Development of a neural network model to predict distortion during the metal forming process by line heating

Resumen Desarrollo de un modelo de red neuronal para la predicción de distorsión durante el proceso de formado metálico utilizando líneas de calentamiento Date Received: December 2nd, 2010 Fecha de recepción: 2 de Diciembre de 2010 Date Accepted: March 6th, 2012 Fecha de aceptación: 6 de Marzo de 2012 Development of a neural network model to predict distortion during the metal forming process by line heating 1 Specialized Research Center for Manufacturing and Joining Processes (LEPUM) – School of Mechanical Engineering – Technological University of Panama. e-mail: adan.vega@utp.ac.pa 2 Member of the National Researcher Systems (SNI) – SENACYT, Panama. e-mail: adan.vega@utp.ac.pa Ship Science & Technology Vol. 6 n.° 12 (41-49) January 2013 Cartagena (Colombia) 42 The process of forming three-dimensional surfaces via line heating has been used for several decades in shipbuilding. The procedure is performed manually and it is considered one of the most difficult to understand techniques. Thus, it becomes difficult for shipyards to maintain a sufficient number of technicians, who are able to do this work without many complications [1]. This leads us to conclude that it is necessary to automate the system in order to restore the small number of skilled workers. To achieve this goal, many investigation projects have been carried out. They can be divided into three main groups: those seeking to understand the situation [2, 3, and 4], those wishing to develop automatic machines that can bend a plate [5 and 6], and finally those seeking to understand the relationship between heating conditions and the deformation to be obtained [7 and 8]. The last group being the most difficult task found until now. Predicting the distortion produced by heating a plate is confusing because it is influenced by many factors like the amount of heat, speed of the heat source, thickness of the plate and by secondary factors like the cooling method, the initial deformation of the plate, residual stresses, cooling method, etc., [9 and 10]. Given the complexity of the problem, it requires using mathematical tools that involve all these variables when analyzing the problem; being the finite element method (FEM), through a threedimensional thermal-elastic-plastic analysis, the method offering the best performance [11]. On the other hand, FEM is limited to analyzing small plates without complex heating patterns. This is far from the reality of shipbuilding where large and complex geometries must be developed. With the purpose of contributing to the solution of this problem, the authors propose a neural network model trained by using data obtained from FEM studies and validated experimentally. In this neural network, the prediction of deformation produced during the process of metal forming by line heating is achieved with considerable accuracy, for heat conditions and sizes of plates that have not been possible to achieve with traditional methods of prediction distortions. The process of plate forming by line heating can be seen as a method of forming a flat plate into a three-dimensional object, with a desired shape, by using the angular shrinkage and distortion produced by plastic stress, induced by heating and subsequent cooling [1]. The deformation is largely dependent on the heat input, the speed of the heat source and the plate thickness. According to [2], heat-induced deformation can be represented by four main components of inherent distortion, as shown in Fig. 1, and expressed by Equations 1 to 4 as follows; Where δx is the longitudinal inherent shrinkage, δy is the transverse inherent shrinkage, θx is the longitudinal inherent bending, and θy is the transverse inherent bending. εx* and εy* are the components of the inherent strain. The xy plane is in the direction of the heating line, and the zy plane is transverse to the heating line, h is the plate thickness. These relationships can be used to predict deformation with remarkably good results, but it is demonstrated that the deformation given by Equations 1 to 4 are affected by other factors, so they are not as effective when the process is complex [4]. Introduction Analysis of the distortion due to heating lines


