An integrated FEMANN model for laser bending process with inverse estimation of absorptivity
 Ravi Kant^{1},
 Shrikrishna N. Joshi^{1}Email author and
 Uday S. Dixit^{1}
DOI: 10.1186/s4075901500061
© Kant et al. 2015
Received: 19 March 2015
Accepted: 8 October 2015
Published: 16 October 2015
Abstract
Background
Absorption of laser energy into the worksheet surface during laser bending process is an important and critical factor for accurate computation of the bend angle. This paper presents an integrated FEMANN approach to compute accurate value of bend angle during laser bending process.
Methods
Initially, a finite element method (FEM) based threedimensional nonlinear transient thermomechanical numerical model is developed using ABAQUS package. Using FEM model and data obtained in actual experiments, the proper values of absorptivity for various sets of process conditions are computed by inverse analysis technique. Based on the proper values of absorptivity, an artificial neural network (ANN) model is developed for accurate and quick prediction of absorptivity for given input process conditions. The predicted absorptivity is then employed in the FEM model for accurate computation of bend angle.
Results
The performance of the integrated approach is verified by conducting experiments.
Conclusion
The verification results showed that the proposed approach is able to compute the bend angle with a very good accuracy (average prediction error of 4.14 %). The proposed approach can also be suitable for the numerical simulations of other laser based manufacturing processes.
Keywords
Laser bending Absorptivity Inverse analysis Bend angle Artificial neural network (ANN) Finite element method (FEM)Background
In laser bending process, a controlled heating of the worksheet surface is carried out by laser beam irradiation. This induces nonuniform thermal stresses that plastically deform the worksheet (Li and Yao 2000). The deformation is the result of complex interaction between laser process parameters, viz. laser power, scanning velocity and beam diameter, worksheet material properties and the worksheet geometry. Due to uneven mechanical restraint and temperature distribution, the bend angle is not uniform along the scanning line which is called as ‘edge effect’ (Shen et al. 2010). The important features of laser bending process include absence of springback, possibility of forming at inaccessible areas, creation of complex shapes with different irradiation strategies, possibility of bending brittle materials and the easy control of the process (Wu et al. 2010, Kant and Joshi 2013, Kant and Joshi 2014).
Literature reports various numerical investigations on the effect of various parameters such as laser power, scanning velocity, beam diameter, sheet geometry, cooling conditions, number of irradiations and clamping on the process mechanism, bend angle and edge effect for a wide range of materials. Li and Yao (2000) studied the effect of strain rate by taking various combinations of laser power and scanning velocity in such a way that the peak temperature at the upper surface of the worksheet remains constant. They observed that the bend angle, residual stresses and hardness decrease with increase in the strain rate. Hu et al. (2001) studied the effect of process parameters on laser bending of stainless steel. Zhang et al. (2002) developed an efficient numerical model for pulsed laser bending of fullhard 301 stainless steel. The computation time was significantly reduced by the proposed methodology. Yanjin et al. (2005) studied the effect of various thermal and mechanical properties of the workpiece material on bend angle. The bend angle decreased with increase in Young’s modulus, yield strength, thermal conductivity, specific heat and density of the material while it increased with increase in the coefficient of thermal expansion. Shi et al. (2006) explained various laser forming mechanisms (temperature gradient mechanism, buckling mechanism and upsetting mechanism) in terms of temperature and stress–strain distributions. Authors proposed a new coupling mechanism based on the combination of temperature gradient and buckling mechanism. Shen et al. (2009) numerically studied the laser bending of metal/ceramic bilayer materials sheet. Pitz et al. (2010) proposed the moving mesh methodology approach to reduce the computational time involved in the numerical simulations of laser bending process. They used a fine mesh around the laser beam and coarse mesh in the remaining part. The mesh moved synchronously with the laser beam. Jamil et al. (2011) studied the effect of laser beam geometries on the laser bending process governed by the buckling mechanism. Gollo et al. (2011) performed a statistical analysis to study the effects of material, laser power, beam diameter, scan velocity, sheet thickness, number of scans and pulse duration on bend angle.
