*3.3. Calibration and Validation of the Model with a Modified Method of Heat Input in the Disk Laser Bead-On-Plate Welding Process*

To confirm the assumptions of the proposed simulation method, a TruDisk 3302 disk laser bead-on-plate welding process on 6.0-mm-thick AISI304 steel samples was carried out (Table 3). Data from metallographic tests of the obtained fusion beads were used to calibrate the heat source model. To validate the calibrated model, 4 samples with a thickness of 4.0 mm from AISI304 steel were also analyzed, to which the bottom surface was welded with three K-type thermocouples (Ni–NiCr wire with diameter of 0.2 mm) to record the heat cycles of the welding process. Thermocouples with accuracy ±0.0075 × T in a temperature range from −40 to +1200 ◦C (according to EN 60584-1 standard) were connected via compensation cables with an Agilent 34970A recorder by Agilent Technologies Inc., Santa Clara, CA, USA [42]. The arrangement scheme was as follows: the TC3 thermocouple was placed on the bead axis in the middle of its length, the TC2 thermocouple was placed 5.0 mm from the bead axis in the middle of its length, and the TC1 thermocouple was also placed in the bead axis at 5.0 mm from TC3 (Figure 10).

**Figure 10.** Diagram of the location of thermocouples relative to the bead axis at the bottom of the test sample.


**Table 3.** Parameters of bead-on-plate laser welding of the 4.0- and 6.0-mm-thick AISI304 plates with the TRUMPF TruDisk 3302 laser.

Remarks—Argon flow rate via cylindrical nozzles: 15 L/min, laser spot diameter: 200 µm, location of the laser beam focus: on the upper surface of the welded sample. TC1\_4 samples: 4.0-mm-thick samples equipped with sets of 3 thermocouples welded to the bottom of the sample according to Figure 10.

Then the symmetrical three-dimensional discrete model, containing 28,080 3D elements and 30,622 nodes for 6.0-mm-thick samples and 21,518 3D elements (6 node linear wedges and 8 node linear bricks) and 23,706 nodes for 4.0-mm-thick samples was prepared. Mesh was refined in the weld and heat-affected zone (HAZ). The applied boundary conditions simulating sample attachment corresponded to its free placement on the laser table throughout the whole process of welding and then its cooling. In addition, a boundary condition of symmetry was added to the surface of the model division (Figure 11). The boundary condition describing heat dissipation to the environment was implemented by defining convective heat dissipation to the environment at 20 ◦C and radiation on all external surfaces of the model. As a heat source model, proposed as volumetric, a conical model with a modified "LOAD" area was used (Figure 11). The calculations to define the material database contained the thermal, mechanical, and metallurgical material properties, which were dependent on temperature and metallurgical phase proportions.

**Figure 11.** View (**a**) of the symmetrical three-dimensional model of the laser bead-on-plate welding process together with a conical model of the heat source and the "LOAD" area, and (**b**) boundary clamping conditions.

Based on the results of metallographic tests, the calculation models were calibrated, and the final results of numerical analyses were compared with the corresponding shapes of the actual beads obtained in the welding tests (Figures 12–14). On this basis, a comparison was also made of the measured and calculated basic geometric dimensions of the beads obtained (Table 4).

**Figure 12.** View (**a**) of the method of loading the mesh elements and placement of the conical model of the heat source, comparing (**b**) the shape of the bead on the cross-section, (**c**) the shape and size of the weld pool, and (**d**) the longitudinal section of DL\_T1 test bead with the results of numerical analyses (Tables 3 and 4).

**Figure 13.** View (**a**) of the method of loading the mesh elements and placement of the conical model of the heat source, comparing (**b**) the shape of the bead on the cross-section, (**c**) the shape and size of the weld pool, and (**d**) the longitudinal section of DL\_T3 test bead with the results of numerical analyses (Tables 3 and 4).

**Figure 14.** View (**a**) of the method of loading the mesh elements and placement of the conical model of the heat source, comparing **(b)** the shape of the bead on the cross-section, (**c**) the shape and size of the weld pool, and (**d**) the longitudinal section of DL\_T5 test bead with the results of numerical analyses (Tables 3 and 4).

