Energy-efficient laser welding with beam oscillating technique – A parametric study

This study investigates the oscillating laser welding process of Ti Grade 5 plates using a design of experiments (DoE) approach. It aims at developing empirical models to correlate the weld qualities with the process parameters. For continuous-wave (CW) mode, three parameters: the peak power, linear velocity, and oscillating velocity were considered. For pulse mode, five parameters: the peak power, number of turns per pulse, duty cycle, linear velocity and oscillating velocity were considered. The responses were multiple but can be divided in three groups: the fusion zone, the weld properties and the energy efficiency of the process. Polynomial models were developed for the surface width and depth of the fusion zone as well as the energy efficiency of the process while no reliable correlations were observed for the microhardness and porosity. The Predicted-R2 values of the models were higher than 0.82, indicating the possibility to predict the quality of the joints with adequate accuracies. DoE results revealed that the parameters interact in a complex manner and depends on the responses. Within the considered ranges of process parameters, the laser power has the largest influence on the depth, surface width and energy efficiency of the fusion zone in CW mode. In pulse mode, the laser power is still the dominant factor on the surface width of the fusion zone but the linear velocity has the largest influence on the other responses. The development of the empirical models is beneficial for the optimization of the welding process due to their simplicities and adequate accuracies. An optimization procedure was demonstrated with a specific set of requirements on the fusion zone’s geometry while reaching highest energy efficiency.