Literarure also contains studies on the effect of forced cooling and curvilinear irradiations on the performance of laser bending. Cheng and Yao (2001) found that the forced cooling can significantly reduce the total forming time in a multipass laser bending process. They also found that the forced air cooling does not have any harmful effect on microstructural and mechanical properties of the workpiece. Shen et al. (2011) studied the effect of forced cooling on laser bending of steel plates using finite element based numerical simulations. The forced cooling was applied either on top or bottom or on both the surfaces. It was moving simultaneously with laser beam by an offset of about one second. The forced cooling did not have a significant effect on the bend angle and edge effect. Chen et al. (2004) observed that in a curvilinear laser bending, the deformation occurs on one side of the laser scanning path along which the rigid constraint is lower. Zhang et al. (2007) showed that the peak temperature at the scanning surface and warping increases with the increase in scanning path curvature. Kant and Joshi (2014) observed that the worksheet does not bend over the laser scanning path in the curvilinear laser bending process. The bend line differs from the scanning path and the bending offset increases with the increase in laser power and beam diameter.
The bend angle per laser scan is small and therefore, the laser bending is also carried out with the assistance of mechanical load. Yanjin et al. (2003) showed that preloading can increase the bend angle significantly. Yao et al. (2007) presented effect of various preloads on laser bending process. They employed pure compression, pure tension and pure bending as preloading and observed that preloading does not cause any external edge effect. Kant and Joshi (2013) proposed a laser assisted bending with moving mechanical load for bending of large sized worksheets. Gisario et al. (2015a) studied laser assisted bending of AISI 304 stainless steel sheets of 1 mm thickness to obtain a large bend angle upto 140° with a small fillet radius of 2 mm. Gisario et al. (2015b) studied laser assisted origami bending of the stainless steel sheets for shaping the sheets into three dimensional items such as cubes and chairs. They found that the manufactured items have good precision, accuracy and aesthetic appearance.
A few attempts have been reported on artificial neural network (ANN) based modeling of laser bending process. Cheng and Lin (2000) used three supervised neural networks to estimate the bend angles, viz., (1) back propagation neural network (BPNN) with hyperbolic tangent function, (2) BPNN with logistic function and (3) radial basis neural network (RBFN). They found that RBFN outperforms the other two. Dragos et al. (2000) demonstrated the use of ANN in automatic control of process. Casalino and Ludovico (2002) developed an ANN model to predict the process conditions for laser bending of SAE 1020 steel sheets. Maji et al. (2014) developed an ANN model to predict the process conditions for obtaining a particular dome height in three dimensional laser bending of a sheet. ANN model was able to predict the process conditions with good accuracy. Gisario et al. (2011) used neural network solutions to predict, control and manage the springback in laser assisted Vshape bending of thin aluminum sheets.
The review of literarture reveals that hard computing based modeling, viz., finite element method (FEM) has limited applications due to difficulty in realistic modeling of laser beamwork surface interaction and unavailability of nonlinear material properties at higher temperatures. Limited attempts have been reported on inverse computation of the unknown process and material parameters. Mishra and Dixit (2013) determined absorptivity, thermal diffusivity and laser beam diameter by an inverse heat conduction method. They measured temperature at the centroid of bottom surface at different time intervals. The estimation of thermal properties from temperature measurements was constructed as a problem of minimization of the objective function with absorptivity, thermal diffusivity and beam diameter as decision variables. The implementation of inverse analysis provided good results. However, more than one combination of parameters provided same temperature variation with the time at a particular location. Therefore, Eideh and Dixit (2013) improved this model and carried out inverse determination by measuring temperature at two different locations. The objective was to minimize errors between predicted and measured temperature at two locations. The proposed methodology was found to be efficient and robust.
