# Revealing the effects of laser beam shaping on melt pool behaviour in
conduction-mode laser melting

Amin Ebrahimi  Department of Materials Science and Engineering, Faculty of
Mechanical, Maritime and Materials Engineering, Delft University of
Technology, Mekelweg 2, 2628CD Delft, The Netherlands Mohammad Sattari  Chair
of Laser Processing, Department of Mechanics of Solids, Surfaces and Systems
(MS3), Faculty of Engineering Technology, University of Twente, Drienerlolaan
5, 7522NB Enschede, The Netherlands Aravind Babu  Department of Materials
Science and Engineering, Faculty of Mechanical, Maritime and Materials
Engineering, Delft University of Technology, Mekelweg 2, 2628CD Delft, The
Netherlands Arjun Sood  Department of Materials Science and Engineering,
Faculty of Mechanical, Maritime and Materials Engineering, Delft University of
Technology, Mekelweg 2, 2628CD Delft, The Netherlands Gert-Willem R.B.E. Römer
Chair of Laser Processing, Department of Mechanics of Solids, Surfaces and
Systems (MS3), Faculty of Engineering Technology, University of Twente,
Drienerlolaan 5, 7522NB Enschede, The Netherlands Marcel J.M. Hermans
Department of Materials Science and Engineering, Faculty of Mechanical,
Maritime and Materials Engineering, Delft University of Technology, Mekelweg
2, 2628CD Delft, The Netherlands

###### Abstract

Laser beam shaping offers remarkable possibilities to control and optimise
process stability and tailor material properties and structure in laser-based
welding and additive manufacturing. However, little is known about the
influence of laser beam shaping on the complex melt-pool behaviour, solidified
melt-track bead profile and microstructural grain morphology in laser material
processing. A simulation-based approach is utilised in the present work to
study the effects of laser beam intensity profile and angle of incidence on
the melt-pool behaviour in conduction-mode laser melting of stainless steel
316L plates. The present high-fidelity physics-based computational model
accounts for crucial physical phenomena in laser material processing such as
complex laser-matter interaction, solidification and melting, heat and fluid
flow dynamics, and free-surface oscillations. Experiments were carried out
using different laser beam shapes and the validity of the numerical
predictions is demonstrated. The results indicate that for identical
processing parameters, reshaping the laser beam leads to notable changes in
the thermal and fluid flow fields in the melt pool, affecting the melt-track
bead profile and solidification microstructure. The columnar-to-equiaxed
transition is discussed for different laser-intensity profiles.

Keywords: Fusion welding and additive manufacturing, Laser beam shaping, Melt
pool behaviour, Microstructural grain morphology, Bead profile, High-fidelity
numerical simulation

##  1 Introduction

Laser beam melting offers unique controllability and high spatial and temporal
precision in manufacturing high-integrity metallic products and has been
utilised in many industries [1]. Melting and solidification in laser welding
and additive manufacturing considerably affect local microstructural features
(i.e. solidification morphology, grain size and crystallographic texture) [2,
3, 4, 5] and the macroscale structure and properties of the material [6, 7,
8]. Studies have demonstrated that internal flow behaviour in melt pools has a
significant influence on melting and subsequent solidification [9, 10], hence
it needs to be controlled effectively to achieve the desired properties [11].
However, controlling internal flow behaviour in laser welding and additive
manufacturing is challenging mainly due to the complexity of molten metal flow
dynamics and high sensitivity to changes in process parameters [12, 13, 14].
Therefore, understanding underlying physical phenomena in laser melting (i.e.
laser-matter interaction, heat and molten metal flow, and solid-liquid phase
transformation) is crucial for effective control of the melt-pool behaviour.

Previous studies suggest laser intensity profile as a critical factor
influencing the processing window, melt-pool shape and dimension, and product
quality in laser welding and additive manufacturing [15, 16, 17, 18, 19].
Changes in the laser intensity profile in such processes affect heating and
cooling cycles by influencing convection in the melt pool [20] and hence can
lead to changes in thermal gradients and solidification growth rates [21, 22,
23]. Advancements in optics and laser technology have opened up new
possibilities for modulating spatial and temporal laser intensity profiles
with high resolution, which is beneficial to enhancing process stability and
control in laser material processing. Various techniques have been developed
to tailor laser beam shape, which are reviewed comprehensively elsewhere [24,
25]. However, the influence of tuning spatial laser intensity profile
(commonly known as laser beam shaping) on internal flow behaviour has been
overlooked and only sparsely investigated, particularly concerning conduction-
mode laser melting, which serves as the foundation for numerous welding and
additive manufacturing techniques. Moreover, the design and optimisation of
laser intensity profile usually relies on trial-and-error experiments that are
generally expensive and time-consuming [26, 27]. This highlights the necessity
of developing reliable predictive models that facilitate design-space
exploration and process optimisation within a reasonable timeframe.

