1 Introduction

Roll formed parts have many applications in several industries, such as the automobile industries, ship construction, and aerospace industries, to reduce the material weight of fabricated parts without degrading their performance [1]. In conventional roll forming processes for the fabrication of parts with constant cross sections, the initial blank is converted gradually to the target profile using a series of rotary rolls, as shown in Fig. 1a. In many applications, products with variable cross sections require weight loss and performance enhancement; this weight loss is possible by using forming processes such as roll forming, flexible roller forming, and hydro-forming [2,3,4,5]. Flexible roll forming was developed to overcome the limitations of the traditional roll forming process, and several studies have been performed on shape analysis and process-induced fracture [6, 7]. In flexible roll forming (FRF), the forming rolls have a linear and rotational motion on the bend line to produce the desired profile with the variable cross section, as illustrated in Fig. 1b.

Fig. 1
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a Traditional roll forming process. b Flexible roll forming

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Shape defects may occur owing to complex deformation in FRF. In general, non-uniform elongations of the material and the related non-homogeneous deformations over the thickness or width of the blank may result in waviness or curvatures. One of the major defects caused by FRF is longitudinal bow, which is the creation of a height deviation at the profile web in the longitudinal direction. The cause of the bow is the non-homogeneous distribution of the longitudinal strain or non-uniform elongation of the blank, as shown in Fig. 2a and b [8].

Fig. 2
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Longitudinal bow caused by flexible roll forming: a photograph and b schematic illustration

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Another major defect caused by FRF is wrinkling, which is instable deformation owing to increase the compression at the longitudinal direction of the flange. This is due to the complex deformations that occur in an area where there is a change in cross section, resulting in some portions of the flange being compressed in the longitudinal direction of the profile [9]. The occurrence of winkling during a FRF process is illustrated in Fig. 3a and b.

Fig. 3
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Wrinkling caused by flexible roll forming process: a photograph and b schematic illustration

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Most previous studies on traditional roll forming and FRF focused on investigating process parameters, the causes of defects in the product, and the effects of the process parameters on the occurrence of defects. Farzin et al. [10] presented the buckling limit of strain (BLS) as an important factor of the roll forming process. Their results showed that a product has no edge buckling when the longitudinal strain remaining in the longitudinal direction is 0, and the buckling rate is independent from the value of bend angle and is a function of the sheet material properties and the ratio of thickness to flange length. Salmani Tehrani et al. [11] analyzed the localized edge buckling in the cold roll forming process. They resulted that the cause of this defect is the development of the longitudinal strain in the process, reducing the local tensile strength, and increasing the compressive deformation. Gulceken et al. [12] simulated the FRF process using the finite element software MSC Marc. The roll design to produce the profiles with variable cross section in FRF was characterized. Larenga and Galdos [13] used local heating during FRF to reduce web warping. Kasaei et al. [9] used finite element analysis to investigate wrinkling in flange of the profile. According to the results of the study, wrinkling occurs when the longitudinal compressive strain is less than the compressive strain required to obtain the target geometry in the transition zone as calculated by mathematical modeling. Mohammadi et al. [14] used finite element simulation to examine the occurrence of web warping. The results showed that the main cause of this defect was insufficient strain at the edge of the wings of the profile in the transition zone.

Jiao et al. [15] studied the FRF process by developing an analytical model for predicting twisting web warping. The obtained results revealed that the longitudinal strain was inversely correlated with web warping. Woo et al. [16] investigated the shape defects in automotive parts produced by FRF. They concluded that the use of an FRF machine with a leveling roll would reduce longitudinal bow and edge wave defects. Ona et al. [17] examined the cause of web warping during FRF. They observed that the main cause of the phenomenon near a transition area was the occurrence of shrinkage during the forming of the area. Park et al. [18] investigated numerically and experimentally the defect of web warping. To reduce this defect, they proposed a new process called incremental counter forming (ICF). They controlled the longitudinal strain distribution on the profile flange by combination of forming and ICF parameters. They found that by increasing the longitudinal strain in convex and concave areas, the web warping defect can be reduced.

In previous studies that utilized optimization algorithms, efforts were made to achieve the optimal conditions for the fabrication of a product with minimal defects. Laouissi et al. [19] optimized the affected parameters of machining of gray cast iron using artificial neural networks (ANN), response surface methodology (RSM), and genetic algorithm (GA) methods. They showed the developed model for machining parameters using ANN and RSM is very useful tool for prediction propose, and the comparison of ANN with RSM showed that the prediction capacity of ANN method is more effective than the RSM approach. The optimal values of the machining parameters obtained from GA were close to the experimental results. Wiebenga et al. [20] controlled the occurrence of defects in the roll forming process using robust optimization. They showed that the scatter in the material properties is a significant factor in controlling the dimensional quality of the product.

