampling calculation and optimization calculation is large. The parameter SA method can eliminate the parameters that have little influence on the result. Assuming that there arenparameters, there is the following functional relationship between thei-th parameterpiand the response r:(1)r=rp1,p2,p3,…,pn.The sensitivityεi of different parameters can be expressed by the following formula:(2)εi=∂rp1,p2,p3,…,pn∂pi,i=1,2,3,…,n.It is often a discrete variable in engineering applications. Its sensitivityεi can be expressed by the finite difference method:(3)εi=rpi+Δpi−rpiΔpi,i=1,2,3,…,n.However, when calculating the sensitivity through the above formula, the influence of the parameter scale and the variation range is ignored. When the parameter scale is large, the calculated sensitivity is often small. Therefore, the original parameter valuepi0 and the original response value r0 are used as scale factors, and the sensitivity parameters are normalized:(4)εi=rpi+Δpi−rpi/r0Δpi/pi0,i=1,2,3,…,n.In order to improve the accuracy of the sensitivity parameter, takem different difference step size Δpij (j=1,2,3,…,m, including the original value, Δpi1=0) within the parameter change range. The least square method is used to fit it into a proportional function relationship of ri=kipi, and the absolute value of the slope ki is taken as the sensitivity parameter:(5)εi=ki=∑j=1mRijPij∑j=1mRij2,i=1,2,3,…,n,j=1,2,3,…,m,where(6)Rij=rpi+Δpij−rpir0,Pij=Δpijpi0.


### 2.3. Sample Acquisition Method Based on FE Model

According to the optimized parameters and their ranges, a set of representative input parameters needs to be extracted. The optimal Latin hypercube sampling method has strong filling ability, uniform sample distribution, and good representativeness. Assuming that a total ofm samples are required, m small hypercubes are randomly selected in the n-dimensional hypercube space composed of n design variables to ensure that each small hypercube is unique in each design variable interval. That is, each design variable will be sampled only once at each level. In this way, a Latin hypercube design matrix with m samples under n design variables is obtained. At the same time, through the optimization algorithm, the Euclidean distance between different sample points is minimized, and the evenly distributed and representative sampling results are obtained. This paper uses the optimal Latin hypercube method to sample the parameters. Generate the optimal Latin hypercube sampling results in Isight software as input samples.The FE model is established based on the input parameters, and the output parameters can be obtained through FE simulation. ABAQUS has a powerful secondary development function. Using Python programs, it can perform secondary development on the preprocessing and postprocessing parts of ABAQUS, achieve automatic modeling, automatic submission calculations, and automatic postprocessing batch calculations, which can save simulation time.Figure2 shows the parameter flow of using the Python program to obtain the output. First read a set of input parameters from the input parameter set and import the ABAQUS library function into the program, and the ABAQUS library function contains members that can be used for modeling. The members in the Model object under the Mdb object can be used for preprocessing work such as building 3D models, defining material properties, meshing, and defining loads and constraints. The Job object can submit calculations to the model. Each member under the Odb object can read the field output and history output results in the calculated odb result file to obtain the required output parameters. In this loop, the output parameters under different input parameters can be obtained, and the output parameter set can be summarized. The input parameter set and the output parameter set are combined as surrogate model training samples.Figure 2
Sample acquisition process based on FE parameter model.


### 2.4. Surrogate Model Establishment Method

Isight software provides a variety of surrogate model fitting algorithms, including response surface model (RSM), radial basis function neural network (RBFNN) model, Kriging model, and so on.


#### 2.4.1. Response Surface Model (RSM)

Taking the fourth-order response surface model as an example, the surrogate model fitting formula is shown in the following equation:(7)y^=β0+∑i=1Mβixi+∑i=1MβM+ixi2+∑i=1Mβ2M+ixi3+∑i=1Mβ3M+ixi4+∑i≠jMβijxixj,where M is the number of input variables; xi represents the input variables; y^ is the output value; and βi represents the coefficients.


#### 2.4.2. Kriging Model

The expression of the Kriging model is shown in the following equation:(8)y^=fTxβ+Zx,where fTxβ is the global regression model; Zx is the random fluctuation; and y^ is the output value.


#### 2.4.3. Radial Basis Function Neural Network (RBFNN) Model

In the RBFNN model, the radial function is the basis function of the model, and its independent variable is the Euclidean distance between the measured point and the input point. (3) is the basic form of the radial basis function.(9)y^=∑j=1MHjrwj=HTrw,where wj is the weight; M is the number of samples; Hjr is the radial function; and y^ is the output value. By adjusting the weights, the radial basis function can be used to fit different models.There are many indicators for evaluating the goodness of fit of a surrogate model. The normalized root mean square error (NRMS) and certainty coefficient (R2) are commonly used to evaluate the global goodness of fit [32]. The expression of the NRMS is(10)NRMS=∑i=1Nyi−y^i2∑i=1Nyi2.The value range of NRMS is [0, 1]. The closer its value is to 0, the better the goodness of fit is. Its threshold value is 0.1.The expression ofR2 is(11)R2=1−∑i=1nyi−y^i2∑i=1nyi−y¯2.The value range ofR2 is [0, 1]. The closer its value is to 1, the better the goodness of fit is. Its threshold value is 0.9.yi is the sample value, y¯ is the sample mean, and y^i is the surrogate model predicted value.


