1 Introduction

Electron beam welding (EBW) has been applied in many sectors of modern industry, owing to several special advantages for joining metals, such as high-power density, deep weld penetration, low contamination, good weldability and high energy efficiency [1]. The quality of electron beam welds, for example the shape of the weld bead, residual strains and degree of porosity, are largely correlated with the penetration depth [2]. In practice, a trial-and-error approach is usually adopted to tune the electron beam parameters to achieve the desired penetration depth, on a test piece, before moving to production. This approach can be very time consuming and costly, especially in the case of welding expensive high-grade materials, and it makes the EBW process very hard to standardise and be fully automated. In order to move towards ‘right first time’ manufacturing, EBW engineers will need to upgrade the conventional welding process and adopt computer-based tools to predict the EBW penetration depth before conducting the weld.

Recent developments in machine learning show the potential for fast and accurate predictive models. For example, machine learning models based on trained artificial neural networks (ANN) enable rapid prediction of weld profile dimensions. Shen et al. [3] used an ANN to predict the penetration depth for EBW of stainless steel, with an average absolute error of 5.57%. A similar approach was taken by Mladenov and Koleva [4], using an ANN to optimise EBW parameters for austenitic stainless steel using 73 training data and 8 verification data. Their neural network was able to predict weld penetration depth with a root mean squared error of 1.52 mm (approximately 6% of the mean weld depth). Jha et al. [5] used both a back-propagation neural network (BPNN) and a genetic algorithm-tuned neural network (GANN) to predict the EBW penetration depth of SS304 plates, trained with inputs of beam current, accelerating voltage and welding speed. They augmented 51 experimental data points with 949 virtual ones, generated from a second-order regression of the 51 experimental points. Although the effect of adding the virtual data was not reported, the average absolute percentage deviation of their predictions was under 5% (for a linear EB power density ranging from 136.5 to 253.1 J/mm), demonstrating the potential for good accuracy with limited experimental data. Similar methods to predict the EB weld bead profile were also applied for joining materials such as an aluminium alloy [6,7,8] and zircaloy-4 [9]. According to these studies, the predictive performance of a BPNN is more stable compared to other neural network types and is thus recommended. Choudhury and Chandrasekaran [10] compared the EB weld prediction methods based on a response surface methodology (RSM) and Bayesian regularisation back-propagation neural network. They found that the BPNN shows a better performance than the RSM method in EB weld area prediction for Inconel 825, with a 2% average absolute percentage deviation from the predicted weld bead area. Similar approaches have been applied to other welding technologies, such as laser welding [11, 12] and tungsten inert gas welding [13, 14], with typical average prediction errors between 6 and 20%. In addition to predicting fusion zone dimensions, ANN and regression methods have also been used to determine EBW parameters [15], predict weld quality [16] and avoid weld defects [17].

However, such statistical and machine learning methods make the weld depth prediction a black box and there is a lack of physical understanding of EBW fusion zone generation. In real EBW process, with same beam current, accelerating voltage and focusing current, the beam power density can still vary with machine parameters, such as electron beam gun working distance [18]. For a given beam power, the electron beam spot size on the workpiece surface will directly determine the energy density and can significantly affect the welding quality. When the beam power and welding speed are fixed, excessive energy density, i.e. a small beam spot size, can result in increased internal porosity [19] and, if insufficient, may lead to shallow penetration or an uneven weld surface. The beam focal position affects the electron absorption significantly [20]. Compared with an over-focus condition, the electrons can be better absorbed with an under-focussed beam and the energy transfer efficiency is higher, which induces a deeper penetration depth.

More comprehensive characterisation of the electron beam before each weld run could offer the potential to improve the predictive accuracy of statistical and machine learning models. One approach that has been widely employed is to measure the electron beam spot via a laser beam standard, but the accuracy of different beam characteristics may differ compared to that of a laser beam owing to the higher noise floor level of electron beam signals [21]. Even though the beam radius has been measured by a number of different methods [22,23,24,25], these improved characterisation approaches have not been used to improve EBW penetration depth prediction methods.

