Numerical Simulation of Balling in Metal Powder Bed Fusion Using Particle Method

ORIGINAL ARTICLE
The International Journal of Advanced Manufacturing Technology (2026) 142:6329–6344
https://doi.org/10.1007/s00170-025-17381-y
1 Introduction
Additive manufacturing (AM), a process for manufacturing
three-dimensional components by layering of raw materials
[1], has been the subject of much research and development
in recent years because it is expected to provide impor-
tant benets: direct manufacture of functional components
with complex internal and external shapes from shape data
designed using a design system.
Laser-based powder bed fusion of metal (PBF-LB/M)
is an extremely well-known AM method. As presented in
Fig. 1, for PBF-LB/M, which is based on sliced shape data
generated from 3D shape data, a 3D shape is obtained by
repeated deposition of metal powder, melting by selective
irradiation with a laser, solidication of the molten part, and
lowering of them all by one layer height [2].
This PBF-LB/M composite process accommodates mate-
rial states such as metal powder, liquid, solid–liquid, solid,
Hitoshi Tokunaga
[email protected]
1 Integrated Research Center for Advanced Manufacturing,
National Institute of Advanced Industrial Science and
Technology (AIST), 1-2-1 Namiki, Tsukuba,
Ibaraki 305-8564, Japan
2 Department of Materials Science, Faculty of Science and
Engineering, Waseda University, 3-4-1 Okubo, Shinjuku,
Tokyo 169-8555, Japan
3 Department of Applied Mechanics and Aerospace
Engineering, Faculty of Science and Engineering, Waseda
University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
4 Kagami Memorial Research Institute of Materials Science
and Technology, Waseda University, 2-8-26 Nishiwaseda,
Shinjuku, Tokyo 169-0051, Japan
5 Department of Information Science and Mechatronics
Engineering, Institute of Technologists, 333 Maeya, Gyoda,
Saitama 361-0038, Japan
Abstract
A new computational method is proposed to simulate the balling phenomenon during powder bed fusion (PBF) metal
additive manufacturing processing. The method combines the discrete element method (DEM), which simulates powder
behavior, and smoothed particle hydrodynamics (SPH) method, which simulates uid and elastoplastic behavior. Balling
is a phenomenon by which spherical defects form on a product surface because of surface tension and poor wettability
between the molten metal and the surrounding powder or solidied material. This phenomenon is particularly evident in
materials with high surface tension, which can cause discontinuities in the melting path and which can aect manufactur-
ing quality. Simulating such complex behavior necessitates consideration not only of the molten metal behavior (including
surface tension), the metal powder and solidied metal behaviors, and the melting and solidication phenomena: it also
includes their mutual interactions. The proposed method can simulate these processes. Moreover, performing simulations
under conditions of various combinations of laser power and scanning speed conrm qualitatively that the proposed
method captures the balling phenomenon. The proposed method can capture trends of experimentally obtained results
obtained from earlier studies, i.e., under the same laser energy per unit area, the balling phenomenon becomes more likely
to occur as the scanning speed and laser power increase. This approach can be a valuable tool for optimizing manufactur-
ing conditions and for improving the quality of metal additive manufacturing processes.
Keywords Additive manufacturing · Balling · Laser powder bed fusion · Smoothed particle hydrodynamics · Discrete
element method · Thermo-uid simulation
Received: 7 July 2025 / Accepted: 30 December 2025 / Published online: 30 January 2026
© The Author(s) 2026
Numerical simulation of balling behavior in metal powder bed fusion
process using particle method
HitoshiTokunaga1· YukiWakai1,2· NaokoSato1· NaokiSeto1· SatoshiKajino1· ShinsukeSuzuki2,3,4·
YuichiMotoyama1· ToshimitsuOkane1,5
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The International Journal of Advanced Manufacturing Technology (2026) 142:6329–6344
and gas, as presented in Fig. 2(a). Their behaviors in each
state and their mutual interactions strongly aect the resul-
tant fabrication quality. Furthermore, changes in state such
as melting and solidication of metal powder occur because
of rapid heating and cooling caused by laser irradiation. For
these reasons, if appropriate processing conditions are not
set in PBF-LB/M, then defects will occur because of vari-
ous factors. Major examples of defects include formation of
spherical particles and discontinuous shapes on the manu-
factured object surface (balling phenomenon), scattering
of the molten part (spattering), insucient fusion (lack of
fusion), and warping deformation caused by residual stress
in the solidied part. This study specically examines one
of these defects: the balling phenomenon.
Balling is a phenomenon by which the molten part is inter-
rupted by surface tension when low anity exists between
the molten part and the metal powder, solidied part, or
substrate. Discontinuous spherical solidied parts form on
the product surface. Because this phenomenon occurs as a
result of a balance of various factors, it is extremely useful
to apply simulation techniques to ascertain the appropriate
processing conditions which prevent the balling phenom-
enon from occurring.