### 2.5. Optimization Method

The optimization problem with constraints can be expressed as(12)minxfx,x∈Rn,(13)s.t.gix≤0,hix=0.(12) is the optimization objective function. (13) is the optimization constraint, where gix is an inequality constraint and hix is an equality constraint. When solving the above two equations, in order to obtain reliable optimization results, it is often necessary to resort to optimization algorithms. Optimization algorithms can be divided into numerical and exploratory types. Among them, the numerical optimization algorithm has a faster calculation speed and fewer iterations, but it is easy to fall into a local minimum. Representative algorithms include NLPQLP (continuous quadratic programming method), LSGRG (generalized gradient descent method), and so on. Exploratory optimization technology can search the whole world and is not easy to fall into the local minimum, but there are relatively many iterations. Among them, the representative algorithm is genetic algorithm, and the process is shown in Figure 3. Commonly used genetic algorithms include MIGA (multi-island genetic algorithm), NSGA-II (nondominant sorting genetic algorithm), and so on.Figure 3
The basic process of genetic algorithm.


## 2.1. Parameter Optimization Process Based on FE Model

In the FE model, due to the model scale, contact nonlinearity, material nonlinearity, and other issues, it often takes a long time to simulate. At the same time, optimization research needs to modify the parameters multiple times for iterative calculations, so there are many repeated calculations. If the FE model is directly called, it will consume a lot of time and be inefficient. Therefore, in order to improve the optimization efficiency under the premise of ensuring accuracy, the optimization method shown in Figure1 can be used for the complicated FE model.Figure 1
Bottom cover optimization method.According to Figure1, firstly, the SA method is used to screen the optimized parameters, and then the optimal Latin hypercube sampling or orthogonal test is used to sample in the range to obtain a uniform and representative parameter sample set. Use the Python program to read the samples in order, modify the parameters of the FE model in ABAQUS, submit calculations, and read the stresses at the key points. Summarize the stress values under different samples and use them as input and output parameters. Use the above input and output parameters as samples and import them into the Isight software to fit the surrogate model and evaluate the goodness of fit. Based on the surrogate model, optimization algorithms such as steepest descent method and genetic algorithm are used for optimization. Then, the optimal solution of the model is obtained and imported into ABAQUS for verification.


## 2.2. Parameter SA Method

When there are many optimization parameters, the amount of calculation in sampling calculation and optimization calculation is large. The parameter SA method can eliminate the parameters that have little influence on the result. Assuming that there arenparameters, there is the following functional relationship between thei-th parameterpiand the response r:(1)r=rp1,p2,p3,…,pn.The sensitivityεi of different parameters can be expressed by the following formula:(2)εi=∂rp1,p2,p3,…,pn∂pi,i=1,2,3,…,n.It is often a discrete variable in engineering applications. Its sensitivityεi can be expressed by the finite difference method:(3)εi=rpi+Δpi−rpiΔpi,i=1,2,3,…,n.However, when calculating the sensitivity through the above formula, the influence of the parameter scale and the variation range is ignored. When the parameter scale is large, the calculated sensitivity is often small. Therefore, the original parameter valuepi0 and the original response value r0 are used as scale factors, and the sensitivity parameters are normalized:(4)εi=rpi+Δpi−rpi/r0Δpi/pi0,i=1,2,3,…,n.In order to improve the accuracy of the sensitivity parameter, takem different difference step size Δpij (j=1,2,3,…,m, including the original value, Δpi1=0) within the parameter change range. The least square method is used to fit it into a proportional function relationship of ri=kipi, and the absolute value of the slope ki is taken as the sensitivity parameter:(5)εi=ki=∑j=1mRijPij∑j=1mRij2,i=1,2,3,…,n,j=1,2,3,…,m,where(6)Rij=rpi+Δpij−rpir0,Pij=Δpijpi0.


## 2.3. Sample Acquisition Method Based on FE Model

According to the optimized parameters and their ranges, a set of representative input parameters needs to be extracted. The optimal Latin hypercube sampling method has strong filling ability, uniform sample distribution, and good representativeness. Assuming that a total ofm samples are required, m small hypercubes are randomly selected in the n-dimensional hypercube space composed of n design variables to ensure that each small hypercube is unique in each design variable interval. That is, each design variable will be sampled only once at each level. In this way, a Latin hypercube design matrix with m samples under n design variables is obtained. At the same time, through the optimization algorithm, the Euclidean distance between different sample points is minimized, and the evenly distributed and representative sampling results are obtained. This paper uses the optimal Latin hypercube method to sample the parameters. Generate the optimal Latin hypercube sampling results in Isight software as input samples.The FE model is established based on the input parameters, and the output parameters can be obtained through FE simulation. ABAQUS has a powerful secondary development function. Using Python programs, it can perform secondary development on the preprocessing and postprocessing parts of ABAQUS, achieve automatic modeling, automatic submission calculations, and automatic postprocessing batch calculations, which can save simulation time.Figure2 shows the parameter flow of using the Python program to obtain the output. First read a set of input parameters from the input parameter set and import the ABAQUS library function into the program, and the ABAQUS library function contains members that can be used for modeling. The members in the Model object under the Mdb object can be used for preprocessing work such as building 3D models, defining material properties, meshing, and defining loads and constraints. The Job object can submit calculations to the model. Each member under the Odb object can read the field output and history output results in the calculated odb result file to obtain the required output parameters. In this loop, the output parameters under different input parameters can be obtained, and the output parameter set can be summarized. The input parameter set and the output parameter set are combined as surrogate model training samples.Figure 2
Sample acquisition process based on FE parameter model.