Most of the presented work considered constant absorptivity for various set of process conditions. However, in real practice, the absorptivity is not constant and depends on various parameters such as workpiece surface condition, properties of laser beam, temperature at the irradiating surface and the properties of workpiece material. Modeling of real behavior of absorptivity during laser bending process is difficult due to complex interaction between various parameters. In this work, an integrated FEMANN methodology for accurate prediction of laser bending for a wide range of process conditions is presented. It is assumed that the absorptivity depends on the set of laser parameters, viz. laser power, scanning velocity and beam diameter. Using FEM model, the absorptivity is predicted by inverse analysis technique for various set of process conditions. The predicted absorptivity and the corresponding laser parameters are used to train an artificial neural network. ANN model is developed to predict the absorptivity for a set of process conditions. The ANN model is integrated with the FEM model to incorporate predicted absorptivity for accurate simulation of laser bending process for a wide range of process conditions. Details of the methodology, results and discussion are presented in the next section.
Methods
Development of integrated FEMANN model for laser bending process
 1.
At first, a finite element model is developed for laser bending of magnesium alloy M1A sheet. The finite element model is used to simulate the laser bending process for various set of process conditions.
 2.
In second step, numerical results are compared with the experiments.
 3.
In third step, absorptivity is estimated by studying the difference between numerical and experimental results (error). A number of numerical simulations are carried out by varying the absorptivity such that the prediction error lies within an error band of ±5 %.
 4.
In fourth step, an artificial neural network (ANN) model is developed to predict the absorptivity. Process parameters, viz., laser power, scanning velocity and beam diameter are considered as input data while the estimated absorptivity obtained in Step 3 is used as the target data.
 5.The developed ANN model is integrated with the FEM model as shown in Fig. 2. Absorptivity predicted by the ANN is used as input to the FEM model for accurate modeling of laser bending process. The proposed methodology is verified the experiments.
Development of numerical model for laser bending process
Threedimensional nonlinear transient thermomechanical finite element method based numerical model is developed for the laser bending process. Coupled thermomechanical analysis is carried out by employing various input process parameters such as worksheet geometry, material properties, laser process parameters and the absorptivity. The computed plastic strains are used to obtain the deformation in the work sheets. The bend angle was computed by measuring distorted coordinates on the deformed work sheets.The details of the numerical model are presented below.
Sheet geometry and material properties
The numerical model was developed for magnesium alloy M1A. It is the MgMn alloy with good corrosion resistant properties. The temperaturedependent and strain rate–dependent material properties available in Avedesian and Baker (1999) were employed in the this work. The geometry of the sheet was taken as 60 mm length, 40 mm width and 1.90 mm thickness.
Assumptions

The worksheet material is isotropic and homogeneous.

The worksheet is considered to be flat and free of residual stresses.

Out of the laser heat applied, the energy stored due to plastic deformation is neglected.

The vonMises criterion is used for plastic yielding.

The metals lose mechanical properties at melting point. Therefore, stiffness of material is taken very close to zero when temperature exceeds the melting point. As such the melting needs to be avoided.
Heat flux modeling
Thermal analysis
Mechanical analysis
Solution methodology
The equations solved in coupled thermomechanical analysis are equations of motion and heat conduction equation. These equations were solved by using Full Newton technique. In this technique, target stiffness matrix is evaluated at each iteration. This helps in getting proper convergence even when the guess value is far away from the solution. Also, the stiffness matrix is more accurate in each iteration which provides a better prediction accuracy.However, computationally, it is not as efficient as Modified Newton method, in which sometimes, convergence is not achieved (Bathe 1996). The automatic time step was selected with maximum time increment of 0.02 s. The maximum temperature change in a step was taken as 30 °C.