**Table 4.** Comparison of geometric dimensions of selected beads on the bead-on-plate laser welding process performed on 6.0-mm AISI 304 steel samples using the Trumpf TruDisk 3302 disk laser with values obtained from numerical analysis (values in brackets) (Table 3, Figures 12–14).


Following the assumptions of the proposed model preparation method, re-calculations were also performed for the DL1.5\_8.33 sample analyzed in sub-chapter 3.1 (Figure 7). The results of the repeated numerical analysis showed that there was a significant improvement in the quality of the fusion line mapping using the same model of a conical heat source but with a modified area of the loaded elements of the mesh (previously all mesh elements covered by the cone were loaded), (Figures 7 and 15).

**Figure 15.** View (**a**) of the method of loading the mesh elements and placement of the conical model of the heat source, comparing (**b**) the shape of the DL1.5\_8.33 bead on the cross-section with the results of numerical analysis (Table 1, Figure 7).

TC\_1-TC\_4 samples were prepared to collect information for model validation using thermal cycle runs. After calibrating the calculation model (Figure 16), the registered and calculated points corresponding to the location of thermocouples were also compared, as was the course of the process thermal cycles (Figure 17, Table 3).

**Figure 16.** Comparison of the bead shape in the cross-section of sample (**a**) DL\_TC3 and (**b**) DL\_TC4 with the results of numerical analyses (Table 3).

**Figure 17.** Comparison of recorded and calculated thermal cycles with the bead-on-plate laser welding process performed on 4.0-mm AISI 304 steel samples using the Trumpf TruDisk 3302 disk laser for sample (**a**) DL\_TC3 and (**b**) DL\_TC4. TCx: cycles recorded during the welding process; SIM\_TCx: cycles obtained from numerical analyses.

To show the purpose of such precise calibration of the model and reproduction of the actual fusion line shape, for selected DL\_T3 beads the von Mises reduced stresses and cumulative plastic strain distribution were also calculated for the simulation case with a standard heat source model and unmodified and modified load element area (Figures 18 and 19).

**Figure 18.** View of the molten pool and von Mises stresses distribution in the case of calculations for (**a**) the model without modification of the area of mesh elements loaded with the heat source model and (**b**) the model based on the proposed calculation modification for the DL\_T3 sample (Table 3).

**Figure 19.** View of the molten pool and the distribution of cumulative plastic strains in the case of calculations for (**a**) the model without modification of the area of mesh elements loaded with the heat source and (**b**) the model developed based on the proposed modification for the DL\_T3 sample (Table 3).

#### *3.4. Assumptions and Construction of a Modified Numerical Model in the Case of a High-Power Diode Laser*

The second laser device analysed for numerical simulations in the presented research was a high-power diode laser. This source is an interesting device that is noteworthy from a numerical simulations point of view as due to its internal structure (laser diode packages in the form of laser rods) it emits a rectangular laser beam, with beam spot dimensions of 1.8 mm × 6.8 mm at a focal length of 82 mm or 1.8 mm × 3.8 mm using an additional focusing lens and focal length of 32 mm. Due to the multimode energy distribution on the surface of the focus of the laser beam, the value of this energy is almost constant on the entire surface (Figure 20).

**Figure 20.** The 2D beam profile of the used HPDL laser in the focal plane [43]. (Reprinted with permission, copyright 2017, EBSCO Industries).

This type of distribution caused the power density on the surface of a relatively large focus of the laser beam to be much lower than on the previously used disk laser (maximum 3.2 <sup>×</sup> <sup>10</sup><sup>4</sup> <sup>W</sup>/cm<sup>2</sup> as compared to 1.05 <sup>×</sup> <sup>10</sup><sup>7</sup> <sup>W</sup>/cm<sup>2</sup> ). The inability to conduct the welding process with the "keyhole" technique in the case of these lasers is compensated by their advantages when used in surface treatment and surfacing processes (wide beam with uniform energy distribution) [43].

