**Tomasz Kik**

Department of Welding Engineering, Silesian University of Technology, Konarskiego 18A, 44-100 Gliwice, Poland; tomasz.kik@polsl.pl; Tel.: +48-32-237-1681

Received: 25 May 2020; Accepted: 5 June 2020; Published: 10 June 2020

**Abstract:** The article presents new possibilities for modifying heat source models in numerical simulations of laser welding processes conducted using VisualWeld (SYSWELD) software. Due to the different power distributions and shapes of a laser beams, it was necessary to propose a modification of heat source models and methods of defining the heat introduced into a welded material in the case of simulations of welding processes using solid-state and high-power diode lasers. A solution was proposed in the form of modification of predefined heat source models in the case of simulations of welding processes using solid-state disc lasers and high-power diode lasers (HPDL). Based on the results of metallographic tests and the acquisition of thermal cycles of real laser welding processes, the process of calibration and validation of the proposed models of heat sources depending on the type of device used as well as the obtained shapes of fusion beads was carried out. The purpose and assumptions of this approach towards creating heat sources were also reported, comparing exemplary stresses and cumulative plastic strain distributions for the calculation variant using a standard and modified heat source model.

**Keywords:** finite element method (FEM); welding; laser; heat source model; keyhole welding; thermal analysis

#### **1. Introduction**

The basics of lasers were presented in 1917 by Albert Einstein in his theory of stimulated emission, where it was stated that stimulated radiation and stimulating radiation have the same properties, i.e., the same direction of propagation and the same phase of vibration and polarization. These properties therefore indicate the fact that with a beam of laser light, a concentrated portion of energy can be transported. However, a long time passed before this could be applied at an industrial scale. This was mainly due to the problem of building devices that would be able to provide adequate power so that the laser beam focused on the surface of the material would be able to heat it to a temperature suitable for the selected technological processes. At present, as this possibility now exists, the undoubted advantages of concentrated, monochromatic light combine the precision of the process with product quality and efficiency inaccessible to traditional arc welding methods. Joining material technologies using a laser beam are characterized by smaller deformations and the effect of heat on the joined elements. The use of laser devices has caused many technologies to change and has expanded the areas of application, offering an up-to-now unattainable level of product quality. With their advantages, laser technologies somehow meet ever-increasing quality requirements and are also able to address the increasing complexity of the welding technologies used [1,2].

Depending on the internal design of the device and the principle of operation, it is an option to use devices with different radiation intensity distributions across the laser beam cross-section [2]. The power density of the device depends mainly on the energy distribution in the focus of the laser beam. The shape of the electromagnetic field generated on the beam cross-section is referred to as

transverse electromagnetic mode (TEM). The multitude of energy distributions that it is possible to obtain on the surface of the laser beam and in devices used in the industry, on the one hand, introduces incredible flexibility in adapting these devices to a specific application. On the other hand, it allows for many devices with significantly different properties despite, e.g., the similar maximum laser beam power offered. The existing variety of types of devices as well as beam focus shapes, radiation wavelengths, and power distributions on the laser beam focus surface means that the modern engineer has considerable possibilities for controlling the dimensions of the beads and the penetration shape obtained in processes using laser devices. However, this also causes a problem in choosing the right device for a particular application, especially taking into account the constantly shortening response time required by the market. The degree of complexity of current, modern welding technologies based on this type of equipment, along with the mentioned advantages and requirements, forces engineers and constructors to take a different approach to the design stage and search for modern tools supporting the preparation of welding and heat treatment technologies themselves [2–7].

One modern tool supporting their work is (among other programs for numerical simulation of manufacturing processes) based on the finite element method (FEM). The idea of supporting calculations was created quite a long time ago. The first attempts at prediction of residual stresses and deformations distributions were presented in 1938 by Rodger and Fletcher [8]. However, the insufficient computing power at that time blocked the use and development of these techniques for years. In 1970, Brust et al. presented the first welding numerical simulations using the finite element method [9–11]. Over the past several years, mainly due to the development of computers and the significant increase in available computing power, there has also been a huge improvement in computational techniques in the field of welding and heat treatment simulation. New computational techniques and a method for the mathematical description of the heat introduced into the welded material were developed. This resulted in a significant extension of the possibility of using modern computational programs in the everyday engineering practice of designers and welding engineers in industry [12–17].