Introduction
Economic and environmental sustainability is a major priority of manufacturing industry. The increasing energy prices and public environmental awareness are the main drivers of this movement. A statistical study from the International Energy Agency (IEA) indicates that the energy consumption of the manufacturing industry amounts to over 30% of the global electricity production and 36% of the global carbon dioxide emission (International Energy Agency, 2017). However, a significant amount of up to 40% of the energy input does not add value to the final product and is only used to establish stable processes (Allen et al., 2002). Therefore, systematic studies on the processes, with a particular focus on the energy efficiency, are required. Yan et al. (2017) estimated that a decrease of 1% in the energy consumption of the arc welding process alone could lead to a total energy reduction of more than 2 billion kilowatt per year. Unfortunately, energy efficiency as a function of process parameters is scarcely reported in the literature, especially for non-conventional processes such as laser-material processing (Pastras et al., 2017;Wei et al., 2015;Yan et al., 2017).
This work presents an investigation on laser beam oscillation welding, a recently developed method for laser welding. Laser welding, as one of the most popular applications of laser technology in industrial manufacturing (Katayama, 2013), have several advantages over conventional ones (e.g. arc welding) such as low energy consumption, little wear materials due non-contact processing, high precision, and low heat input (Dawes, 1992;Katayama, 2013). This process is so highly dynamic that its final quality is influenced by numerous parameters from both the material and the laser radiation (Fabbro, 2010;Katayama, 2013;Le-Quang et al., 2018). This is particularly true for the keyhole regime, which is a deep laser welding regime during which material is intensively vaporized and where the ratio between the welding depth (D) and laser beam diameter (∅ laser ) is D/∅ laser > 1 *. The complexity increases even further in the case of laser beam oscillation welding. This process involves a laser beam oscillating at high frequencies either along or across the weld seam instead of a linear trajectory as in the case of conventional welding (Berend et al., 2005;Hagenlocher et al., 2018;Vakili-Farahani et al., 2016). The resulting high speed of the laser beam was shown to enhance the stability of the process, i.e. reduce defects, in particular, during the keyhole regime (Berend et al., 2005;Fetzer et al., 2018;Müller et al., 2018). This recently developed technology has showed potential in welding highly reflective materials such as copper and aluminum (Fetzer et al., 2016;Hagenlocher et al., 2018;Li et al., 2020;Müller et al., 2018), indicating a possibility to replace other more energy demanding methods. Another advantage of this technique is its energy efficiency. Firstly, with a suitable oscillating trajectory, it is possible to cover a larger surface area with a much smaller laser beam, allowing a significant reduction of the laser power to obtain a similar weld depth (Müller et al., 2018;Vakili-Farahani et al., 2016). Secondly, the overlapping of the beam path during the oscillating movement allows each surface location to interact with the laser beam multiple times. This process can potentially enhance the coupling of laser energy in the workpiece thanks to the modification/pre-heating of the surface by the previous scan (Hagenlocher et al., 2018;Sommer et al., 2017).
Despite the promising results, establishing the dependencies between the process parameters, weld qualities, and energy efficiency is challenging due to the complexity of the process and the involvement of many factors. As a result, no work has investigated the energy efficiency of laser beam oscillation welding in terms of process parameters. 1 Similarly, the correlations between processing parameters and weld qualities are not yet successfully formulated. There are recently indeed a few works on this topic, in which the one-factor-at-a-time method was used (Li et al., 2020;Wang et al., 2019). Unfortunately, this method has a major drawback, which is the lack of information about possible interactions among factors (Li et al., 2020;Vakili-Farahani et al., 2016;Wang et al., 2019). To overcome this issue, design of experiment (DoE) method is often exploited to investigate the effects of the main factors (e. g. laser process parameters), their interactions on selected responses (e. g. weld qualities) (Montgomery, 2017). Such statistical approaches have been applied to conventional laser welding with suitable results. In particular, Benyounis et al. (2005) and Khan et al. (2011) formulated polynomial models to correlate the width and depth of laser weld seam with laser power, welding speed and focus position. Vakili-Farahani et al. (2016) showed that the correlations could also be reliably represented by linear models containing also the interactions between the process parameters. In general, the advantage of this statistical approach lies on the capacity to develop semi-empirical models to approximate highly complex processes that are not thoroughly understood (Saeidi et al., 2016;Vakili-Farahani et al., 2016;Wasmer et al., 2017). It must be emphasized that the DoE method is to provide simplified presentations of the process outcomes. Besides the possibility to predict the weld quality, the developed models can allow optimizing the process parameters to meet specific requirements (Vakili-Farahani et al., 2016;Wasmer et al., 2017), which is of a great interest for industrial applications.
This paper presents an investigation of the laser beam oscillation welding of Ti Grade 5 plates by means of a parametric experimental approach using the design of experiments (DoE) methodology (Montgomery, 2017). To guarantee the integrity of finished product, the laser process must meet the imposed dimensional and mechanical specifications; in particular it must fulfil requirement in terms of weld width, depth, hardness and porosity of the fusion zone as well as the process energy efficiency of the process.
The article consists of 5 Sections. Section 2 presents the laser setup used for the experiments, the designs of the models as well as the measurement procedure for the chosen responses. Section 3 presents the DoE models and demonstrates an optimization of the process based on the developed models and for specific requirements of the joint qualities. Section 4 discusses the models. Section 5 summarizes the findings of the investigation.

Laser setup and materials
Welding experiments were performed with a fiber laser StarFiber 150P (Coherent Switzerland AG, Switzerland). The laser could operate in either continuous-wave (CW) or pulse modes. The laser beam was guided via a fiber with 12 μm core diameter to a laser head Scanner hurrySCAN 30 (Scanlab GmbH, Germany), which was equipped with a 170 mm focal length lens. This configuration resulted in a laser spot of 30 μm in diameter at the focal point (1/e 2 ). The laser head allowed a great variety of laser beam trajectories, but only circle oscillating was considered in the present work.
The weld samples were 2 mm thick Ti Grade 5 plates. This material was chosen since its fusion zone can be easily observed with an optical microscope after a metallography preparation procedure. The samples were placed in a controlled environment chamber filled with argon at ambient pressure (approximately 0.945 bar) to avoid oxidation. The laser beam entered the chamber via a viewport consisting of a 5 mm thick glass plate with a transmission of approximately 90%.

