Modelling of Thermal Quantities for Quick Process Assessment and Process Capability Space Re nement at an Early Design Phase During Laser Welding

This research aims at understanding the impact of welding process parameters and beam oscillation on the weld thermal cycle during laser welding. A three-dimensional heat transfer model is developed to simulate the welding process, based on the finite element (FE) method. The calculated thermal cycle and weld morphology are in good agreement with experimental results from literature. By utilizing the developed heat transfer model, the effect of welding process parameters such as heat source power, welding speed, radius of oscillation, and frequency of oscillation on the intermediate performance indicators (IPIs) such as peak temperature, heat-affected zone volume (HAZ), and cooling rate is quantified. Parametric contour maps for peak temperature, HAZ volume, and cooling rate are developed for the estimation of the process capability space. An integrated approach for rapid process assessment, process capability space refinement, based on IPIs is proposed. The process capability space will guide the identification of the initial welding process parameters window and help in reducing the number of experiments required by refining the feasible region of process parameters based on the interactions with the IPIs. Here, the peak temperature indicates the mode of welding performed while the HAZ volume and cooling rate are weld quality indicators. The regression relationship between the welding process parameters and the IPIs is established for quick estimation of IPIs to replace time-consuming numerical simulations. The proposed approach provides a unique ability to simulate the laser welding process and provides a robust range of process parameters.


Introduction
Current environmental regulations and policies aimed at reducing carbon emissions have accelerated the need to develop industry capabilities for a higher uptake of lightweight structures for e-mobility and fuel efficiency [1,2]. In turn, these requirements have led to the development of joining processes with capabilities for: (1) joining multi-material [3,4]; (2) high welding speed for high volume manufacturing of multiple product variants; and, (3) rapid development and deployment of new joining processes as required by shortened lead time and increased product variants [5]. Remote Laser Welding (RLW) is one of the joining processes which can potentially meet the aforementioned requirements due to several advantages such as imposing heat in a very controlled manner by modulating power and location of the heat source which provides flexibility in terms of thermal cycle developed. It provides flexibility and control over process parameters and other process variables like beam oscillation to deliver heat to the needed area of the workpiece. It offers a non-contact singlesided joining technology that allows high-speed welding for high volume manufacturing [6].
However, a key challenge is shortening the new process development and selection of key process parameters to meet welding thermal outputs pertaining to weld quality. Currently, the development of the RLW process is time-consuming due to the interactions between those factors that provide flexibility. Consequently, process development at present has relied on physical experimentations. Since modelling of weld thermal cycle, is becoming more advanced, there is an opportunity to augment some of the physical experimentation through computer simulations.
The heat-treatment induced during the welding thermal cycle causes significant metallurgical changes to the surroundings of the weld area (heat affected zone, HAZ) which determines the quality and mechanical properties of the weld. Therefore, the understanding and selection of the welding thermal cycle is a crucial task in obtaining desired weld quality as described by microstructure and mechanical properties. To be able to do this, it is important to model the relations between welding thermal cycle; welding process parameters (WPPs); and welding thermal outputs. It is worth noting that these relations are crucial not only to select WPPs but also to estimate welding process capability, to understand process variation and then be able to control the process during the production phase. In the case of RLW, the key WPPs which govern the local temperature distribution during the weld thermal cycle are welding speed, heat source power modulation and beam oscillation [6]. These WPPs are highly coupled with varying sensitivity to IPIs. For example, the adjustment of beam oscillation modifies the weld thermal cycle and results in changes of weld IPIs such as the peak temperature, HAZ volume and cooling rate which affects the weld microstructure and the mechanical properties [7,8].
Overall, there is a strong need for both welding process parameter selection and understanding their sensitivity to IPIs that need to be determined during the process design phase ideally via simulations or with a minimum number of physical experiments. Physical experiments alone are not sufficient to survey the entire design space, as it takes weeks or months to run them. Efficient simulation of weld thermal cycle provides option to be (i) run in the entire parameters space (high fidelity simulations which replace physical experimentations); or (ii) integrated with physical experiments in such a way that simulations are run in un-tested settings of parameters space decreasing the number of experiments required (lower fidelity simulations which reduce the number of physical experiments).
Experimental measurements of peak temperature and cooling rate are difficult due to very high peak temperature and small weld pool formed, however, they can be quantified by using simulation and modelling tools [9].
The local temperature distribution during the weld thermal cycle is important for process assessment. Moreover, with the advent of high computational advancements, numerical approaches such as finite element (FE) methods have gained importance as these can model different material combinations, heat source types, geometrical configuration, and boundary conditions for laser welding. The state of the art for numerical simulation of the welding processes can be found in [10][11][12]. To evaluate the thermal cycle, a precise elucidation of the heat source is crucial. Several heat transfer models have been developed for different dimensions and shapes of the heat source for the determination of the thermal cycle, by setting appropriate energy balance. Rosenthal et al. [13,14] proposed a point heat source moving in an infinite material and Hook et al. [15] approximated the solution to predict power needed to cause melting. Various one-dimensional and two-dimensional heat source models have been developed considering that the keyhole formed is circular and isothermal [16][17][18]. Heat transfer models with several designs of the heat source(s) from the very basic cylindrical to the most realistic conical/elliptical, double ellipsoidal or the combination of them are developed [19][20][21]. A numerical model for the three-dimensional (3D) transient heat source to calculate the temperature distribution is developed for full and partial penetration during welding [22][23][24][25]. Bianco et al. [28] showed that a quasi-steady state of the temperature profile is reached and Shanmugan et al. [27,28] consider the heat source as a 3D conical Gaussian heat source to calculate the distribution of temperature in the laser-welded T-joint. The Gaussian distribution is one of the most used methods to describe the continuous heat distribution from the laser beam [29]. However, only a few studies have addressed the modelling of beam oscillation and its interactions on the weld thermal cycle, but the interest is on the rise.
In welding simulation, inputs are understood as welding process parameters (WPPs) and the quality performance is evaluated by using multiple outputs which are called Intermediate Performance Indicators (IPIs) in this paper. It was observed that these process parameters not only affect the welding thermal cycle IPIs, but also the key performance indicators of other downstream processes. However, the current literature has focussed mainly on single or multiple inputs and single-output scenarios which are not sufficient as laser welding is characterised as multiple inputs and multiple outputs (MIMO) [30]. In this paper, a comprehensive and detailed process variation has been performed and addresses the MIMO scenario. To the best of our knowledge, there are not many conclusive studies reported in the literature so far which describe the interactions between the welding speed, heat source power and beam oscillation.
In addition to the existing numerical models, it is needed to study the effect of beam oscillation on the weld thermal cycle. For this reason, this paper introduces a simulation methodology to develop (i) a numerical model for the welding process and design space refinement considering multiple inputs (i.e. key WPPs) and multiple-outputs (i.e. IPIs) system by conducting computer simulations; (ii) process capability space (Pc-space) depicted by parametric contour maps based on the developed numerical model that allows the estimation of a desired process fallout rate in the case of violation of process; and, (iii) a surrogate model to develop the relationship between the WPPs and IPIs and also for the quick estimation of IPIs without running time-consuming numerical simulations. The Pc-space is the subset of the design space which includes all the selected WPPs producing all the accepted IPIs. The proposed methodology provides a unique ability to: (1) simulate the effect of process variation as generated by the manufacturing process; (2) generate the relationship between the WPPs and the IPIs and assess the sensitivity of process parameters on the IPIs; and (3) design space refinement under quality requirements.