Currently, numerical analyses of welding processes are usually used to optimize and validate these processes both from a technological and economic point of view, but in some cases they are used for the modification of the welded structure itself. Thanks to the continuous development of this field of knowledge, it is now possible to conduct local welding analyses, and then based on the received heterogeneous distributions of structures and material properties determine stresses and plastic deformations as well as the loads of the analyzed structure with external forces, pressures, and deformations. Based on the results obtained, it is possible to determine potential places of fatigue damage in the analyzed welded constructions. Modern software packages for numerical analyses create completely new opportunities for the modern engineer. The engineer may receive help that allows for a significant reduction in the time of production preparation or the introduction of changes in the technological process, thanks to the possibility of simulating their impact on the process result. In addition, this operation is completely safe because it takes place in the computer's memory and does not affect the actual technological process or security. These possibilities are however also associated with a certain degree of inconvenience, which is possible to resolve. The abovementioned possibilities for conducting analyses and the amount of input data and related information make numerical simulations of welding processes difficult and complicated to carry out using FEM simulations [17].

#### **2. Description of the Problem**

#### *2.1. Basics of Formulating the Problem of Temperature Distribution Calculations in Numerical Simulations*

Before FEM computer simulations of welding processes came into general use, the only way to present temperature fields during welding was through analytical methods. However, the emergence of modern technologies, new materials, and the increasing complexity of the designed structures have made the application of analytical methods very difficult or even impossible in some cases. The development of packages for computer simulation of welding processes based on

the finite element method has opened up new possibilities in the field of design support for these processes. Since numerical simulations of temperature distributions are calculated on the basis of Fourier's differential formula, to perform calculations it is necessary to obtain the values of the heat conduction coefficient, specific heat, and density depending on the temperature [18–20]:

$$\frac{\partial T}{\partial t} = \frac{\lambda}{c\rho} \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) = a\nabla^2 T \tag{1}$$

where *T* is the temperature, *t* is time, *x, y,* and *z* refer to the point coordinates, *a* is the thermal diffusivity coefficient, λ is the heat conductivity coefficient, *c* is specific heat, and ρ is mass density.

In the analyses presented later in the article, the VisualWeld 15.5 (SYSWELD) package of the ESI Group (Paris, France), was used, where the coupled thermo-metallurgical analysis is described by a modified heat convection equation [19,20]:

$$\left(\sum\_{i} P\_i(\rho \mathcal{C})\_i\right) \frac{\partial T}{\partial t} - \nabla \left(\left(\sum\_{i} P\_i \lambda\_i\right) \nabla T\right) + \sum\_{i$$

where *Pi,j* refers to the proportions of phases with indexes *i* and *j*, respectively, *Lij(T)* is the latent heat of *i* to *j* transformation, *Aij* is the proportion of phase *i* transformed to *j* in a time unit, and *Q* is the total amount of introduced heat.

The consequence of using Fourier's differential equation is the non-linearity of the calculated temperature field distributions. It is also necessary to consider that the temperature field distributions are calculated based on the mathematical description of a heat source model represented by thermal flow density into material. In many cases it is a moving heat source and the coordinates describing its location are related not only to the space coordinates but also to the duration of the process. The formation and development of welding strains and stresses is associated with several very closely related factors such as clamping conditions, thermo-mechanical material properties, type of used welding technology, process parameters, ambient temperature, and cooling conditions, among others [14–19,21,22].

