![Melt pool widths from FEM and experimental re- sults [27] with a fixed laser power of 50 W and different scan speeds. Difference (Diff.)=FEM-Experiments (Exper.). Error=|FEM-Exper.|/FEM.](https://www.researchgate.net/profile/Dan-Wang-171/publication/343808256/figure/tbl1/AS:927143060058113@1598059539080/Melt-pool-widths-from-FEM-and-experimental-re-sults-27-with-a-fixed-laser-power-of-50_Q320.jpg)
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Closed-loop High-fidelity Simulation Integrating
Finite Element Modeling with Feedback Controls
in Additive Manufacturing
Dan Wang
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: daw1230@uw.edu
Xu Chen∗
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Email: chx@uw.edu
A high-precision additive manufacturing process, powder
bed fusion (PBF) has enabled unmatched agile manufactur-
ing of a wide range of products from engine components to
medical implants. While finite element modeling and closed-
loop control have been identified key for predicting and
engineering part qualities in PBF, existing results in each
realm are developed in opposite computational architectures
wildly different in time scale. This paper builds a first-
instance closed-loop simulation framework by integrating
high-fidelity finite element modeling with feedback controls
originally developed for general mechatronics systems. By
utilizing the output signals (e.g., melt pool width) retrieved
from the finite element model (FEM) to update directly the
control signals (e.g., laser power) sent to the model, the pro-
posed closed-loop framework enables testing the limits of ad-
vanced controls in PBF and surveying the parameter space
fully to generate more predictable part qualities. Along the
course of formulating the framework, we verify the FEM by
comparing its results with experimental and analytical so-
lutions and then use the FEM to understand the melt-pool
evolution induced by the in- and cross-layer thermomechan-
ical interactions. From there, we build a repetitive control
algorithm to attenuate variations of the melt pool width.
1 Introduction
Distinct from conventional subtractive machining, addi-
tive manufacturing (AM) builds a part directly from its dig-
ital model by joining materials layer by layer. In particular,
applying high-precision lasers or electron beams as the en-
ergy source, powder bed fusion (PBF) AM has enabled un-
precedented fabrication of complex parts from polymeric and
metallic powder materials. However, broader adoption of
the technology remains challenged by insufficient reliability
and in-process variations. These variations are induced by,
for example, uncertain laser-material interactions, environ-
mental vibrations, powder recycling, imperfect interactions
∗Corresponding author
of mechanical components, and complex thermal histories
of materials [1–3].
PBF builds a typical part from many thousands of thin
layers. Within each layer (Fig. 1), the energy beam is regu-
lated to follow trajectories predefined by the part geometry in
a slicing process. After the printing of one layer, a recoater
will spread a new thin layer of powder over the just-fused
layer, and then another cycle will begin. Current researches
employ finite element modeling and feedback controls to un-
derstand the energy-deposition mechanisms and to regulate
the in-process variations in PBF. For instance, [4–7] adopt
finite element modeling to investigate how various scan pat-
terns, scan speeds, number of lasers, overhanging structures,
and underlying physics affect the thermal fields of the pow-
der bed, the geometries of the melt pool, and the mechanical
properties of the printed parts. Specially, [8] brings up the
idea of building digital twins of AM machines by integrat-
ing various high-fidelity multiphysics models. Existing feed-
back control strategies usually implement low-order system
models obtained from system identification techniques, such
as [2,9] in PBF and [10–12] in laser metal deposition (LMD),
an AM technology analogous to PBF. A nonlinear memory-
less submodel [11, 13] and a spatial-domain Hammerstein
model [12] are further built in LMD to cover more compli-
cated process dynamics. From there, PID control, sliding
mode control, predictive control, and iterative learning con-
trol have proved their efficiencies in improving the dimen-
sional accuracy of the printed parts in PBF [2, 9, 14,15] and
in LMD [10, 13, 16–19].
Although finite element model (FEM) and feedback
control have been identified key for predicting and engi-
neering part qualities in PBF, existing results in each realm
are developed in separate computational architectures due
to their different time scales. Feedback controls are imple-
mented in real time, while FEM takes hours or even days to
simulate the printing of a few layers that finishes in seconds
in reality. If we can integrate FEM with feedback controls
directly in a closed loop, however, we can 1) combine afore-
1 Copyright c
by ASME
Vt
Vs
W
L
Laser
beam
Powder bed
Unfused
powder
Fused tracks
B
Scanning
direction
Traveling
direction
W
Energy
beam
x
1 2 345678 9 10
Track No.
Track direction
x
T
xodd track
even track
00
Heat
conduction
direction
n-1 n
Track No.