Modelling Approach
The approach proposed in this study addresses two main limitations of the current literature as mentioned in the introduction section by taking into consideration (i) oscillating beam as a process parameter for modelling welding cycle, (ii) MIMO scenario, and (iii) process variation over the design space to reduce the number of physical experimentations required.
The Pc-space is defined to address the limitations by establishing a set of IPIs to extensively evaluate the laser welding process and refinement of the design space based on the given quality requirements. The flowchart of the three main objectives in this study are: (i) weld thermal cycle estimation; (ii) process capability space definition and visualization; (iii) process parameters selection which is shown in Figure 1 along with the necessary sub-steps. Fig. 1 Flowchart of the main objectives of this study along with necessary sub-steps and mentioning respective sections, figures, tables, and equations used in thFe paper.

Transient FEM heat transfer model for weld thermal cycle estimation
The first objective of this study is to estimate the weld thermal cycle using a transient heat transfer model which enables to calculate the spatial and temporal distribution of temperature (T(x,y,z,t)) during welding. The geometry with the mesh element size and distribution is shown in Figure 2. To simplify the calculations, several assumptions were considered for developing the model. A Gaussian distributed heat source was considered and assumed perpendicular to the surface of the workpiece. No gap between the two metal plates was considered. Material properties such as thermal conductivity (k), heat capacity (c p ) and density (ρ) were assumed to be temperature dependent. No convection in the melt pool was considered in the model. Additional contributions to the calculated temperature profile as originating from oxidation (i.e., exothermal reactions) and local effects of the shielding gas near the melt pool were not considered in this research work.

Geometry
A 3D Cartesian coordinate system is used. The heat source is moving in the x-direction and it is incident in the z-direction at z = 0. In the 3D model, the size of the plate was 100 mm × 50 mm with a thickness of 3 mm. For the better mesh distribution near the weld centre line, each plate was divided into three domains having dimensions 100 mm × 49 mm × 3 mm, 100 mm × 1 mm × 3 mm and 100 mm × 1 mm × 3 mm, respectively. The tetrahedral mesh design was used for the simulation. For the validation of the model, material properties and geometry was following Sonia et al. [31] having a dimension of each plate as 100 mm × 50 mm with a thickness of 2 mm for the Titanium alloy and 3 mm for the Aluminium alloy. The material properties are reported in Table A.1 and Table A

Governing Equations
The heat source is considered striking normally to the work-piece surface, along the z-axis and moving with constant welding speed in the x-axis. The time-dependent temperature field is a function of position (x, y, z) and time (t) which is determined by the non-linear energy conservation Eq. (1) [32].