#### *2.2. History of Heat Source Models in Numerical Simulations*

Models of heat sources used in numerical simulations of welding processes can be generally divided into two groups:


In 1941, Daniel Rosenthal was the first person to apply Fourier's law to a moving heat source [24]. This resulted in the first model of a heat source with a homogeneous distribution. Unfortunately, the biggest disadvantage of this solution was the fact that all the energy of the process was concentrated in one point, which in effect meant that the use of this model did not in any way allow for the determination of the shape of the molten pool and the depth of penetration. This solution was then modified, but an acceptable compliance of temperature distribution prediction was obtained only at a distance from the heat source itself. A certain extension was the linear model of the heat source proposed in 1969 by Pavelic [25]. Thanks to this solution, an attempt was made to determine the temperature distribution in a two-dimensional element. Still, a problem was the inability to precisely map the temperature gradient over the thickness of the model. A few years later, Friedmann proposed a heat source model that had a volume distribution. Equations describing cylindrical and square models of the heat source were successively developed. Thanks to these activities, the results obtained became closer and closer to those obtained from real tests. It should be noted that at that time processes with high power density were already being described using rectangular (electron beam) and cylindrical heat source models (laser beam). Nevertheless, the degree of simplification of these models did not ensure a complete mapping of the nature of these processes [23,26]. In 1983, Eagar and

Tsai introduced the first model with the heterogeneous distribution. The proposed model to some extent allowed for the weld pool geometry to be shown by using the Gaussian heat distribution on the surface [27]. On this basis, a disk model was later created. Its use was still somewhat limited due to the inability to determine the depth of operation of the model [23].

Over the years, in association with the development of numerical analyses of welding processes, many mathematical descriptions have been created regarding models of heat sources. These range from the simpler point models that limit all energy of the welding process to one point (mesh node), through to linear and circular models, up to more advanced forms. However, it was in the early 1980s that John Goldak proposed a widely used model in the shape of a double ellipsoid. Although it is currently mainly used for simulation of arc welding processes, i.e., a technique with a liquid metal pool, after appropriate modifications it can also be used to simulate selected laser welding processes, as shown later in the article [16,21–23,28,29].

#### *2.3. Typical Models of Heat Sources Used in Numerical Simulations*

In commercially available software dedicated to solving welding process problems, three types of predefined heat source models are usually used:


The Gaussian distribution model, in which energy is distributed according to the course of the Gaussian curve, is usually used in simulations of welding processes with high power densities, i.e., laser, plasma, and microplasma or electron beam welding (Figure 1). With the help of this heat source model it is possible to map deep penetration while maintaining a small width, characteristic of "keyhole" welding techniques. In the VisualWeld (SYSWELD) environment, a normal distribution heat source is a predefined and recommended model used in simulations of a surface heat treatment processes, including laser beam hardening [20]. The Gaussian surface heat source model is described as in [20]:

$$Q(x, y, z) = Q\_0 \exp\left(-\frac{x^2 + y^2}{r\_\varepsilon^2}\right) \tag{3}$$

where *Q<sup>0</sup>* is the amount of heat introduced by the source, and *x, y,* and *r<sup>e</sup>* are the geometrical parameters describing the heat source position.

**Figure 1.** Gaussian surface heat source model [20].

The most popular heat source model currently used in commercial programs for numerical analysis of welding processes is the volumetric double-ellipsoid heat source model, also known as Goldak's model. This model is built of two ellipsoids placed perpendicular to each other, allowing the size of the source to be determined in the plane resulting from the position of the source perpendicular to the direction of welding (Figure 2).