Experimental details
Bend angle and absorptivity for various process conditions
Data No.  Power (W)  Velocity (mm/min)  SD (mm)  Beam Diameter (mm)  A: Average Bend Angle Experimental study (°)  Coefficient of Variation (%)  B: Bend Angle Numerical simulation (°)  Absolute Error = \( \left\frac{\left(AB\times 100\right)}{A}\right \) (%)  Absorptivity 

1  300  1000  20  3.87  1.07  4.11  1.10  2.40  0.63 
2  300  1000  30  5.81  0.73  3.14  0.72  0.75  0.67 
3  300  1000  40  7.74  0.49  4.6  0.47  3.25  0.72 
4  300  1000  50  9.68  0.21  24.86  0.20  3.18  0.725 
5  300  2000  20  3.87  0.84  3.96  0.84  0.04  0.72 
6  300  2000  30  5.81  0.31  3.77  0.31  1.40  0.74 
7  300  2000  40  7.74  0.12  18.09  0.13  4.17  0.8 
8  300  2000  50  9.68  0.05  29.31  0.05  3.54  0.875 
9  300  3000  20  3.87  0.6  19.31  0.62  3.83  0.83 
10  300  3000  30  5.81  0.14  18.62  0.15  4.44  0.84 
11  300  3000  40  7.74  0.09  20.56  0.10  4.15  0.95 
12  300  3000  50  9.68  ^{a}  ^{a}  ^{a}  ^{a}  0.98 
13  300  5000  20  3.87  0.26  2.46  0.25  3.35  0.81 
14  300  5000  30  5.81  0.16  33.25  0.17  4.87  0.98 
15  300  5000  40  7.74  ^{a}  ^{a}  ^{a}  ^{a}  0.99 
16  300  5000  50  9.68  ^{a}  ^{a}  ^{a}  ^{a}  1.0 
17  500  1000  20  3.87  0.99  7.31  0.95  4.35  0.825 
18  500  1000  30  5.81  0.94  1.44  0.95  0.57  0.8 
19  500  1000  40  7.74  0.75  18.69  0.78  4.11  0.6 
20  500  1000  50  9.68  0.53  12.94  0.55  4.08  0.54 
21  500  2000  20  3.87  1.42  15.45  1.35  4.77  0.52 
22  500  2000  30  5.81  1  7.58  0.97  2.54  0.59 
23  500  2000  40  7.74  0.72  13.98  0.71  1.24  0.66 
24  500  2000  50  9.68  0.4  0.67  0.41  3.30  0.68 
25  500  3000  20  3.87  1.27  6.63  1.14  1.91  0.57 
26  500  3000  30  5.81  0.84  15.93  0.85  0.74  0.67 
27  500  3000  40  7.74  0.34  12.29  0.38  1.10  0.69 
28  500  3000  50  9.68  0.14  0  0.14  2.26  0.715 
29  500  5000  20  3.87  0.74  11.42  0.74  0.65  0.625 
30  500  5000  30  5.81  0.26  22.3  0.34  3.43  0.67 
31  500  5000  40  7.74  0.12  19.97  0.12  3.00  0.785 
32  500  5000  50  9.68  0.05  33.44  0.05  1.58  0.88 
33  700  1000  20  3.87  0.35  4.89  0.34  4.16  0.8 
34  700  1000  30  5.81  0.57  12.18  0.56  2.36  0.79 
35  700  1000  40  7.74  0.75  3.75  0.72  4.51  0.8 
36  700  1000  50  9.68  0.64  17.7  0.67  4.69  0.9 
37  700  2000  20  3.87  1.05  6.32  1.02  2.39  0.8 
38  700  2000  30  5.81  1.3  13.67  1.33  2.29  0.78 
39  700  2000  40  7.74  0.96  5.85  1.00  4.34  0.57 
40  700  2000  50  9.68  0.8  18.81  0.78  2.78  0.64 
41  700  3000  20  3.87  1.68  2.02  1.61  4.16  0.67 
42  700  3000  30  5.81  1.16  9.42  1.18  1.72  0.55 
43  700  3000  40  7.74  1.1  16.09  1.05  4.84  0.68 
44  700  3000  50  9.68  0.52  28.03  0.52  0.36  0.75 
45  700  5000  20  3.87  0.94  30.25  0.92  2.24  0.5 
46  700  5000  30  5.81  0.9  5.45  0.89  1.41  0.63 
47  700  5000  40  7.74  0.72  6.99  0.73  0.88  0.76 
48  700  5000  50  9.68  0.18  15.48  0.18  2.16  0.71 
Average error =2.77 % 
Inverse computation for prediction of absorptivity
Absorptivity is an important characteristic of the material which can be obtained easily by some experimental techniques usually based on spectroscopy (Stuart, 2004). A spectrophotometer can find out the ratio of reflected to incident radiation for a particular wavelength, from which the absorptivity can be calculated. However, usually the commercially available spectrophotometers have provision to find out the reflectivity or absorptivity only when the specimen is at room temperature. In laser bending, a high temperature profile generates into the workpiece, which depends on the complex interaction of laser power, scanning velocity, beam geometry, number of scans, workpiece surface condition, wavelength of the laser beam, phase transformation and surrounding conditions. It is not possible to measure the absorptivity by considering all these realtime conditions during laser bending process. In this scenario, inverse analysis is one of the best methodologies to predict the unknown parameters. The use of inverse analysis to predict the unknown parameter is widely used and accepted by the research community for many realtime problems (Mishra and Dixit 2013, Eideh and Dixit 2013).