Studies on the effect of laser beam shaping in fusion welding and additive
manufacturing, to a great extent, are experimental and focus primarily on
measuring melt-pool shape and the resulting solidification texture and
microstructural properties (see for instance [2, 28, 29, 15, 18]). High-
fidelity numerical simulations supplement experimental studies by providing a
better insight into the complex heat and fluid flow in melt pools and
associated melting and solidification, supporting the development of process
design and optimisation for laser-based manufacturing. Han and Liou [30]
developed a three-dimensional computational model and studied the effects of
employing a rectangular and three different axisymmetric laser intensity
profiles on the evolution of thermal and fluid flow fields in stationary laser
melting of a stainless steel alloy (AISI 304). In their model, Han and Liou
[30] neglected the influence of surface-active elements on Marangoni flow
pattern. Safdar et al. [31] conducted three-dimensional numerical simulations
to study the effects of different rectangular beam shapes on thermal and fluid
flow fields in laser melting of a mild steel alloy (EN-43A). They neglected
the influence of surface-active elements on the Marangoni flow pattern in
their model and assumed that the melt-pool surface is flat and does not deform
during the process. Numerical predictions of laser beam welding of aluminium-
copper alloys reported by Rasch et al. [32] showed that the average surface
temperature and resulting recoil pressure changes notably when using different
beam shapes with comparable laser powers. Roehling et al. [3, 4] studied the
effects of elliptical laser beam shapes on the resulting microstructure of
stainless steel 316L in laser powder-bed additive manufacturing and
demonstrated experimentally that microstructure grain nucleation and
morphology could be controlled using the laser beam shaping technique. Later,
Shi et al. [5] developed a cellular automaton model to predict solidification
grain structure using elliptical laser beam shapes in laser powder-bed
additive manufacturing and employed a thermal-fluid model to predict the
evolution of the thermal field during the process. The influence of surface-
active elements on Marangoni flow pattern was neglected in their thermal-fluid
simulations. Focusing on conduction-mode laser melting of Ti6Al4V alloy, Abadi
et al. [33] numerically studied the effects of using elliptical and circular
beams with a Gaussian intensity profile on molten metal flow behaviour and
melt-pool shape. In these studies, the effects of surface morphology, laser
incidence angle and temperature on local energy absorption are neglected and
the value of absorptivity is assumed to be constant. Neglecting these factors
limits the general applicability of the model in predicting melt-pool
behaviour for different laser intensity profiles and various laser material
processing scenarios. Despite the great potential of laser beam shaping for
improving process control and final product quality, very little is known
about the influence of various laser beam shapes on heat and fluid flow in
melt pools and the resulting bead quality.

The present work focuses on studying laser beam melting of metallic substrates
using different beam shapes, intensity profiles and laser beam incidence
angles. The analysis primarily concentrates on conduction-mode laser melting,
which forms the basis for a wide variety of welding and additive manufacturing
techniques. Due to highly transient and complex molten metal flow behaviour,
involving multiple physical effects such as solid-liquid phase transformation,
laser-material interactions and free-surface oscillations, it is challenging
to investigate the evolution of melt-pool behaviour using merely trial-and-
error experiments. In the present work, high-fidelity three-dimensional
numerical simulations are performed to describe geometrical evolution and
molten metal flow dynamics of the melt pool. An enhanced laser absorptivity
model [13] is utilised that takes into account the effects of surface
morphology, material composition and surface temperature on local energy
absorption. The model has been rigorously validated with experimental data
acquired using various laser beam shapes. Subsequently, the validated model is
employed to predict the effects of different laser intensity profiles on the
melt-pool behaviour and resulting solidification microstructure. The present
work offers a simulation-based approach to predict the effects of laser beam
shaping in conduction-mode laser melting and to explore the design space for
process optimisation at a reduced cost compared to experiments.

##  2 Methods

Three-dimensional physics-based numerical simulations were performed to study
different beam shapes, intensity profiles and laser beam incidence angles in
conduction-mode laser melting. The geometrical configuration of the problem,
defined in a Cartesian coordinate system, together with the prescribed
boundary conditions and the beam shapes studied in the present work are shown
schematically in figure 1. For beam shapes shown in figure 1(b), a uniform
intensity profile was employed. Beam shapes shown in figure 1(c) were produced
by changing the laser beam incidence angle about the $x$ or $y$ axes, and the
beam had a (pseudo-)Gaussian intensity profile. An Yb:YAG laser with an
emission wavelength ($\lambda$) of
$1.030\text{\times}{10}^{-6}\text{\,}\mathrm{m}$, a laser power
($\mathscr{P}$) of $700\text{\,}\mathrm{W}$ and a focal spot diameter
($d_{\mathrm{b}}$) of $1.5\text{\times}{10}^{-3}\text{\,}\mathrm{m}$ provides
the thermal energy flux required to locally melt the plate that is made of
stainless steel AISI 316L and is initially at room temperature
($T_{\mathrm{i}}=$300\text{\,}\mathrm{K}$$). The laser beam scans the surface
at a fixed travel speed ($\mathscr{V}$) of
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,
unless stated otherwise. To protect the molten material from oxidation, an
argon shielding gas was employed.

Figure 1: (a) Schematic of laser beam melting, plate dimensions and the
boundary conditions prescribed in numerical simulations. (b) different beam
shapes with uniform intensity profile (the laser beam is perpendicular to the
$x$-$y$ plane), and (c) different beam shapes obtained by inclining the laser
beam with (pseudo-)Gaussian intensity profile. Values shown above each laser
beam shape indicate the area of laser focal spot $\mathscr{A}$.