The shape of the laser beam focus and the even distribution of energy of the laser radiation on the surface of this focus also require a special approach in the case of numerical analyses due to the fact that models that are usually not well represented are not available. While the historical rectangular heat source model described here corresponds to the beam shape, as already mentioned the user of modern programs dedicated to welding and heat treatment processes usually has three predefined models of heat sources: the Gaussian surface model, the double-ellipsoid model, and the conical model. As before, in this case it is possible to load the model's heat source with selected elements of the model's mesh and utilize its dimensions to control the width and depth of bead fusion.

Taking into account the internal structure of this type of laser (single diode emitters with single-mode distribution), the first approximation of this heat source can be a model in the form of two rows (due to the longitudinal dimension of the laser beam focus of 1.8 mm) of individual normal (Gaussian) distributions moving along the trajectory placed in the load area, specifying the bead obtained with some approximation (Figure 21a). This brings with it the aforementioned disadvantage associated with the need to define a large number of trajectories and heat source models as well as to divide the total amount of heat introduced by the appropriate number of sources used. Such an action does not reflect the construction of the laser rod in such a way where there are emitters with a power of 1–2 W. Therefore, with a laser power of 2000 W the grid of the discussed model and the number of heat sources necessary to define the model would cause the model to cease to be real due to the quantity of grid elements used in the numerical model (hundreds or even thousands of small heat source models corresponding to individual diode emitters). The tests carried out showed that even with a much rarer mesh of this type (and a reduced number of used heat sources), the solution ensured high compatibility of the shapes of the calculated beads with those obtained as a result of real tests (Figure 22, Table 5). The aforementioned much lower power density on the surface of the laser beam focus and the use of processes with the "weld pool" technique (conduction mode) in a similar way to arc welding inspired an attempt to adapt the most popular model, which is the double-ellipsoid model. Due to the ease of use in modeling the "weld pool" technique, the need to define only one trajectory of heat source model movement, and the ability to determine the depth of model interaction, attempts were made to confirm the usefulness of this solution. For this purpose, a model was built in which, as in the previous case, the "LOAD" area roughly corresponded to the dimensions of the obtained bead, while the double-ellipsoid heat source model was much wider than this area to ensure uniform energy distribution on the surface of the loaded area (Figure 21b). Other dimensions of the heat source model are related to the depth of penetration of the bead and the longitudinal dimension of the laser beam focus.

**Figure 21.** View of loading of the elements of the model mesh and placement of (**a**) heat source models with a normal distribution (Gaussian surface heat source model) and (**b**) a double-ellipsoid heat source model.

To verify the assumptions of the presented model, a three-dimensional discrete model in VisualWeld (SYSWELD) package was created, containing 11,120 3D elements and 13,243 nodes. The boundary conditions used, simulating the clamping of the sample, corresponded to its free placement on the table of the laser stand throughout the welding process and then its cooling. The boundary condition regarding heat dissipation was met by convective dissipation to the environment at 20 ◦C and radiation was defined on all external surfaces of the model. Numerical analyses were carried out for a model consisting of 12 pieces of the Gaussian surface heat source models (in two rows placed behind each, 6 models in each) and 6 trajectories determining the movement of these sets, located at a distance of 1.0 mm from each other. The second model was a model consisting of 22 Gaussian surface heat source models (in two rows placed behind each with 11 models in each) and 11 trajectories determining their movement at a distance of 0.5 mm from each other (Figure 21a). In the case of the described model, built using a single double-ellipsoid heat source model, analyses of 6 variants differing in the dimensions of the heat source model itself were performed (Figure 22, Table 5). In each of the analyzed cases the values of laser beam power and welding velocity were the same and were respectively 1400 W and 2.5 mm/s (Table 5).

**Table 5.** Comparison of geometric dimensions of selected beads on the bead-on-plate laser welding process performed on 6.0-mm AISI 304 steel samples using the ROFIN DL020 high-power diode laser with values obtained from numerical analyses (values in brackets) (Figures 21 and 22).


\* Dimensions for the Gaussian surface heat source model: diameter/distance between trajectories (mm); for the double-ellipsoid model: width/length/height of model. \*\* Heat transfer efficiency coefficient: 0.5 (determined on the basis of model calibration based on actual bead dimensions).

**Figure 22.** Comparison of the view of the molten pool surface and cross-sections (**a**) when using a model consisting of normal distributions and (**b**) a double-ellipsoid heat source model.