**Figure 2.** Double ellipsoid heat source models [17,18,20,21].

Unlike other volumetric models used for arc welding simulations, Goldak's model is described by two equations individually for each part of the ellipsoid separately [17,18,20,21]. For the front part of the ellipsoid, the equation is:

$$Q\_f(\mathbf{x}, y, z) = \frac{6\sqrt{3}f\_f Q}{ab\mathbb{C}\_f \pi \sqrt{\pi}} \exp\left(\frac{-\mathbf{k}\mathbf{x}^2}{a^2}\right) \exp\left(\frac{-\mathbf{l}y^2}{b^2}\right) \exp\left(\frac{-\mathbf{m}z^2}{c^2}\right) \tag{4}$$

and for the rear part of the ellipsoid, the equation is:

$$Q\_r(\mathbf{x}, y, z) = \frac{6\sqrt{3}f\_l Q}{ab\mathbf{C}\_l \pi \sqrt{\pi}} \exp\left(\frac{-\mathbf{k}x^2}{a^2}\right) \exp\left(\frac{-\mathbf{l}y^2}{b^2}\right) \exp\left(\frac{-\mathbf{m}z^2}{c^2}\right) \tag{5}$$

where *Q<sup>f</sup> , Q<sup>r</sup>* are the volumetric heat flux density in front and rear part of the model, respectively, (W/m<sup>3</sup> ), *Q* is the total introduced power, and *a b, c<sup>f</sup> ,* and *c<sup>r</sup>* are respectively the width, depth, and length of the front and the rear part of the estimated molten pool.

The coefficient of heat transfer efficiency to the welded material is determined depending on the welding method being analyzed [17,20].

How much of the total heat is transferred to the front and back of the ellipsoid is determined by the parameters *f<sup>f</sup>* and *f<sup>r</sup>* . These are constants that affect the distribution of energy flow to the material, which can somewhat compensate, e.g., the tilting effect of the welding torch. As practice has shown, the relation of *f<sup>f</sup>* to *f<sup>r</sup>* is usually accepted as a ratio of 60:40. It is important to remember that these parameters must satisfy the dependence *f<sup>f</sup>* + *f<sup>r</sup>* = 2 [17,30]. Determining the values of coefficients *k*, *l*, and *m* is a much more complex issue. When using an unmodified heat source model, all three factors assume a value of 3. An example of such an application can be e.g., simulation of the Metal Manual Arc (MMA) welding process. However, in the case of simulation, e.g., of the Gas Metal Arc (GMA) welding process, the values of these coefficients often must be modified experimentally to obtain the correct shape of the molten pool. Currently, this model so far best describes the actual state of arc welding methods [16–18,20–22].

In analyses of welding processes characterized by high energy densities, i.e., laser, electron beam welding or plasma welding, a conical source model with a normal distribution is usually used (Figure 3).

**Figure 3.** Three-dimensional conical heat source models [17,20].

The mathematical description of volumetric heat flow density into the material is described as follows [17,20]:

$$Q(x,y,z) = Q\_0 \exp\left(-\frac{x^2 + y^2}{r\_0^2(z)}\right) \tag{6}$$

$$r\_0(z) = r\_\varepsilon + \frac{r\_i - r\_\varepsilon}{z\_i - z\_\varepsilon}(z - z\_\varepsilon) \tag{7}$$

where *Q<sup>0</sup>* is the maximum value of volumetric heat flux density, *r<sup>e</sup> ,* and *r<sup>i</sup>* are respectively the upper and lower cone radius dimensions parameters, *z<sup>e</sup> ,* and *z<sup>i</sup>* are the cone length parameters, and *x, y,* and *z* are point coordinates (Figure 3).

The use of a moving heat source model in numerical simulations of welding processes requires the use of an appropriate calibration procedure. Only in this case is it possible to correctly determine the stresses and deformation levels and distributions, calculated based on previously obtained results of temperature fields and individual metallurgical phase distributions. Incorrectly performed calibration of the heat source model usually leads to significant differences in the results obtained as compared to the reality. The use of predefined models does not mean, however, that the user cannot create a combination of these sources to obtain e.g., a set suitable for the simulation of hybrid welding consisting of a conical model for the laser beam and the double-ellipsoid model for mapping the effect of arc welding [31]. It is possible to use self-defined heat source models as well as other computational techniques that do not use moving heat source models. Unfortunately, they often require initial calibration based on the analyses performed using them [17].