Heat source design and boundary conditions
A 3D cylindrical and conical volumetric heat sources are the most used heat source designs for the modelling of thermal quantities during laser welding [18,33]. It has gaussian-shaped energy distribution, and the maxima are at the nominal power of the heat source. A 3D conical heat source with origin (x 0 ,y 0 ,z 0 ) is defined by a Gaussian distribution as shown in Eq.

Welding
Direction (3). Note that this relationship represents the 3σ range (99.7% of total heat input) of a heat source with radius r and penetration depth d [34].
Where, P (W) is the power of the heat source beam, r (mm) is the heat source spot radius, d (mm) is the maximum depth of penetration. In the conical design of the heat source, the temperature is maximum at the surface and minimum at the bottom which is a simplified representation of the keyhole formation during laser welding. To define the moving heat source, the position of its centre needs to change with time. Hence, we write (x (t) , y (t) , z (t) ) for the position of the heat source centre at time t. The position at the start of the simulation (t=0) was chosen to be at the origin (x 0 ,y 0 ,z 0 ).
The derived trajectory of an oscillated heat source is considered to be at the surface, starting at the point connecting the two plates for welding (z (t) = z 0 ). Circular oscillated heat sources in the xy-plane are used because it has been found that less welding defects were observed experimentally [12,35,36]. The trajectory of the moving heat source is defined by Eq. (4) Where x (t) , y (t) and z (t) are time-dependent heat source spot coordinates, where α is the electrical resistivity of the workpiece and λ is the wavelength of the laser. Eq.
(5) represents the heat input by the moving conical heat source with and without beam oscillation. The initial temperature of the workpiece is assumed to be maintained at room temperature (T 0 ). The energy absorbed by the workpiece is transferred by conduction, which further is partially lost due to the contact with the ambient atmosphere and by radiation leading to cooling. The heat loss due to the contact with the ambient atmosphere and by radiation during welding are governed by Newton's law of cooling and Stefan-Boltzmann relation [38]. Convective and radiative heat loss is considered for the top and bottom surfaces which follows Eq. (7) The governing energy conservation equation is discretized using the finite element method.
The solution to the above energy conservation equation gives the spatial and temporal distribution of the temperature yielding weld thermal cycle. The input process parameters are welding speed, heat source power, radius of oscillation, frequency of oscillation and heat source spot size as depicted in Eq. (4) and Eq. (5). The peak temperature, HAZ volume and cooling rate is calculated from the weld thermal cycles as the outputs.

Formulation of Surrogate Model for regression relation between WPPs and IPIs
A surrogate model is developed to formulate an analytical relationship between the WPPs and the IPIs using a second order and third-order polynomial expression. The linear expression is not sufficient to model a non-linear complex relationship whereas higher-order polynomial expressions would lead to over-fitting. The general formulation of the second order and third-order polynomial expressions is shown in Eq. (8).
Where x 1 , x 2 ,…,x k are independent variables and β 0 , β i , β ij , β ijl … are the regression coefficients. The regression model is developed by writing codes in Python. The coefficient of determination (R-square) is calculated and t-test, p-value were evaluated to assess the goodness of surrogate models. The dispersion between the predicted and measured values was plotted to visualize the closeness in the prediction to the actual value from the numerical model. R-square depicts the goodness and relative measure of fit. To evaluate the significance of regression coefficients t-statistics and p-value are calculated with a 95% confidence interval. A 10-Fold Cross Validation has been performed to assess the predictive performance of the surrogate model. A 10-Fold R-square is calculated to judge the model performance to new data outside the model data set.