## 2.4. Surrogate Model Establishment Method

Isight software provides a variety of surrogate model fitting algorithms, including response surface model (RSM), radial basis function neural network (RBFNN) model, Kriging model, and so on.


### 2.4.1. Response Surface Model (RSM)

Taking the fourth-order response surface model as an example, the surrogate model fitting formula is shown in the following equation:(7)y^=β0+∑i=1Mβixi+∑i=1MβM+ixi2+∑i=1Mβ2M+ixi3+∑i=1Mβ3M+ixi4+∑i≠jMβijxixj,where M is the number of input variables; xi represents the input variables; y^ is the output value; and βi represents the coefficients.


### 2.4.2. Kriging Model

The expression of the Kriging model is shown in the following equation:(8)y^=fTxβ+Zx,where fTxβ is the global regression model; Zx is the random fluctuation; and y^ is the output value.


### 2.4.3. Radial Basis Function Neural Network (RBFNN) Model

In the RBFNN model, the radial function is the basis function of the model, and its independent variable is the Euclidean distance between the measured point and the input point. (3) is the basic form of the radial basis function.(9)y^=∑j=1MHjrwj=HTrw,where wj is the weight; M is the number of samples; Hjr is the radial function; and y^ is the output value. By adjusting the weights, the radial basis function can be used to fit different models.There are many indicators for evaluating the goodness of fit of a surrogate model. The normalized root mean square error (NRMS) and certainty coefficient (R2) are commonly used to evaluate the global goodness of fit [32]. The expression of the NRMS is(10)NRMS=∑i=1Nyi−y^i2∑i=1Nyi2.The value range of NRMS is [0, 1]. The closer its value is to 0, the better the goodness of fit is. Its threshold value is 0.1.The expression ofR2 is(11)R2=1−∑i=1nyi−y^i2∑i=1nyi−y¯2.The value range ofR2 is [0, 1]. The closer its value is to 1, the better the goodness of fit is. Its threshold value is 0.9.yi is the sample value, y¯ is the sample mean, and y^i is the surrogate model predicted value.


## 2.4.1. Response Surface Model (RSM)

Taking the fourth-order response surface model as an example, the surrogate model fitting formula is shown in the following equation:(7)y^=β0+∑i=1Mβixi+∑i=1MβM+ixi2+∑i=1Mβ2M+ixi3+∑i=1Mβ3M+ixi4+∑i≠jMβijxixj,where M is the number of input variables; xi represents the input variables; y^ is the output value; and βi represents the coefficients.


## 2.4.2. Kriging Model

The expression of the Kriging model is shown in the following equation:(8)y^=fTxβ+Zx,where fTxβ is the global regression model; Zx is the random fluctuation; and y^ is the output value.


## 2.4.3. Radial Basis Function Neural Network (RBFNN) Model

In the RBFNN model, the radial function is the basis function of the model, and its independent variable is the Euclidean distance between the measured point and the input point. (3) is the basic form of the radial basis function.(9)y^=∑j=1MHjrwj=HTrw,where wj is the weight; M is the number of samples; Hjr is the radial function; and y^ is the output value. By adjusting the weights, the radial basis function can be used to fit different models.There are many indicators for evaluating the goodness of fit of a surrogate model. The normalized root mean square error (NRMS) and certainty coefficient (R2) are commonly used to evaluate the global goodness of fit [32]. The expression of the NRMS is(10)NRMS=∑i=1Nyi−y^i2∑i=1Nyi2.The value range of NRMS is [0, 1]. The closer its value is to 0, the better the goodness of fit is. Its threshold value is 0.1.The expression ofR2 is(11)R2=1−∑i=1nyi−y^i2∑i=1nyi−y¯2.The value range ofR2 is [0, 1]. The closer its value is to 1, the better the goodness of fit is. Its threshold value is 0.9.yi is the sample value, y¯ is the sample mean, and y^i is the surrogate model predicted value.


## 2.5. Optimization Method

The optimization problem with constraints can be expressed as(12)minxfx,x∈Rn,(13)s.t.gix≤0,hix=0.(12) is the optimization objective function. (13) is the optimization constraint, where gix is an inequality constraint and hix is an equality constraint. When solving the above two equations, in order to obtain reliable optimization results, it is often necessary to resort to optimization algorithms. Optimization algorithms can be divided into numerical and exploratory types. Among them, the numerical optimization algorithm has a faster calculation speed and fewer iterations, but it is easy to fall into a local minimum. Representative algorithms include NLPQLP (continuous quadratic programming method), LSGRG (generalized gradient descent method), and so on. Exploratory optimization technology can search the whole world and is not easy to fall into the local minimum, but there are relatively many iterations. Among them, the representative algorithm is genetic algorithm, and the process is shown in Figure 3. Commonly used genetic algorithms include MIGA (multi-island genetic algorithm), NSGA-II (nondominant sorting genetic algorithm), and so on.Figure 3
The basic process of genetic algorithm.


## 3. Establishment and Verification of the Equipment Cabin Bottom Cover FE Model

The equipment cabin bottom cover structure is shown in Figure4. The main body is a flat plate, which is reinforced by three rectangular tube beams. The plate size is 1144 × 576 mm. The flat plate and rectangular tube beams are formed by bending 2 mm thick cold-rolled SUS304 stainless steel plates. The bottom cabin is fixed to the side beam of the equipment cabin by a total of eight bolts and backing plates at both ends.Figure 4
Structure of bottom cover.