They also showed that adjustment of the final rolls significantly improved the product quality by reducing the occurrence of defects and minimizing the destructive effects of scattered variables. Radovanovic [21] optimized the turning operation of AISI 1064 steel using three optimization approaches including Ip multi-media subsystem (IMS), multi-objective genetic algorithm (MOGA), and GA. Comparison of the three algorithm results with experiments showed that the IMS algorithm provides more appropriate solutions. Doriana et al. [22] optimized the hot forging process by combining of two methods of ANN and multi-objective optimization. They concluded that it was necessary to consider multiple processes when using global optimization methods to obtain optimal results. They also found that the simulation of tests using the finite element method could be replaced by the use of an artificial neural network (ANN), which is less sensitive to the problem dimensions compared with the design of experiment (DOE). Alizadeh and Omrani [23] integrated the robust multi-response Taguchi neural network with a CO2 laser cutting process. They used the robust optimization to control the uncertainty of the neural network results. Bacanin and Tuba [24] and Yazdi et al. [25] presented a modification of the artificial bee colony (ABC) algorithm, referred to as genetically inspired artificial bee colony (GI-ABC). They showed that GI-ABC afforded an improvement on the performance of the ABC algorithm by applying uniform crossover and mutation operators obtained from GA. Yaghoobi et al. [26] optimized the pressure path in the hydroforming process using the neural fuzzy method and GA. They investigated the effect of the pressure path on the maximum thinning on the critical areas of the product by developing the adaptive neurofuzzy inference system (ANFIS) model based on simulation results and then the developed model is used as the objective function in the optimization process. High speed and avoiding trial and error and multiple simulations are the most important advantages of this method.

As mentioned above, most previous studies on FRF focused on a general analysis of the process, the conditions under which defects occur, and the parameters that affect the defects. Because of the complexity of the process and the need to achieve products with minimal geometric defects, multi-objective optimization was combined with ANN in the present study. The shape defects were minimized by optimizing the parameters that affected them, thereby enabling the achievement of a product with the desired bend angle. Numerical simulation of the FRF was used to investigate the effect of the process parameters on two defects. The numerical results were verified by experiments performed using a lab-scale FRF machine. The ANN was used to predict two defects as objective functions, and non-dominated sorting genetic algorithm II (NSGA-II) was subsequently used to optimize the input parameters for each target bend angle under the minimum conditions of the objective functions.

2 Finite element simulation

To investigate the longitudinal bow and wrinkling defects, FRF processes with and without leveling roll were simulated using the FEM code ABAQUS™ Implicit 6.14. The parameters that affect longitudinal bowing and wrinkling include the bend angle, sheet thickness, yield strength, and blank shape. Three blank shapes were considered in the analyses, namely, trapezoidal (tr), convex (cv), and concave (cc). To investigate the effects of the above parameters, an experiment was designed using the perfect factorial method. The experiment plan included 27 tests using target bend angles of 15–30° for two modes of FRF with and without leveling roll, which are 108 tests in total, as detailed in Table 1.

Table 1 Experiment plan
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In the simulation, the sheets were modeled as shells and the rolls were considered to be rigid. A pair of forming rolls moved in the y-direction and rotated around the z-axis to shape an entire sheet. The results of the uniaxial tensile testing of the sheets were applied to the simulation. The stress and strain results are given in Table 2. An elastic-plastic material model and the von Mises yield criterion were employed, with the assumption of piecewise linear isotropic hardening.

Table 2 Stress-strain values of three simulation materials
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For the simulation of the process, the rolls were considered as rigid body models, and the blank as a deformable part. Figure 4a-c show the geometries and dimensions of the three different blanks considered in this study.

Fig. 4
figure 4

Different blank geometries: a trapezoidal, b convex, and c concave (unit: mm)

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The blank was modeled as a shell using type S4R shell elements. Figure 5a shows the simulated profiles, while Fig. 5b shows the blank meshing. Because of the greater deformation in the bending and flange zones, 1 mm × 3 mm meshing was applied to both, while larger meshing between 1 mm × 3 mm and 9 mm × 3 mm was applied to the web zone. For definition of the contact condition between the blank and the rolls, the penalty approach was used.

Fig. 5
figure 5

Flexible roll forming process: a schematic of the process simulation and b areas with different meshing densities

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The CPU time to complete the FE simulation was approximately 4 h. The Coulomb friction model was used, with the coefficient of friction considered to be 0 [27]. Table 3 summarizes the simulation conditions.