The computational model was developed based on the finite-volume approach
using a proprietary general-purpose solver, ANSYS Fluent [34]. The volume-of-
fluid (VOF) method [35] was employed to model the movement of metal-gas
interface. Details of the computational model are explained in our previous
works [13, 12, 36, 37], and are not repeated here. In the present work,
temperature-dependent material properties are employed for AISI 316L, and the
values are taken from [13]. The present model predicts multiple strongly-
coupled physical processes that occur simultaneously during laser melting,
such as variable laser absorptivity, molten metal flow due to Marangoni and
thermal buoyancy forces, solid-liquid phase transformation and melt-pool
surface deformation caused by the violent internal flow and the recoil
pressure of metal vaporisation. Thermal and fluid flow fields and associated
melt-pool surface movements were predicted by solving the conservation
equations of mass, momentum, energy and a scalar field, called volume fraction
$\phi$ to capture the position of the melt-pool surface, assuming that both
the molten metal and argon are Newtonian fluids and that their densities are
independent of any pressure variations that occur during processing. These are
defined as follows:

$\frac{D\rho}{Dt}+\rho\left(\nabla\cdot\mathbf{u}\right)=0,$ (1)

$\rho\frac{D\mathbf{u}}{Dt}=\mu\nabla^{2}\mathbf{u}-\nabla p-C\
\frac{(1-\psi)^{2}}{\psi^{3}+\epsilon}\ \mathbf{u}+\mathbf{F}_{\mathrm{s}},$
(2)

$\rho\frac{Dh}{Dt}=\frac{k}{c_{\mathrm{p}}}\nabla^{2}h-\rho\frac{D\left(\psi\mathcal{L}_{\mathrm{f}}\right)}{Dt}+S_{\mathrm{q}}+S_{\mathrm{l}},$
(3)

$\frac{D\phi}{Dt}=0.$ (4)

Here, $\rho$ is the density, $\mathbf{u}$ the fluid velocity vector, $t$ the
time, $\mu$ the dynamic viscosity, $p$ the pressure, $C$ the mushy-zone
constant, equal to
$10^{7}\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-2}\text{\,}{\mathrm{s}}^{-2}$
[38], $\epsilon$ a constant incorporated to avoid division by zero, equal to
$10^{-3}$, $h$ the sensible heat, $k$ the thermal conductivity,
$c_{\mathrm{p}}$ the specific heat capacity, $\mathcal{L}_{\mathrm{f}}$ the
latent heat of fusion, $\psi$ the local liquid volume-fraction, and
$\left(\psi\mathcal{L}_{\mathrm{f}}\right)$ the latent heat of the material.
The value of $\phi$ in a computational cell ranges between 0 and 1,
representing the volume-fraction of the metal phase. Marangoni shear force,
capillary force and recoil pressure are modelled by incorporating the source
term $\mathbf{F}_{\mathrm{s}}$ into the momentum equation (equation 2) [13].
The energy input from the laser beam and heat losses from the workpiece due to
convection, radiation and vaporisation are modelled by adding the source term
$S_{\mathrm{q}}$ and the sink term $S_{\mathrm{l}}$ to the energy equation
(equation 3) respectively [13, 12]. The boundary conditions prescribed at the
outer boundaries of the computational domain are shown in figure 1. Spatial
and temporal variations of the laser absorptivity during the process resulting
from changes in surface temperature and laser beam incidence angle were
modelled using an enhanced absorption model that is described in our previous
work [13]. User-defined functions (UDFs) were employed to implement the source
and sink terms in the governing equations, the laser absorptivity and surface
tension models. The accuracy of the present computational model in predicting
thermal profile and molten metal flow dynamics in laser melting is thoroughly
investigated in our previous works [36, 12, 13, 37] by comparing the numerical
results with experimental and theoretical data for different laser systems and
various laser beam intensity profiles and shapes. The validity of the model is
further investigated in the present work, and the results are presented in
section 3.1.

The computational mesh consists of about $2\text{\times}{10}^{6}\text{\,}$
non-uniform hexahedral cells with minimum cell spacing of
$4\text{\times}{10}^{-5}\text{\,}\mathrm{m}$ close to the gas-metal interface.
The computational cell spacing was chosen sufficiently fine to achieve at
least $35$ cells along the melt-pool width, which is sufficient to obtain
grid-independent numerical predictions [12, 13, 37]. The second-order central
differencing scheme and the first-order implicit formulation were employed for
spatial and temporal discretisation respectively. An explicit VOF method [39]
was employed to capture the gas-metal interface. The PISO (pressure-implicit
with splitting of operators) scheme [40] was employed for resolving the
pressure-velocity coupling, and the PRESTO (pressure staggering option) scheme
[41] was applied for the pressure interpolation. To keep the value of the
Courant number $(\mathrm{Co}=\lVert\mathbf{u}\rVert\Delta t/\Delta x)$ less
than $0.25$, a fixed time-step size of $\Delta
t=$1\text{\times}{10}^{-6}\text{\,}\mathrm{s}$$ was employed.

