1. Introduction
An electron beam (EB) is an ideal heat source for metal joining in various scenarios, as its large aspect ratio and the distortion accumulation caused by electron beam welding (EBW) can be significantly reduced compared with plenty of traditional joining methods [
1]. A notable molten pool is often generated during EBW, and in fully penetrated welding, there is less solid support under the hot liquid metal, which potentially leads to collapse of the weld joint. Partially penetrated welds are an effective approach to prevent collapse caused by liquid metal gravity, and provide a suitable fusion zone pattern. The post process of partially penetrated EBW largely depends on the weld penetration depth, and the penetration depth can be much different with varied beam parameters or welding settings such as the beam current, accelerating voltage, travelling speed, focal spot size, etc. [
2]. For EBW operators, accurately estimating the penetration depth based solely on these figures and settings is challenging, especially when welding a new material for the first time. Therefore, there is a direct demand of a developing reliable EBW penetration depth prediction approach.
Numerous studies have been dedicated to using computational fluid dynamics (CFD) models for replicating the physical phenomena of EBW and guiding manufacturing processes. Taking into account the purpose of the models and complexity of boundary conditions, both 2D versions and 3D versions are frequently used, all of which exhibit the potential to predict penetration depth. While the 2D models can provide basic simulation results, they are particularly useful in specific EBW scenarios.
Liu and He [
3] introduced a keyhole tracking method, and applied it to simulate aluminium alloy spot weld pool dynamics. The model successfully reproduced the formation of vapour cavities caused by keyhole collapse. However, it was limited to spot welds without a moving heat source, making it difficult to verify its results in traditional butt weld situations. To address this limitation, they simplified the model to a 2D version in 2017, and also found that keyhole collapse was the main reason for penetration stopping [
3].
In another study, Tomashchuk et al. [
4] developed a 2D CFD model to simulate the propagation of dissimilar liquid metals during EBW of pure copper with AISI 316 austenitic stainless steel. This model successfully predicted different types of molten pool morphologies, and provided calculated copper/steel fractions in the melted zone. The study also revealed that residual stresses decreased when copper filler was added during the Ti-15-3 alloy to 304 stainless steel EB welding process. However, due to the lack of calculation in one flow direction, predicting penetration depth efficiently is challenging with 2D models when compared to 3D models.
The 3D CFD models in previous studies open up new possibilities for predicting EBW penetration depths. For instance, the 3D CFD method has been successfully applied to various materials, as demonstrated by Borrmann et al. [
5], who used it to simulate EB-welded TRIP/TWIP steels with different thermophysical properties. Liang et al., employed a CFD–FEM model to simulate the residual stress in EB-welded Ti-6-Al-4V alloy caused by spiking defects [
6], and Chen et al. utilised CFD simulations to study porosity defects during the EBW process of the 2A12 aluminium alloy [
7,
8]. Therefore, with proper material properties, the welding process of most metal types can be simulated using 3D CFD models.
Moreover, 3D CFD models have successfully simulated the generation of keyholes inside the molten pool and the physical phenomena during the EBW process. For instance, B. Huang’s 3D CFD model based on the level set (LS) method effectively simulated weld pool dynamics and keyhole generation [
9]. The simulation revealed a 1–2 kHz keyhole oscillation, with certain high welding speeds stabilising the keyhole and flow speeds inside the molten pool reaching 5 m/s. Other studies have also employed similar LS methods to simulate the EBW process of Ti-6Al-4V alloy [
10,
11]. One of these studies focused on keyhole and molten pool dynamics during scanning electron beam welding, finding that high-frequency beam scanning resulted in better uniformity of the keyhole and fewer defects compared to low-frequency welds. The other study investigated vapour plume dynamics inside the EBW keyhole, reporting a maximum velocity of 1500 m/s based on simulated results.
Furthermore, 3D CFD modelling successfully replicated various fusion zone patterns of EBW. Wang et al. [
12] used a combined point-linear heat source to simulate the temperature field of an electron beam-welded titanium alloy, successfully replicating various fusion zone patterns such as the nail pattern, bell pattern, funnel pattern, and chock pattern.
Additionally, 3D CFD simulations have demonstrated their potential in guiding weld parameter selection for real EBW production. D. Trushnikov and G. Permyakov introduced a 3D CFD model in 2017 to investigate the influence of focus position on the molten pool dynamics of steel welded by oscillated EBW [
13]. The results, both experimental and simulated, showed that the amplitude of convection caused by the Marangoni effect increased when the beam focusing position moved downward. Yin et al. [
14] utilised the CFD method to simulate the weld bead shape of an electron beam-welded pure niobium plate, aiming to obtain dependable electron beam oscillation parameters for use in manufacturing superconductivity cavities.