In general, the researchers use a constant value of absorptivity for all sets and ranges of process parameters. It is a common practice in modeling and simulation of the laser bending process using physics based hard computing methods viz. FEM (Hu et al. 2001, Zhang et al. 2002, Yanjin et al. 2005, Kant and Joshi 2013). This equivalent absorptivity depends on many factors such as amount of burnt coating, temperature of the surface, plasma generated, melting condition, phase transformation, contact time and contact area of laser beam etc. For a system, these factors depend on the set of process parameters and mainly on the laser power, scanning velocity and beam diameter. Therefore, the equivalent absorptivity is determined by inverse analysis technique using developed finite element model. The schematic of the inverse computation is shown in Fig. 1. The bend angle obtained from FEM model was compared with the bend angle obtained during experimental studies. The absorptivity was tuned up using a bisection method (Rao 1984) to get the FEM results within ±5 % error as compared with the experimental results. Finally, the tuned absorptivity was considered as the equivalent absorption for the particular set of process parameters. The predicted absorptivities are shown in Table 1. It is seen that the process conditions have significant effect on the absorptivity. The computed/obtained absorptivity varies from 0.50 to 1.
In order to assess the effectiveness of proposed method, a few measurements of absorptivity were carried out using Fourier Transform InfraredSpectrometer (model: Spectrum BX, make: Perkin Elmer). The normalized absorbance, A _{ b }, is expressed as the common logarithm of the ratio of incident radiation to reflected radiation. The samples were placed in the spectrometer after laser scanning was complete. The surface property may undergo some change after laser scanning; hence the comparison of computed absorptivity and absorbance by Spectrometer should be treated as qualitative rather than quantitative. It is to be emphasized that when the measurement of absorptivity during the process is not possible, it is better to measure the absorptivity at the end of the process rather than at the beginning of the process. The measurement at the end of process will be closer to that during the process compared to measurement at the beginning of the process.
Comparison of inversely computed absorptivity and absorbance by spectrometer
P (W)  V (mm/min)  D (mm)  Inversely calculated absorptivity  Absorbance by spectrometer 

300  2000  3.87  0.72  0.727 
300  1000  3.87  0.63  0.675 
700  2000  9.68  0.64  0.687 
700  5000  7.74  0.76  0.770 
700  3000  3.87  0.67  0.690 
500  3000  5.81  0.67  0.662 
300  3000  3.87  0.83  0.805 
ANN model for estimating absorptivity
Training dataset
Data set No.  Power (W)  Velocity (mm/min)  Beam Diameter (mm)  Absorptivity 

1  300  1000  3.87  0.63 
2  300  1000  5.81  0.67 
4  300  1000  9.68  0.725 
5  300  2000  3.87  0.72 
6  300  2000  5.81  0.74 
7  300  2000  7.74  0.8 
9  300  3000  3.87  0.83 
10  300  3000  5.81  0.84 
11  300  3000  7.74  0.95 
12  300  3000  9.68  0.98 
14  300  5000  5.81  0.98 
15  300  5000  7.74  0.99 
16  300  5000  9.68  1.0 
17  500  1000  3.87  0.825 
18  500  1000  5.81  0.8 
19  500  1000  7.74  0.6 
20  500  1000  9.68  0.54 
21  500  2000  3.87  0.52 
22  500  2000  5.81  0.59 
23  500  2000  7.74  0.66 
25  500  3000  3.87  0.57 
26  500  3000  5.81  0.67 
27  500  3000  7.74  0.69 
29  500  5000  3.87  0.625 
31  500  5000  7.74  0.785 
32  500  5000  9.68  0.88 
34  700  1000  5.81  0.79 
35  700  1000  7.74  0.8 
36  700  1000  9.68  0.9 
37  700  2000  3.87  0.8 
38  700  2000  5.81  0.78 
39  700  2000  7.74  0.57 
41  700  3000  3.87  0.67 
42  700  3000  5.81  0.55 
44  700  3000  9.68  0.75 
46  700  5000  5.81  0.63 
47  700  5000  7.74  0.76 
48  700  5000  9.68  0.71 
Testing datasets
