1. Introduction
Sheet metal bending is a representative forming craft in manufacturing industries [
1]. “Springback” refers to the elastically driven change in shape that occurs following a sheet bending when forming loads are removed from the work piece, which causes problems such as increased tolerance and variability in subsequent forming operations, in assembly, and in the final part(s) [
2]. In air bending, therefore, precise bending has to be guided by a springback prediction model that represents the accurate relationship between punch stroke and forming angle [
3]. However, the factors for springback of sheet metals in air bending are so complicated that the springback prediction models have a certain degree of error no matter how accurate they are [
4]. In the air bending process, the forming angles need to gradually approach the expected value via repeating trial bending; that is, the punch stroke keeps being corrected until the tolerance of the forming angle is reached.
The punch stroke correction model, which affords a relationship between deviation of the bending angle and correction of punch stroke, is also critical for sheet bending and has been paid much less attention than the springback prediction models [
5,
6]. The deviation of the bending angle and the correction of punch stroke should be the differential or variational perturbation of the bending angle and punch stroke, respectively. Consequently, the punch stroke correction model could be regarded as the differential or variational form of the springback prediction model. If the springback prediction model presents an explicit analytical formula, the punch stroke correction model can be obtained by differentiation calculation. Some springback prediction models have been analytically deduced by means of mechanical analysis, considering the geometrical dimensions of forming dies and work pieces, the mechanical properties of sheet metals, processing parameters, etc. [
7,
8,
9]. However, these analytical models are not accurate enough, due to simplifications and assumptions during mechanical analysis because the influence of springback in sheet bending is highly nonlinear, involving many complicated factors. Thus, their differential forms—for instance, the punch stroke correction model—are also inaccurate, leading to poor efficiency in trial bending.
With the development of machine learning, data-driven statistical models have been proposed for the prediction of springback in sheet metal bending [
10,
11,
12,
13,
14]. The accuracy of the data-driven models depends on the large scale of data. Based on the design of the experiments, the data acquisition is generally implemented through a large number of real/virtual tests of air bending, considering the variations in each factor related to springback. Compared with real tests, the virtual bending tests by finite element modeling are suitable for the data-driven statistical models, due to their higher efficiency and lower costs [
15]. In practical application, the data-driven statistical models present high accuracy in the prediction of springback. The accurate springback prediction models were built by approximation methods, such as response surface methodology [
16,
17], artificial neural network [
18,
19,
20,
21,
22], and Kriging [
23]. The authors proposed a springback prediction model for air bending through the combination of genetic algorithm and backpropagation neural network (GA–BPNN), presenting high accuracy and great versatility.
However, it is infeasible to deduce the punch stroke correction model through differentiation calculation from the springback prediction models, which have remarkably complicated statistical forms. In the study, therefore, we aimed to propose a punch stroke correction model for trial tests in metal sheet bending, which would be able to guide the accurate correction of punch stroke with high efficiency. The model would be built from big data on springback in air bending through a GA–BPNN approach. Firstly, a large number of virtual bends would be implemented to obtain the big data. For comparison, additionally, another punch stroke correction model would be proposed based on dimensional analysis; that is, a semi-analytical method [
24]. Finally, practical sheet metal trial bending tests would be implemented in order to investigate and contrast the correction accuracy of punch stroke in the two models.
2. Research Methods
2.1. Modeling Principle
In order to establish a large-scale dataset, the virtual bending finite element tests were simulated. On the basis of the acquired dataset, the GA–BPNN prediction model was trained. The input parameters consisted of bending angle , elastic modulus E, yield strength , hardening coefficient K, hardening exponent n, thickness t, and groove width . The output parameter was punch stroke D.
Random angle deviations were generated and used to alter the original bend angles in the established dataset. Corresponding with new angles, the new punch strokes were obtained through calculation of the prediction model. Punch stroke compensation was defined as the offset before and after alteration. In this way, another large sample dataset was created that contained angle deviations and punch stroke compensations . According to the new dataset, a GA–BPNN and dimensional analysis were used to build the correction model.
2.1.1. GA–BPNN Model
Machine learning is a statistical modeling technique that enables a computer system to learn or recognize implicit relationships in given data without any explicit description. Abstract information or undiscovered phenomena can theoretically be modeled by means of machine learning as a result of its data-driven properties [
25,
26,
27].