Process capability space
The quality performance of laser welding is governed by multiple thermal outputs which are called as IPIs in this paper. The input WPPs considered in this study are welding speed (S), heat source power (P), the radius of oscillation (R) and frequency of oscillation (f). The four WPPs (S, P, R, f) are defined in Eq. (9), where i represents the index of WPP, n represents the total number of WPPs.
The upper limits (UL) and lower limits (LL) of input process parameters have been contour maps are developed for three IPIs (1) peak temperature (T peak ), (2) HAZ volume (V HAZ ) and (3) cooling rate (C rate ). The T peak (maximum) is the maximum temperature recorded during welding as given in Eq. (10).
HAZ is a solid portion of the workpiece that undergoes metallurgical transformation due to exposure to the critical temperature ranges during welding. In general, the temperature range for the HAZ is high enough to cause metallurgical transformation, grain growth but not sufficient to melt the metal. This temperature range is calculated using the phase diagram of the alloy. Therefore, the temperature range for the present study is taken between solidus and lower temperature limit where grain growth can be neglected which in this case was very close to the 933 K-470 K temperature range. The V HAZ was calculated by volumetric integration of the workpiece having HAZ temperature region defined in Eq. (11).
The calculated weld thermal cycle is utilised to calculate C rate (dT/dt), which is defined as the change in temperature from the melting point to the 470 K upon the time taken to cool from the melting point to this temperature. The upper and lower limits of the IPIs are given in Table 2. The three IPIs (T peak , V HAZ and C rate ) are formulated in Eq. (12)(13), where j represents the index of IPI, and m represents the total number of IPIs.  Cooling rate C rate (K min -1 ) Maximum C rate at LL of T peak Maximum C rate at UL ofT peak The second objective of the work is to develop process capability space. A numerical model is computed to estimate the IPI values over the design space. The model that describes this relationship is given as: If the estimated IPI violates the allowance limits, WPPs is considered unfeasible. For the j th IPI, process capability space is defined as expressed in Eq. (15), where k is the indices of individual process capability space and m is the total number of IPIs It defines a set of IPIs which evaluates the laser welding process and identifies the process parameters within the design space satisfying all the quality requirements. The final process capability space is the aggregation of process capability space (α j ) of individual IPI as defined in Eq. (16) The final process capability space envelops all the feasible WPPs values inside the design space. It provides the estimated feasible range of WPPs which is given as:

Process capability space refinement
The last objective of this study is the refinement of process capability space depending upon the requirements of the downstream processes. This refinement of process capability space can have either of the two objectives: (i) completely replace physical experimentations by providing appropriate process parameters over the entire design space; and, (ii) reduces the number of experiments required by providing IPIs over un-tested settings of design space or refines the initial range of design space to be tested for early design of experiments. It is important to note down that the methodology for process parameters refinement provided in this study are for the early design phase and process assessment.
The process parameters refinement is performed on the final Pc-space. In this work, sequential refinement strategies are developed based on three constraints: (i) IPIs constraints which address the proper coalescence of metals produced by heating to a suitable temperature; (ii) objective function which addresses the functional and strength requirements to control weld defects and strength of the weld; and, (iii) process parameters constraints which address the key targets of the joining industries to have high production rate, low cost and aesthetic requirements of welded joints. Sequential refinement strategies for some of the downstream processes are given in Table 3 Mechanical stirring by beam oscillation helps in escaping trapped gasses reducing porosity Partial penetration Partial penetration requires peak temperature above the boiling

Results and discussion
The transient heat transfer model is validated with the experimental results from the literature. A total number of around 800 numerical simulations were carried out which serve as a base for the discussed results. Each simulation took 25 minutes to solve for no oscillation condition and 105 minutes for oscillation condition. This is due to the increased complexity introduced with the heat source oscillation (Eq. 4) and an increase in the number of mesh elements as the heat source traversing area has been increased. The effect of WPPs on the IPIs and the process capability space has been discussed. The process capability space refinement is illustrated with the help of an example followed by establishing the regression relationship between the WPPs and the IPIs using a surrogate model for the quick estimation of IPIs.

Model Validation point
The transient heat transfer model is validated by comparing the experimental thermal cycle generated from the thermocouple and the weld morphology from the optical micrographs with the model. The thermal cycle is validated with experimental results reported for the Al-5754 and Ti 6 Al 4 V alloy systems [31] where an offset of 1 mm was taken along the titanium side and the thermocouple was placed at 2 mm from the welding line on both sides. The thermal cycle generated from the simulation for the validation is at the same position where the thermocouple is placed for recording as shown in Figure 4. The thermal cycle obtained from the simulated model showed a similar tendency to the experimentally obtained thermal cycle from the thermocouple which illustrates the rationality of the numerical model. The obtained values of the peak temperatures were within the acceptable range as the difference is less than 0.1% and is constant throughout the model. The calculated heating cycle is in good agreement with the experimental results having less than the 0.5% difference. While the cooling cycle for both the metals was found to deviate. The discrepancy regarding the cooling cycle is due to the assumption, especially not considering convective heat transfer. The cooling rate calculated from the peak temperature to the minimum temperature was found to deviate by less than 8% for Titanium alloy and less than 2% for Aluminium alloy. This is due to the high thermal conductivity of Aluminium (158.7 W m -1 K -1 ) as compared to Titanium