This paper aims to improve the accuracy in predicting the weld depth in EBW, for several different prediction approaches, through a deeper understanding of the electron beam characteristics and the impact on the welding process. In doing so, it will identify the merits of introducing other metrics to characterise the electron beam, on modelling accuracy and, for machine learning approaches, will shed light on the effect of data volume and the addition of virtual data.

2 Experimental methods

2.1 Electron beam probing

The electron beam was characterised using the BeamAssure™ 4-slit probe, developed by The Welding Institute (TWI), UK, as shown in Fig. 1a, which consists of slits in two arms of the probe, along the x and y-directions. A Faraday cup is positioned under these slits, as shown in Fig. 1b, to measure the voltage [21]. The mutually perpendicular slits b and c in Fig. 1a measure the focal spot diameter and the other two slits a and d calculate the beam deflection speed in the x and y-directions using Eqs. (1) and (2), where \({S}_{x}\) and \({S}_{y}\) are the beam deflection speeds in the x- and y-direction, respectively, \({D}_{s}\) is the distance between the two slits in each direction and \({t}_{x}\) and \({t}_{y}\) are the time gaps between the two signal peaks. At the beginning of each measurement, the electron beam is positioned at the intersection point of slits b and c, and is deflected to rotate along the dotted circular path shown in Fig. 1a. The method for signal processing is shown in Fig. 2. Figure 2a illustrates a typical 4-slit probe voltage signal. Peaks 1, 2, 4 and 5 are the signal received from the four slits and peak 3 is used to determine the beam current level. Peaks 2 and 4 refer to the inner two mutually perpendicular slits b and c in Fig. 1a. Peaks 1 and 5 are the signals received from slits a and d. The beam width profile of peak 4 is shown in Fig. 2b, with the x-axis transformed into distance by multiplying by the speed \({S}_{y}\) in the y-direction deflection. The beam spot characteristic, i.e. the width of the beam measured at a position 1/e2 of the maximum value of the peak (1/e2 width), can be determined from this profile. In this study, the 1/e2 width was adopted in all prediction methods to avoid any deviations caused by different beam radius definitions.

Fig. 1
figure 1

(a) The structure of the 4-slit probe. The electron beam is deflected to draw an anti-clockwise circle via Y slits and X slits. (b) A schematic diagram of the slit probe

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Fig. 2
figure 2

The signal analysis procedure for the 4-slit electron beam probe. (a) The voltage signal received by the probing system. (b) Definition of 1/e2 width

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$${S}_{x}=\frac{{D}_{s}}{{t}_{x}}$$
(1)
$${S}_{y}=\frac{{D}_{s}}{{t}_{y}}$$
(2)

Beam characterisation can be conducted before welding and in process. During operation with a chosen beam current and accelerating voltage, if the focusing current is changed, the new beam shape can be categorised with the BeamAssure™ system along with the new beam radius. More detail on the operation of the device can be found in [21].

2.2 Welding experiments

Welding experiments were carried out using an EB machine developed by Cambridge Vacuum Engineering (serial no. CVE 661, maximum power 4 kW and maximum voltage 60 kV). Experimental conditions were varied between 40 and 60 kV accelerating voltage, 25–45 mA beam current and a welding speed of 500–700 mm/min, at a vacuum level of < 10−3 mbar. The linear electron beam power densities were controlled in the range between 86 and 324 J/mm. The focusing currents were varied between 277 and 369 mA, leading to over-focus, sharp-focus and under-focus conditions. Over-focus, sharp-focus and under-focus refer to the beam focal position above, at and below the level of sample surface, respectively. The welding distance, which was measured from the chamber roof to the work piece surface, was 157 mm. The beam radius, as defined earlier, was measured by the 4-slit probe. The mean beam radius can be calculated using Eq. (3):

$$\sigma=\sqrt{x_{1/\mathrm e^2}\cdot y_{1/\mathrm e^2}}$$
(3)

where \({x}_{1/{\mathrm{e}}^{2}}\) and \({y}_{1/{\mathrm{e}}^{2}}\) are the beam 1/e2 radii in the x and y directions.