Scan
direction
Heat
conduction
direction
n-2
n-3
Unfused
powder
𝜉
𝑦
A
C
Figure 1: Schematic of in-layer PBF process.
mentioned knowledge from each realm, 2) test the limit of
advanced control in PBF, 3) survey the parameter space fully
to generate more predictable part qualities, and 4) quickly
design controllers and update parameters for novel materi-
als and printer settings. These benefits are more prominent
when the experiments are costly and time-consuming.
In pursuit of the above benefits, this paper builds, in
the first instance to our best knowledge, a closed-loop high-
fidelity simulation framework that leverages modern archi-
tectures of finite-element-modeling tools and the power of
data processing and advanced controls. In particular, we
build a bidirectional architecture so that the output signals
(e.g., melt pool width or peak temperature) retrieved from
the FEM can be utilized to directly update the FEM process
parameters (e.g., laser power or scan speed) in external con-
trol toolboxes (e.g., MATLAB). In this way, we design an
interface that enables simulating closed-loop controllers for
PBF thermal processes. Along the course of formulating the
framework, we validate the FEM by comparing its results
with experimental and analytical solutions and then use the
verified FEM to investigate the periodic in- and cross-layer
thermal interactions. From there, we justify the effectiveness
of the repetitive control (RC) in attenuating the periodic vari-
ations of the melt pool width.
The major contributions of this paper are:
◦developing a closed-loop simulation architecture for im-
plementing and testing out feedback control strategies in
FEM;
◦building an RC algorithm and an FEM of the PBF ther-
mal process, as applications of the proposed architec-
ture;
◦verifying the effectiveness of RC in regulating the peri-
odic thermal interactions in PBF.
The remainder of this paper is structured as follows. Section
2 frames the main closed-loop high-fidelity simulation with
the bidirectional architecture between FEM and controller.
As application examples, Section 3 develops an FEM of the
thermal fields in PBF, and Section 4 builds an RC design to
regulate the periodic thermal cycles. Under the architecture
of the closed-loop simulation, Section 5 evaluates the perfor-
mance of RC. Section 6 concludes the paper.
2 Architecture of Proposed Closed-Loop Simulation
with Bidirectional Communication Between FEM
and Feedback Controls
We propose in this section the main closed-loop simula-
tion that integrates FEM with feedback controls directly in a
closed loop. The key idea is to use the output signals (e.g.,
melt pool width) retrieved from the FEM to update through
the controller the control signals (e.g., laser power) sent back
to the FEM. The core of the FEM is the numerical simulation
of a partial differential equation (PDE) of the general form:
f(x1,...,xn;u,∂u
∂x1
,..., ∂u
∂xn
;∂2u
∂x1∂x1
,..., ∂2u
∂x1∂xn
;...)=0,
where uis the output signal predicted by the FEM and
x1,...,xnare the independent variables. Examples of PDEs
in PBF include the governing equations for heat transfer,
thermal stress, and fluid flow; the output signals are thereby
the temperature, stress, and melt pool velocity, respectively
[20, 21]. On the other hand, in feedback control, an output
signal is compared with a reference signal to generate an er-
ror signal, which is then filtered by a controller to produce
a control signal. By reacting to the in-process change of the
output signal, feedback control compensates for model inac-
curacies and unmeasured disturbances.
Leveraging the power of FEM and feedback controls,
we develop the closed-loop simulation algorithm and outline
its workflow as follows:
1. Designing and initializing the FEM in an FEM soft-
ware (e.g., COMSOL) and the controller in a control-
oriented scripting programming environment (abbrevi-
ated as programming environment, e.g., MATLAB). For
the focused closed-loop FEM in this paper, we set the
computation time of the FEM as ts, where tsis the sam-
pling time of the discrete-time feedback loop;
2. Building an interface to connect the FEM software with
the programming environment. This interface enables
simulating closed-loop controllers for FEM and taking
advantage of the programming environment in prepro-
cessing, model manipulation, statistical analysis, and
post-processing;
3. Through the interface, the programming environment
retrieves the output signal from the FEM software.
Thereafter, the controller processes the output signal to
get the new control signal, which is then sent back to
the FEM. With the updated control signal, a new FEM
computation begins;
4. Repeating step 3 until the whole simulation ends.
With the bidirectional architecture between the FEM and
controller, the closed-loop simulation enables updating di-
rectly the control signals of the FEM, retrieving signals hard
to reach by experiments, and surveying the parameter space
fully before real PBF experiments. Moreover, the proposed
architecture is agnostic to the types of programming envi-
ronment and FEM software, facilitating incorporating vari-
ous control methods into different multiphysics simulations
2 Copyright c
by ASME
C(z)FEM or P(z)
d(k)
+
r(k)+e(k)u(k)
+
y(k)
−
Figure 2: Block diagram of general feedback control.