The effects of laser beam shaping on microstructural grain structure are
studied by computing the values of solidification growth rate ($R$) and
thermal gradient ($G$) at the computational grid points located in the
solidifying region (i.e. mushy region) as follows [5]:

$G=\lVert\nabla T\rVert=\sqrt{\nabla T\cdot\nabla T},$ (5)

$R=\mathscr{V}\cdot\cos\left(\alpha\right).$ (6)

Here, $\alpha$ is the angle between the maximum heat flow direction and the
scanning direction, and is determined as follows:

$\alpha=\arccos\left(\frac{\partial T/\partial y}{G}\right).$ (7)

The values of solidification growth rate ($R$) and thermal gradient ($G$)
obtained from the simulations are plotted in the so-called ‘solidification
map’. The solidification map is divided into three regions according to the
analytical model proposed by Hunt [42], indicating the nucleation propensity
of fully columnar (characterised by high $G/R$ values), fully equiaxed
(characterised by low $G/R$ values) and mixed columnar-equiaxed grains.
According to the Hunt’s model, the critical gradient conditions for fully
columnar and fully equiaxed grain growth can be approximated respectively by

$G>\xi\,\Delta T_{\mathrm{c}}\,\sqrt[3]{100\cdot
N_{0}}\,\left(1-\left(\frac{\Delta T_{\mathrm{N}}}{\Delta
T_{\mathrm{c}}}\right)^{3}\right),$ (8)

and

$G<\xi\,\Delta T_{\mathrm{c}}\,\sqrt[3]{N_{0}}\,\left(1-\left(\frac{\Delta
T_{\mathrm{N}}}{\Delta T_{\mathrm{c}}}\right)^{3}\right),$ (9)

where, $\xi$ is a constant equal to 0.617 [42], $\Delta T_{\mathrm{c}}$ the
undercooling for dendrite formation, $N_{0}$ the nucleation density and
$\Delta T_{\mathrm{N}}$ the critical undercooling for nucleation. Assuming
that the tip velocity of the dendrite is equal to the solidification growth
rate $R$ [43], the undercooling for dendrite formation is determined by an
empirical correlation as follows [44, 5]:

$\Delta T_{\mathrm{c}}=\left(\frac{R}{a}\right)^{1/b},$ (10)

where, $a$ and $b$ are constants whose values depend on the material and are
equal to $7.325\text{\times}{10}^{-6}\text{\,}$ and $3.12\text{\,}$
respectively [45] for the alloy employed in the present study. The values of
nucleation density $N_{0}$ and the critical undercooling for nucleation
$\Delta T_{\mathrm{N}}$ are taken from the literature [17, 5, 4] for stainless
steel AISI 316L and are equal to $10^{15}\text{\,}{\mathrm{m}}^{-3}$ and
$10\text{\,}\mathrm{K}$ respectively.

##  3 Results and discussion

###  3.1 Model validation

Four different laser beam shapes were studied to examine the validity of the
present numerical simulations in predicting the melt-pool shape and
dimensions. An Yb:YAG laser (Trumpf TruDisk 10001) with an emission wavelength
of $\lambda=$1.030\text{\times}{10}^{-6}\text{\,}\mathrm{m}$$ in combination
with three optical transport fibres (i.e. a
$600\text{\,}\mathrm{\SIUnitSymbolMicro m}$ circular core fibre, a
$620\text{\,}\mathrm{\SIUnitSymbolMicro m}$ square core fibre and a
$400/100\,$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ 2-in-1 core fibre) was
employed in the experiments. The focusing optics (BEO D70 Trumpf) consist of a
collimator with a focal length of $2\text{\times}{10}^{-1}\text{\,}\mathrm{m}$
and lenses with two different focal lengths of
$4\text{\times}{10}^{-1}\text{\,}\mathrm{m}$ and
$6\text{\times}{10}^{-1}\text{\,}\mathrm{m}$. Employing the lens with a focal
length of $6\text{\times}{10}^{-1}\text{\,}\mathrm{m}$ in combination with the
2-in-1 fibre, a beam with a focal spot size of
$d_{\mathrm{b}}=$1.2\text{\times}{10}^{-3}\text{\,}\mathrm{m}$$ and a uniform
intensity profile in the focal plane, as shown in figure 2, was produced. The
laser power ($\mathscr{P}$) was set to $500\text{\,}\mathrm{W}$ to locally
melt a stainless steel AISI 316L plate at a fixed travel speed of
$\mathscr{V}=$2\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$.
The plate employed in the experiments had a cuboid shape with dimensions ($x$,
$y$, $z$) of $100\times 250\times$10\text{\,}{\mathrm{mm}}^{3}$$. Each
experimental trial was conducted with a minimum of three repetitions to
establish the replicability of the outcomes. To capture the experimental melt-
pool shapes, macrographs were extracted ex-situ from middle of the melting
track and photographed using a digital microscope. Uncertainties in
experimental measurements of the melt-pool depth and width are about $4$–$7\%$
and $2$–$4\%$ respectively. Iso-surfaces of solidus temperature
($T_{\mathrm{s}}=$1658\text{\,}\mathrm{K}$$) obtained from the numerical
simulations were used to visualise the melt-pool shape. A comparison is made
between the numerically predicted and experimentally measured melt-pool shapes
for different beams, and the results are shown in figure 2. The maximum
absolute error in the prediction of melt-pool depth and width are less than
$7\%$ and $5\%$ respectively, demonstrating a reasonable agreement without
exercising any parameter tuning. Such an error is attributed to the
simplifying assumptions made to develop the model, uncertainties associated
with modelling material properties at elevated temperatures and experimental
variations.

Figure 2: Comparison of the numerically predicted melt-pool shapes using the
present computational model with the experimental measurements for (a)
circular, (b) annular, (c) square and (d) diamond laser beam shapes. Red
symbols on each subfigure show the laser beam shape. Blue curves show the
numerically predicted melt-pool shape.