### *2.4. The Shapes of Beads Geometry Obtained in the Case of Laser Welding Processes—Formulation of the Research Problem*

As mentioned before, the conical heat source model is the most commonly used heat source model in the case of the "keyhole" welding technique. However, the model does not always allow for correct results as compared to the reality. While it works relatively well when calculating a process partial penetration, when there is full penetration obtained it is not possible to model the bead expansion characteristic of the laser welding process in the root area when welding thicker parts. In the laser welding process, essentially three different bead geometry shapes can be obtained. In the case of welding without full penetration, the shape is usually narrow, cylindrical, or slightly expanded upwards—the so-called bell shape (Figure 4a).

**Figure 4.** Examples of macrographs performed with disc laser welding (**a**) with partial penetration (bell shape), and (**b**) and (**c**) with full penetration (with a narrow and wide root known as the hourglass shape).

The situation changes after achieving full penetration of the bead. In this case, basically two shapes of the melted metal area are obtainable. One has a wide face area and slight widening in the area of the root (Figure 4b). The situation changes after achieving full remelting. In this case, two shapes of the melted metal area are obtainable. The first is due to wide melting in the face area and slight expansion in the ridge area. The second, in turn, shows wide remelting in both the face and root areas, with a narrow area melted in the middle of the bead (hourglass shape) (Figure 4c). The bead shape obtained when welding with full penetration depends on the type of device used, the process parameters, and sheet thickness as well as the type of material being welded [29,31–35]. The information found in the literature shows that the nature of the bead geometry changes at full penetration results from several complex phenomena that determine the flow of liquid metal in the molten pool and the convective heat that affects the shape of the bead [29,34–36]. A complete solution to this type of problem requires numerical simulations using complex fluid mechanics models. Unfortunately, the aforementioned simple models used in commercial software for simulation of welding processes are models based on thermal conductivity, where this flow is not taken into account. Several studies have appeared in the literature on attempts to solve this problem. In 2017, Flint proposed a model that is an extension of the double-ellipsoid model that can be used for narrow-groove and keyhole weld configurations [37]. Wu et al. also proposed some modifications of the conical model where a modified three-dimensional cone model (TDC) was used to simulate the plasma welding process with the mesh technique [38]. Similarly proposed by Farrokhi et al. in the case of hybrid welding, combined models based on a double three-dimensional conical heat source model for the reconstruction of "keyhole" welding conditions with partial and full penetration indicate the fact that in order to obtain the correct shape of the melted area, it is necessary to combine two or more volumetric models of heat sources [29]. In 2018, for the investigation of a dual-beam laser welding of aluminum sheets, He et al. proposed modification of the cone-shaped heat source model for the lap joint configuration. Based on the proposed model and the experimental validation procedure, investigations of hot cracking were conducted [39].

From a scientific point of view, this is the most correct reasoning aimed at precisely and accurately reproducing the obtained bead shape in the heat source model. For a typical commercial user of mathematical notation of this type, proposed models of heat sources can be a real obstacle to everyday use. Not every user has the programming knowledge needed to create their own heat source model description. Therefore, research should be continued to propose the simplest and most effective way to reproduce the actual geometry of the bead during numerical simulations of welding processes.

#### **3. Modification of Predefined Heat Source Models in the Case of Laser Bead-On-Plate Welding**

#### *3.1. Test Stand and Preliminary Attempts to Use Predefined Heat Source Models*

To propose a solution that would be simple to use for a typical commercial user, welding tests were carried out on two diametrically opposed laser devices. The first test stand was equipped with a Trumpf TruDisk 3302 (Ditzingen, Germany) disk laser with a maximum beam power of 3300 W (Figure 5a). This laser allowed us to obtain a high quality beam with a wavelength of 1030 nm while maintaining a beam spot

diameter of 200 µm. The laser head was equipped with a 200-mm collimator lens and a 200-mm focusing lens. The beam parameter product (BPP), as claimed by the manufacturer of this laser, was <8.0 mm × mrad.