Among diverse machine learning algorithms, artificial neural networks (ANNs) are popular because of their excellent modeling performance and wide range of applications for general approximation [
28,
29,
30]. The backpropagation neural network (BPNN) is a sort of ANN that is widely applied at present; its basic idea is to adjust and modify the connection weights and thresholds of the network through the reverse propagation of network output errors, so as to minimize the mean squared error of the output.
In general, the initial weight and bias of the network are generated randomly at the beginning of network training, which may result in local optima of the network. In this paper, a genetic algorithm (GA) was used for optimization the initial weight and bias of the network [
31]. Based on a GA–BPNN, the prediction model and correction model were obtained in this paper. The network model is shown in
Figure 1.
- (1)
Forward propagation of signals
The input variables need to be normalized to avoid adverse factors in the optimization process, which is denoted as
. The feedforward neural network propagates information by iterating the following formulae:
where
is the net input of neurons in layer
;
is the weight matrix from layer
to layer
;
is the output of neurons in layer
;
is the bias from layer
to layer
; and
is the activation function of neurons in layer
l. Equations (1) and (2) can also be combined and written as:
The final output of the network
can be obtained through the layer-by-layer transmission of information in the feedforward neural network. The whole network can be regarded as a compound function
. The input at the first level is defined as
, and the output of the whole function is
.
where
represent the weights and bias, respectively, of all layers in the network.
- (2)
Backpropagation
Each sample
in the given training set
is input to the feedforward neural network, and then
can be obtained, whose structural risk function of the dataset
is defined as:
where
and
are all the weight matrices and bias vectors in the network respectively;
is the regularization term to prevent overfitting;
is the super parameter, and the larger
is, the closer
is to 0.
The parameters of network can be learned through the gradient descent algorithm. In each iteration of the gradient descent algorithm, the parameters
and
of the
layer are updated as follows:
where
is the learning rate.
2.1.2. Mathematical Model of Dimensional Analysis
Dimensional analysis is an analytical method to establish mathematical models in the field of physics [
32,
33]. Based on dimensional analysis, the laws of physics can be explained by comparing the dimensions of independent and dependent variables. According to the principle of homogeneity of dimensions, the dimensions on both sides of the equals sign must be the same when mathematical expressions are used to express physical relations. The Buckingham π theorem can be expressed in a physical equation with n variables as:
where
variables are independent of one another, and the remaining (n − m) variables are independent. The physical relations can be expressed by (n − m) dimensionless variables as follows:
where
are (n − m) dimensionless variables. The main factors that affect the physical process should be ascertained. According to the dimensionless method, the functional relationship between the factors can be established. Through the combination of experiments and functional relationships, the exact mathematical expression is obtained.
2.2. Sample Range Definition for Dataset
The springback of sheet metal is affected by many factors, including material parameters, dimensions of V-dies and blanks, and processing parameters. To simplify the modeling process, in this paper, it is assumed that the sheet metal materials are independent of strain rate and strain path, obeying the Hill’48 anisotropic yield criterion and the Hollomon hardening model. The functional form of the hardening model can be expressed as follows:
where
E is the elastic modulus,
K is the hardening coefficient, and
n is the hardening exponent. Three above and the yield strength
were regarded as material factors. For a common bending process, an 88° V-die is usually desirable. The processing parameters include width of slot
and punch stroke
D, and the product factor is the thickness of sheet metal T. As mentioned above, 7 affecting factors were involved.
According to the distribution of sheet metal properties, the conventional working conditions of the bending process, and the standard thicknesses of sheet metals, the variation ranges of the 7 factors were determined under various conditions, as shown in
Table 1.
A Latin hypercube design was used to determine the sample distribution of the 7 factors. A total of 1732 combinations were obtained for the springback prediction model. It is important to note that the sample size of this article is only an example of the large sample number, and the actual sample can choose a larger number.
2.3. Acquisition of Finite Element Sample for Training Data
As the output of the prediction model, the springback angle of each combination was obtained via the finite element simulations of sheet metal bending. In order to simulate V-bending and springback, a combination of explicit and implicit methods was used. The element type of the sheet metal part was four-node shell element (S4R), and the friction coefficient was 0.1.