Table 3 Simulation conditions
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3 Experiments

To verify the numerical results, experiments were performed using SPCC blanks and the conditions detailed in Table 4. The laboratory-scale three-stand FRF machine shown in Fig. 6 was employed for the experiments.

Table 4 Experimental conditions
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Fig. 6
figure 6

Laboratory-scale FRF machine

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The chemical composition and mechanical properties of SPCC, which is a type of aluminum, are summarized in Tables 5 and 6, respectively.

Table 5 Chemical composition of SPCC (wt%)
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Table 6 Mechanical properties of SPCC sheet
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To confirm the FEM simulation results, the longitudinal strain at the flange edge was measured for three different conditions (flange width of 35 mm; sheet thickness of 2 mm; and bend angles of 15°, 30°, and 45°). The experiments shown in Fig. 7a-c were performed using the specifications in Table 4. The longitudinal strain during FRF was measured using a resistance strain gauges.

Fig. 7
figure 7

Measuring of the longitudinal strain by a strain gauge and data logger: a FRF experimental setup, b location of the strain gauge, and c data logger set

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4 Optimization

In this study, the inputted effective parameters of the FRF process, such as the sheet thickness, yield strength, and blank shape, were optimally determined for each target bend angle. This was done to minimize the longitudinal bow and wrinkling, which were used as the two objective functions of the ANN-based multi-objective optimization. The ANN was first used to find both objective functions, and Non-dominated Sorting Genetic Algorithm II was then used to optimize the input parameters to minimize the objective functions for each target bend angle.

4.1 Design of back-propagation neural network

A artificial neural network (ANN) is a parallel operating system that simulates the human neuron. Each neuron receives information from neurons in a previous layer and then distributes it to neurons in the next layer. As shown in Fig. 8, the neural network designed to find the objective functions (longitudinal bow and wrinkling) in this study has three layers, namely, the input, hidden, and output layers. Based on the number of inputs, the input layer has four units.

Fig. 8
figure 8

Topology of the designed neural network

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Under the condition that the issue determines the number of specific training patterns, the number of the hidden layer units can be specified by trial and error. To do this, a back-propagation neural network (BPNN) was developed in MATLAB. The network was tested using a number of hidden layer units between 3 and 40 and with 24 training patterns. Considering that the purpose of the network was to find two objective functions, the number of output layer units was considered to be 1. It was thus not necessary to assume the input and output of the neural network. To accelerate the convergence of the network, different momentums and learning rates were used simultaneously. During the construction of the BPNN, many parameters had to be set. The initial learning rate was set to 0.001, while the number of hidden layer units was set to 1. Out of 108 data sets, 96 sets were selected as training data and 12 were used as test data. The effects of the number of hidden layer nodes, conversion function of the hidden layer, decrease and increase ratios of the training rate, and momentum on the quality of the network training were investigated. The training process was terminated when one of the two following convergence criteria was achieved:

  1. 1.

    Mean squared error (MSE) < 0.001.

  2. 2.

    Number of repetitions = 50,000.

The quality of the training was specified by the mean squared error and the average mean squared error for all the repetitions, which are given by Eqs. 1 and 2, respectively.

$$ MSE=\frac{1}{n}\sum \limits_{j=1}^m\sum \limits_{i=1}^n{\left({T}_{ij}-{P}_{ij}\right)}^2 $$
(1)
$$ \overline{MSE}=\frac{1}{r}\sum \limits_{q=1}^r MS{E}_q $$
(2)

A simple and effective means of increasing and improving the learning rate to prevent instability and oscillation of the network is the addition of a momentum sentence to the Storm Descending (SD) algorithm. The idea of the back-propagation algorithm is the addition of an intersection or movement size to each parameter of the perceptron multi-layer network, so that the parameter tends to move in a direction that reduces the energy function.

4.2 Non-dominated sorting genetic algorithm for optimization

4.2.1 Definition of optimization

Objective optimization involves the optimization (i.e., minimization or maximization) of a single or several objectives using a number of inequality or equality constraints simultaneously [28]. The problem is described in detail as follows:

Find x = (xi) ∀ i = 1, 2, …, Nparam such that

fi(x) is a minimum (or maximum) ∀i = 1, 2, …, Nobj.

subject to

$$ {g}_j(x)=0\forall j=1,2,\dots, M, $$
(3)
$$ {h}_k(x)\le 0\forall k=1,2,\dots, K, $$
(4)

where x is a vector include the Nparam design parameters, (fi)i = 1, . …, Nobj is the objective function, and Nobj is the number of objectives.