**Figure 5.** View of test stands equipped with (**a**) TruDisk 3302 disk laser with 3300 W laser beam power and (**b**) a Rofin DL020 high-power diode laser with 2200 W beam power.

The second type of laser used in the research was a high-power diode laser (HPDL) (Figure 5b). The ROFIN DL 020 direct diode laser from ROFIN Sinar (Hamburg, Germany) allows for the following rectangular beam spot dimensions to be obtained: 1.8 mm × 6.8 mm at 82 mm focal length or 1.8 mm × 3.8 mm at 32 mm focal length (with an additional focusing lens). The maximum output power of the HPDL ROFIN DL 020 is 2200 W with the wavelength at 808 nm, with very even energy distribution across the beam spot (multimode TEM).

For preliminary tests using a disk laser, bead-on-plate welding tests were performed on a 6.0-mm-thick sample of S355 steel at 1500 W with a welding speed of 8.3 mm/s, and on a 4.0-mm sample of AISI304 steel (X5CrNi18-10) at 1000 and 1500 W and 8.3 mm/s welding speed (Tables 1 and 2).


**Table 1.** Chemical composition of S355 (EN10025-2) and X5CrNi18-10 (EN10088-3) steel [40,41].

To simulate bead-on-plate welding tests, a three-dimensional numerical model of a 6.0-mm-thick flat sheet was prepared. To ensure the correct mapping of heat dissipation conditions, a boundary condition was adopted describing heat exchange with the environment on all external surfaces of the model. Boundary conditions for clamping were adopted as standard conditions corresponding to welding of the sample placed without additional fastening on the laser station table. The calculations were performed in the "heat source input fitting" module as a "steady state"-type analysis (analysis carried out at only one moment of time after the process was established), using a normal distribution heat source model (Gaussian surface heat source model) with the following parameters: laser beam power, 1500 W; welding speed, 8.33 mm/s; *r<sup>0</sup>* = 2.0 mm (Figure 1). As a result of tests carried out at the laser stand, a bead width of 2.08 mm and a maximum penetration depth of 5.20 mm were obtained. For comparison, the results of numerical simulation allowed the determination of these dimensions at the level: bead width 2.02 mm and maximum penetration depth 5.31 mm (Figure 6). In the case of the parameters used, a similar molten pool shape was obtained in the numerical simulation. The tests carried out showed that as the laser beam power (and penetration) increased, the compatibility of the numerical simulation with the reality for this heat source model decreased significantly.

**Figure 6.** View of the macrostructure of the bead (**a**) in bead-on-plate welding tests performed on a 6.0-mm-thick sample of S355 steel using the Trumpf TruDisk 3302 disk laser at laser beam power of 1500 W and a welding speed of 8.3 mm/s; (**b**) cross-sectional view of the molten pool obtained as a result of numerical simulation; and (**c**) the 1450 ◦C iso-surface.

In the case of AISI304 steel samples, a three-dimensional numerical model of a flat sheet with a thickness of 4.0 mm was made analogously. The boundary conditions were also set to simulate no sample fixing. The calculations were performed using a conical source heat source model using the following parameters of the heat source model (Figure 3):

DL1.0\_8.33—Laser beam power: 1000 W; welding speed: 8.33 mm/s; r<sup>e</sup> = 2.44 mm; r<sup>i</sup> = 1.12 mm; z<sup>e</sup> = 0; z<sup>i</sup> = 3.16 mm

DL1.5\_8.33—Laser beam power: 1500 W; welding speed; 8.33 mm/s; r<sup>e</sup> = 3.1 mm; r<sup>i</sup> = 2.4 mm; z<sup>e</sup> = 0; z<sup>i</sup> = 4.0 mm.

As a result of the tests, beads with the aforementioned shapes were obtained, with bell and hourglass shapes with the geometrical dimensions presented in Table 2 and Figure 7. While there is considerable agreement between the dimensions measured on real samples and the same dimensions obtained from the results of numerical analyses carried out using a conical model of the heat source, the shape of the real and calculated beads varied significantly (Figure 7).