The numerically predicted $G$-$R$ pairs for different laser beam shapes are
plotted in the solidification map, as shown in figure 3. The solidification
map is divided into three regions according to the analytical model proposed
by Hunt [42]. EBSD (electron backscatter diffraction) technique was used to
visualise microstructural grain structure on the transverse cross-sections of
the melting track after the experiments, and the results are shown in figure
3. For EBSD measurements, the samples were further polished with colloidal
silica for 50 minutes. A Thermo Scientific Helios G4 PFIB UXe SEM equipped
with an EDAX detector was employed for EBSD analysis. A specimen tilt angle of
60∘ along with an accelerating voltage of $20\text{\,}\mathrm{kV}$ and a
current of $3.2\text{\,}\mathrm{nA}$ was used for data acquisition. A step
size of $0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ was employed for the
measurements. EBSD data processing was performed using EDAX OIM Analysis 8
software. The examination of EBSD data shows that changes in beam shape result
in variations in the proportion of columnar grains. The EBSD-derived
percentages of columnar grains for circular, annular, square, and diamond-
shaped beams are approximately 88%, 85%, 86%, and 95%, respectively. This
observation demonstrates a satisfactory agreement between the numerically
predicted grain morphologies following Hunt’s model and the EBSD-derived
measurements.

Figure 3: The distribution of numerically predicted grain morphologies on the
solidification map based on Hunt’s model (left column) and the EBSD
measurements (right column) for different laser beam shapes. Orange symbols
show the laser beam shape.

###  3.2 Melt-pool behaviour

Figure 4 shows three snapshots of thermal and fluid flow fields over the melt-
pool surface for circular beams with Gaussian and top-hat intensity profiles
after reaching the quasi-steady-state condition. The results show that the
molten metal flows from the melt-pool rim towards its central part because of
the Marangoni shear force, which agrees with experimental observations of
Mills et al. [46]. The molten metal temperature in the central part of the
pool is above the critical temperature for which the value of the temperature
gradient of surface tension ($\mathrm{d}\gamma/\mathrm{d}T$) is positive,
resulting in an outward Marangoni flow from the central part towards the melt-
pool rim. A complex unsteady flow pattern forms in the central part of the
pool where the inward and the outward streams interact. Although the melt-pool
resembles an almost symmetric shape with respect to the $y$-$z$ plane, the
molten metal flow and the temperature distribution in the pool are inherently
unsteady and asymmetric due to hydrodynamic instabilities, requiring a
transient and fully three-dimensional model to simulate transport phenomena in
the pool properly. The thermal energy absorbed by the material diffuses
through the material by conduction and advects with fluid flow in the pool.
The overall energy transfer due to heat conduction and advection in
combination with heat loss from the material determines the melt-pool shape;
however, heat loss from the material due to radiation and convection is about
$0.3\%$ of the absorbed laser optical power, and thus its effect is
negligible. Maximum fluid velocity in the pool is in the order of
$\mathscr{O}($0.6\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$)$, resulting
in a Péclet number ($\mathrm{Pe}=\rho
c_{\mathrm{p}}\mathscr{D}\lVert\mathbf{u}\rVert/k$) of about 100 that
signifies the dominant effect of advection in total energy transfer.

For the process parameters studied here, the melt-pool dimensions obtained
using circular beams with top-hat and Gaussian intensity profiles are
comparable but the peak temperature obtained using a beam with a Gaussian
intensity profile is about $7\%$ higher than that using a beam with a top-hat
intensity profile. This is because the peak beam intensity of a Gaussian
profile is higher than that of a top-hat profile for similar laser power and
focal spot diameter. However, the intensity of the Gaussian beam rapidly
reduces to values less than that of the top-hat beam when moving away from the
pool centre. The greater peak temperature at the pool centre obtained using
the Gaussian beam also results in higher absorptivity of laser energy that
increases with temperature [13]. Accordingly, the magnitudes of temperature
gradient and Marangoni shear force for the case with a Gaussian intensity
profile are greater than those with a top-hat intensity profile, resulting in
more sensitivity to spatial disturbances when using a beam with a Gaussian
intensity profile. The maximum fluid velocity predicted for the case with a
Gaussian intensity profile seems to be generally lower than that with a top-
hat intensity profile, which is attributed to the greater outward Marangoni
force in the former case.

Figure 4: Snapshots of thermal and fluid flow fields over the melt-pool
surface at three different time instances and the predicted melt-pool
dimensions for circular laser beams with Gaussian (a-c) and top-hat (d-f)
intensity profiles. The laser power is $700\text{\,}\mathrm{W}$ and the travel
speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$. The
laser beam is perpendicular to the plate surface.

