**2. Numerical Simulations**

### *2.1. Simulations and Influence of Scanning Speed*

The powder layer on the target surface is considered thermally thin and is not taken into account. Laser radiation is supposed to be absorbed on the surface. In the case of partial reflection, the laser power in the equations mentioned above means the absorbed part of the laser beam radiation. In the coordinate system moving with the scanning speed, the steady-state heat diffusion equation is [43]:

$$
\kappa \Delta T + \mu\_s \frac{\partial T}{\partial x} = 0,\tag{5}
$$

where *us* is scanning speed, m/s; *α* is the thermal diffusivity, m2/s; and Δ is the Laplace operator. Equation (5) is solved by numerical or analytical methods where possible, with boundary condition:

$$T \to T\_a \text{ at } x \to \pm \infty, \ y \to \pm \infty, \ z \to \infty,\tag{6}$$

where *T*a is the ambient temperature. The target surface *z* = 0 is adiabatic, excluding the laser spot where

$$-\lambda \frac{\partial T}{\partial z} = q. \tag{7}$$

The temperature fields are presented in Figures 2 and 3.

**Figure 2.** Normalized distributions: flux density of the absorbed laser energy *q* over the target surface *z* = 0 (top row); temperature *T* over the target surface (second and third rows); temperature *T* over the vertical plane of mirror symmetry *y* = 0 formed by the beam axis and the scanning line (two rows on the bottom). Red in the *q*/*q*0 graph indicates the approach to the area of the discontinuity at the beam boundary where *r* = *r*0.

**Figure 3.** 3D plot of the implicit function *q*0 (**a**); implicit function *q*0 for various values of *r*0 (**b**); normalized implicit function *q*/*q*0 (TEM00 profile) (**c**); normalized implicit function *q*/*q*0 (TEMFT profile) (**d**); normalized implicit function *q*/*q*0 (TEM01\* profile) (**e**); and temperature distributions along the direction of the scanning speed on the surface, when *y* = 0, *z* =0(**f**). The beam boundary where *r* = *r*0 is marked red in graphs (**<sup>c</sup>**–**<sup>e</sup>**).

The scanning speed is specified by the thermal Péclet number:

$$\text{Pe} = \frac{2r\_0 u\_s}{a}.\tag{8}$$

The temperature rise relative to the ambience ( *T* − *T*a) is normalized by *T*0 specified by Equation (3). Normalizing coordinates by *r*0 makes the obtained results universal for a linear conductive medium. The results significantly depend on the Péclet number. The top row in Figure 2 shows two-dimensional views of laser profiles (1), (2), and (4) normalized by [43] (Figure 3a,b):

$$
\eta\_0 = \frac{P}{\pi r\_0^2}.\tag{9}
$$

The normalized graphs of profiles are as follows (Figure 3c–e):

*q*TEM00 *q*0 = *e* (− *r*2 *r*2 0 ), (10)

$$\frac{\eta\_{\rm TEM\_{\rm FT}}}{q\_0} = \frac{1}{2 \cdot \sqrt{1 - \frac{r^2}{r\_0^2}}} \,\,\,\tag{11}$$

$$\frac{q\_{\rm TEM\_{01\*}}}{q\_0} = \frac{r^2}{r\_0^2} \cdot \mathbf{e}^{\left(-\frac{r^2}{r\_0^2}\right)}.\tag{12}$$

The other rows in Figure 2 are two-dimensional temperature distributions over two characteristic planes. The 3D plot of the implicit function is shown in Figure 3a.

Figure 3f shows all the obtained results as profiles of the surface temperature along line *y* = 0, *z* = 0. For all laser profiles, the temperature profiles decrease with the increase of Pe that corresponds to the increase of the scanning speed. The forward temperature front becomes sharper with the increase of Pe, and the backward temperature front is insensible to Pe, according to the well-known asymptotics:

$$\frac{T - T\_a}{T\_0} = \frac{2}{\pi} \frac{r\_0}{R} \exp\left(\frac{\text{Pe}}{4} \frac{\text{x} - R}{r\_0}\right),\tag{13}$$

with *R*<sup>2</sup> = *x*2 + *y*2 + *z*2, shown by dashed lines in Figure 3f. In the case of mode TEM00, all three numerically calculated temperature profiles are bell-like. At Pe = 0, the maximum is in the origin. The numerically obtained maximum value is about the analytical result *T*max,

$$\frac{T\_{\text{max}} - T\_a}{T\_0} = \frac{2}{\sqrt{\pi}},\tag{14}$$

shown by a horizontal dash in Figure 3f. The increase of Pe slightly shifts the position of the temperature maximum in the direction opposite to that of the scanning speed vector that is explained by the thermal inertia of the target.