### 3.1. Static Strength Test of Bottom Cover

The static strength test of the equipment cabin bottom cover is carried out under the conditions of the indoor fatigue test bench, and the stress measurement points are arranged at the key positions of the bottom cover as shown in Figure5. Vacuum suction cups are uniformly arranged on the surface of the bottom cover to simulate the aerodynamic uniform load. The MTS fatigue testing machine is used to load the aerodynamic load under tension and compression at 4 kPa. The static strength test method and tooling under aerodynamic load are as described in the literature [33]. After the loading force is stable, use TDS-530 static data acquisition instrument to collect strain data. The loading tooling is shown in Figure 6.Figure 5
Bottom cover measurement point layout.Figure 6
Bottom cover loading tool.


### 3.2. Bottom Cover FE Model

Based on the 3D model of the bottom cover, the FE model of the bottom cover is established in ABAQUS. The flat plate and rectangular tube beam of the bottom cover are extracted from the midsurface and divided into shell elements. Divide the backing plate into solid elements. Both ends simplify the equipment cabin bracket to a discrete rigid body, imitating the fixing of the equipment cabin bracket to the bottom cover. The FE model is shown in Figure7.Figure 7
FE model of bottom cover.The position of the rivet hole between the flat plate, rectangular tube beam, and backing plate adopts beam element and rigid element to imitate riveting, and the contact pairs are set between each surface as shown in Table1. A fixed constraint is applied to the equipment cabin bracket and backing plate bolts, and a uniformly distributed aerodynamic load of ±4 kPa is applied to the surface of the bottom cover. Simulations show that the model can converge reliably.Table 1
Contact attribute settings for each contact pair.
No.Master surfaceSlave surfaceDiscretization methodContact property options1Flat plate lower surfaceRectangular tube beam upper surfaceSurface to surfaceHard contact friction coefficient: 0.152Flat plate lower surfaceBacking plate upper surface3Equipment cabin bracket lower surfaceFlat plate upper surface


### 3.3. Validation of the FE Model

The measurement points with relatively large stress values and relatively small stress gradients are selected to prevent the effect of stress concentration and the zero point error of the strain gauge from affecting the results. The results are compared with the output results of the FE model to verify the FE model. The comparison between the FE simulation results and the test results of some measuring points is shown in Table2:Table 2
Comparison of FE simulation and test results of some measuring points.
Measurement pointPull 4 kPa uniform loadPush 4 kPa uniform loadMeasured stress/MPaSimulation stress/MPaMeasured stress/MPaSimulation stress/MPa2−10.4−9.05.76.63−14.9−12.712.312.24−7.6−9.06.06.712−31.8−37.122.426.113−55.7−64.340.448.914−36.8−36.824.627.315−36.0−36.224.027.716−58.1−65.241.548.017−37.9−36.723.727.6In Table2, measuring points 2, 3, and 4 are the corner positions of the edge of the flat plate, and measurement points 12 to 17 are the process hole edges of the middle rectangular tube beam. In comparison, with the exception of the relatively large gap between measurement point 13 and measurement point 16, the measured stresses of the remaining measuring points are relatively close to the simulated stresses, indicating that the boundary condition settings of the FE model are basically consistent with the actual model. Measuring point 13 and the measuring point 16 are located at the edge of the center process hole rectangular tube beam, where the stress gradient is relatively large due to the stress concentration effect. Therefore, the attachment position of the strain gauge has a greater influence on the final result. The test result is different from the result of the FE model, but the trend is the same, which does not affect the direction and accuracy of the optimization.Through FE analysis, under aerodynamic loads of ±4 kPa, the maximum normal stress of the bottom plate appears on the edge of the center process hole rectangular tube beam (its location is marked by the box in Figure8), and the direction is along the length of the rectangular tube beam. The stress distribution at the edge of the process hole and the maximum nodal stress value are shown in Figure 8.Figure 8
Stress distribution on the edge of the center process hole rectangular tube beam. (a) Push working condition. (b) Pull working condition.
(a)(b)Under pull working conditions, the maximum nodal stress is −130.7 MPa, and the maximum integration point stress is −138.2 MPa read by Python program; under push working conditions, the maximum nodal stress is 114.1 MPa, and the maximum integration point stress is 119.6 MPa. The nodal stress is obtained by extrapolating the integration point stress. In the following optimization calculations, the integration point stress is used.In summary, the FE model can imitate the actual model well, and it can converge reliably, ensuring the accuracy of the optimization.


### 3.4. Calculation of Equivalent Stress under Symmetrical Cycle

Due to the influence of factors such as contact nonlinearity, when the bottom cover is subjected to a symmetrical cycle of pull and push load, its stress response is not necessarily a symmetrical cycle. The FE method is used to obtain the stress peak-valley value, average stress, and stress amplitude of the bottom cover under ±4 kPa aerodynamic load. The equivalent stressσ−1 under symmetrical cycles can be calculated by the Goodman equation (hereinafter referred to as equivalent stress):(14)σ−1=σa1−σm/σu,where σa is the stress amplitude, σm is the average stress, and σu is the ultimate strength of the material. Tensile test is performed on the small sample of the bottom plate on the MTS-810 universal material testing machine, and the mechanical properties are shown in Table 3.Table 3
Mechanical properties of SUS304 cold-rolled stainless steel plate.
MaterialYield strengthσb/MPaUltimate strengthσu/MPaSUS304320660In ABAQUS, the integration point stress of each element under pull and push conditions can be read through the Python program. Then, the equivalent stress of each element at the integration point can be calculated by (14).