Design of experiments
In laser welding, a large number of factors may potentially influence the joint quality. They can be divided into two main groups: (1) process parameters and (2) material factors. Each group can be divided into subgroups, e.g. process parameters include wavelength of the laser, laser and sample velocities, etc … To reveal which of these factors most strongly affect the weld properties and quality, we adopt a DoE approach. For details information about DoE methodologies as well as all terminologies (e.g. confidence interval, null hypothesis, etc.), the readers can consult Montgomery (2017).
In this contribution, it is assumed that interactions between the various factors have a significant influence on the responses. Consequently, a two-level factorial design or 2 k factorial design with k corresponding to the number of independent factors, has been chosen. This approach operates with the consideration that each factor k has two levels (Montgomery, 2017). It is commonly used for parameter screening (Saeidi et al., 2016;Vakili-Farahani et al., 2016). The first degree polynomial model for the effects of k factors and their mutual interactions can be express by equation (1): where Y is the experimental response, a 0 is a constant, X i is the factor, a i is the main effect coefficient (half-effect) associated to the factor X i , a ij is the interaction effect coefficients (half-effect) and ε (also referred to as the residual) the error observed in the response Y. Coefficients or halfeffects can be calculated using equation (2): where α is the model coefficient matrix, N the total number of runs and X T the transpose of the model matrix.
The model coefficient matrix is a matrix containing the coefficients/ factors for the empirical models. The matrix form is employed to facilitate the solving of the set of linear equations obtained using the experimental results. Compared to the one-factor-at-a-time approach, factorial design provides a better insight into the influence of the experimental factors on the output results since the possible interactions between the factors are also taken into consideration (Montgomery, 2017).
The required number of experiments increased exponentially with the number of factors (Montgomery, 2017). Thus, to avoid an inacceptable number of experiments and an extremely complex model with many parameters from which most of them would have limited to no impact on the laser process, it is necessary to predefine the experimental factors of interest. In order to limit the number of investigated factors, only circle trajectory was considered where the diameter of the circle was fixed at 0.5 mm. The trajectory of the beam is presented in Fig. 1a. The factors of interest were chosen depending on the welding mode and the justification are given in Sections 2.2.1 and 2.2.2.

Continuous-wave mode
Three factors are considered for the CW mode: the laser power (P p ), linear velocity (V l ) and oscillating velocity (V w ). The velocity terms are indicated Fig. 1b, where V l is also the velocity of the welded plate with respect to the laser head. The low and high levels of the chosen parameters were determined through preliminary tests and the actual values are listed in Table 1. This table contains also their coded values; low level (− 1) and high level (1). The normalization of parameter values to − 1 and 1 is done to facilitate the evaluation of factors' significance, which might be difficult due to the differences in value ranges of different parameters.
The ranges of processing parameters were chosen based on the requirement that the weld seams are without significant loss of materials via spattering which usually happens at high laser power (Katayama, 2013;Le-Quang et al., 2018). The maximum oscillating velocity was at 600 mm/s due to mechanical limitations of the scanner. To investigate the CW mode, we opted for a two-level full factorial design. A total number of 2 3 = 8 experiments were performed in randomized orders to minimize systematic errors.