Characterization of peak temperature, HAZ volume and cooling rate for no oscillation condition
For the no-oscillation condition, Figure 3(e) shows the simulated peak temperature contours for different combinations of heat source power and welding speed. For laser welding in conduction mode, the peak temperature must remain between the melting and the boiling point of the metal. For the keyhole mode, the peak temperature must be above the boiling point of the metal being welded. In this study, for the full penetration welding, the peak temperature at which the temperature at the bottom surface is just below the melting point leading to full penetration during welding is found to be 3500 K. The upper limit of peak temperature is 3500 K and the lower limit is the melting point to define the process capability space for welding as illustrated in Figure 3(e). The space above the upper peak temperature limit will lead to the cutting of the workpiece and the space below the lower limit will lead to only heating of the workpiece without any melting. The upper and the lower limit of the peak temperature for the process capability spaces can be achieved by multiple possible combinations of process parameters. The cooling rate and HAZ volume are calculated for all these WPPs combinations. The maximum value of the cooling rate is selected from the possible WPPs combination corresponding to the upper limit of the peak temperature. This value is selected as the upper limit of the cooling rate for the Pc-space and similarly, for the lower boundary of the cooling rate, the maximum value is selected from the WPPs combination corresponding to the lower limit of the peak temperature. For the upper and lower limits of HAZ volume for the Pc-space, a minimum value is selected from the possible WPPs combinations corresponding to the upper and lower limit of the peak temperature. The maximum value of the cooling rate and minimum value of HAZ volume is selected because the aim is to maximise the cooling rate and minimise HAZ volume.

Fig. 4 Calculated temperature distribution of Al-Ti welded joint, investigated in this study (top). The position of the weld central line and the thermocouple (top) is indicated by the arrow (top). Comparison between experimental and simulated thermal cycle at the Titanium (bottom left) and Aluminium side (bottom right)
. Experimental data has been reproduced from [31].
It was observed that with the increase in power at constant speed the peak temperature increases due to the local increase in net heat input per unit length to the workpiece defined as (ηP)/S. Whereas when the welding speed is increased at constant power, the net heat input to the workpiece decreases, leading to a decrease in peak temperature. The simulated cooling rate and heat-affected zone volume profiles are shown in Figures 3(f) and 3(g), respectively, for different combinations of heat source power and welding speed. It was observed that at constant welding speed with the increase in power HAZ volume increases due to the higher heat input to the workpiece while the cooling rate decreases. Whereas when the welding speed is increased at constant power, the HAZ volume decreases as net heat input to the workpiece decreases, causing a decrease in peak temperature which leads to an increase in cooling rate. An important characteristic of the results in Figure 3 is that a particular weld attribute, such as the peak temperature or cooling rate or HAZ volume can be achieved by multiple combinations of welding process parameters. This existence of multiple paths to achieve the desired weld requirements exhibits the flexibility of the laser welding process. For the estimation of Pc-space, the peak temperature is the most important IPI and is chosen first which followed by either of the other two depending upon the requirements of the downstream processes as given in Table 3. The final Pc-space is the intersection of the individual Pc-spaces for all the IPIs as shown in Figure 3(d) and the shape of the final process capability space will remain constant irrespective of the order of IPIs considered.

Characterization of peak temperature, HAZ volume and cooling rate for beam oscillation condition
Process variables that correspond to the shape and overlapping of the heat source are the radius of oscillation, frequency of oscillation, and welding speed which is explained in detail in Appendix B. Heat input per unit length in case of oscillating heat source is defined as [(ηP)/(S + 2πRf)]. There are two possibilities during circular beam oscillation: (i) no overlapping of the path traversed by the oscillating heat source and, (ii) overlapping of the path traversed by the oscillating heat source as shown in Figure 14 in Appendix B. The onset of overlapping during oscillating heat source occurs when (4Rf/S) = 1 as derived in Appendix B. The term 2πRf denotes the circular motion of the oscillating heat source and S denotes the welding speed of the heat source in the welding direction. A higher value of (2πRf/S) implies that the circular motion is more dominant compared to the linear forward motion of nonoscillation welding, leading to more overlapping as shown in Figure 14 (b) and (c). Lower value implies forward motion is dominantly leading to either no overlapping or few overlaps.
The number of overlapping points is calculated graphically as explained in Appendix B.

Effect of frequency of oscillation
There are two opposing factors governing heat uptake during oscillating heat source: (i) decrease in heat input per unit length due to increase in effective speed (S + 2πRf) i.e., linear plus circumferential speed and, (ii) increase in heat input due to increase in the number of overlapping points. Figure 6 illustrates that the range of frequency, welding speed and radius of oscillation investigated in this study shows that the frequency has a negligible effect on the heat uptake during welding.
At a constant frequency, with an increase in the value of (2πRf/S) heat input increases. In this case, the increase in heat input is due to the increase in overlapping points as R increases or decrease in S which leads to an increase in heat input per unit length. This explicit solution is supported by the transient heat transfer simulations as shown in Figure 6. Figures 6(b) and 6(c) show the simulated peak temperature for different combinations of frequency, welding speed and heat source power, it is evident that the effect of frequency of oscillation is negligible on the peak temperature in both cases as contour lines are parallel to the y-axis showing the frequency of oscillation. Similar trends are found for the cooling rate and HAZ volume. With an increase in oscillation frequency, the number of overlapping points for every rotation will increase which leads to re-heating and also the effective speed of the heat source increases as it has to rotate more times in the same amount of time. This re-heating should lead to a rise in temperature, but it is compensated by a decrease in heat energy absorbed per unit length due to the increase in the actual speed of the heat source. The increase in the actual distance travelled by the heat source with an increase in f is demonstrated in Figure 14 in Appendix B.