## 3.1. Static Strength Test of Bottom Cover

The static strength test of the equipment cabin bottom cover is carried out under the conditions of the indoor fatigue test bench, and the stress measurement points are arranged at the key positions of the bottom cover as shown in Figure5. Vacuum suction cups are uniformly arranged on the surface of the bottom cover to simulate the aerodynamic uniform load. The MTS fatigue testing machine is used to load the aerodynamic load under tension and compression at 4 kPa. The static strength test method and tooling under aerodynamic load are as described in the literature [33]. After the loading force is stable, use TDS-530 static data acquisition instrument to collect strain data. The loading tooling is shown in Figure 6.Figure 5
Bottom cover measurement point layout.Figure 6
Bottom cover loading tool.


## 3.2. Bottom Cover FE Model

Based on the 3D model of the bottom cover, the FE model of the bottom cover is established in ABAQUS. The flat plate and rectangular tube beam of the bottom cover are extracted from the midsurface and divided into shell elements. Divide the backing plate into solid elements. Both ends simplify the equipment cabin bracket to a discrete rigid body, imitating the fixing of the equipment cabin bracket to the bottom cover. The FE model is shown in Figure7.Figure 7
FE model of bottom cover.The position of the rivet hole between the flat plate, rectangular tube beam, and backing plate adopts beam element and rigid element to imitate riveting, and the contact pairs are set between each surface as shown in Table1. A fixed constraint is applied to the equipment cabin bracket and backing plate bolts, and a uniformly distributed aerodynamic load of ±4 kPa is applied to the surface of the bottom cover. Simulations show that the model can converge reliably.Table 1
Contact attribute settings for each contact pair.
No.Master surfaceSlave surfaceDiscretization methodContact property options1Flat plate lower surfaceRectangular tube beam upper surfaceSurface to surfaceHard contact friction coefficient: 0.152Flat plate lower surfaceBacking plate upper surface3Equipment cabin bracket lower surfaceFlat plate upper surface


## 3.3. Validation of the FE Model

The measurement points with relatively large stress values and relatively small stress gradients are selected to prevent the effect of stress concentration and the zero point error of the strain gauge from affecting the results. The results are compared with the output results of the FE model to verify the FE model. The comparison between the FE simulation results and the test results of some measuring points is shown in Table2:Table 2
Comparison of FE simulation and test results of some measuring points.
Measurement pointPull 4 kPa uniform loadPush 4 kPa uniform loadMeasured stress/MPaSimulation stress/MPaMeasured stress/MPaSimulation stress/MPa2−10.4−9.05.76.63−14.9−12.712.312.24−7.6−9.06.06.712−31.8−37.122.426.113−55.7−64.340.448.914−36.8−36.824.627.315−36.0−36.224.027.716−58.1−65.241.548.017−37.9−36.723.727.6In Table2, measuring points 2, 3, and 4 are the corner positions of the edge of the flat plate, and measurement points 12 to 17 are the process hole edges of the middle rectangular tube beam. In comparison, with the exception of the relatively large gap between measurement point 13 and measurement point 16, the measured stresses of the remaining measuring points are relatively close to the simulated stresses, indicating that the boundary condition settings of the FE model are basically consistent with the actual model. Measuring point 13 and the measuring point 16 are located at the edge of the center process hole rectangular tube beam, where the stress gradient is relatively large due to the stress concentration effect. Therefore, the attachment position of the strain gauge has a greater influence on the final result. The test result is different from the result of the FE model, but the trend is the same, which does not affect the direction and accuracy of the optimization.Through FE analysis, under aerodynamic loads of ±4 kPa, the maximum normal stress of the bottom plate appears on the edge of the center process hole rectangular tube beam (its location is marked by the box in Figure8), and the direction is along the length of the rectangular tube beam. The stress distribution at the edge of the process hole and the maximum nodal stress value are shown in Figure 8.Figure 8
Stress distribution on the edge of the center process hole rectangular tube beam. (a) Push working condition. (b) Pull working condition.
(a)(b)Under pull working conditions, the maximum nodal stress is −130.7 MPa, and the maximum integration point stress is −138.2 MPa read by Python program; under push working conditions, the maximum nodal stress is 114.1 MPa, and the maximum integration point stress is 119.6 MPa. The nodal stress is obtained by extrapolating the integration point stress. In the following optimization calculations, the integration point stress is used.In summary, the FE model can imitate the actual model well, and it can converge reliably, ensuring the accuracy of the optimization.


## 3.4. Calculation of Equivalent Stress under Symmetrical Cycle

Due to the influence of factors such as contact nonlinearity, when the bottom cover is subjected to a symmetrical cycle of pull and push load, its stress response is not necessarily a symmetrical cycle. The FE method is used to obtain the stress peak-valley value, average stress, and stress amplitude of the bottom cover under ±4 kPa aerodynamic load. The equivalent stressσ−1 under symmetrical cycles can be calculated by the Goodman equation (hereinafter referred to as equivalent stress):(14)σ−1=σa1−σm/σu,where σa is the stress amplitude, σm is the average stress, and σu is the ultimate strength of the material. Tensile test is performed on the small sample of the bottom plate on the MTS-810 universal material testing machine, and the mechanical properties are shown in Table 3.Table 3
Mechanical properties of SUS304 cold-rolled stainless steel plate.
MaterialYield strengthσb/MPaUltimate strengthσu/MPaSUS304320660In ABAQUS, the integration point stress of each element under pull and push conditions can be read through the Python program. Then, the equivalent stress of each element at the integration point can be calculated by (14).