4.2.2 Non-dominated sorting and Pareto front

In this method, unlike single-objective optimization, there is no single optimal solution, but a set of solutions is created that none are dominate to the other, and this is called pareto optimal solutions.

Here, the constraint violation ℓ(X) of an individual X is defined as the sum of the violated constraint function values [29]:

$$ \ell (X)={\sum}_{j=1}^B\gamma \left({g}_j(X)\right){g}_j(X) $$
(5)

where γ is the Heaviside step function. A set of non-dominated individuals is used to form a Pareto-optimal front [29].

4.2.3 Tournament selection

Each individual competes in exactly two tournaments with randomly selected individuals, a procedure that imitates the survival of the fittest in nature [28].

4.2.4 Controlled elitism sorting

To preserve diversity, the effect of elitism is controlled by using geometric distribution to choose the number of individuals from each subpopulation [29]:

$$ {S}_q=S\frac{1-c}{1-{c}^w}{c}^{q-1} $$
(6)

To form a parent search population Pt + 1 (t denotes the generation) of size S, where 0 < c < 1 and w is the total number of ranked non-dominated individuals.

4.2.5 Crowding distance

The crowding distance of each individual with its nearest neighbors is shown by this parameter. The crowding distance parameter is calculated for each member of the group and computes the density of the solutions around a specific point in the population. In fact, in order to calculate the density of solutions around a specific point in a population, the average distance between two points is taken from both sides and along each of the targets [29]. Hence, if the two individuals have the same rank, each one has a larger crowding distance than each other is better [30].

4.2.6 Crossover and mutation

Uniform crossover and random uniform mutation were used to obtain the offspring population Qt + 1. The integer-based uniform crossover operator takes two distinct parent individuals and interchanges their corresponding binary bits with a probability 0 < pc ≤ 1. After this crossover, the mutation operator also interchanges the binary bits with a mutation probability 0 < pm < 0.5 [28].

5 Results and discussion

5.1 Finite element model validation

Figure 9a-c compare the longitudinal strains determined by simulation and experiment. As can be observed from the figure, the maximum longitudinal strain occurs at the edge of the sheet, specifically near the forming stand, due to the elongation of the sheet before it reaches the forming stand. As can also be observed, there is good agreement between the simulation and experimental results, confirming the accuracy of the model used for the simulation.

Fig. 9
figure 9

Comparison of longitudinal strains determined by simulation and experiment; a convex blank, a thickness of 0.5 mm, a flange width of 35 mm, and bend angles of a 15°, b 30°, and c 45° for SPCC and bend angle of 30° for d SPFC590 and e SPFC1180

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5.2 Effects of network structural parameters on training quality

5.2.1 Selection of initial weights and biases of layers

The first step in applying the error-back propagation algorithm is the determination of the initial weights and biases of the layers. The use of appropriate values would enable faster convergence of the post-propagation algorithm. The initial values used for the present network were generated by the init (net) function, which randomly creates weights and initial biases. The effects of the activation functions of the hidden layer nodes on the network learning quality are given in Table 7.

Table 7 Effects of the activation functions of the hidden layer nodes
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Because the values of \( \overline{MSE} \) and MSEmin for the sigmoid function are less than those for the other two functions, the sigmoid function was selected as the hidden layer activity function.

5.2.2 Momentum

The momentum allows the network to neglect the small-scale features at the error level. Table 8 gives the effects of the momentum on the network training quality.

Table 8 Effects of the momentum on the network training quality
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The results show that the errors are minimum for a momentum of 0.85.

5.2.3 Performance of BPNN

The network performance significantly depends on the selection of the test couples. In this study, the number of performance tests was systematically increased. Each network performance test was performed using three sets of data randomly selected from Table. 2. To evaluate the network performance, the mean squared error for each test was calculated, and the network corresponding to the training and test data sets that produced the lowest MSE was selected as the final network. Obviously, the weights and biases of the final network were related to the performance test that produced the least MSE. Figure 10 shows the network error curve obtained from the performance test that produced the least MSE.

Fig. 10
figure 10

Network error curve obtained from the performance test with the least MSE

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A data set consists of 96 set for each output that were unseen by the ANN during training, and they were used to test the trained ANN. The qualitative accuracy spread of the training and test samples are shown in Fig. 11a-d. The points are clustered along the 45° line, indicating that the predicted values are close to the true values.

Fig. 11
figure 11

Accuracy spread of the ANN predictions (prediction vs. true value): a training and b test longitudinal bow data and c training and d test wrinkling data

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Table 9 compares the neural network training outputs and the desired outputs for the longitudinal bow and wrinkling. As can be observed, there is good agreement between the predictions of the neural network and the desired outputs.