Alloy steel grade S275JR was used as the substrate material, with dimensions of 100 × 75 × 20 mm. Its chemical composition is shown in Table 1 [26]. Two groups of bead-on-plate welds were produced. A sketch of the welding trials is shown in Fig. 3. The welding parameters are listed in Tables 3 and 4. As specific values of beam radii in the x and y direction are difficult to achieve in practise, the four processing parameters, i.e. accelerating voltage (40 kV, 50 kV and 60 kV), beam current (25 mA, 30 mA, 35 mA, 40 mA and 45 mA), welding speed (500 mm/min, 550 mm/min, 600 mm/min, 650 mm/min and 700 mm/min) and focusing current (− 8 mA, − 4 mA, 0 mA, 4 mA and 8 mA) were varied and the resulting radii measured. To be as efficient and transparent as possible, an orthogonal experimental design approach [27] was used to reduce the number of tests from 375 to 69 (T1–T69) using the commercial statistical software SPSS. For ANNs with relatively simple inputs and outputs, it has been suggested that training data sizes can be between tens and one hundred [4, 17, 28].

Table 1 Chemical composition of S275JR (all values are in mass %)
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Fig. 3
figure 3

Schematic of EB welds on a S275JR mild steel plate

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Additional welds (C1–C30 in Table 4) were produced to test the accuracy of the different predictive methods. The welding parameters in these trials were selected to cover a wide range of new welding parameter combinations.

To reduce the potential impact of substrate temperature variation, and to leave some space for attaching thermocouples, four welds were conducted on each steel plate following the weld path shown in Fig. 4, with a 2-min delay between each weld to minimise heat build-up. The resulting difference in substrate temperature from the first to last weld, on a given plate, was less than 60 °C. The welded work pieces were sectioned along the cut paths shown in Fig. 4, ground, polished and etched, using a 5% nital solution, to reveal the fusion zone and allow the penetration depth to be measured manually from images taken on an optical microscope, processed using ImageJ.

Fig. 4
figure 4

Schematic of the layout of the weld paths and the five sectioning lines

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2.3 Prediction methods

Several different approaches can be used to predict EBW penetration depth; this paper considers the most common and relevant methods, an empirical equation, a regression fitting model and an artificial neural network. These methods are illustrated in Fig. 5 and the key approaches are summarised below:

  1. 1)

    An empirical equation for depth prediction, tuned with data T1–T69 [29, 30].

  2. 2)

    Prediction based on a second-order regression, tuned by data T1–T69.

  3. 3)

    A back-propagation neural network (BPNN) trained with data T1–T69.

Fig. 5
figure 5

An overview of methods for penetration depth prediction

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The penetration depth can be predicted using an empirical Eq. (6) developed in [30] from Eqs. (4) and (5), as follows:

$$Y=\delta\cdot X^\varepsilon$$
(4)
$$X=\frac{S\cdot\sigma}\alpha$$
(5)
$$Y=\frac{Q_{in}}{D_{pe}\cdot k\cdot\theta_M}$$
(6)

where \({Q}_{in}\) is the beam power (U, the accelerating voltage multiplied by I, the beam current), σ is the beam radius, \({D}_{pe}\) is the predicted penetration depth, \(\alpha\) and \(k\) are the thermal diffusivity and thermal conductivity, \({\theta }_{M}\) is the difference between the melting temperature and ambient and \(S\) is the welding speed. \(\delta\) and ε are constants and can be determined by fitting a power law to the experimental data. The relevant material properties used in this empirical equation are listed in Table 2 and are values at room temperature.