Pulse mode
The difference between CW and pulse modes is the turning ON and OFF of the laser beam in the latter process. Therefore, additional factors need to be considered, for which the duty cycle Cd and number of turn per pulse NTP were chosen. The former can be expressed as Cd=T p ⋅f in which T p and f are the duration and repetition rate of the laser pulse, respectively. This factor represents the actual irradiation time within 1 s. It was chosen as an independent factor instead of T p and f due to the constraints set by the laser source. NTP, on the other hand, is related to T p and V w , oscillating velocity, according to NTP = Tp⋅Vw π⋅∅ , where ∅ is the oscillating diameter, which was fixed at 0.5 mm in this work. To explain this variable, it is worth emphasizing that the melt pool obtained in this experimental configuration is smaller than the oscillating diameter. Thus, one of the requirements for a continuous joint is that the laser beam completes at least one turn during each laser pulse, i.e. NTP ≥ 1. The selected ranges of parameters for pulse mode are listed in Table 2. A full factorial design based on Table 2 would require at least 2 5 = 32 experiments. In order to reduce the number of experiments, we selected a fractional factorial design with the interaction P p ⋅NTP⋅Cd⋅V l ⋅V w being the generator and V w = P p ⋅NTP⋅Cd⋅V l . This choice resulted in the requirement of 2 (5-1) = 2 4 = 16 experiments (Montgomery, 2017). In this condition, each main effect is aliased with a four-factor interaction (e.g. P p → P p + NTP⋅Cd⋅V l ⋅V w ) and every two-factor interaction with a three-factor interaction (e.g. P p ⋅NTP → P p ⋅NTP + Cd⋅V l ⋅V w ) (Montgomery, 2017). This is defined as a design with a resolution R = V. It is expected to provide significant information regarding the main effects and the two-factor interaction effects.

Experimental responses of interest and sample analysis
Representative optical images from the top of the weld seam as well as at the lateral cross-section are presented in Fig. 2. The weld seams were sectioned, followed by polishing and etching with Kroll reagent for 15 s (Vander Voort, 2004). Due to the melting and re-solidification processes, the fusion zone (FZ) has the highest resistance against etching with the Kroll reagent, followed by the heat affected zone (HAZ) (Meylan et al., 2019) and finally the base material (BM). The three different regions are indicated in the optical images in Fig. 2b.
In the present work, the geometry of the fusion zone (FZ), particularly the surface width (SW), depth (D), and the microhardness Ha were of interest. SW and D are indicated in Fig. 2. They were determined by optical observations with an AxioPlan Zeiss microscope (Carl Zeiss AG, Germany). SW was measured directly on the surface of the seams (Fig. 2a). D was defined as the depth of the FZ at the center of the weld line as observed at the lateral cross-section of the seam (Fig. 2b).
The microhardness Ha of the FZ was measured via micro-indentation   tests on lateral cross-section of the welds using a Micro-indentation Tester (Anton Paar GmbH, Austria). The hardness indenter was a Vickers with a load of 0.98 N (HV0.1). For each cross-section, three indentations were made approximately at half-value of the FZ lateral depth: one in the center of the weld and the other two 100 μm apart from it on each side, indicated by the red arrows in Fig. 3. The diagonals of each indentation were measured ten times to have a statistically reliable average of the Vickers hardness for each indentation. The values of the three indentations for one weld were then averaged to give a mean Vickers hardness of the FZ. In addition, as mentioned in the introduction, the laser beam oscillation welding technology has high potential to energy reduction.
Hence, in this study, the energy efficiency of the process η was also considered. This response was defined as the a fusion zone's volume created with 1 J of output laser energy (Fuerschbach, 1996). Accordingly, η can be mathematically expressed by.
for CW mode and for the pulse mode, where A FZ is the area of the fusion zone measured at the lateral cross-section (See Fig. 3b). To determine the energy efficiency η, the area of the fusion zone A FZ was measured on the optical images using the software ImageJ (Schneider et al., 2012). Afterwards, η was calculated using equation (3) or (4). The chosen responses and their symbols are listed in Table 3.

Results
This section is organized four sub-sections. Section 3.1 presents the experimental results for both CW (Table 1) and pulse modes (Table 2). These results are not discussed in details since the factors and responses are only used to develop the different models. The DoE analysis was performed using the software Design-Expert® Version 9 by Stat-Ease. For the sake of clarity and concision, Section 3.2 describes in detailed the procedure and analysis of variance (ANOVA) only for SW in pulse mode as an example. The identical procedure and analysis were carried out for all other models and the results are presented in Section 3.3. Finally, Section 3.4 reports a process optimization of the process in order to minimize the laser power (P p ) and duty cycle (C d ), maximize the linear velocity (V l ) and energy efficiency (η), and achieve a welding surface width (SW) of 730 ± 10 μm and a depth (D) of 100 ± 20 μm.