To facilitate such simulations, eorts have been under-
taken to elucidate various phenomena associated with
PBF-LB/M and to develop evaluation indices. Li et al.
used stainless steel as a representative material to inves-
tigate eects of processing conditions (oxygen content,
scan speed, laser power, scan interval, layer thickness,
and surface remelting) on the onset of ball formation [3].
They found that balling occurs because of poor wettability
between the molten pool and the substrate. Using an analyti-
cal thermal model, Wang et al. proposed a method to predict
the molten pool shape and to characterize the occurrence
of insucient fusion, balling, and keyholes from the shape
[4]. Yadroitsev et al. investigated the capillary instability of
a cylindrical shape formed by melting and investigated the
eects of process parameters such as scan speed and laser
power on the formation of a single track [5]. Gu et al. inves-
tigated phenomena, including the balling phenomenon, by
investigating the phases and microstructures for various
PBF-LB/M process parameters (scanning speed, linear
energy densities, etc.) used for the processing of Ti powder
[6]. Lindström et al. investigated the ball formation mech-
anism while considering the thermal balance between the
molten powder and the solid surface. Based on their nd-
ings, they proposed a criterion for predicting ball formation
in PBF-LB/M at low to medium laser scanning speeds [7].
Liu et al. analyzed the Mg surface morphology during laser
processing and examined the balling phenomenon mecha-
nism [8]. They reported that balling occurs because increas-
ing the scan speed reduces the energy input, lowers the
molten pool temperature, and increases the liquid surface
tension and viscosity, thereby preventing the liquid from
owing smoothly and moving to its surroundings. They
therefore claried that the balling phenomenon is aected
Fig. 2 PBF-LB/M process and
calculation method in this paper
Fig. 1 Laser-based powder bed
fusion of metal (PBF-LB/M)
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The International Journal of Advanced Manufacturing Technology (2026) 142:6329–6344
by the balance (competition) between diusion and solidi-
cation in the molten area. Narra et al. investigated defects
(keyholes, balling, and lack of fusion) in parts that had been
printed under various conditions. They also performed four-
point bending fatigue tests on the parts, and established a
process window [9]. Based on their ndings, they reported
that balling occurs when bonding between the molten
part and the previous layer is insucient because of poor
wettability, or when the relation between the molten pool
length and width are inappropriate and when melting into
the metal powder layer is insucient. After numerically and
experimentally investigating defect generation for process
parameter settings, Promoppatum et al. constructed a pro-
cess window [10]. They demonstrated that the occurrence or
non-occurrence of balling is determined by the combination
of scanning speed and laser power. They particularly noted
that even with the same energy density, increasing the speed
and output engenders balling. Ogura et al. used in-situ X-ray
and thermal imaging to elucidate the Ti powder layer melt-
ing behavior caused by in-situ laser irradiation. Particularly,
they investigated the mechanism underlying formation
of the keyhole depth and width [11]. They also observed
melting behavior related to the balling phenomenon: they
observed that the metal powder melts immediately after
the start of laser irradiation. Then droplets are formed, sup-
ported by the surrounding metal powder. The recoil pressure
acts as a driving force to propel the droplets into the mol-
ten and solidied parts. Wakai et al. used a similar setup to
clarify the mechanism of continuous and discontinuous melt
track generation experimentally under support-free printing
conditions with constant energy density [12]. Similarly to
Promoppatum et al., their results demonstrated that even
under support-free conditions, discontinuous melt tracks
are generated when a high laser power and a high scanning
speed are combined, even at the same energy density [10].
Therefore, comprehensive consideration of the behav-
iors of the metal powder, molten part, and solidied part,
the melting and solidication phenomena of the metal
powder, and the surface tension eects is important to
produce a PBF-LB/M simulation that includes the balling
phenomenon. Moreover, one must coordinate and facilitate
their interactions and balance. Some mesh-based additive
manufacturing simulations use conventional methods such
as nite element method (FEM), nite dierence method
(FDM), and nite volume method (FVM). However, these
methods are not good at accommodating various states that
dier from place to place, or with complex shapes, discon-
tinuities, multiple states, and state changes accompanied by
large deformation, or with complex interface movements.
Such methods are therefore inadequate for analyzing addi-
tive manufacturing processes including the balling phenom-
enon. Some studies of additive manufacturing simulations
are being conducted using particle methods [13, 14], which
are thought to be suitable for such analysis subjects [15].