Development of a neural network model to predict distortion during the metal forming process by line heating
The process of forming three-dimensional surfaces via line heating has been used for several decades in shipbuilding.The procedure is performed manually and it is considered one of the most difficult to understand techniques.
Thus, it becomes difficult for shipyards to maintain a sufficient number of technicians, who are able to do this work without many complications [1].
This leads us to conclude that it is necessary to automate the system in order to restore the small number of skilled workers.To achieve this goal, many investigation projects have been carried out.
They can be divided into three main groups: those seeking to understand the situation [2, 3, and 4], those wishing to develop automatic machines that can bend a plate [5 and 6], and finally those seeking to understand the relationship between heating conditions and the deformation to be obtained [7 and 8].The last group being the most difficult task found until now.
Predicting the distortion produced by heating a plate is confusing because it is influenced by many factors like the amount of heat, speed of the heat source, thickness of the plate and by secondary factors like the cooling method, the initial deformation of the plate, residual stresses, cooling method, etc., [9 and 10].
Given the complexity of the problem, it requires using mathematical tools that involve all these variables when analyzing the problem; being the finite element method (FEM), through a threedimensional thermal-elastic-plastic analysis, the method offering the best performance [11].On the other hand, FEM is limited to analyzing small plates without complex heating patterns.This is far from the reality of shipbuilding where large and complex geometries must be developed.
With the purpose of contributing to the solution of this problem, the authors propose a neural network model trained by using data obtained from FEM studies and validated experimentally.In this neural network, the prediction of deformation produced during the process of metal forming by line heating is achieved with considerable accuracy, for heat conditions and sizes of plates that have not been possible to achieve with traditional methods of prediction distortions.
The process of plate forming by line heating can be seen as a method of forming a flat plate into a three-dimensional object, with a desired shape, by using the angular shrinkage and distortion produced by plastic stress, induced by heating and subsequent cooling [1].The deformation is largely dependent on the heat input, the speed of the heat source and the plate thickness.
According to [2], heat-induced deformation can be represented by four main components of inherent distortion, as shown in Fig. 1, and expressed by Equations 1 to 4 as follows; Where δx is the longitudinal inherent shrinkage, δy is the transverse inherent shrinkage, θx is the longitudinal inherent bending, and θy is the transverse inherent bending.εx* and εy* are the components of the inherent strain.The xy plane is in the direction of the heating line, and the zy plane is transverse to the heating line, h is the plate thickness.These relationships can be used to predict deformation with remarkably good results, but it is demonstrated that the deformation given by Equations 1 to 4 are affected by other factors, so they are not as effective when the process is complex [4].

Introduction
Analysis of the distortion due to heating lines In addition to the information mentioned above, the relation between heating condition and rate of the heating source for independent plate thicknesses, as shown in Fig. 6 [3], was incorporated, in the analysis.Based on the heating equation (Equation 5) [8], we can obtain the current and voltage, required to generate the heating conditions.Here, it is necessary to mention that the plate surface temperature is 800 °C, which is the recommended temperature to achieve plate forming by line heating.
An outstanding characteristic of neural networks is their ability to generalize to future data.To achieve maximum development of this characteristic, the network needs to be trained by using as many details as possible so that it can find the characteristics that distinguish the behavior of the data.
To enhance the model of interest, we first specified the information that we would use as training for the network.To do this, we used results published in [3].In that work, the relationships described in Equations 1 to 4, for thick plates, for a wide range of heating conditions are plotted.Figs. 2 to 5 show a summary of the data.As seen in the figures, the relationship between inherent deformation and applied heat can be described by mathematical relationships.However, for conditions of heating out of the range used in that study, there is no reference in the present literature.
To obtain the largest possible number of data from the figures, information points, from each graph, are extracted.Then, regression, to achieve smooth behavior of the data while understanding the deformation behavior, for a given applied heating requirement, is performed.
Extraction of data for the development of the neural network model   After gathering the information we want to emulate, we continue developing the neural network, to accurately replicate the information.To develop the network, we must first create architecture (layers, neurons, and training functions) that derives the characteristic patterns of the training data, thus, ensuring that the network has the best performance.

Network architecture
A multilayer network, from a host of others, was chosen because we wanted the network to get the best generalizations of the results.The training algorithm used is the back-propagation model [12], given its good performance in predicting nonlinear functions.

Number of layers
One of the common uses of neural network is to approximate functions or specific patterns.However, a single layer may not be sufficient to represent a function because the behavior may not be fully captured.In the case of the network to predict the distortion, to avoid any possible lack of accuracy, three layers were used.