## 4. Structural Parameter Optimization Design of Equipment Cabin Bottom Cover

### 4.1. Parameter Selection and SA

Combined with the survey results, cracks mostly appeared in the weak points of the hole edge of the rectangular tube beam. After FE analysis and static strength bench test, the edge of the center process hole rectangular tube beam (measurement points 13 and 16 in Figure5) has relatively high stress, which is likely to be the initiation of cracks. Taking into account factors such as bottom cover assembly and manufacturability, six parameters are selected as shown in Figure 9.Figure 9
Parameter determination.Among them,d is the distance between the rivet hole and the plate end, d1 is the lateral hole distance of the rivet hole, p is the longitudinal spacing of the rectangular tube beam, w is the width of the rectangular tube beam midsurface, h is the height of the rectangular tube beam midsurface, and t is the thickness of the rectangular tube beam. The parameters d and d1 are related to the lateral number of rivets b:(15)d1=dc−2db,where dc is the lateral length of the flat plate, and the original values and value ranges of each parameter are shown in Table 4.Table 4
The original value of each parameter and its range.
No.ParameterOriginal valuesLower limitUpper limit1d60 mm30 mm100 mm2p193 mm168 mm243 mm3w35 mm35 mm50 mm4h28 mm12 mm50 mm5t2 mm1 mm3 mm6b9711Using the SA method described in Section2.2, the stress-influence sensitivity of each parameter under the two working conditions of pull and push at 4 kPa aerodynamic loads is shown in Figure 10.Figure 10
Sensitivity index of different parameters.It can be seen from Figure10that the sensitivity of the parametersh, t, andwis high, indicating that their changes have the most significant impact on the stress. These three parameters are selected as the optimization parameters, and the other parameters have less impact and are ignored in the optimization.


### 4.2. Parameter Sampling and Calculation

Based on the selected parameters, the range of values is the same as that shown in Table3, and the optimal Latin hypercube method is used to sample them. In order to meet the fitting needs of the surrogate model, the sampled data are initially set to 50 groups. The distribution in the parameter space is shown in Figure 11.Figure 11
Parameter sample space.It can be seen from Figure11 that the distribution of each sample in the parameter space is relatively uniform and representative. Use the Python program to input these 50 groups of samples into ABAQUS to establish a parameter model, complete the preprocessing of the model, and run it in batches. Read the maximum normal stress of the process hole edge of the square tube beam under two different working conditions of the flat plate under push and pull of 4 kPa aerodynamic load and calculate the equivalent stress as the output parameter in combination with (14). The 50 groups of input and output parameters after all simulations are completed and are shown in Table 5.Table 5
Surrogate model training samples.
No.w/mmh/mmt/mmEquivalent stress/MPa119.762.5548.7878.0236.042.1440.2067.2332.161.5738.67115.6446.901.3739.9085.9536.821.0840.82129.0638.371.6142.6573.9712.781.7436.53218.2823.631.9047.5595.8950.002.4746.3331.31012.002.1846.63127.81143.802.9637.7637.41234.492.6337.1459.91325.182.4339.5982.41428.291.9442.9686.51535.271.8246.9465.81629.062.5144.1862.21715.102.8440.51113.21817.431.2938.98211.31931.392.3548.4754.12015.881.0443.57241.22130.611.4550.0088.72224.411.4935.31191.92320.532.7135.92116.22427.511.0038.06199.82542.241.4149.3978.02644.571.1245.1084.92718.202.2242.96110.12845.352.0249.08113.22918.982.8044.8044.93025.961.3342.3579.33141.472.5949.69142.53222.861.1647.24162.83313.552.3138.3733.23433.711.2546.02151.53516.651.6543.88104.03626.733.0041.73150.83739.922.2744.4961.33839.141.2036.2246.43947.671.9841.12124.54043.021.7836.8448.14114.331.5348.1669.84221.312.0635.00160.34349.222.6742.04161.04440.692.8845.7131.84532.162.0635.6135.54622.081.8639.2997.54737.592.7641.43121.44848.451.6945.4145.34946.122.3937.4556.25029.842.9247.8644.5The 50 sets of input and output parameters in Table5 can be used as training samples for the surrogate model and input into Isight for approximate fitting.


### 4.3. Surrogate Model Fitting

Use different fitting models to fit the training samples in Table5. Based on the cross-validation method, two indexes of NRMS and R2 are used to evaluate the goodness of fit. The goodness of fit indexes under different models are obtained as shown in Table 6.Table 6
Goodness of fit under different fitting algorithms.
Fitting modelGoodness of fitNRMSR2Second-order RSM model0.0330.982Third-order RSM model0.0320.984Fourth-order RSM model0.0320.984RBFNN model0.0280.988Kriging model0.0800.900Chebyshev polynomial model0.0320.984From Table6, the indexes of the RBFNN model are all the best, so the RBFNN model is used as the surrogate model of the original model. The comparison between the predicted value of the output and the original value is shown in Figure 12.Figure 12
Comparison between predicted value and original value.It can be seen from Figure12that the distribution of the predicted value and the original value points are all around the straight line y=x, indicating that the fitting accuracy is high. The fitted surrogate model can be used as a simplified model to replace the original model for optimization.