Some methods use the smoothed particle hydrodynam-
ics (SPH) method [13] to calculate all processes including
melting and solidication [16, 17]. Others use discrete ele-
ment method (DEM) [14] for powder analysis to analyze
metal powder behaviors during recoating, use the SPH
method [18, 19] for the melting process, or use the lattice
method CFD (Optimal Transportation Meshfree, OTM)
[20]. Some models calculate the solidication part as a rigid
body [21, 22]. Nevertheless, these methods do not simulta-
neously analyze the metal powder movement, metal powder
melting by the laser, the molten part ow and solidication,
and the solidied part movement and elastic–plastic behav-
ior. These methods are therefore inadequate for PBF-LB/M
simulations that include the balling phenomenon. In other
words, without simultaneous analysis, analyzing the eects
of metal powder movement on the molten and solidied part
movement, or vice versa, is dicult. Moreover, analyzing
eects of these movements on the continuity of the molten
parts and other features during the process is dicult.
As described herein, a method combining a simula-
tion method for metal powder behavior using DEM and a
simulation method for melting and solidication of metal
powder by laser using the SPH method, as presented in Fig.
2(b), is proposed to support analysis that can simultaneously
address the various behaviors described above. This simula-
tion method is based on the ow and solidication simula-
tion method, which the authors developed using the SPH
method [23–25]. Based on the settings reported by Ogura
et al. [11] and by Wakai et al. [12], the proposed method
presented herein is applied to simulation of the settings
depicted in Fig. 3, for which a holder is lled with pure tita-
nium powder, moved at a specied speed, and then irradi-
ated with a laser having specied power. The possibility and
eectiveness of its application to simulating the balling phe-
nomenon is demonstrated by executing simulations under
several processing conditions with various laser outputs and
scanning speeds (holder speeds). It is noteworthy that this
study does not incorporate consideration of metal powder
Fig. 3 Schematic illustration of PBF-LB/M process simulation
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dρ
i
dt
=
∑j
mj(vi
−
vj)
·∇
iWij (3)
dv
i
dt
=
−
∑
jmj
[(
pi
ρ2
i
+pj
ρ2
j
)
−ξ
ρiρj
4ηiηj
ηi+ηj
(vi−vj)·rij
r2
ij +ζ2
]
∇iWij
+f
pi
+f
i
+f
gi
(4)
dvi
dt =−
∑
jmj
(
pi
ρ2
i
+pj
ρ2
j
)
∇iWij
+
∑
jmj
(
Si
ρ2
i
+Sj
ρ2
j
)
·∇
iWij
+f
pi
+f
i
+f
gi
(5)
Therein, ρi, mi, vi, pi, ηi, and Si respectively denote the
density, mass, velocity, pressure, dynamic viscosity, and
deviatoric stress tensor of the ith particle. In addition, rij rep-
resents the relative position of the ith particle with respect to
the jth particle. rij denotes |rij|. fpi is the acceleration vector
representing potential forces such as surface tension, fi is
the acceleration vector representing external forces, and fgi
is the gravitational acceleration vector. Also, ξ is the viscous
scaling factor. In addition, ζ is a small parameter used to
smooth out the singularity. Wij is the kernel function cen-
tered on the ith particle position. This study uses the quintic
function presented by Wendland [27] as the kernel function
to correct tensile instability.
The equation of state, dened as Eq. (6), is used to calcu-
late pressure from the density value [13].
p
=
100
ρ0v
2
max
γ
{(
ρ
ρ
0)γ
−1
}
(6)
Therein, ρ0 stands for the reference density. In addition,
v
max
represents the maximum ow velocity, which is regarded as
a constant for the simulation. The exponent
γ
= 7 is used.
The combination of Eqs. (3), (4), and (6) ensures that the
density variation is less than 1% and that the ow is nearly
incompressible [13]. The mass of each particle is calculated
by multiplying the density by the volume calculated from
the particle size set at the start of the simulation. It is main-
tained until the end of the simulation.
To prevent molten metal particles from penetrating
through the structure wall, the repulsive force per unit area
dened as Eq. (7) [13] is applied from the wall particles to
the molten metal particles. Equation (7) shows the repul-
sive force based on intermolecular forces, in Lennard–Jones
form.
f
i
=∑j
fij
f
ij =
5
v
2
max
2
{(
r0
rij
)n1
−
(
r0
rij
)n2}
rij
r2
ij
(7)
evaporation when melted by the laser, or recoil pressure
associated with evaporation.
2 Simulation technique and simulation
condition
2.1 Simulation technique
This paper presents a proposal of a method to analyze ow
and solidication behavior in the molten zone, elastic–plas-
tic behavior in the solidied zone, and powder behavior in
the metal powder zone. Furthermore, the method calculates
their mutual interactions. The proposed method also calcu-
lates heat conduction and heat transfer simultaneously, and
accommodates state determination and changes. An over-
view of the proposed method is presented in Figure 2(b).
Each is calculated as described below.
(1) Governing equations for molten and solidied metal,
discretization
As described herein, the SPH method [13] is used to sim-
ulate molten and solidied metal behaviors. In fact, SPH
method is fully Lagrangian, mesh-free, and particle-based.