At Pe = 0, the flat-top laser beam profile forms steady-state temperature distribution

$$\frac{T - T\_a}{T\_0} = \frac{2}{\pi} \arcsin \frac{2r\_0}{\sqrt{(r - r\_0)^2 + z^2} + \sqrt{(r + r\_0)^2 + z^2}} \tag{15}$$

where *r*2 = *x*2 + *y*2, with an exactly horizontal plate over the laser spot. When Pe increases, this plate inclines towards the scanning speed vector and slightly sags. In the case of donut mode, the surface temperature distribution inherits the ring-like ridge. The ridge becomes more asymmetric with the increase of Pe (Figure 3f).

### *2.2. Temperature and Energy Flux Profiles*

Temperature distribution in a cross-section perpendicular to the scanning direction cannot objectively characterize the temperature conditions for laser powder bed fusion because retarding the maximum target temperature relative to the central cross-section *x* = 0. The retardation depends on the scanning speed value and the distance from the scanning axis (*X*). The most representative quantity is the maximum temperature along axis *X* for threshold-like and Arrhenius temperature dependencies of the process kinetics. Figure 4a shows the transverse profile of the quantity on the surface [43]:

$$\max\_{x} T(x, y, 0),$$

**Figure 4.** Maximum temperature *T* on the target surface *z* = 0 versus distance y from the scanning axis (**a**); the testing profiles (*q*/*q*0) and estimation of their radii at half-width *<sup>r</sup>*1/2 (**b**).

The asymptotics at Pe = 0 are given by Equation (13) at *x* = 0. At Pe = 0.71 and Pe = 2.86, the asymptotics are obtained by numerical treatment of Equation (13) by Equation (16). The widths of the re-melted zone on the surface often estimate the contact's width between the consolidated powder, and the substrate can be deduced by this profile.

The transverse profiles of the surface temperature shown in Figure 4a present the thermal conditions for laser powder bed fusion. They cannot be compared with the tested laser beam profiles because all the obtained temperature profiles have different absolute maxima. The tentative laser-beam radius *r*0 is not an objective measure of its width applicable to various beams' radial profiles. Thus, beam TEM01\* in Figure 4b seems wider than beam TEM00 at the same *r*0. Let us estimate the width of a laser profile by its diameter at half-maximum *d*1 2 that is conventional in laser technology applications. The scheme for estimating the corresponding radius at half-maximum *r* 12 = *d*12 /2 is shown in Figure 4b and Table 1.


**Table 1.** Calculated absolute maximum of temperature *T*max versus Péclet's number Pe.

It should be noted that temperatures above *T*max are unallowable because of material evaporation or chemical decomposition. Temperatures below the minimum *T*min are not sufficient to complete the specified physical or chemical processes. The boiling point is specified as *T*max, and the melting point is *T*min for laser powder bed fusion of pure metals [46,47]. For alloys, *T*max and *T*min are determined by the component with the lowest boiling and melting points, correspondingly.

The temperature dependencies of the kinetic constants can be taken into account to define the laser powder bed fusion interval (*T*min, *T*max). The maximum temperature in the laser-processing zone and the width of the laser beam characterized by *d*1/2 or *<sup>r</sup>*1/2 can be effectively controlled by variation of the laser power or by laser beam expansion. The former quantity can be set at *T*max. The latter quantity can be set at the specified dimensional uncertainty.

Figure 5 shows the same temperature profiles as in Figure 4a to apply the chosen criterion for evaluating the laser beam profiles. However, these profiles are renormalized by their absolute maxima, height, laser beam radii at half maximum, and width. The normalizing constants for all the nine testing profiles are obtained from the data shown in Figure 4a,b and Table 1. A qualitative review of the temperature profiles shown in Figure 5 indicates that laser profile TEMFT results in the broadest top of the temperature profile, as expected. Laser profile TEM00 results in the broadest base of the temperature profile. This means that evaluating the three tested laser profiles is not straightforward and depends on the acceptable temperature range of laser treatment *T*max − *T*min relative to the maximum temperature increment *T*max − *T*a. If the acceptable temperature range is narrow, the treated band of the surface is near the top level of the temperature profile. In this case, theoretically, the flat-top profile provides the widest laser-treated band, which means the most effective use of the laser energy. If the acceptable temperature range is wide, the most effective profile seems to be TEM00.

**Figure 5.** Normalized transverse profiles of the maximum surface temperature and the definition of the widths of laser-treated band (*B*1/2 and *B*0.9).