Melt-pool behaviour in laser melting cannot be characterised solely based on
laser power, spot size, travel speed and material properties; and information
about beam shape and laser intensity profile is also required. Reshaping the
laser beam can lead to changes in power density $\mathscr{P}_{\mathrm{d}}$ and
interaction time ($t_{\mathrm{i}}$), the product of which results in energy
density $\mathscr{E}_{\mathrm{d}}$ [47] that critically affects the melt-pool
behaviour [48]. Figure 5 shows the numerically predicted temperature
distribution and velocity vectors over the melt-pool surface and melt-pool
shape for different laser beam shapes with uniform intensity profiles. For all
the beam shapes studied here, an inward flow from the melt-pool rim is
observed over the surface that collides with an outward flow close to the pool
centre, the interaction of which leads to the formation of vortex flow over
the surface that enhances fluid mixing in the pool. Although the total heat
input to the material is almost identical for different beam shapes111The
value of laser absorptivity depends on surface temperature, material
composition and laser beam incidence angle [37]. Such effects are accounted
for in the absorptivity model employed in the present numerical simulations.
Hence, the total energy input to the material is not necessarily the same for
different cases studied in the present work., the predicted melt-pool
dimensions differ considerably. The results show that there is not necessarily
a one-to-one relationship between energy density and melt-pool volume, hence
employing the concept of energy density is insufficient to explain changes in
melt-pool dimensions obtained using different beam shapes. Assuming that the
average laser absorptivity is similar for different beam shapes, it can be
argued, based on the conservation of energy principle, that the material’s
total enthalpy must be identical when employing different beam shapes because
the total heat input is the same. Therefore, variations in the thermal field
should also be accounted for in addition to the energy density to explain the
difference between melt-pool shapes obtained using different beam shapes.
Changes in the pool shape also lead to changes in the direction and magnitude
of temperature gradients, affecting the grain morphology of the material after
solidification [49].

Elliptical beam shapes with different energy densities can be produced by
changing the incidence angle of circular laser beams. A change in the laser
beam incidence angle not only distorts the beam shape but also affects the
laser absorptivity and hence the total energy input to the material. Figure 6
shows the numerically predicted temperature distribution and velocity vectors
over the melt-pool surface and the pool shape for different laser beam
incidence angles with (pseudo-)Gaussian intensity profiles. The flow pattern
obtained using inclined beams seems similar to that obtained using the
respective circular beam, and the molten metal flows from the melt-pool rim
toward its centre. The melt-pool shape projected on the $x$-$z$ plane
resembles a symmetric shape for all the laser beam inclination angles except
those where the beam is inclined about both $x$ and $y$ axes (see figure
6(b)). For the cases studied in the present work where an Yb:YAG laser with an
emission wavelength of $1.030\text{\times}{10}^{-6}\text{\,}\mathrm{m}$ is
employed, the average absorptivity of stainless steel AISI 316L increases by
2–3% and 8–12% by increasing the inclination angle from $0^{\circ}$ to
$45^{\circ}$ and $60^{\circ}$ respectively [13]. Although total energy input
to the material increases by inclining the laser beam due to enhanced laser
absorptivity, power density decreases with increasing the inclination angle
because of the considerable increase in the spot area $\mathscr{A}$ (41.4% for
an inclination angle of $45^{\circ}$ and 99.4% for an inclination angle of
$60^{\circ}$). When the laser spot is elongated in the transverse direction
(i.e. inclining the beam about the $y$-axis), the interaction time remains the
same as that of the circular beam; therefore, the overall effect of inclining
the laser beam is a decrease in the energy density and thus the melt-pool
depth and peak temperature. Similarly, the power density decreases when the
spot is elongated in the scanning direction because of the increased spot
area; however, decrease in the energy density is less than that of the case
with a laser spot elongated in the transverse direction due to the increased
interaction time. Hence, deeper melt-pools are observed when the beam is
elongated in the scanning direction compared to those where the spot is
elongated in the transverse direction. The present numerical results agree
with the experimental observations reported in the literature [50, 15, 18, 8]
for conduction-mode laser melting.

Figure 5: Thermal and fluid flow fields over the melt-pool surface and the
predicted melt-pool shapes for different laser beam shapes after reaching the
quasi-steady-state condition ($t=$0.65\text{\,}\mathrm{s}$$). The laser power
is $700\text{\,}\mathrm{W}$, the beam has a uniform intensity profile and the
travel speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.
Green symbols on each subfigure show the laser beam shape. Figure 6: The
effects of laser beam incidence angle on thermal and fluid flow fields over
the melt-pool surface and the pool shapes after reaching quasi-steady-state
condition ($t=$0.65\text{\,}\mathrm{s}$$). The laser beam is perpendicular to
the $x$-$y$ plane in subfigure a. Beam shape in subfigure b is produced by
inclining the beam about both $x$ and $y$ axes by $45^{\circ}$. Beam shapes in
subfigures c and d are produced by inclining the beam about the $y$-axis by
$45^{\circ}$ and $60^{\circ}$ respectively. Similarly, beam shapes in
subfigures e and f are produced by inclining the beam about the $x$-axis by
$45^{\circ}$ and $60^{\circ}$ respectively. The laser power is
$700\text{\,}\mathrm{W}$, the beam has a (pseudo-)Gaussian intensity profile
and the travel speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.
Green symbols on each subfigure show the laser beam shape.

###  3.3 Bead profile

Bead profile and surface quality after laser melting are influenced by the
thermal and fluid flow fields in the melt-pool. Figure 7 shows the surface
profile of the bead measured in the $z$ direction with reference to the un-
melted surface and the arithmetic mean of the absolute bead deformation
($\overline{\left|z_{\mathrm{b}}\right|}$) for different laser beam shapes
with uniform intensity profiles. The bead surface is elevated in the central
region of the melt track due to inward molten metal flow during laser melting
for all the beam shapes shown in figure 7. Since the mass of the workpiece is
almost unchanged during the process (i.e. vaporisation of the material and
changes in the material density are negligible in conduction-mode laser
melting), the bead surface is depressed in regions close to the outer
boundaries of the track. The mean bead deformations obtained using circular,
annular and square beam shapes appear to be similar and in the order of
$12\text{\,}\mathrm{\SIUnitSymbolMicro m}$ that is lower than those obtained
using diamond and triangular beam shapes, which can be attributed to the lower
energy densities of the former cases. For some cases, the crest trail of the
bead shows a wavy pattern that is due to molten metal flow instabilities
during laser melting. Although no humping defect is observed for the cases
studied in the present work, the surface profile obtained using the diamond
beam shape (figure 7(d)) suggests that the bead could be prone to a humping
defect.