Table 2 Material properties for S275JR
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A second-order regression analysis of the experimental data was carried out using commercial software SPSS to establish the input–output relationships. The form of the regression, Eq. (7) [31], is given by:

$$Y={\beta }_{0}+\sum_{i=1}^{k}{\beta }_{i}{X}_{i}+\sum_{i=1}^{k}{\beta }_{ii}{X}_{i}^{2}+\sum_{i=1}^{k}{\sum_{i<j}^{k}{\beta }_{ij}X}_{i}{X}_{j}+\varepsilon$$
(7)

where \(\varepsilon\) represents the error in fitting and the \(\beta\) terms are coefficients.

A back-propagation neural network (BPNN) was used and written using Python (Keras). An indicative schematic of the neural network is depicted in Fig. 6, but a series of networks were built based on different combinations of input variables from Table 3. Cross-verification was performed to determine the coefficients for each BPNN, using experimental output data from Table 3 (T1–T69), split into four equal sets. Three sets were selected as training data to predict the remaining data and this procedure was repeated four times by rotating the test data sets.

Fig. 6
figure 6

Schematic of the back-propagation neural network structure

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Table 3 Welding parameters and measured penetration depth for each trial in group 1. Penetration depth varies from 3.12 to 10.73 mm
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3 Results and discussion

3.1 Beam characteristics

Figure 7 shows the effect of focussing current on achieving a minimum in the beam diameter, in this instance for a beam current of 25 mA. Important to note is that as the accelerating voltage increases, the beam radius decreases and the focussing current increases. Although not shown, the focussing current required to achieve the minimum (focussed) beam was approximately constant over the range of beam currents investigated. From plots such as these, the sharp focus positions were identified, for a given accelerating voltage (approximately 290, 320 and 360 mA for 40, 50 and 60 kV, respectively) and from these points under and over focussed conditions were achieved by set mA differences in focussing current above and below these positions.

Fig. 7
figure 7

The effect of accelerating voltage and focussing current on the beam radius (for a beam current of 25 mA)

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Figure 8 plots the focussed beam sizes for all beam currents and accelerating voltages from which it is clear that the beam size increases as the beam current increases and the accelerating voltage decreases. The approximate range of the beam radius was 0.2 to 0.85 mm, for conditions of 60 kV, 25 mA and 40 kV, 45 mA, respectively. Figures 7 and 8 highlight the potential problems arising when beam dimensions are not measured, as there is considerable sensitivity of the beam radius to inaccurate or inconsistent focussing.

Fig. 8
figure 8

The effect of accelerating voltage and beam current on the focussed beam radius

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3.2 Weld geometry measurements

Examples of weld profile dimension measurements taken at each of the 5 cutting positions within one sample are shown in Fig. 9. The mean penetration depth and the standard deviation were determined based on these measurements. Weld profile dimension data are summarised in Tables 3 and 4. The average, maximum and minimum absolute deviation of measured penetration depth are 2.62%, 7.39% and 0.94%, respectively.

Fig. 9
figure 9

Weld profile dimension measurements at each sample point for a given sample, taken from microscope images

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Table 4 Welding parameters and measured penetration depth for each trial in group 2. Penetration depth varies from 3.2 to 10.42 mm
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The total error in the measured penetration depth, for a given set of welding parameters, has been estimated from the sum of the individual errors. These errors originate from the variance in machine input and achieved parameters (± 2–3%), the measurement method from the microscope images (± < 0.5%), the average variation measured at different weld positions (± 2–3%) and the variation estimated from fluctuations in substrate temperature (± 1.5–2%). The overall accumulated error was estimated to be approximately ± 4%.

3.3 Depth prediction using the empirical equation

Based on fitting to the T1–T69 data, shown in Fig. 10, the expanded empirical equation can be given by:

Fig. 10
figure 10

Data fitting using Eq. (4) from data T1–T69

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$$D_{pe}=4.8383\cdot10^6\cdot\frac{\alpha^{0.5682}}{k\cdot\theta_M}\cdot\frac{Q_{in}}{{(S\cdot\sigma)}^{0.5682}}$$
(8)

The penetration depths predicted using Eq. (8) are compared with measured depths from group 2 (C1–C30) in Fig. 11. The average absolute percentage deviation is 6.71% and the maximum absolute percentage deviation is 20.74%.