Number of Neurons
The hidden neurons play a vital role in the operation of a neural network given that they work as feature detectors.Once the learning process begins, the hidden neurons gradually begin to recognize outstanding features presented by the training data.
It should be noted that using few neurons may cause poor fit, while the opposite may result in a tight network.These characteristics are undesirable because they reduce the generalization ability of the network, yielding incorrect results and increasing error in the results.
To determine the optimal number of neurons, it is necessary to establish a criterion in assessing where the smallest amount of hidden neurons permits the best performance closest to the known response.
The mean square error (MSE) is a standard that provides details of the error of the output response generated by the neural network with respect to the desired output response.The MSE is defined as: Where h ci is the desired response from the network (targets), h i is the neural network response (output), and m is the number of data analyzed.This rule becomes necessary because it evaluates the accuracy of the results, as the fixed weights, at each step of the iteration, during the learning process [13].
Fig. 7 shows that by increasing the number of neurons, MSE varies and, therefore, the neural network error.

Network Training Process
The standard back-propagation algorithm was used in developing the network for distortion prediction.This is an algorithm of descent gradient in which the network weights moves along the negative value of the gradient of the function of execution.
The training of neural networks can be more effective if certain processes are performed on the input and output data.The processes performed on the input data improve the way the network reads the input information.Before training the network, inputs and outputs are scaled to keep them within a range: from -1 to 1. Thus, the activation functions of the neurons are automatically sensitized, obtaining a better performance of the network.

Generalization of the neural network
One of the largest features that occur during the development of neural networks seeks to know how good the response of the network will be for inputs to which the network has not been trained.Some techniques exist in the literature commonly Neural network model to predict the distortion due to heating lines (6) Development of a neural network model to predict distortion during the metal forming process by line heating used to improve the generalization of a neural network, these are: early detection (crossvalidation), removing the weights, and inclusion of noise to the data during training [14].In this article, we used the cross-validation technique [15] to generalize the network because it resulted as the most appropriate after several trials.In the cross-validation used to determine the generalized network data correctly, we divided the training data into three subsets: a training subset, a validation subset, and a testing subset, which was the one selected for this network.
Using as the selection criterion the cross-validation techniques, we selected the neural network from a number of possible networks.Then, by using the MSE, we computed the error of the entries that have not been seen by the neural network; thereby, the network selected is the one that presents the lowest validation error.
To evaluate the network that best fits the data, 20 different neural networks were developed.By varying the number of layers and neurons of each one, the best one was chosen after comparing the performance of each.
Table 2 shows the most significant networks obtained and how using MSE selection criterion, we selected the 12-8-4 network.Fig. 8 shows the results of the validation test of the 12-8-4 network.The figure displays the three subsets used for testing.One can see that beyond 469 days, both the test error and the validation error tend to increase, while the training error decreases as the ages pass.Each network has its characteristic curve, which served as a basis for selecting the network.
The 12-8-4 network has a total of three layers.
The first layer has 12 neurons, the second eight and the third (the output layer) has four neurons.Furthermore, the cross-validation test also analyzed the network response to the training data.Figs. between the inherent deformation of the study and the network response is shown in Table 4.By comparing results, we concluded that the error is negligible; given that these are cases for which the network had never been trained, the accuracy of the network could be considered as sufficiently good.
In this paper, we propose using neural networks to predict distortions induced by heating during

Validation of the results
After obtaining the deformation components by using the neural network, we should confirm the accuracy of the network for arbitrary data, those which the network has never learned (generalization).For that, the network was trained using additional cases obtained from the study presented in [8] (Table 3).The percentage of error

Fig 9 .Fig 11 .Fig 12 .Fig 10 .
Fig 9. Comparison between the network response and the training data for Transverse Inherent Shrinkage )

Table 1 .
Model of individual cases used as input of the networkFig 6. Relationship between speed of the heating source and heat input for different plate thickness called the vector "target".These four components are the known response extracted from Figs. 1 to 5, also associated to the heating conditions.Ship Science & Technology -Vol.6 -n.° 12 -(41-49) January 2013 -Cartagena (Colombia) Pinzón, Plazaola, Banfield, Fong, Vega

Table 2 .
Examples of networks presenting the best performances Vega after the network training.It seems that the values calculated by the network are close to the training values, so we can say that the network has precisely calculated the deformation.