Using this method, the governing equations are discretized
using a nite set of discrete values dened at interpola-
tion points. Each point, representing a particle, is assigned
physical properties such as density, velocity, or enthalpy.
By updating the property values of every particle using
discretized equations, one can calculate the overall uid or
elastoplastic behavior.
To model the molten and solid metal behavior, the fol-
lowing governing equations are used.
Dρ
Dt
=
−
ρ
∇·
v (1)
Dv
Dt
=
−1
ρ∇
p+
1
ρ∇·
τ+fp+f+f
g
(2)
In those equations, ρ signies density
v
, denotes velocity,
p represents pressure, τ is the deviatoric stress tensor, fp is
the acceleration vector representing potential forces such
as surface tension, f is the acceleration vector representing
external forces, and fg is the gravitational acceleration vec-
tor. Equations (1) and (2) respectively represent the continu-
ity equation and the momentum equation.
This study uses a combination of the discretization form
of the continuity equation (Eq. (3)) and the momentum
equation (Eq. (4) for a liquid state or Eq. (5) for a solid state)
[13, 26].
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For this study, a solidied metal is modeled as an elastic
perfectly plastic body by introducing the algorithm pro-
posed by Gray et al. [26]. The deviatoric stress tensor Si in
Eq. (5) is calculable by integrating the Jaumann stress rate
presented in Eq. (11).
˙
S
αβ =2µs
(
˙εαβ
−1
3
δαβ ˙εγγ
)
+Sαγ Ωβγ +Ωαγ Sγβ (11)
˙
ε
αβ =
1
2(
∂v
α
∂xβ+∂v
β
∂xα
)
(12)
Ω
αβ =
1
2(
∂v
α
∂x
β
−
∂v
β
∂x
α
)
(13)
In those equations, α, β, and γ respectively represent
indices of tensor and vector components,
˙εαβ
stands for
the strain rate tensor, Ωαβ denotes the rotation rate ten-
sor, and μs signies the shear modulus. Plastic deforma-
tion can be computed according to the von Mises yield
criterion.
(2) Governing equations for metal powder, discretization
For this study, the metal powder behavior is calculated using
DEM [14]. The governing equations for the metal powder
particles are presented in Eqs. (14) and (15).
dv
i
dt
=
∑j
fcij =
∑j(
fnij +fsij
)
(14)
where
fnij
=(
fcij
·
nij
)
nij ,fsij
=
fcij
−
fnij ,nij
=r
ij
rij
dω
i
dt
=
m
i
Ii
d
pi
2∑
j
(
nij
×
fsij
)
(15)
As shown there, fcij denotes the acceleration represent-
ing the contact force acting from particle j to i. More-
over, it is divisible into a component fnij in the direction
from particle j to i (unit vector nij) and a perpendicular
component fsij. In addition, ωi is the angular velocity
vector, Ii denotes the moment of inertia, and dpi repre-
sents the diameter of the particle when it is assumed to
be a sphere. For this study, fcij is defined by Eq. (16) as
presented below.
f
cij
=
fij
−c
mi
vij (16)
where
vij =vi
−
vj+nij
×(
dpi
2
ωi+dpj
2
ωj
)
In those equations, r0 represents the reference particle spac-
ing. For calculation of the repulsive force between the struc-
ture wall and the molten metal particles, the initial particle
spacing rini is set as r0. In addition, n1 and n2 are, respec-
tively, 4 and 2.
For this study, surface tension is calculated using the
surface tension model based on the interparticle potential
(expressed in Eq. (8)) developed by Kondo et al. [28].
f
pi =
∑
j
C
d
dr fp(rij )=
∑
j
C(rij −rini)(rij −re
)
(8)
where
fp(r)=
1
3(
r
−3
2
rini +
1
2
re
)
(r
−
re)2
Therein, C is the potential coecient, fp(r) represents the
potential, and re is the eective radius, which is usually set
as 3.2 times the value of rini. The potential coecient C
between uid particles is calculable using Eq. (9), which
shows the work necessary to detach particles existing in
area A from particles existing in area B to generate two sur-
faces of area rini
2, as portrayed in Fig. 4. By calculating this
coecient C in advance, it is possible to use Eq. (8) to cal-
culate the uid behavior corresponding to surface tension σf.
2
σfr
2
ini
=∑
i∈A,j∈B
Cfffp
(
rij
)
(9)
In addition, by calculating Eq. (8) between uid particle and
wall particle using the potential coecient Cfs calculated
using Eq. (10) derived from Young’s equation, the behavior
of a uid with a contact angle θ relative to a wall surface can
be calculated.
C
fs =
1
2
(1 + cos θ)C
ff
(10)
Fig. 4 Calculation of work needed to create two surfaces of area rini
2
[28]
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