The earlier study highlighted that CFD modelling holds promise in predicting penetration depth and even other dimensions of the molten pool. Moreover, the 3D models can offer a more dependable prediction, as they can incorporate a greater amount of boundary information. However, the CFD models mentioned above are mostly focused on quality variations in the EBW process rather than on providing specific quantities to guide EBW parameter selections. Due to the complexity of the solid–liquid–gas transformation during keyhole generation, there is a lack of reliable CFD models capable of predicting the penetration depths for EBW processes that last a few seconds.
On the other hand, obtaining a prediction result from CFD modelling frequently takes hours or even days. Additionally, adept skills in model setup are crucial for achieving a highly dependable solution. These factors render the application of CFD prediction challenging in numerous industrial scenarios, especially those necessitating real-time solutions in production. As a result, for predicting EBW penetration depth, the prevailing approach remains reliant on statistical tools like regression fitting and artificial neural networks (ANN).
For example, Dey et al. [
15] in 2008 utilised a backpropagation neural network (BPNN) and a genetic algorithm-tuned neural network (GANN) to achieve the EB weld bead profile predictions of stainless steel 304. Moreover, they applied a similar methodology to predict the EB weld bead profile of other materials, including aluminium alloy [
16,
17,
18] and reactive material zircaloy-4 [
19]. According to the studies conducted by Shen et al. in 2009 [
20], they also utilised BPNN to predict weld bead dimensions for 1cr18ni9ti stainless steel. The maximum absolute value error for the predicted penetration depth was 6.6%.
Such methods offer the advantage of being straightforward to implement, and generally provide reliable results when appropriate tests are conducted and prediction parameters are within a specified range. However, when applying these methods to a different material, statistical approaches require extensive training data, which is typically expensive and time-consuming.
Several studies have also used empirical equations to predict the penetration depth of EBW. However, some of these equations are not available before conducting the weld [
21], while others are challenging to implement in real industrial production due to their unclear beam radius characterisation method [
22,
23].
Combining numerical simulations with artificial intelligence algorithms has been demonstrated as an efficient approach for addressing numerous industrial challenges. These challenges often involve either the unavailability of extensive experimental data or complexities influenced by various factors. Examples include predicting the size of flammable vapour clouds caused by liquid hydrogen release [
24], solving problems related to natural convection and conjugate heat transfer [
25], predicting solid particle erosion in the oil and gas industry [
26], and enhancing the design of centrifugal pumps [
27], etc. Based on these studies, the integrated model has shown promising potential in guiding manufacturing processes, explaining physical rules, and enhancing industrial safety.
In this research, a machine learning (ML) enhanced CFD model is introduced that accurately and efficiently predicts the penetration depth during electron beam welding. Several improvements have been made in comparison to previous studies:
Characterisation of electron beams: The electron beams are characterised using the method described in [
2], allowing the beam radius in both the welding and cross-sectional directions to be identified. As a result, the beam characteristics can be easily incorporated into the CFD model as inputs, following a standard protocol.
Simulation of a 2-s welding process: The CFD model has been adjusted to simulate a 2-s welding process, incorporating an efficient and easily understandable heat generation algorithm.
Applications for neural network training: the CFD model is also suitable for training a neural network model. This allows for rapid and reliable penetration depth predictions in real industrial environments.
The predicted data are compared with experimental data, and mild steel S275JR was chosen for the partially penetrated experiments to validate the feasibility of this method. Through these measures, this approach can provide accurate penetration depth predictions, can be adapted to different materials, eliminates the need for expensive and time-consuming trial and error tests, and can be integrated into an ML model to meet the requirements of Industrial 4.0.
3. EBW Process Monitoring
The welding experiments were conducted using an EB machine developed by Cambridge Vacuum Engineering (serial no. CVE 661), with a maximum power of 4 kW and a maximum accelerating voltage of 60 kV. The experiments were carried out under a vacuum level of 10
−3 mbar. The experimental conditions were varied within the following ranges: 40–60 kV for the accelerating voltage, 25–45 mA for the beam current, and a welding speed of 500–700 mm/min. The EB linear power was controlled in the range of 86 J/mm to 324 J/mm. The focusing currents were adjusted between 277 mA and 366 mA, resulting in three different focal conditions: over-focus, sharp-focus, and under-focus. Over-focus refers to the beam focal position above the sample surface, sharp-focus at the surface level, and under-focus below the surface level. The beam radii were characterised using the 4-slit probe developed by The Welding Institute (TWI), UK [
28]. The working distance, measured from the chamber roof to the workpiece surface, was 157 mm.