Experimental results
The matrices of experiment are presented in Table 4 and Table 5 for the CW and pulse modes, respectively. They contain columns of the experimental parameters in coded values as defined in Tables 1 and 2 and the order of the experiments. In order to avoid systematic errors, the experiments were performed in a randomized run order. The tables present the measured values of the examined responses.    Table 6. The former was calculated using equation (2) while the latter was defined as the ratio of the HE to the factor a 0 . For example for the factor P p , the RHE is obtained by 100* 45.88 740.42 = 6.2 % (See Table 6). The half-normal distribution for this response is plotted in Fig. 4a with negative (blue squares) and positive (orange squares) standardized effects. In this plot, the negligible factors have a normal distribution centered close to zero while factors with a significant effect show a normal distribution centered at their corresponding large but undefined effect values (Montgomery, 2017). Consequently, the terms lying along the pink line are considered to have negligible effects on the response whereas those being off the line have are significant. Accordingly, four main effects P p , V w , V l and Cd are considered to have significant effects on SW and are displayed as hollow squares in Fig. 4a. A Shapiro-Wilk test was done on the unselected terms, giving a Shapiro-Wilk p-value of 0.3. This test is commonly used in statistics to check if the samples are from a normally distributed population (Shapiro and Wilk, 1965). It is used in the software Design-Expert® to check if the experimentally determined values of factors in the models are normally distributed, i.e. representing noise. If the p value is higher than 0.1, then the considered factors are insignificant. The value of 0.3 obtained in the present analysis confirms that the unselected points are approximately normally distributed (Shapiro and Wilk, 1965). The half-effects for all terms (main factors and interactions) are plotted in Fig. 4b in decreasing order of their absolute values. The sequence of significant terms is as follows: P p (|HE| = 45.88) > V w (25.37) > V l (12.92) > Cd (10.87) (See also Table 6).

Analysis of variance
The significance of the selected terms in Fig. 4 and Table 6 are also evaluated using an ANOVA with a 95% confidence interval, and the results are presented in Table 7. In this case, the terms with the probability of error (p-value) less than 0.05, which gives a 95% confidence interval, are significant and those with p-value bigger than 0.05 can be neglected (Montgomery, 2017). A 95% confidence interval in the estimation of a parameter is the value range, which, if measured repeatedly, has 95% confidence level to contain the true value of the parameter. The F-value presented in Table 7 is defined as: where the mean square of terms (MS Terms ) is the ratio of the sum of squares within terms to its degree of freedom (SS Terms /DF Term ), and, similarly, MS Residual = SS Residual /DF Residual . The F-value can be linked to the lack of fit sum of squares for the linear model. A lack-of-fit error significantly larger than the pure error indicates that something remains in the residuals that can be removed by a more appropriate model. Equation (5) is a statistical test to verify the "null hypothesis", which is the absence of factors that are statistically significant to the corresponding response. Normally, for effects having an F-value greater than 3 times their standard error (residual), the null hypothesis is rejected. As seen from Table 7, the model built based on the selected main factors and interactions has an F-value of about 39, which implies that the model is significant. The small p-value (<0.0001) indicates that the probability that this model occurs due to noise is only lower than 0.01%.
The R-square value (R 2 ), Adjusted-R 2 and Predicted-R 2 of this model are 0.9334, 0.9091, and 0.8590, respectively. The high value of predicted-R 2 is desirable since it indicates that the model can predict SW with an adequate precision. Fig. 5 plots the predicted SW (calculated from equation (6)) versus the experimentally measured values. We can observe a high correlation between the model and the measured values.