Perpendicular to the direction of the bending line, the minimum mesh size was 0.2 mm and the maximum size was 2.0 mm. Parallel to the direction of the bending line, the mesh size was 0.6 mm. Five integration points were set in the direction of thickness. The simulation process of bending and springback is shown in
Figure 2. Each angle after springback corresponding to each combination of factors was obtained. A total of 20 samples were randomly selected as test samples; samples for modeling and for testing are shown in
Table 2 and
Table 3, respectively.
All calculations were performed on a personal computer (LATOP-FM0CIDDQ Intel(R) Core(TM) i7-108575H CPU @ 2.30GHz(16CPUs), ~2.3GHz). Dataset was obtained through integration of ABAQUS and Isight. The GA-BPNN was established using MATLAB 2016a.
2.4. Mechanical Tests and Bending Tests
Five different sheet metals were selected for the tests, including mild steel HC220YD, stainless steel 304, aluminum alloy 5182, high-strength steel DP980, and copper H62. The widths of samples were processed to 20 mm. The sheets mentioned above were used for uniaxial tensile tests and bending experiments. The uniaxial tensile tests were performed on the electronic universal material testing machine (AGS-100kN, Shimadzu, Suzhou, China). The material performance parameters were obtained as shown in
Table 4.
Sheet metal air bending experiments were carried out with a computerized numerical control bending machine (WDB100-3100, JFMMRI-JIEMAI, Jinan, China). The V-shaped slot angle was 88°, the punch radius r was 1 mm, the V-shaped slot width
was 12 mm, and the punch round radius r was 1 mm. The punch stroke was measured with a grating ruler whose accuracy was within 2 µm. Bending molds and parts after forming are shown in
Figure 3.
Each sheet metal was subjected to four bending tests with different punch strokes. After bending, the springback angles were measured by angle ruler with ±0.08° accuracy, the data from which are listed in
Table 5. The accuracy of the simulations was verified by the comparison with the actual bending tests.
2.5. Data Acquisition for the Correction Model
The remaining samples were divided into 85% and 15% as training samples and verification samples, respectively. Then, the GA-BPNN springback prediction model with 7 factors was established.
E,
,
K,
n,
t,
and
were input parameters, and punch stroke D was the output parameter. The mean squared error (MSE) was used to measure the accuracy of the network:
The conclusion of the network structure research showed that the error was minimal in the network with the [
1,
2,
3,
4,
5,
6,
7] structure. The comparisons between the simulation samples (
Table 3) and the network prediction results, as well as the bending tests (
Table 5) and the network prediction results, are shown in
Table 6. As can be seen from
Table 6, the prediction deviation of the punch stroke prediction model was within 0.16 mm.
In general, to achieve a target bending angle, a trial bending needs to be carried out. Then, according to the difference from the target value, the forming angle is adjusted by the correction of the punch stroke and, therefore, the difference from the target value can be reduced. It usually takes three or four attempts to reach the target angle. Our punch stroke prediction model could ensure that the error of stroke prediction was within 0.2 mm. It was our aim to control the final sheet’s forming angle by fine-tuning the punch stroke, which is also the significance of the correction model. Based on the prediction model, the punch strokes
were obtained. A total of 1732 data points as
were randomly generated and evenly distributed in [−3°, 3°]. Each
was changed to
correspondingly. New angles were input to the prediction model, and the new punch strokes
were generated. Stroke difference
was defined as:
According to
with the other corresponding factors as input and
as output, the correction model could be established. Some training samples are shown in
Table 7, and testing samples are shown in
Table 8.
3. Results and Analysis
3.1. Punch Stroke Correction Model Based on a GA-BPNN
To optimize the topology of the neural network, different hidden layers and different neurons in hidden layers were studied (random weights and bias were used tentatively). First, we focused on the springback prediction model. The MSE values (sum of the mean squared errors of the training set and verification set) under different structures were obtained, as shown in
Figure 4. Early stopping was used to ensure that the model was not overfitting.