Figure 7: Surface profile of the bead obtained from the numerical simulations
for different laser beam shapes ($t=$0.65\text{\,}\mathrm{s}$$). The laser
power is $700\text{\,}\mathrm{W}$, the beam has a uniform intensity profile
and the travel speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.
Orange symbols on each subfigure show the laser beam shape. Negative values of
$z$ indicate surface elevation. Bead deformation values over the melt track,
excluding the molten region, were employed to compute mean bead deformations.

Figure 8 shows the surface profile of the bead for different laser beam shapes
with (pseudo-)Gaussian intensity profiles. The bead surface produced using the
circular beam with a Gaussian intensity profile shows solidified waves that
result from pulsating molten metal flow instabilities [37, 12, 51]. Elongation
of the laser spot in the transverse direction increases the bead width and
seems to have an insignificant influence on the mean bead deformation. In
contrast, elongation of the laser spot in the scanning direction does not
change the bead width and reduces the mean bead deformation. The results
indicate that inclining the beam about the $y$-axis by $60^{\circ}$ (figure
8(f)) results in irregular surface deformations that can be ascribed to the
flow instabilities in the melt pool. The bead profile obtained using a beam
inclined about both $x$ and $y$ axes (figure 8(d)) differs from the other
cases and is elevated in regions close to one edge of the melt track and is
depressed in regions close to the other edge. The results show that the laser
beam inclination angle can be utilised as a potential means to control bead
profile in laser melting processes.

Figure 8: Surface profile of the bead obtained from the numerical simulations
for different laser beam incidence angles ($t=$0.65\text{\,}\mathrm{s}$$). The
laser power is $700\text{\,}\mathrm{W}$, the beam has a (pseudo-)Gaussian
intensity profile and the travel speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.
Orange symbols on each subfigure show the laser beam shape. Negative values of
$z$ indicate surface elevation. Bead deformation values over the melt track,
excluding the molten region, were employed to compute mean bead deformations.

###  3.4 Microstructural grain morphology

The present high-fidelity numerical predictions revealed that laser beam
shaping significantly affects the melt-pool shape and thermal and fluid flow
fields in conduction-mode laser melting. This can lead to changes in the
material’s microstructural grain structure and macroscopic properties after
solidification. The distribution of $G$-$R$ pairs on the solidification map
are shown in figures 9 and 10 for different laser beam shapes and inclination
angles respectively. The results suggest that changing the laser beam shape
leads to the variation of the ratio of columnar to equiaxed grains. Since the
travel speed was the same in all the simulations, the value of solidification
growth rate $R$, which scales with the travel speed (see equation 6), is
similar for all the cases. However, the value of temperature gradient $G$
ranges between
$4\text{\times}{10}^{5}\text{\,}\mathrm{K}\text{\,}{\mathrm{m}}^{-1}$ and
$7\text{\times}{10}^{6}\text{\,}\mathrm{K}\text{\,}{\mathrm{m}}^{-1}$ for
different laser beam shapes and inclination angles studied in the present
work. Due to the large magnitude of temperature gradients in conduction-mode
laser melting ($\mathscr{O}(10^{6})$ $\mathrm{K}$), it is unlikely that
equiaxed grains nucleate homogeneously during solidification [52, 53]. Grid
points located at the melt-pool tail on the track centreline are the farthest
points from the centroid of the laser spot and have the lowest temperature
gradient $G$ and the highest solidification growth rate $R$ (i.e. the lowest
$G/R$); hence, equiaxed and mixed columnar-equiaxed grains are more likely to
form in that region. For partially penetrated melt-pools, the highest $G/R$
values are found at the fusion boundary and close to the melt-pool bottom and
sides since the solidification growth rate $R$ is minimum at those locations;
thus, non-equiaxed (i.e. planar, cellular and columnar) grains are expected to
form in those regions [3].

Figure 9: Scatter plot of $G$-$R$ data on the solidification map for different
laser beam shapes with uniform intensity profiles after reaching quasi-steady-
state condition ($t=$0.65\text{\,}\mathrm{s}$$). The laser power is
$700\text{\,}\mathrm{W}$ and the travel speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.
Orange symbols show the laser beam shape.