Table 9 Comparison of the ANN training outputs with the desired outputs for longitudinal bow and wrinkling
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5.3 Optimal values of design parameters for minimizing objective functions and minimum values of objective functions

In the present study, the objective functions were the longitudinal bow and wrinkling, which were to be minimized. Three design parameters, namely, the sheet thickness, yield strength, and blank shape, were considered for each target bend angle. The lower and upper bounds of these parameters are listed in Table 10.

Table 10 Decision variables and their ranges
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Multi-objective optimization involves the simultaneous optimization of several fitness parameters. For this purpose, the Pareto front theory is used instead of single-objective optimization concepts, and the final solution is chosen from the optimal Pareto front collection. A design vector x∗ is a Pareto optimum if and only if there is no other feasible vector x such that [31]

$$ {\displaystyle \begin{array}{c}{f}_i(x)\le {f}_i\left({x}^{\ast}\right)\\ {}i=1,2,\dots, n\end{array}} $$
(7)

and

$$ {f}_j(x)<{f}_j\left({x}^{\ast}\right)\ \mathrm{for}\ \mathrm{at}\ \mathrm{least}\ \mathrm{one}\ j,1\le j\le n. $$
(8)

After the execution of the optimization algorithm, the Pareto optimal solutions were obtained for all the target bend angles. Figure 12a and b show the Pareto-optimal curves obtained using NSGA-II for a target angle of 15° in FRF with and without leveling roll, respectively.

Fig. 12
figure 12

Pareto-optimal curves obtained using NSGA-II for a target angle of 15° in FRF a with leveling roll and b without leveling roll

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The best results were selected through ranked selection and identification of the optimal Pareto front for the decision variables and objective functions.

The optimal values of the design parameters and the obtained values of the objective functions of the longitudinal bow and wrinkling for different target bend angles in FRF with and without leveling roll are presented in Tables 11 and 12, respectively.

Table 11 Optimal values of the design parameters and the obtained values of the objective functions of the longitudinal bow and wrinkling for different target bend angles in FRF with leveling roll
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Table 12 Optimal values of the design parameters and the obtained values of the objective functions of the longitudinal bow and wrinkling for different target bend angles in FRF without leveling roll
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The distributions of the three design parameters considered for the optimization are presented with respect to the generation in Fig. 13a-c. From the distributions of all the parameters and their scattering over their allowable variation ranges, it can be concluded that the sheet thickness has the greatest effect on the minimization of the two objective functions for all the target bend angles, followed by the yield strength and the blank shape, respectively.

Fig. 13
figure 13

Distributions of the design parameters with respect to their population index: a sheet thickness, b yield strength, and c blank shape

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After the optimization, the results were compared with those of the experiments and numerical simulations. Figure 14a-d compare the experimental, numerical, and optimization results of the longitudinal bow and wrinkling for FRF with and without leveling roll. Good agreement can be observed between the results.

Fig. 14
figure 14

Comparison of the experimental, numerical simulation, and optimization results of the longitudinal bow a with leveling roll, b without leveling roll and wrinkling, c with leveling roll, and d without leveling roll

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6 Conclusion

In this study, ANN-based multi-objective optimization (NSGAII) was used to optimize the effective parameters of the FRF process, such as the sheet thickness, yield strength, and blank shape, with respect to the target bend angle, to minimize the longitudinal bow and wrinkling of the product. Following is a summary of the findings of the study and the conclusions drawn from it.

  1. 1)

    Because the neural network is based more on statistics than a physical model, it is suitable for modeling the multi-parameter FRF process. The BPNN training algorithm was thus used to predict the two defects as the objective functions.

  2. 2)

    To increase the learning rate of the conventional BPNN and prevent instability and oscillation of the network, a momentum sentence was added to the Storm Descending (SD) algorithm.

  3. 3)

    Among the activation functions of the hidden layer nodes, the sigmoid function was selected as the function of the hidden layer nodes owing to its lower \( \overline{MSE} \) and MSEmin values.

  4. 4)

    The network prediction capabilities were confirmed by network prediction error analysis, and the BP network prediction results were found to be in good agreement with those of numerical simulation.

  5. 5)

    In the use of NSGAII to optimize the input parameters for minimization of the objective functions with respect to the target bend angle, the best results were chosen from the optimal Pareto front collection. The choice was based on ranked selection, the optimal values of the design parameters, and the minimum values of the objective functions of the longitudinal bow and wrinkling for FRF with and without leveling roll.

  6. 6)

    The distributions of all the parameters and their scattering over their allowable variation ranges showed that the sheet thickness had the greatest effect on the minimization of the two objective functions, followed by the yield strength and blank shape, respectively.