where S is the weld/HAZ cross-section surface, B is the weld/HAZ width and the coefficient k=2.5 for SSt and k=3, when the width at the top of the weld or the HAZ (St45) is used. The coefficients are estimated by ordinary least squares method.
Figure 62 presents superimposed the experimentally observed form of the weld cross-section and the approximation made by eq. (6) for SSt weld for P=4.2 kW, v=80 cm/min and dZ=-60 mm.
Figure 62. Approximation of the form of the cross-section of the weld for stainless steel
N
EURALN
ETWORKM
ODELING OFEBW P
ROCESSOne of the most promising fields of the Artificial Intelligence is related to the Neural Networks [63] that has the ability to learn and approximate any functional relationship. The
NN integration in intelligent control system [64] is based on such characteristics of connectionist systems as: availability of learning, generalization, classification; stability in relation to partial faults in the network and the noise; improving performance with increased experience; associative memory. The advantages of NN are demonstrated when the mathematical description of the plant is very complex or the computational task is not completely defined. In relation to control systems NN are attractive tools for solving problems in which classical analytic methods are difficult to be applied. It is appropriate to use neural network for process modeling and control, pattern recognition, fault diagnosis.
Moreover, despite the possibility of equally comparable solutions to a given problem, several additional aspects of a neural network solution are appealing, including parallel implementations that allow fast processing; less hardware which allows faster response time, lower cost, and quicker design cycles; and on-line adaptation that allows the networks to change constantly according to the needs of the environment.
A number of process engineering problems have been studied and solved using the neural networks approach that exploits symbolic processing and knowledge representation [65÷69].
The majority of the neural networks utilized in the applications are the multilayered feed-forward networks. First and still widely used method of training the neural networks is the so-called back propagation method (BPM) [70]. It requires a preliminary generated (usually experimentally obtained) set of training data containing sets of input-output data for the neural network.
An example of a model structure in the form of a Neural Network is shown in Figure 63.
Further a procedure of creating neural network-based models and their application to the prediction of the electron beam welding (EBW) performance characteristics and to the parameter optimization are presented.
Figure 63. Neural network structure
The proposed methodology for developing NN-based models for EBW performance characteristics consists of the following general steps:
1. Construction of the neural network model structure.
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 151 2. Training of the created neural network by using the back propagation method [70]
and experimentally obtained (and/or numerically simulated) set of training data to a satisfactory accuracy.
3. Recall of the trained neural network for prediction and parameter optimization.
The modelled EBW process parameters define the input-output structure of the neural network-based model used, i.e. the neural network should consist of 4 input neurons and 1 output neuron. NN models for each output (weld depth H and mean weld width B) are considered (illustrated in Figure 64).
The best results for Neural network models for the weld depth H and mean width B were obtained with 5 hidden units and different number of iterations for training (above 10000 iterations). For the purpose of validation the data were split into two parts: training datasets containing 73 observations and the testing datasets limited to 8 observations each (for H and for B). For each performance characteristic randomly were chosen 10 datasets (73 training and 8 test observations) and for each dataset the best network model was obtained and verified. For comparison of the models the absolute value of the error calculated as the difference between the predicted and the measured values of the weld geometry characteristics, as well as root mean squared error (RMSE) and the non-dimensional error index (NDEI) are used. The last two are calculated by:
RMSE =
n y
yˆ 2 ; NDEI =
RMSE,where yˆ and y is the predicted and the experimental values, n is the number of data and is the standard deviation of the data points. These error measures are defined on the basis of the training error (average loss over the training sample) and the generalization error (expected prediction error on an independent sample). Their values are minimized during the neural network training.
Figure 64. Neural networks input-output parameters for the weld depth H and mean width B
The experimental results (marked with points) and the predicted results (connected with the straight lines) using the estimated best model for the weld depth H using the training dataset (73 observations) are presented in Figure 65.
The absolute value of the errors, presented as the difference between the predicted and the measured values of the weld depths, are calculated and graphically presented in Figure 66,
connected with lines. Generally, the error values are situated in the region (-22 mm) with the exception of only 5 errors. The model precision is estimated quantitatively by RMSE and NDEI and the results are presented in Table 17.
Figure 65. Predicted end experimental values for the weld depth H – training
Figure 66. Absolute error values (the differences between the experimental and the predicted weld depths H) – training
In Figure 67 and Figure 68 are presented the results from the training of the best neural model for the weld mean width B.
A comparison between trained neural networks (Figure 69), describing the relationship of the thermal efficiency and different combination of factors: a) depth H and mean width B of the welds (2 factors); b) electron beam power P, welding velocity v, the distances between the
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 153 main surface of the magnetic lens of the gun to the beam focus zo and to the surface of the sample (4 factors) and c) all considered factors (6 factors). The results from the training and the cross-validation are presented in Table 17. It can be seen, that the trained neural network models with 4 factors give very good results. Visualization of the experimental and the predicted results for the thermal efficiency in this case are presented on Figure 70 and Figure 71.
Figure 67. Predicted end experimental values for the weld mean width B – training
Figure 68. Absolute error values (the differences between the experimental and the predicted weld mean widths B) – training
a) b)
Figure 69. Neural networks input-output parameters for the thermal efficiency – inputs and outputs Table 17. RMSE and NDEI error measures
Process Parameters
Training (73 experiments)
Testing (validation) (8 experiments)
Performance Characteristic
4 RMSE 1.33382 1.52107
NDEI 0.141456 0.162708 H
4 RMSE 0.226097 0.131611
NDEI 0.231885 0.116459 B
2 RMSE 0.0531979 0.0814766
T
NDEI 0.908591 0.875771
4 RMSE 0.0290363 0.0273294
T
NDEI 0.47571 0.3782120
6 RMSE 0.0253802 0.0222612
T
NDEI 0.397551 0.557775
Figure 70. Predicted end experimental values for the thermal efficiency T – training
Process Parameter Optimization and Quality Improvement at Electron Beam Welding 155
Figure 71. Absolute error values (the differences between the experimental and the predicted the thermal efficiency T) – training