### 4.4. Parameter Optimization Solution and Verification

Based on the above surrogate model, it can be optimized and solved in Isight. The optimized value range of each parameter is shown in Table4. Determine the optimization goal to minimize the equivalent stress:(16)minσ−1.In order to control the weight of the optimized bottom cover, corresponding constraint conditions need to be set. Since the above parameters only relate to the cross-sectional area of the rectangular tube beam, the weight of the bottom cover can be controlled by controlling the optimized cross-sectional area of the rectangular tube beam. In the original structure, the square tube beam midsurface heighth=28mm, midsurface width w=35mm, and thickness t=2mm; ignoring rounded corners, its area is S=35+28×2×2=252mm2. Therefore, the optimization constraints can be expressed as(17)s.t.S=2tw+h≤252.Using the original parameters as the initial values, after calculation, the iteration times and optimization results of the NLPQLP and MIGA optimization algorithms are obtained as shown in Table7. It can be seen that the convergence results of the two optimization methods are similar, so the optimization calculation results are credible. Among them, the numerical optimization method is less time-consuming than the exploratory optimization method. The optimization history of the equivalent stress under the two algorithms is shown in Figure 13 where the red point is the point that does not meet the constraint, the black point is the general point, and the green point is the optimal point.Table 7
Optimization results under different optimization algorithms.
Optimization algorithmsIteration timesOptimization resultw/mmh/mmt/mmNLPQLP7443.33339.1511.5276MIGA sub-population size: 40; number of generations: 50; number of islands: 102000144.34239.3081.5063Figure 13
Optimization iteration process. (a) NLPQLP. (b) MIGA.
(a)(b)From the above optimization process, it can be seen that the search speed of the NLPQLP algorithm is faster, but the number of iterations has reached 74. The optimization speed of the MIGA method is slower, but it can perform a global search in the parameter space.Based on the optimization results in Table7, round the parameters and input them into ABAQUS for FE simulation verification. The output and the original model are shown in Table 8.Table 8
Comparison between the optimized model and the original model.
w/mmh/mmt/mmS/mm2σ−1/MPaOriginal model35.028.02.0252.0126.7Optimized model44.039.01.5249.078.9Optimization rate1.2%37.7%It is shown in Table8 that the equivalent stress of the hole edge of the optimized model is significantly optimized compared with the original model, the optimization rate reaches 37.7%, and its weight is also reduced. The optimization effect is significant.


## 4.1. Parameter Selection and SA

Combined with the survey results, cracks mostly appeared in the weak points of the hole edge of the rectangular tube beam. After FE analysis and static strength bench test, the edge of the center process hole rectangular tube beam (measurement points 13 and 16 in Figure5) has relatively high stress, which is likely to be the initiation of cracks. Taking into account factors such as bottom cover assembly and manufacturability, six parameters are selected as shown in Figure 9.Figure 9
Parameter determination.Among them,d is the distance between the rivet hole and the plate end, d1 is the lateral hole distance of the rivet hole, p is the longitudinal spacing of the rectangular tube beam, w is the width of the rectangular tube beam midsurface, h is the height of the rectangular tube beam midsurface, and t is the thickness of the rectangular tube beam. The parameters d and d1 are related to the lateral number of rivets b:(15)d1=dc−2db,where dc is the lateral length of the flat plate, and the original values and value ranges of each parameter are shown in Table 4.Table 4
The original value of each parameter and its range.
No.ParameterOriginal valuesLower limitUpper limit1d60 mm30 mm100 mm2p193 mm168 mm243 mm3w35 mm35 mm50 mm4h28 mm12 mm50 mm5t2 mm1 mm3 mm6b9711Using the SA method described in Section2.2, the stress-influence sensitivity of each parameter under the two working conditions of pull and push at 4 kPa aerodynamic loads is shown in Figure 10.Figure 10
Sensitivity index of different parameters.It can be seen from Figure10that the sensitivity of the parametersh, t, andwis high, indicating that their changes have the most significant impact on the stress. These three parameters are selected as the optimization parameters, and the other parameters have less impact and are ignored in the optimization.


## 4.2. Parameter Sampling and Calculation

Based on the selected parameters, the range of values is the same as that shown in Table3, and the optimal Latin hypercube method is used to sample them. In order to meet the fitting needs of the surrogate model, the sampled data are initially set to 50 groups. The distribution in the parameter space is shown in Figure 11.Figure 11
Parameter sample space.It can be seen from Figure11 that the distribution of each sample in the parameter space is relatively uniform and representative. Use the Python program to input these 50 groups of samples into ABAQUS to establish a parameter model, complete the preprocessing of the model, and run it in batches. Read the maximum normal stress of the process hole edge of the square tube beam under two different working conditions of the flat plate under push and pull of 4 kPa aerodynamic load and calculate the equivalent stress as the output parameter in combination with (14). The 50 groups of input and output parameters after all simulations are completed and are shown in Table 5.Table 5
Surrogate model training samples.
No.w/mmh/mmt/mmEquivalent stress/MPa119.762.5548.7878.0236.042.1440.2067.2332.161.5738.67115.6446.901.3739.9085.9536.821.0840.82129.0638.371.6142.6573.9712.781.7436.53218.2823.631.9047.5595.8950.002.4746.3331.31012.002.1846.63127.81143.802.9637.7637.41234.492.6337.1459.91325.182.4339.5982.41428.291.9442.9686.51535.271.8246.9465.81629.062.5144.1862.21715.102.8440.51113.21817.431.2938.98211.31931.392.3548.4754.12015.881.0443.57241.22130.611.4550.0088.72224.411.4935.31191.92320.532.7135.92116.22427.511.0038.06199.82542.241.4149.3978.02644.571.1245.1084.92718.202.2242.96110.12845.352.0249.08113.22918.982.8044.8044.93025.961.3342.3579.33141.472.5949.69142.53222.861.1647.24162.83313.552.3138.3733.23433.711.2546.02151.53516.651.6543.88104.03626.733.0041.73150.83739.922.2744.4961.33839.141.2036.2246.43947.671.9841.12124.54043.021.7836.8448.14114.331.5348.1669.84221.312.0635.00160.34349.222.6742.04161.04440.692.8845.7131.84532.162.0635.6135.54622.081.8639.2997.54737.592.7641.43121.44848.451.6945.4145.34946.122.3937.4556.25029.842.9247.8644.5The 50 sets of input and output parameters in Table5 can be used as training samples for the surrogate model and input into Isight for approximate fitting.