Fig. 11
figure 11

Depth calculated using Eq. (7) compared with measured depths (C1–C30)

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Hemmer and Grong [32] applied a similar regression method to predict the welding depth of grade 2 titanium, aluminium alloy AA 5052 and duplex stainless steel SAF 2507, and the maximum uncertainty in their predictions varied from 15 to 25%, similar to those found in this study. The use of the beam width, instead of the weld bead top width, does lead to significant improvements in the accuracy of the predictions. Our own use of this approach, substituting the weld bead top width (measured as shown in Fig. 9) for the beam width in Eq. (5), resulted in an average deviation of 14.5% and a maximum deviation of 47.5%. Similar high maximum deviations (40%) were also observed for this same approach in [29]. Possible explanations for this are, the physical relationship between weld width and penetration depth is not strong, and the depth/width ratio can vary depending on the weld cross section [33]. The beam width is, however, strongly correlated with beam power density, which plays an important role in penetration depth. Additionally, for a single weld with constant weld parameters, the deviation in weld bead top width observed in this study is around 7%. Considering the high depth/width ratio, the 7% fluctuation can account for much of the depth prediction deviation observed. These observations demonstrate the clear benefit of adopting techniques to measure the beam dimensions and the need to measure them in-process.

3.4 Depth prediction using second-order regression

An example for the depth prediction based on a second-order regression of T1–T69 data, for variables of accelerating voltage \(U\), beam current \(I\), welding speed \(S\) and 1/e2 radius in the x direction and y directions, ignoring the insignificant terms, is given by Eq. (9). The penetration depths calculated by Eq. (9) are compared with the measured data (C1–C30) in Fig. 12. The average absolute percentage deviation is 5.97% and the maximum absolute percentage deviation is 22.28%.

Fig. 12
figure 12

Depths calculated by Eq. (9) compared with measured depths (C1–C30)

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The regression approach allows all combinations of variables to be included in the analysis. Reducing this to the accelerating voltage \(U\), beam current \(I\), welding speed \(S\) and beam radius σ gives an average absolute percentage deviation of 6.63% and a maximum absolute percentage deviation of 27.53%. Increasing it to include the relative focussing current gives an average absolute deviation of 4.88% and a maximum absolute deviation of 14.92%. A summary of the accuracy of all the predictions is presented in Table 5.

Table 5 Summary of absolute prediction deviations using different models and parameters
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$${D}_{p}=6.29\times {10}^{-2}U-2.318\times {10}^{-3}{U}^{2}- 1.283\times {10}^{-2}{I}^{2}-3.525\times {10}^{1}{x}_{1/{\mathrm{e}}^{2}}^{2}+1.53\times {10}^{-2}UI+1.322\times {10}^{-4}US+3.711\times {10}^{-1}U{x}_{1/{\mathrm{e}}^{2}}-9.31\times {10}^{-1}U{x}_{1/{\mathrm{e}}^{2}}-4.277{\times 10}^{-4}IS+9.346\times {10}^{-1}I{x}_{1/{\mathrm{e}}^{2}}+3.832\times {10}^{-1}I{x}_{1/{\mathrm{e}}^{2}}-3.349\times {10}^{-2}S{x}_{1/{\mathrm{e}}^{2}}+4.079\times {10}^{-2}S{y}_{1/{\mathrm{e}}^{2}}$$
(9)

The prediction accuracy for the second-order regression is slightly improved compared to the empirical equation, more so if a greater number of variables are considered. It should be emphasised that the input values for the prediction should not exceed those used in the regression, as the prediction accuracy may drop significantly or even become physically unrealistic. For example, when the accelerating voltage, beam current, welding speed and beam radii are 60 kV, 10 mA, 600 mm/min and 0.3 mm, the depth predicted by Eq. (9) is − 2.47 mm. In contrast, it is 2.489 mm when calculated using Eq. (8).