The results presented in figure 9 indicate that changing the beam shape from
circular shape leads to a decrease in the nucleation propensity of fully-
columnar grains except for the diamond beam shape. Calculating the percentage
of $G$-$R$ pairs located in the mixed columnar-equiaxed region for different
laser beam shapes, it is found that employing backward triangular (7.0%),
annular (6.4%) and square (6.0%) beam shapes results in the highest nucleation
propensity of equiaxed grains. The distribution of $G$-$R$ pairs in figure 10
indicate that elongation of the laser spot in the transverse direction
decreases the nucleation propensity of fully-columnar grains that can be
ascribed to the decrease in the magnitude of temperature gradient $G$ at the
melt-pool tail. Elongation of laser spot in the scanning direction by
inclining the beam by $45^{\circ}$ seems to have insignificant effect on the
nucleation propensity of fully-columnar grains; however further increase in
the inclination angle to $60^{\circ}$ leads to an increase in the nucleation
propensity of fully-columnar grains. This is because melt-pool elongation in
the scanning direction results in an increase in the average angle between the
maximum heat flow direction and the scanning direction ($\alpha$), leading to
a decrease in the solidification growth rate ($R$) and an increase in the
value of $G/R$.

Figure 10: Scatter plot of $G$-$R$ data on the solidification map for
different laser beam inclination angles with Gaussian intensity profiles after
reaching quasi-steady-state condition ($t=$0.65\text{\,}\mathrm{s}$$). The
laser power is $700\text{\,}\mathrm{W}$ and the travel speed is
$1.5\text{\times}{10}^{-2}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.
Orange symbols show the laser beam shape.

##  4 Conclusions

The effects of laser beam shaping on melt-pool behaviour, solidified bead
profile and microstructural gain morphology were studied comprehensively
through high-fidelity numerical experiments for conduction-mode laser melting
of stainless steel AISI 316L. Three-dimensional unsteady numerical simulations
were performed to examine the effects of laser beam intensity profile, shape
and inclination angle on the melt-pool behaviour. Critical physical phenomena
in laser melting such as temporal and spatial variations of absorptivity, heat
and fluid flow dynamics, solidification and melting, and free-surface
oscillations were accounted for in the present computational model. Moreover,
experiments were performed using different laser beam shapes and the validity
of the present numerical predictions was demonstrated. Based on the results of
the present work, the following conclusions are drawn.

  * •

Altering the laser intensity profile leads to changes in the thermal and fluid
flow fields in the melt pool, which in turn affect the melt-pool dimensions,
melt-track bead profile and grain morphology. Moreover, the sensitivity of
melt-pool responses to spatial and temporal disturbances, and hence process
stability, depend on the laser-intensity profile.

  * •

For identical processing parameters, reshaping the laser beam can change the
energy density that affects the melt-pool behaviour. However, variations in
the melt-pool shape and volume caused by laser beam shaping cannot be
described solely based on the concept of energy density, and the material’s
total enthalpy should be taken into consideration.

  * •

Inclining the laser beam not only distorts the beam shape but also affects the
laser absorptivity and hence total energy input to the material. However, the
increase in the absorptivity due to laser beam inclination is considerably
less than the increase in the spot area; thus, the overall effect of laser
beam inclination is a reduction in the power density. Employing identical
processing parameters and inclination angles, a deeper melt-pool was observed
when the laser spot was elongated in the scanning direction compared to the
case in which the spot was elongated in the transverse direction, which can be
ascribed to the increased interaction time in the former case.

  * •

Laser beam shaping has a notable effect on melt-track bead profile and can be
utilised as a potential means to control bead profile in laser welding and
additive manufacturing.

  * •

Laser beam shaping significantly affects microstructural grain morphology by
influencing the thermal gradient and solidification growth rate. The present
findings demonstrate that the material’s microstructure, and thus its
macroscopic properties, can be adjusted locally through laser beam shaping.

The findings of the present work provide an improved insight into the effects
of laser beam shaping in laser welding and additive manufacturing. The
simulation-based approach developed in the present work offers new routes for
design space exploration and process optimisation.

Future research in this area could focus on advancing the optimisation of
laser beam shaping for laser-based additive manufacturing. One promising
avenue is to explore multi-objective optimisation techniques that consider not
only the melt-pool behaviour and bead profile but also the microstructural
grain morphology. This would enable the development of tailor-made
microstructures with desired properties for specific applications. Data-driven
approaches, combining experimental data with numerical simulations, could be
employed to create predictive models that guide the selection of optimal laser
beam intensity profiles for achieving precise control over microstructural
characteristics. Additionally, investigating the influence of various process
parameters and materials on the interaction between laser beam shaping and
microstructure could open new avenues for tailoring material properties in
additive manufacturing, enhancing the design and performance of additively
manufactured components.

## Acknowledgement

This research was carried out under project number S17024 in the framework of
the Partnership Program of the Materials innovation institute M2i (www.m2i.nl)
and the Netherlands Organisation for Scientific Research (www.nwo.nl). This
research project is part of the AiM2XL program (www.m2i.nl/aim2xl).

## Author Contributions

Conceptualisation, A.E.; methodology, A.E.; software, A.E.; validation, A.E.,
A.B., A.S. and M.S.; formal analysis, A.E.; investigation, A.E.; resources,
A.E., G.R.B.E.R. and M.J.M.H.; data curation, A.E.; writing—original draft
preparation, A.E.; writing—review and editing, A.E., A.B., M.S., G.R.B.E.R.
and M.J.M.H.; visualisation, A.E.; project administration, A.E. and M.J.M.H.;
and funding acquisition, G.R.B.E.R. and M.J.M.H.

## Conflict of interest

The authors declare no conflict of interest.

## Data availability

The raw/processed data required to reproduce these findings cannot be shared
at this time due to their large size, but representative samples of the
research data are presented in the paper. Other datasets generated during this
study are available from the corresponding author on reasonable request.

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