Compared with the nets containing one or two hidden layers, the accuracy of the network with three hidden layers is higher, and is also more stable. The MSE of the network with a [
1,
2,
3,
4,
5,
6,
7] architecture can be less than 1.258 (°^2). According to the parameters of the network structure study, we determined a [
1,
2,
3,
4,
5,
6,
7] fully connected architecture as the BPNN structure of the prediction model. The total number of network parameters thus determined was 545 (128 + 128 + 272 + 17). Then, a genetic algorithm was used to optimize the initial weights and bias of the network.
Based on experience from other studies and trial training, the following parameters worked well. A summary of the GA parameters is shown in
Table 9.
The same strategy and parameters were adopted for the punch stroke correction model, which is also a network with seven inputs and a single output. To illustrate the advantage of the GA in the stability of optimization, the decreasing loss trend of the training and validation datasets in the GA-BPNN correction model is shown in
Figure 5.
The GA-BPNN combines the advantages of efficiency and accuracy. After initial parameter optimization, only 5838 epochs are trained to reach the target, which is just half of the training process before GA optimization (12,483). After network training, the MSE could be less than 2.6872 × 10−4 mm2.
The regression coefficient of network training is shown in
Figure 6. Comparison between testing samples and network-predicted values is shown in
Figure 7a, and deviation of punch stroke compensations is shown in
Figure 7b.
As shown in
Figure 6 and
Figure 7, the deviation of the punch strokes can be controlled within 0.05 mm. From the above results, it can be concluded that the neural network model can correct the punch stroke with sufficient accuracy.
3.2. Punch Stroke Correction Model Based on Dimensional Analysis
In this work, the functional relationship between punch stroke compensation
and angle deviation
with elastic modulus
E, yield strength
, hardening coefficient
K, hardening exponent
n, sheet thickness
t, and groove width
could be expressed as:
The basic dimensions—including length (L), mass (M), and time (T)—were used.
and
are dimensionless, while the other physical quantity can be expressed as:
According to the π theorem, four dimensionless variables could be obtained from four fundamental solutions:
And
and
were written as:
Therefore, there was a function
as:
whose specific expression form was:
In order to avoid errors caused by different orders of magnitude between factors, each dimensionless variable was normalized before calculation. The least squares method was used for fitting. The parameters obtained by fitting were
= 0.41426,
= 0.01508,
= 0.0285,
= 0.0302,
= 0.00994, and
= −0.1584. The specific function was obtained as follows:
The model was tested with the data shown in
Table 8. Comparison between the test samples and the predicted values of the model is shown in
Figure 8a, and the deviation of the punch stroke compensation value is shown in
Figure 8b.
Comparing the results, using the punch stroke correction model based on dimensional analysis, the deviation of punch stroke compensation can be kept within 0.15 mm, while the punch stroke correction model based on a GA-BPNN can keep it within 0.05 mm. The GA-BPNN model can predict punch stroke more accurately and control the forming angle to be closer to the target angle.
3.3. Application Examples
Three kinds of sheet metal were chosen for bending experiments with the universal testing machine (WQ4200, Changchun Kexin instrument institute, Changchun, China)—HC220YD mild steel, 304 stainless steel, and 5182 aluminum alloy—to further illustrate that the GA-BPNN punch stroke correction model could control the forming angle accurately by adjusting the punch stroke. The mechanical property parameters of the three materials are shown in
Table 4.
Three target angles were chosen for each material. The initial strokes were obtained according to the target angle by calculation of the GA-BPNN punch stroke prediction model. Based on the GA-BPNN punch stroke correction model, the strokes were adjusted by the deviation of the angles. The bending tests were performed with the universal material testing machine, as shown in
Figure 9, so that the punch stroke could be freely controlled. The punch strokes were measured on a grating scale. For the measurement of the forming angles, a digital protractor was used. The radius of punch
R was 1 mm; the width of the V-shaped groove
was 12 mm, and the radius of the punch fillet R was 1 mm.
The bending samples are shown in
Figure 10, and the experimental data are shown in
Table 10. Some samples could reach the target angles directly through the prediction model, while most samples could reach the target angles through one use of the correction model. All samples from the tests could achieve error precision within 0.5°.
It should be noted that the main purpose of this paper is to provide a correction method for studying the sheet metal bending springback, which cannot represent optimal accuracy. In practice, the accuracy of the correction model can be improved by further improving the finite element simulation accuracy and machine learning fitting accuracy to meet the requirements of actual conditions.