Fig. 6
Heat input per unit length with varying frequency and ratio of circular speed and linear forward speed (welding speed) (a). Figure (b) shows the parametric contour maps for peak temperature depending on the frequency of oscillation and welding speed at a constant power of 3500 W and figure (c) shows the parametric contour maps for peak temperature depending on the frequency of oscillation and heat source power at a constant welding speed of 2500 mm min -1 . The oscillation radius R was kept constant at 0.3 mm. The parametric contour maps are developed for the joining of Al-5754 with the Al-6061 alloy system for butt welding configuration.

Effect of the radius of oscillation
To examine the effect of radius of oscillation during laser welding, two types of parametric contour maps are generated: (i) contour maps of IPIs between the radius of oscillation and heat source power at a constant welding speed and frequency of oscillation; and (ii) contour maps of IPIs between the radius of oscillation and the welding speed at a constant heat source power and frequency of oscillation. Figure 7 shows the simulated parametric contour maps of the peak temperature, HAZ volume and the cooling rate for different combinations of the radius of oscillation and heat source power at a constant welding speed of 2500 mm min -1 and frequency of 200 Hz. As shown in Figure 7(a), at a constant power with an increase in the radius of oscillation, the peak temperature decreases as the effective heat source speed increases which results in the decrease in the heat input per unit length. However, for the HAZ volume, it depends on the relative length of heat source spot radius (r), which is 0.2 mm in this study and the radius of oscillation. When R ≤ r, the HAZ volume remains constant with the increase in R as it is equivalent to the situation with a heat source having a larger spot radius. So, an increase in the width of HAZ is compensated by the decrease in depth leading to constant HAZ volume.
When R > r, there is a sharp decrease in HAZ volume when R is just above r which is depicted in Figure 7(b) at R = 0.3 mm. With an increase in R, HAZ volume increases until it reaches R = 3r. This is due to the delay in the cooling process due to the repeated heating.
Though the maximum value of HAZ volume is still smaller than for R ≤ r. When R > 3r, the heat input rate decreases leading to quick removal of heat which decreases the HAZ volume.
So, it can be concluded that HAZ volume decreases with the application of heat source oscillation as in practical cases R > 2r. Similarly, the cooling rate remains constant when R ≤ r, there is a sharp increment when R > r which is shown in Figure 7(c) at R=0.3 mm. It again decreases with an increase in R till R = 3r. When R is further increased it decreases due to the decrease in heat input rate as the velocity of the heat source increases. At constant power, the lowest cooling rate is found when R ≤ r. So, the cooling rate increases when beam oscillation is applied (as for practical cases R > 2r). A resonance effect is found for the HAZ volume and cooling rate which is a function of oscillation radius and heat source spot size.  The generated parametric contour maps (Figure 8) show similar trends as in Figure 7, but the maxima and minima are exchanged from right to left as both welding speed and heat source power are related to the heat uptake. An increase in heat source power is like a decrease in welding speed leading to an increase in heat input per unit length. This effect is also depicted in Figures 6(b) and 6(c).  Figure (a) shows a contour map for peak temperature, (b) shows HAZ volume, and (c) shows for cooling rate. The parametric contour maps are developed for the joining of Al-5754 with the Al-6061 alloy system for butt welding configuration.

Effect of beam oscillation on the process capability space
From section 3.3.1, it was concluded that the effect of the frequency of oscillation is negligible on the IPIs for the range of frequencies investigated in this study. So, the Pc-space depends on only three process parameters yielding a 3D response surface, where x, y, and z axes are the three process parameters which are heat source power, welding speed and radius of oscillation respectively. This response surface can be visualized by stacking up 2D contour plots in the xy plane with a constant value for the third process parameters, as shown in figure   9. The size of the Pc-space increases and is applicable for broader process parameter ranges with increasing radius of oscillation as depicted in Figure 9. This demonstrates the increase in flexibility of the laser welding process due to the application of beam oscillation. The upper limit of the allowed peak temperature ceases to exist at the highest radius of oscillation as shown in Figure 9 (f). For the same combination of welding speed and heat source power, the peak temperature decreases though the net heat input to the workpiece is the same when compared to the non-oscillating heat source. To attain the same peak temperature oscillating heat source requires more heat input as compared to a non-oscillating condition which is shown in Figure 9(a) and 9(b) where to achieve the upper limit of peak temperature of 3500 K in oscillation condition requires more power at the same welding speed.