Data set No.  P (W)  V (mm/min)  D (mm)  Absorptivity 

3  300  1000  7.74  0.72 
8  300  2000  9.68  0.875 
13  300  5000  3.87  0.81 
24  500  2000  9.68  0.68 
28  500  3000  9.68  0.715 
30  500  5000  5.81  0.67 
33  700  1000  3.87  0.8 
40  700  2000  9.68  0.64 
43  700  3000  7.74  0.68 
45  700  5000  3.87  0.5 
Results and discussion
Verification of the developed FEM–ANN methodology
Results of verification experiments
S. No.  Power (W)  Velocity (mm/min)  SD (mm)  Beam diameter (mm)  ANN predicted absorptivity  Computed bend angle with the numerical model (°)  Average value of bend angle in Experiments (°)  Error in bend angle (%) 

1  650  2000  45  8.710  0.638  0.856  0.816  4.90 
2  500  2500  30  5.807  0.648  0.885  0.863  2.55 
3  450  4500  30  5.807  0.847  0.473  0.512  7.62 
4  400  3500  35  6.775  0.86  0.314  0.305  2.95 
5  600  1500  35  6.775  0.773  1.189  1.202  1.08 
6  350  3000  40  7.743  0.938  0.179  0.171  4.68 
7  700  4000  20  3.872  0.551  1.546  1.469  5.24 
8  500  3500  50  9.678  0.777  0.109  0.118  7.62 
Average error = 4.14 % 
The proposed methodology is very suitable for the shop floor applications. With the development of newer materials and variation in the quality of the surface as well as coating, a handy method for the estimation of absorptivity is always useful. Even if the absorptivity data is available, it may be required to fine tune that data in real applications depending on the processing conditions. The main contribution of this work is to provide a suitable methodology, which can be easily employed in industrial practice. With correct values of absorptivity, FEM model can obtain the stress, strain, strainrate and temperature distribution. This also helps in developing microstructural model for the process.
Conclusions
In this work, an FEM–ANN based integrated approach is developed for accurate computation of bend angle during laser bending process. Initially threedimensional nonlinear thermomechanical FEM based numerical model is developed for computation of bend angle for given process conditions. The absorptivity used in the numerical model was then tuned by using inverse analysis technique. In this technique, FEM simulations were repeated by varying the values of absorptivity. The value of absorptivity for each set of process condition was estimated by comparing the bend angle obtained in the numerical and experimental studies so that the computational error of bend angle is less than 5 %. Based on the estimated values of absorptivity and the corresponding input process conditions, an ANN based model was developed for fast and accurate prediction of absorptivity. Then, the ANN based absorptivity was employed in the FEM model for computation of bend angle. Within the scope of process conditions used in the present work, the proposed approach was found to be accurate in the computation of bend angle with an average error of 4.14 %. The estimated absorptivity is a fitting parameter as it serves the purpose of correctly estimating the bend angle. However, it also matches qualitatively with the experimentally determined absorbance. This approach can be used for accurate computation of process responses in the other laser based manufacturing processes viz. laser welding, laser machining, etc.
Declarations
Acknowledgements
Partial funding from the Engineering and Physical Sciences Research Council (UK) through grant EP/K028316/1 and Department of Science and Technology (India) through grant DST/RCUK/14AM/2012, project Modeling of Advanced Materials for Simulation of transformative Manufacturing Processes (MAST) is gratefully acknowledged. The 2.5 kW CO_{2} Laser Cutting Machine was purchased under DST_FIST scheme of Department of Science and Technology, India (Sanction no. SR/FST/ETI244/2008, Date: 17.03.2009).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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