## 4.3. Surrogate Model Fitting

Use different fitting models to fit the training samples in Table5. Based on the cross-validation method, two indexes of NRMS and R2 are used to evaluate the goodness of fit. The goodness of fit indexes under different models are obtained as shown in Table 6.Table 6
Goodness of fit under different fitting algorithms.
Fitting modelGoodness of fitNRMSR2Second-order RSM model0.0330.982Third-order RSM model0.0320.984Fourth-order RSM model0.0320.984RBFNN model0.0280.988Kriging model0.0800.900Chebyshev polynomial model0.0320.984From Table6, the indexes of the RBFNN model are all the best, so the RBFNN model is used as the surrogate model of the original model. The comparison between the predicted value of the output and the original value is shown in Figure 12.Figure 12
Comparison between predicted value and original value.It can be seen from Figure12that the distribution of the predicted value and the original value points are all around the straight line y=x, indicating that the fitting accuracy is high. The fitted surrogate model can be used as a simplified model to replace the original model for optimization.


## 4.4. Parameter Optimization Solution and Verification

Based on the above surrogate model, it can be optimized and solved in Isight. The optimized value range of each parameter is shown in Table4. Determine the optimization goal to minimize the equivalent stress:(16)minσ−1.In order to control the weight of the optimized bottom cover, corresponding constraint conditions need to be set. Since the above parameters only relate to the cross-sectional area of the rectangular tube beam, the weight of the bottom cover can be controlled by controlling the optimized cross-sectional area of the rectangular tube beam. In the original structure, the square tube beam midsurface heighth=28mm, midsurface width w=35mm, and thickness t=2mm; ignoring rounded corners, its area is S=35+28×2×2=252mm2. Therefore, the optimization constraints can be expressed as(17)s.t.S=2tw+h≤252.Using the original parameters as the initial values, after calculation, the iteration times and optimization results of the NLPQLP and MIGA optimization algorithms are obtained as shown in Table7. It can be seen that the convergence results of the two optimization methods are similar, so the optimization calculation results are credible. Among them, the numerical optimization method is less time-consuming than the exploratory optimization method. The optimization history of the equivalent stress under the two algorithms is shown in Figure 13 where the red point is the point that does not meet the constraint, the black point is the general point, and the green point is the optimal point.Table 7
Optimization results under different optimization algorithms.
Optimization algorithmsIteration timesOptimization resultw/mmh/mmt/mmNLPQLP7443.33339.1511.5276MIGA sub-population size: 40; number of generations: 50; number of islands: 102000144.34239.3081.5063Figure 13
Optimization iteration process. (a) NLPQLP. (b) MIGA.
(a)(b)From the above optimization process, it can be seen that the search speed of the NLPQLP algorithm is faster, but the number of iterations has reached 74. The optimization speed of the MIGA method is slower, but it can perform a global search in the parameter space.Based on the optimization results in Table7, round the parameters and input them into ABAQUS for FE simulation verification. The output and the original model are shown in Table 8.Table 8
Comparison between the optimized model and the original model.
w/mmh/mmt/mmS/mm2σ−1/MPaOriginal model35.028.02.0252.0126.7Optimized model44.039.01.5249.078.9Optimization rate1.2%37.7%It is shown in Table8 that the equivalent stress of the hole edge of the optimized model is significantly optimized compared with the original model, the optimization rate reaches 37.7%, and its weight is also reduced. The optimization effect is significant.


## 5. Summary

(1)
Establish the parameterized model of the bottom cover by ABAQUS + Python jointly development method. Obtain the output samples in batches according to the input samples, which eliminates tedious pre and postprocessing and can improve the efficiency of sample acquisition.(2)
Comparing the FE model with the bench test results of the equipment cabin bottom cover, the stress levels at each measurement point are basically the same, indicating that the boundary conditions and contact conditions of the FE model can basically simulate the actual model. Its accuracy can ensure the subsequent optimization design.(3)
The bottom cover is optimized based on the SA method and surrogate model. After optimization, the equivalent stress at the edge of the model hole decreased from 126.7 MPa to 78.9 MPa, which was optimized by 37.7%. The optimization effect was significant.(4)
This study only needs to perform 50 simulations on the FE model to obtain samples, and the surrogate model method can save thousands of repeated optimization calculations on the FE model. Therefore, based on the method in this paper, the optimization efficiency can be improved for complex FE models.



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*Source: 6204795-2022-08-22.xml*