Process capability space refinement
Process capability space refinement is illustrated by using the example of improving mechanical properties for the keyhole-mode of welding for full penetration. In this example, the final process capability space is chosen for no oscillation condition which is illustrated in Figure 10(a). The region above the feasible region (marked yellow) corresponds to the cutting or over weld region as this region causes burn-through due to deeper penetration of the keyhole. The region below the feasible region depicts the under-weld region where due to insufficient penetration leads to a poor coalition of the materials. The last region at the bottom is no welding region/ preheating as the peak temperature is below the melting point of the material. The first step for process capability space refinement depending upon the downstream process requirement is to select the proper peak temperature thus satisfying IPIs constraints as given in Table 3. The lower limit of peak temperature should be greater than the boiling point of the metal as keyhole-mode of welding is considered, and the upper limit should be less than 3500 K (as this temperature corresponds to full penetration welding). The yellow-coloured region in Figure 10(b) represents a feasible region for the keyhole mode of welding. The region just below the feasible region (in between the red and black lines) represents the feasible region for the conduction mode of welding. In the second step, the WPPs combinations are narrowed down based on the objective functions (Table 3) which are minimising HAZ volume and maximising cooling rate. The narrowed down process capability space based on the objective function is shown in Figure 10(c). In the final step, those process parameter combinations are selected from the refined space which has faster welding speed and lower heat source power (WPPs constraints as given in Table 3). For a special case of full penetration keyhole welding, the peak temperature should be 3500 K, so the refined process capability space is just a line in place of an area as shown in Figure 10(d).
Therefore, the final refined process capability space has the full penetration welding with the lowest HAZ volume, highest cooling rate, highest welding speed, and lowest power. The robust process parameters can be selected from the refined process capability space having initial estimates about the quality of the weld from the IPIs without prior experimental characterization and mechanical testing. This approach reduces the number of experiments required. A similar approach can be used for the conduction mode of welding where the peak temperature should be less than the boiling point of the materials.

Surrogate for the quick estimation of peak temperature
The frequency of oscillation has a negligible effect on the IPIs as concluded in section 3.3.1 from the parametric contour maps. Therefore, regression analysis is only performed for welding speed (S), heat source power (P) and radius of oscillation (R) as independent variables. Peak temperature, cooling rate and HAZ volume are evaluated as outcomes (IPIs) depending upon the goodness of fit. The adequacy of the surrogate model and the significance of the coefficients are analysed using the sequential analysis of variance (ANOVA). The codes are written in the Python programming language. A 10-Fold Cross Validation is performed to check the robustness of the model. In this method, fitting is performed ten times, with each fit being performed on a training set which consists of 90% of the total training set selected randomly, with the remaining 10% used as a holdout set for validation. The ANOVA provides which process parameters significantly affect each IPIs. Table 4 presents the analysis-of-variance of the responses (peak temperature, cooling rate and HAZ volume) and provides the F-value, p-value, percentage of contribution (POC), R-square, adjusted R-square and 10-Fold R-square. At 95% confidence interval, a p-value greater than 0.05 is statistically insignificant and a p-value less than or equal to 0.05 is statistically significant. The second-order and third-order polynomial equation was exploited to formulate the thermal responses as defined in equation (8). The number of data points taken to perform multivariate regression analysis was more than 700 following the criteria suggested by Green  = 78 − 186 × + 6.6 × 10 −2 × − 1.788 × 10 −1 × + 523 × 2 + 4.1 × 10 −5 × 2 + 1.02 × 10 −4 × 2 − 1.970 × 10 −1 × × + 2.989 × 10 −1 × × − 1.06 × 26 10 −4 × × − 579 × 3 + 1.511 × 10 −1 × 2 × − 1.313 × 10 −1 × 2 × + 3 × 10 −6 × × 2 − 2.2 × 10 −5 × × 2 (20) Figure 11 a-c shows the dispersion plots representing the relationship between the numerical (from the FEM model) and predicted values (from the surrogate model) of T peak , V HAZ and C rate with a 95% confidence level. These dispersion plots indicate the accuracy and adequacy of the model developed and the predicted results are in good agreement with the measured data. Fig. 10 (a) Final process capability space for welding having cutting/over-welding region, welding region, under-welding region and no-welding/pre-heating region (b) Process capability space refinement based on IPIs constraints i.e. 3500 K ≥ Tpeak > Tbp having cutting/over-welding region, keyhole mode welding region, conduction mode welding region and under-welding region ; (c) Process capability space refinement based on the objective functions i.e. minimising HAZ volume and maximising cooling rate and (d) Final refined process capability space for full penetration welding. The temperature 3500 K shows the estimated peak temperature for full penetration welding.