Modelling of the surface width, depth and energy efficiency for CW and pulse mode
The analysis of other responses was performed following the same procedure. Significant models could be obtained for responses SW, D and η in both weld modes, and the models are summarized in Table 8 and Fig. 6. The latter displays a visual representation of the relative halfeffect of the significant factors and interactions of the models given in Table 8, making a comparison easier. In general, the results confirm that the interactions between the processing parameters can have considerable influences on the joint qualities. The obtained R 2 -values imply that the developed models are statistically significant and can be used to predict the corresponding responses in the examined range of process parameters. On the contrary, the statistical analysis on the experimental data for Ha only lead to models with either p-values larger than 0.05 or R 2 -values below 50%. The results indicate that the models are rather insignificant and therefore are not included in Table 8 and Fig. 6.

Process optimization
In Section 3.3, we developed and validated significant but independent models for the weld qualities. However, in real-life, the requirements are typically for specific width and depth of the joint's fusion zone with minimal energy consumption and processing time (Benyounis et al., 2005;Vakili-Farahani et al., 2016). To find the set of parameters yielding the best local optimum, an optimization procedure to operate within the boundaries of the process parameter space is necessary. In the present work, we will demonstrate this procedure for the optimization of a weld in pulse mode using the developed models. In particular, we defined following set of quality criteria: SW = 730 ± 10 μm, D = 100 ± 20 μm. Furthermore, the laser power as well as the duty cycle must be minimized while the linear velocity must be maximized. These criteria are summarized in Table 9. In this work, the optimization was performed using the software Design-Expert® Version 9, which exploited a hill climbing technique. This iterative process operates within the boundaries of the process parameter space to search for the set of parameters yielding the best local optimum. The number of iterations for the optimization was limited to 15. Fig. 7 displays an overlay of areas in the process parameter space that result in feasible response values. The areas are limited by lines corresponding to the lower and upper limits of the responses, e.g. SW within the range 720-740 μm and D within the range 80-120 μm. The lines were constructed based on the models developed in Sections 3.2 and 3.3. The yellow colored region satisfies the goals for every response. The optimal set of process parameters, as determined by the numerical optimization procedure mentioned in the previous paragraph, is marked by a red dot in Fig. 7. According to the models, this set of parameters results in the responses closest to the predefined criteria. The corresponding process parameters are listed in Table 9. In order to confirm this optimization outcome, twelve additional weld samples were made T. Le-Quang et al. using the optimal parameters. The optical images of a representative sample are shown in Fig. 8. The measured values of SW, D and η are 735.83 ± 15.98 μm, 78.6 ± 14.31 μm and 5.71 ± 1.10*10 − 3 mm 3 /J, respectively.

Discussion
In this section, the influence of different processing parameters on the weld properties are discussed based on their coefficients in the models presented in Section 3.3. Before going into details, it is important to mention that the dimensions of the fusion zone was reliably formulated by linear models containing main factors and interaction terms of up to two factors. The very high values of Predicted-R 2 (above 0.85) indicate that these semi-empirical models can provide adequately precise predictions of the process outcomes (Montgomery, 2017;Vakili--Farahani et al., 2016). Hence, integrating the complex physics phenomena will significantly complicate the models while bringing only a slight improvement of their accuracies. Nevertheless, it needs to be emphasized that the validity of the models developed in this study is only proven for the chosen ranges of processing parameters. For extrapolating outside the processing window regimes, the experimental data used for the statistical analysis need to be updated accordingly. Furthermore, it is worth reminding the implications of the effects' signs in Fig. 6. In particular, a positive effect means that an increase of the corresponding factor results in an increase of the response and vice versa. On the contrary, a negative effect means that an increase of the corresponding factor results in a decrease of the response and vice versa.

Surface width SW
The linear models presented in Table 8 indicate that laser power P p has the most significant influence on this response in both modes. In pulse mode, the effects of P p and Cd are positive while the velocity terms V l and V w have negative effects. This observation can be explained in term of heat input, which can be expressed by following formula (Benyounis et al., 2005): in which V laser is the velocity of the laser beam in relative to the workpiece. In the case of laser beam oscillation welding, this parameter can be expressed as followed: Equation (7) shows that as P p and Cd increase, higher radiation energy is deposited into the material, causing a larger melt pool. On the other hand, an increase of the velocity V laser , which is a combination of V l and V w in case of oscillation welding, reduces the heat input, consistent with the negative effects of those factors for the response SW.
Similar correlation is observed in CW mode, but the effect of V l is, in this case, rather insignificant and is, therefore, not included in the model. Further studies are needed to confirm this observation, with a full factorial design to enhance the resolution of the analysis.