3.5 Depth prediction using a back-propagation neural network

Typical training results, as a function of different network parameters, are shown in Fig. 13. Figure 13a depicts the mean squared error (MSE) of different neural network structures with cross-verification. NX at the abscissa means one hidden layer with X neurons, NXNY means two hidden layers with X and Y neurons, and so on. Figure 13b depicts the MSE of the N20 NN with different iterations. Based on the results of the cross-verification process, the coefficients of the BPNN with lowest MSE were selected for predicting data from the set C1–C30. Typical coefficients for BPNNs with high prediction accuracy are:

  1. 1)

    One hidden layer with 15 neurons.

  2. 2)

    Linear transfer function for input layer and ‘exponential’ activation function for hidden layer.

  3. 3)

    Type ‘Normal’ kernel initialiser for the input layer and the hidden layer.

  4. 4)

    ‘SGD’ optimiser: Initial learning rate is 0.001, decay steps 10,000 and decay rate 0.9.

  5. 5)

    Losses type: mean absolute percentage error.

  6. 6)

    5000 iterations.

Fig. 13
figure 13

(a) MSE of each neural network from cross-verification. (b) MSE of N20 with different iterations from cross-verification

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Figure 14 shows the predictions for data C1–C30 based on a BPNN with inputs of accelerating voltage, beam current, welding speed and 1/e2 radius in the x direction and y directions. For this case the average absolute deviation of the neural network is 5.35% and the maximum absolute deviation is 17.65%. Table 5 summarises the findings and shows that, in general, increased accuracy can be achieved by increasing the number of inputs.

Fig. 14
figure 14

Predicted depths by BPNN compared with measured depths of group 2 (C1–C30)

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The importance of measuring the beam radius is reinforced by these results and is seen to have a greater influence on the accuracy of prediction than the relative focus current (which does affect the EBW penetration depth [20]). Predictions are improved for the BPNN, compared to the other models, but whilst the mean absolute deviations do not significantly improve, when compared on equal terms by inputs, the maximum values are lower and there are no occurrences of unrealistic predictions. It is worth noting that the mean absolute deviations for both the regression and BPNN methods may not decrease much further, given that they are approaching the estimated error for conducting and measuring the welding process.

A study was made of the least accurately predicted depths, for each of the models, listed in Table 6. According to the cases listed in Table 6, all the significant deviations occur with accelerating voltages of 40 kV and 60 kV, and there are some interesting features. Firstly, there was no single case where the magnitude of the deviation was larger than 15% for all of the predictive methods. For the welds with 40 kV accelerating voltage and any focal position, the large deviations for the empirical equation method are all positive, which means the method has over-estimated the penetration depth. When the accelerating voltage is lower, the kinetic energy of a single electron is lower, causing the electron to be more likely to be affected by ambient particles. Therefore, it can be concluded that the efficiency of 40 kV welds should be lower than those of 50 kV and 60 kV welds, but this cannot be reflected in the empirical equation methodology. For the welds with accelerating voltages of 60 kV, the large deviations are strongly correlated with electron beam focal position. Big negative deviations at 60 kV only occur for the under-focus condition, and big positive deviations only occur for the over-focus condition, supporting observations for an increased penetration depth for under-focus conditions [34, 35]. This observation agrees with the inference that electrons can be better absorbed with under-focussed beams and the energy transfer efficiency is higher, emphasising the importance of measuring beam focal position before predicting EBW penetration depth. Even though neural network and second-order regression models show better performance in solving non-linear problems with adequate data, the omission of beam focus information still affects the prediction accuracy.

Table 6 List of cases with absolute deviation of penetration depth larger than 15%. Variables of models are accelerating voltage \(U\), beam current \(I\), welding speed \(S\) and 1/e2 radius
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3.6 Depth prediction using reduced experimental data

The influence of reducing the number of data used to train the model was evaluated. It was facilitated by using a smaller subset of the data in Table 3 and retraining new models. For these models, the inputs were accelerating voltage, beam current, welding speed and 1/e2 radius in the x and y directions. In order to select the parameters from the factors and levels outlined in Table 3, an orthogonal experimental design approach was performed using SPSS software. This yielded a minimum of 25 tests, which was expanded to 36, 47 and 58 by using the ‘holdout cases’ function. The training data used for each of these cases is identified in Appendix Table 7.