Effect of input process parameters on responses
The response profilers for the effect of process parameters on the thermal responses have been constructed according to the surrogate models (Eqs. [17][18][19]. Figure 12 shows the prediction profiler for the responses (T peak , V HAZ and C rate ) for oscillation and no oscillation condition with the variables within the working range. The surrogate model have similar trends as shown by the FEM simulations with respect to the responses due to the variation in process parameters as shown in Figure 12. Beam oscillation has a cyclic effect on the HAZ volume and cooling rate as shown in Figures 12 (f) and (i). It is due to the two competing factors. The heat is distributed in a larger area i.e., the heat input decreases leading to shallow penetration leading to a decrease in HAZ volume but on the other hand, a larger radius leads to a wider melt pool, thus increasing HAZ volume. Similarly for the cooling rate reheating leads to a decrease in cooling rate while due to larger area interaction heat input per unit length decreases leads to an increase in cooling rate. Beam oscillation will always yield lower peak temperature, HAZ volume for the same process parameters combination. While for the cooling rate the application of beam oscillation may decrease or increase the cooling rate depending upon the two competing factors.

Fig. 11
Value of the peak temperature, HAZ volume and cooling rate from the FEM model vs predicted from the surrogate model with a 95% confidence interval to show the accuracy of the polynomial fit function. The dispersion plot was developed for the joining of Al-5754 with the Al-6061 alloy system for butt welding configuration using 740 data points extracted from the physics-based model.
In Pc-space of peak temperature, the combination of the three main welding parameters (welding speed, heat source power, and radius of oscillation) will form a plane in 3D design space representing a specific peak temperature and each point on that plane will give the welding process parameters combination to attain the required peak temperature.   Fig. 12 Peak temperature (a-c), HAZ Volume (d-f) and cooling rate (g-i) prediction profiler to depict the relationship between the welding process parameters and the thermal responses and also to depict the linear/non-linear dependency between them.
Therefore, it provides a finer grid in the Pc-space as compared to the parametric contour plots. This is a more general and robust approach as compared to the binary interactions shown with the parametric contour maps. The 3D graph generated will save time from running many simulations and shows the effect of the three process parameters at once, unlike the contour maps where the effect can be understood for two parameters at a time. One example is shown in Figure 13 where the required peak temperature is set to 2500 K. The red line shows the solution for without oscillation. The 3D graph gives a better visualization of the equivalent effect of process parameters and the flexibility of the laser welding process.

Fig. 13
Response surface generated from the regression model for the effect of welding speed, the radius of oscillation, and heat source power for a peak temperature of 2500 K. The red line shows the response surface for no oscillation conditions. The response surface is developed for the joining of Al-5754 with an Al-6061 alloy system for butt welding configuration.

Conclusions
This paper presents a novel methodology for the process capability space refinement for the laser welding process. During this research, FEM based heat transfer model has been developed to simulate the peak temperature, HAZ volume and cooling rate in laser welding.
Process capability space has been depicted by using parametric contour maps. A semiempirical mathematical model was developed and validated. The main conclusions are as follows:  There is a good agreement between the FEM model and the experiments both in terms of thermal cycle and weld morphology to evaluate thermal responses.  It has been found that the welding speed and heat source power has an opposite effect on the IPIs.  Parametric contour maps show that the effect of oscillation frequency is negligible on the IPIs during beam oscillation. The main parameter affecting the weld thermal cycle during beam oscillation is the radius of oscillation.  The final process capability space increases with the application of beam oscillation which exhibits the increase in flexibilities due to beam oscillation.  Application of beam oscillation always decreases the peak temperature and HAZ volume while it may increase or decrease the cooling rate.  The cooling rate and HAZ volume varies cyclically with the change in radius of oscillation.
Finally, it should be stressed that the presented modelling approach, relies on the thermal aspect of the welding only and all the calculation is based on temperature. Nevertheless, the main features are likely to represent the experimental situation sufficiently for the IPIs studied, especially parametric study for the interdependency of the process parameters for the target welding quantities (IPIs), assessment of the sensitivity of process parameters on the IPIs, and the process capability space refinement for the design of the experiments. The presented approach is general in its nature and can be applied to other welding techniques as it models heat source and heat transfer during welding which is fundamental for every welding technique. Further, the present model can also be used for parameter selection where partial penetration is required or the temperature at the bottom surface must not exceed a particular temperature such as joining battery tabs during battery pack assembly.   and Al-6061used in this study [31,40].  Equation (B.7) gives the condition for the onset of the overlapping. Welding speed greater than this value will have no overlapping points.

B.2 Estimation of overlapping points graphically
The number of overlapping points is estimated graphically in terms of welding speed, radius of oscillation and frequency of oscillation due to the complexity of equations. The graphical solution comprises plotting and counting the number of overlapping points in terms of process variables and finding the general series as given in Table B.1. Using general induction of S = (4/(2n+1))Rf and adding two more overlapping points for each cross-over.
Graphical solutions yield the number of overlapping points as a function welding speed radius of oscillation and frequency of oscillation which is given in Table B.1. Just to note that this relationship has been proved by the method of induction. and (c) overlapping at a higher frequency of oscillation when S ≤ 4Rf. Where P is the power of the heat source, n is the number of overlapping points, l is the distance travelled by the heat source in one revolution, η is the absorption coefficient of material and f is the frequency which is also equal to the time taken for one revolution to take place.