Fig. 7.
Overlay plot showing the optimal processing parameters ranges for the present example with the best solution marked by a red dot. The yellow area satisfies every response in Table 9.

Depth of the fusion zone D
Regarding the FZ's depth D, the constructed models have very good p-values and the R 2 -values, showing that they are rather reliable. In case of pulse mode, the terms P p as well as Cd have positive effects on D while the effects from velocity-related terms are negative, similar to the behavior observed for surface width SW. On the other hand, the model for the CW mode suggests that within the considered ranges of process parameters, D can be adequately predicted using only the laser power P p with predicted-R 2 values as high as 0.97. The inclusion of the velocity terms V l and V w, while increasing the complexity of the models, can only lead to a negligible improvement of the prediction capability. It is surprising as the latter terms should affect directly the heat input as discussed in Section 4.1, which is of utmost importance to the depth (Benyounis et al., 2005;Vakili-Farahani et al., 2016). In order to check the possibility of a polynomial correlation, additional experiments were performed with processing parameters listed in Table 10. The results are displayed together with the initial data in Fig. 9.
A high Pearson correlation coefficient of 0.95 is obtained, implying the likelihood of a linear correlation. Therefore, the observed insignificance of V l and V w is likely limited to the considered parameter ranges and should not be considered as a general trend. Further studies with either a full factorial design or an extended range of process parameters will be conducted to gain a better insight the correlations.

Energy efficiency η
By definition, η is an estimate of the amount of laser energy that really contributes to melting the workpiece. It is unfortunately challenging to formulate this parameter due to the complexity of the lasermaterial interaction, which requires many physical phenomena to be taken into consideration, e.g. heat conduction, melting, Maragoni effect, evaporation and phase transformation (Fabbro, 2006(Fabbro, , 2010(Fabbro, , 2006Otto et al., 2011;Pastras et al., 2017;Shevchik et al., 2020;Wei et al., 2020). The complexity increases further with the oscillating movement of the laser beam due to the overlap of the beam path. Such an overlap results in the workpiece's surface being modified/heated prior to the subsequent exposure to the laser radiation, which can significantly affect the energy coupling of the incoming laser radiation (Hagenlocher et al., 2018;Sommer et al., 2017).
In the present work, empirical models were successfully developed for the correlation between η and the process parameters. For the CW mode, P p , V l and their interaction term have positive influence on the response, i.e. an increase in those factors results in an increase of η and vice versa. The positive influence of P p was also observed in other previous works on conventional laser beam welding (Fuerschbach, 1996;Pastras et al, 2014Pastras et al, , 2017. It must be emphasized that, based on the measured D values presented in Tables 4 and 5, the weld experiments were in keyhole mode. Therefore, increasing laser power leads to growing keyhole channel's depth, which results in an improvement in the laser energy deposition in the workpiece (Fabbro, 2006(Fabbro, , 2010. Furthermore, it is expected that increasing P p causes more significant modifications to the surface roughness of the workpiece, which can enhance the laser absorption during subsequent turns of the oscillating movement (Bergström et al., 2008). On the other hand, the influence of V l , can be simply due to its effect on the total processing time, which affects the total amount of laser energy consumed.
In case of pulse mode, the model is more complicated with the additional contributions from Cd and V w , which have negative influence on η. Pastras et al. (2014) observed a similar dependency of η on Cd in pulse laser drilling. The authors mentioned that the laser energy coupling in the workpiece is not efficient at the beginning of each laser pulse due to part of the energy being reflected (Fabbro, 2006;Pastras et al., 2014). Consequently, decreasing Cd leads to a lower total amount of laser energy loss. Regarding V w , it is unexpected that this factor is rather insignificant in CW mode while having a negative influence in pulse mode. Theoretically, an increase in V w should improve the energy coupling in the workpiece since the process zone is pre-heated by the previous turn of the laser beam, which should lead to better optical absorption (Fabbro, 2006;Hagenlocher et al., 2018). This issue will be addressed in a future work.
Despite the lack of physical interpretation of the developed models, the models are of a significant interest for practical application. The models have Predicted-R 2 values higher than 0.8, indicating that they can be used to estimate η with adequate accuracies without considering complex physical phenomena. Consequently, the estimation of the results can be done without a significant computational demand. The limitation of the models is that they have only been validated on the examined process parameters ranges. When another range of processing parameters is considered, the model needs to be validated and appropriately adjusted.