Figure 15 shows that the average absolute deviation is sensitive to the size of the training data set. When the size of the training dataset is 25, the average deviation is 6.37% (the maximum is 18.12%). If this is increased above 36, the average deviation is relatively stable between 5.3 and 5.6% (and the maximum is below 17%). For the particular case shown here, the number of experimental training data should be greater than 36 to achieve a stable predictive performance for the BPNN.

Fig. 15
figure 15

Prediction accuracy by BPNN with different numbers of experimental training data

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It is worth noting that the average deviation for only 25 tests with detailed beam characteristics could not be matched by the full 69 tests without any beam probing data. The beam probing technology, therefore, significantly reduces the number of tests required to train the models to achieve a target accuracy.

3.7 Depth prediction using experimental and virtual data

In production, EBW trials are costly and it is attractive to use virtual data generated by statistical methods to augment the experimental dataset to aid training of an ANN [5]. There are some limitations to this process based on the differences between the BPNN and the models. The BPNN uses 5 different focal positions as inputs to create the experimental matrix; this had to be substituted by 5 different beam radii for the models (covering the approximate 0.2 to 0.85 mm range). To make it possible to use both the empirical and second-order regression models, the BPNN had to be based on inputs of accelerating voltage, beam current, welding speed and mean beam radius only. For our test cases, we assumed that virtual data would be generated to complete the full factorial experimental matrix, building on test cases using either 25 or 69 experimental data (as defined earlier) and empirical and regression models derived from them.

As a baseline, the BPNN models had mean deviations of 6.25% and 6.00% when trained using only 25 or 69 experimental data, respectively (different to that reported in Fig. 14, as these were for models including x and y beam dimensions). The empirical model has deviations of 6.61% and 6.71% when created with 25 and 69 data, respectively, and the second-order regression 9.27% and 6.83%, respectively, highlighting the sensitivity of the regression model to the volume of data.

Perhaps not surprisingly, when these less accurate models are used to create the remaining (300 +) data for the full factorial matrix, and are incorporated into the BPNN, the mean deviations tend very closely to those for the sources of these additional data (6.7% and 6.8% for data added from the empirical model, 9.15% and 7.1% for the second-order regression). Thus, the predictive accuracies (mean and maximum deviations) for the BPNN based on limited data are not improved.

It was possible to demonstrate that with a model providing virtual data that is more accurate than a BPNN model derived from very limited experimental data points, the accuracy of the BPNN could be improved. This was achieved by having highly scattered and very limited initial data, it could also be the case for data derived from highly accurate numerical simulations or very well established empirical models from the literature. However, with accurate and simple models or simulations available, the rational for developing a BPNN is greatly diminished.

4 Conclusions

This paper shows that beam probing plays an important role in predicting EBW penetration depth. In addition to accelerating voltage, beam current and welding speed, accurate measurement of beam radius and focal position, using beam probing technology, significantly improves the accuracy of model predictions based on an empirical equation, a second-order regression and an artificial neural network. For a linear EB power density ranging from 86 to 324 J/mm, the mean absolute deviations for an ANN model with inputs for beam radius and focal position were as low as 4.5%, lower than those for an analytical model (6.7%) and a second-order regression model (4.9%) and were more reliable. They were found to be close to the estimated 4% variability in the process and its measurement.

The size of the training data set was found to influence the accuracy of model predictions. By using a design of experiment method to choose the parameters, a minimum of 36 data points was deemed necessary, for this problem, to ensure the ANN prediction was as accurate as reasonably achievable (compared to the variability in the process and its measurement). By using the beam probing technology and using the additional input measurements of beam focal position and shape, the number of data points required to train neural network models with the same level of prediction accuracy is reduced, in this case, by more than 60%. Adding virtual data generated by less reliable analytical or regression models did not improve the predictive capability for the ANN developed in this study.