A sketch of the beam probing system setup is illustrated in
Figure 2. The system comprises an electron beam (EB) machine, an EB probe, a PC-based oscilloscope, a computer, and a waveform generator. The EB probe was installed inside the vacuum chamber of the EB machine and connected to an oscilloscope outside the chamber via the chamber interface. The probe’s working surface with slits was positioned under the electron beam gun and perpendicular to the electron beam path. To ensure precise beam parameters, the probe’s working surface was aligned with the surface of the workpiece to be welded. The installed probe is shown in
Figure 3.
The electron beam is rapidly deflected to scan across the slits of the EB probe for beam characterisation, as depicted in
Figure 4. A specific deflection pattern can be designed on the computer and converted into a series of signals to control the deflection coils in the EB machine through a waveform generator. Electrons passing through slits in multiple directions can be converted into a voltage signal by a Faraday cup positioned beneath and subsequently captured by an oscilloscope. A Faraday cup is an instrument used to measure electric charge or the current of charged particles, typically in electron beams. And the voltage signal, which refers to the beam width information, is visualised on the computer screen by the BeamAssure
TM (TWI Ltd., Cambridge, UK) electron beam characterising system shown in
Figure 5 [
28]. In
Figure 5, the ‘System Setup’ and ‘Test Details’ boxes record relevant beam parameters that link beam parameter information with beam radius data. The ‘Test Results’ box displays detected beam radii in the x or y directions. The ‘Beam Caustic’ graph illustrates beam radius changes with varying beam-focusing currents, allowing the determination of the focusing status (‘over-focus’, ‘sharp-focus’, and ‘under-focus’). In the ‘Beam Width’ box, values for differently defined beam radii can be found. The system can issue a warning signal if any detected value exceeds the preset limit. These beam probing data can then be further analysed and applied in the CFD model. Once the beam probing process is complete, the welding process can begin.
S275JR mild steel was used as the substrate material, with dimensions of 100 × 75 × 20 mm. More details of the experiment design, beam probing, sample treatment and measurement data have been published in [
2]. Additionally,
Table 1 presents the measured depths obtained under various parameters.
5. Results and Discussion
The simulated results of the depth were compared with the experimentally measured depth of C1–C30, as shown in
Table 4 and
Figure 11. It is important to note that in some cases, the penetration depth of a single-path weld is not consistent, and may even decrease over time. This inconsistency can be attributed to the potential collapse of the keyhole due to gravitational effects. Hence, it was necessary to record five depths at different times. The simulated average absolute percentage deviation was found to be 8.26%, and the maximum absolute percentage deviation was 26.56%. Taking into account the accumulated error (typically around 4%) caused by variances in machine input, the measurement method from microscope images, measurements at different weld positions, and fluctuations in substrate temperature, the overall deviation of approximately 8% demonstrates that the CFD model is an efficient method for predicting the electron beam weld penetration depth before conducting the weld.
If we set a requirement of less than 15% deviation, 26 out of 30 predictions fall within this range. To be more specific, the predictions with significant deviations are listed in
Table 5.
When compared to ANN with experimental data from previous studies whose average absolute percentage deviations went as low as 5% [
2], the prediction accuracy of the CFD model is notably lower.
Table 5 shows that significant errors occur only in over-focus situations with an accelerating voltage of 40 kV and under-focus situations with 60 kV. For the welds with a 40 kV accelerating voltage, the large deviations are all positive, indicating that the CFD method overestimates the penetration depth of these welds. This is likely due to the lower kinetic energy of single electrons at 40 kV, making them more susceptible to ambient particle effects. As a result, the efficiency of 40 kV welds may be lower than that of 50 kV and 60 kV welds, a feature not reflected in the current CFD model. On the other hand, for the welds with an accelerating voltage of 60 kV, the large deviations are all negative, further supporting the above statement.
Additionally, the under-focus status, which can generate deeper penetration during EBW compared with the over-focus situations [
35,
36], is not appropriately reflected in the current model, thus amplifying the errors in
Table 5. However, there is potential to enhance the CFD model by incorporating the focus status of the electron beam, a topic that will be explored in future studies.