Microhardness Ha
In the present work, no reliable correlation models were obtained for the response Ha. It can be due to the linear assumption as mentioned in Section 4.2 as well as to inaccuracies in the experimental results associated with the measurement approaches used.
Theoretically, this physical property depends strongly on the microstructure of the weld. Recently, it has been shown that the joint's microstructure, including the phase content, grain size and grain orientation, is not uniform along the lateral cross-section of the seam (Hagenlocher et al., 2018;Meylan et al., 2019). In the case of conventional welding, in which the beam moves in a linear path, the weld seam contains mostly equiaxed structure at the edge due to very high cooling rate and columnar structure at the center (Meylan et al., 2019). The oscillation movement of the laser beam in this technique causes re-melting and solidification of the material, leading to higher complexity of the microstructure and, hence, higher fluctuation in micro-hardness values (Hagenlocher et al., 2018). Consequently, a macro scale mechanical test, e.g. tensile test, probably provides a better evaluation of the joint strength. This issue will be also addressed in the future works.

Table 10
Processing parameters of experiments in CW mode for model validation.

Conclusion
The present work exploited the design of experiments approach to study the correlations between joint's properties and processing parameters in case of laser beam oscillation welding. The experimental responses consisted of surface width SW, weld depth D and microhardness Ha of the fusion zone of the weld seam and the energy efficiency of the process η. The study took into consideration both CW and pulse modes. In order to reduce the number of factors for the models, only circle beam path with a diameter of 0.5 mm was considered. For CW mode, laser power P p , linear velocity V l and oscillating velocity V w were considered for a full factorial design. For pulse mode, additional factors duty cycle Cd and number of turn per pulse NTP were taken into consideration. Due to the large number of parameters, only fractional design was used in this case.
The correlations between SW and D with the processing parameters could be reliably represented by linear models containing main factors and two-factor interaction terms for both welding modes. In the developed models, laser power is the most significant factor in both welding modes. It has a positive influence on the responses while the velocityrelated terms have negative influence.
Reliable models were also developed for the energy efficiency of the process. In case of CW mode, the three main terms are P p , V l and their interaction P p .V l , with the first term being the most significant. On the contrary, V l is the most dominant factor in the model for the pulse mode. The development of such empirical models is of a particular interest for the industrial application of the laser beam oscillation welding process due to their simplicity while still achieving adequate accuracies. As a demonstration of the usage of the developed models, an optimization procedure was performed in the CW mode with a specific set of requirements on the weld geometry and optimal energy efficiency.
On the contrary, no reliable models were obtained for the microhardness Ha due to large variation of the experimental data. The variation in the measured values comes mostly from the variation in the weld microstructure. Due to the differences in cooling rate during the resolidification and the oscillating movement of the melt pool, a highly complex distribution of microstructure is expected. It is proposed that macro scale mechanical tests will provide a better evaluation of the joint strength.
Finally, regardless of the promising results, the developed models have only been validated for a narrow range of process parameters. Hence, caution must be taken when using our models outside the process range windows.