**Table 2.** Comparison of geometric dimensions of beads in bead-on-plate welding tests performed on a 4.0-mm-thick sample of AISI304 steel using the Trumpf TruDisk 3302 disk laser with values obtained from numerical analysis (values in brackets) (Figure 7).


**Figure 7.** View of the macrostructure of the bead (**a**, **a'**) in bead-on-plate welding tests performed on a 4.0-mm-thick sample of AISI304 steel using the Trumpf TruDisk 3302 disk laser at laser beam power accordingly 1000 and 1500 W and welding speed of 8.3 mm/s; (**b**, **b'**) a cross-sectional view of the molten pool obtained as a result of numerical simulation; (**c**, **c'**) the 1450 ◦C iso-surface.

In summary, the modeling tests carried out using implemented models of heat sources, i.e., the Gaussian surface and conical heat source model, showed that there is a need to develop modifications to obtain the actual shape of calculated beads.

### *3.2. Assumptions and Construction of a Modified Numerical Model in the Case of a Disk Laser*

It is possible to create a combined model consisting for example of two conical sources or one conical and other cylindrical sources (Figure 8). However, in this case it is problematic to divide the amount of energy for each of these parts (as shown by the author's experience, not from the volume ratio of both models). This is due to different heat distribution conditions in individual batches of materials as well as heat dissipation conditions. Additionally, this requires defining two trajectories along which the heat source model moves (Figure 8). Nonetheless, it should be emphasized that this simulation method allows for a significant improvement in the results obtained and a better mapping of the fusion line shape.

**Figure 8.** A view of heat sources models consisting of (**a**) two conical sources and (**b**) a conical and cylindrical source (green lines correspond to trajectories along which the sources move).

Considering the above, the simplest methods for creating a model with a high correlation coefficient of results with reality are still being sought as simply as possible. To solve this problem, it is possible to use the functionality offered by the VisualWeld 15.5 package from the ESI Group, France. In this software, it is possible to specify elements in the model mesh that will be only affected by a standard heat source model. Simply put, only the selected mesh elements covered by the standard heat source model will be heated. Using this function, it is possible to modify the shape of the fusion line in a relatively convenient and simple way (Figure 9).

**Figure 9.** Examples defining the loaded areas of the model (network) and location of the heat source model in the case of laser welding with the mesh technique with (**a**) incomplete and (**b**) full penetration.

To use this method, however, it must be assumed that in principle, it applies to all numerical analyses of welding processes. The shape of the bead fusion line (based on the macroscopic metallographic picture) should be treated as input data, not the result of the analysis. The task of numerical analysis performed based on the heat conduction equation, not the problem of fluid mechanics, is to obtain additional information on the distribution of metallurgical phases, hardness, stresses, or joint deformations during and after the process. The lack of consideration of complicated mathematical models describing the convective movements of liquid metal in the weld pool itself means that the shape of the fusion line, which is highly influenced, must be set by the user. This is justified by the fact that in simulations with available programs for numerical analyses of welding and heat treatment processes many factors are ignored that in welding processes have a significant impact on the shape of the fusion line. This even includes the impact of the shielding gas used or the method of transferring metal in the arc during GMA welding, which are factors that have a huge impact on the nature and intensity of the abovementioned movements of liquid metal in the pool. Thus, the shape of the fusion line is a very important input parameter at the calibration stage of the calculation model. Following the proposed assumption, the calibration of the heat source model is not only about entering the source power and dimensions of the heat source model corresponding to actual or expected values, but also manipulation of the heat source model shape and the area it affects (red-colored area in Figure 9 marked as "LOAD") so that the obtained result is as close as possible to the result of metallographic tests of the bead used in model calibration. The user, by appropriately selecting the parameters of the heat source model and the mesh elements that he exposes, has much greater possibilities of mapping the actual shape of the melted area.