Despite not reaching the same level of prediction accuracy as the ANN trained with adequate actual data, the CFD model still shows promise as it reduces the cost and time of conducting experiments. It took approximately 30 h to complete the simulation for each model, and multiple models can run simultaneously without affecting each other. However, the ANN model relies on extensive trial-and-error testing, which typically spans a few weeks. According to the data in [
2], when the number of training experiments is below 14, the CFD model performs better. Thus, it can be concluded that CFD models can offer superior prediction accuracy compared to BPNNs trained with limited experimental data.
Based on the established CFD data, the fast-responding ANN model is able to be built based on CFD-generated paired data. When configuring the model, it is essential to define the range of welding parameters, as a specific welding scenario must be provided for making predictions. However, it is not feasible to enumerate all of the potential parameter combinations within that range. Therefore, orthogonal experimental design proves to be an effective method for addressing this issue.
Table 6 displays the CFD-simulated penetration depth using the weld parameters designed via the orthogonal experimental design approach using the commercial software SPSS (version 29), including beam current (25 mA, 30 mA, 35 mA, 40 mA, and 45 mA), accelerating voltage (40 kV, 50 kV, and 60 kV), welding speed (500 mm/min, 550 mm/min, 600 mm/min, 650 mm/min, and 700 mm/min), and beam 1/e
2 radii in both the x and y directions (0.25 mm, 0.35 mm, 0.45 mm, 0.55 mm, 0.65 mm, and 0.75 mm). Tests C1–C30 listed in
Table 4 also fall within the range of these parameters.
Figure 12 illustrates the sectional view of the molten pools in the 52 simulation cases, where the light area represents the liquid phase, and the dark area represents the solid phase.
Utilising the 52 CFD-simulated data from
Table 6, the BPNN model can be trained and the structure confirmed as follows:
- (1)
One hidden layer with 15 neurons.
- (2)
Linear transfer function for the input layer and ‘softplus’ activation function for the hidden layer.
- (3)
Type ‘Normal’ kernel initialiser for the input layer and the hidden layer.
- (4)
‘SGD’ optimiser: Initial learning rate is 0.001, decay steps 10,000, and decay rate 0.9.
- (5)
Losses type: mean absolute percentage error.
- (6)
5000 iterations.
The predictions made by the BPNN were subsequently compared with the actual depths in
Table 1. The results are presented in
Figure 13.
With 52 training data points, the BPNN achieved a prediction with an average absolute percentage deviation of 9.19%. The prediction deviations range from +23.88% to −27.61%. If we set a requirement of less than 15% deviation, 24 out of 30 predictions fall within this range. It can be observed that the prediction accuracy of the BPNN is slightly lower than that of the CFD model. Although it may be possible to improve the BPNN accuracy to closer to that of the CFD prediction by increasing the quantity of paired data or adjusting the neural network structure, surpassing the CFD prediction seems unlikely, as the generated data are already affected by the lack of focal position information.
The ML-enhanced CFD model significantly reduced the calculation time to less than 1 s compared to the 30 h required by the CFD model alone. Remarkably, this efficiency improvement has not led to a significant compromise in the prediction accuracy. The results confirm the feasibility and efficiency of the ML-enhanced CFD model.
6. Conclusions
In this study, a novel CFD approach was employed to predict the penetration depth during electron beam welding. The CFD model yields satisfactory prediction results, with an average absolute percentage deviation of approximately 8%. The advantage of this approach lies in its ability to reduce costs and save time, since it does not require extensive testing before building the model, unlike statistical methods. The prediction method can also be adapted to different materials by modifying the CFD material properties, and to different machines by following the beam characterisation method.
Furthermore, in this study, virtual paired data between weld parameters and penetration depth were generated using the CFD model, allowing the tuning of neural networks for real-time and fast response prediction in real industrial scenarios. The ML-enhanced CFD model demonstrates the ability to achieve rapid predictions, exhibiting an average absolute percentage deviation of roughly 9%, without the need for expensive trial-and-error experimentation.
However, the current ML-enhanced CFD model does have some limitations. The model does not yet incorporate detected beam focal position information, leading to certain prediction deviations. Additionally, errors may arise when the electron beam power density deviates from the ideal Gaussian distribution. The current beam probing method falls short in accurately describing non-Gaussian beams. These challenges persistently affect the development of EBW automation and prediction transfer among different machines. The next objective is to enhance the simulation accuracy of the CFD model by incorporating additional information, such as the focal position and the impact index of the accelerating voltage. Additionally, the aim is to improve the beam probing method by integrating more detection directions and implementing automatic height adjustments. These enhancements will empower the ML-enhanced CFD model to offer more accurate and improved depth prediction results.