(e.g., heat transfer and fluid flow studies). As application ex-
amples, we choose COMSOL Multiphysics, MATLAB, and
LiveLink for MATLAB to build the FEM, controller, and in-
terface, respectively. We provide in this section the skeletons
of an FEM of the thermal fields and a feedback control de-
sign. We will build the detailed FEM and controller in Sec-
tions 3 and 4, respectively.
1) The governing equation for the conduction heat flow
in PBF is
ρcp
dT (x,y,z,t)
dt =∇ · (k∇T(x,y,z,t)) +qs,(1)
where kis the thermal conductivity, cpthe specific heat ca-
pacity, ρthe effective density, tthe time, Tthe temperature,
and qsthe rate of local internal energy generated per unit vol-
ume [22]. When no confusion would arise in the context, we
abbreviate T(x,y,z,t) to Tin the rest of this paper.
The initial condition is defined by setting T(x,y,z,0) as
the ambient temperature T0. When the substrate (Fig. 6) is
designed to be large enough compared to the heat-affected
zone, one boundary condition is established by assuming
the bottom (z=h) of the substrate has no heat loss [4–6]:
−k∂T
∂z
z=h=0. A quantitative way to test and validate the
size of the substrate is that from the FEM results, the bottom
of the substrate should maintain the temperature of T0. The
other boundary condition is established by considering the
top surface (z=0) of the powder bed with input heat flux,
convection heat loss, and radiation heat loss:
−k∂T
∂z
z=0
=−Q+hc(T−T0)+εσB(T4−T4
0),(2)
where Qis the input heat flux, hcthe convection heat transfer
coefficient, εthe emissivity, and σBthe Stefan-Boltzmann
constant. Here, we assume Qhas a Gaussian laser beam pro-
file: Q≈2q
πR2e−2r2
R2, where qis the laser power, Rthe effective
laser beam radius, and rthe radial distance from a certain
point to the center of the laser spot. From the temperature
distribution predicted by the FEM, melt pool width can be
further calculated, as will be elaborated in Section 3.1. From
here on, we select melt pool width was the output signal and
laser power qas the control signal for the subsequent feed-
back control design.
2) A generic feedback system consists of a plant P(z)
and a controller C(z) (Fig. 2). In this study, P(z) is the
plant model identified from the FEM simulation. The signal
r(k) represents the system reference, which here is a desired
Initialization
&*' -, ) ' )*
!*"# $# %# &*+ ' !*
! "# $ # %# &*' ! "# $# % # &+
&*' &+
#(
) ' )*&++End if &+' &,-.
FEM calculation for
&*.&+' &*/ &/
Calculate melt pool
width ,*&++from
!*"# $# %# &++
Apply control
algorithms
(Baseline or RC)
Compare ,
with ,.
Get control signal
)*&++(laser power)
COMSOL MATLAB
Retrieving
!*"# $# %# &++
Figure 3: Schematic of proposed closed-loop simulation.
Algorithm Closed-loop simulation
1
Import classes
2
Open the FEM file
3
Initialize parameters of FEM and Controller
!!" #, !"" !!$ !#, %&!!'" %!, ( " (!
4
While
!") !$%&
:
1
Call FEM software to compute the FEM
2
Get the temperature distribution %*!"+
3
Calculate melt pool width ,*!"+ from %*!"+
4
Apply controller to update laser power -*!"+
5
Update iterative variables in the FEM
!!. !", %&!!'. %*!"+, ( . (*!"+
• The computation time of FEM is 𝑡!, where 𝑡! is the
sampling time in discrete-time feedback control.
Figure 4: Pseudocode of closed-loop simulation algorithm.
melt pool width; u(k) represents the control signal, which
here is the laser power; y(k) represents the system output,
which here is the melt pool width as calculated from FEM-
predicted temperature fields; d(k) represents the input distur-
bance, which here is the in-process melt-pool-width varia-
tions.
Fig. 3 demonstrates the operating procedures of the pro-
posed closed-loop simulation architecture, taking as exam-
ples the FEM of the PBF thermal process and general feed-
back control. We design and initialize the FEM in COM-
SOL while programming in MATLAB the main file of the ar-
chitecture that includes the interface establishment, the con-
troller design, and the recursive calls for COMSOL. First,
we initialize the FEM by setting the start time t0as 0, the
laser power qas an initial value q0, and the temperature dis-
tribution T(x,y,z,t0) as the ambient temperature T0. The
computation time of the FEM is set as one time step from
t0to tf=t0+ts. After the initialization, MATLAB will call
COMSOL to start the FEM computation. After retrieving
the FEM-predicted temperature distribution T(x,y,z,tf), the
main file in MATLAB calculates the melt pool width w(tf)
from T(x,y,z,tf) and, based on the control algorithms, pro-
cesses w(tf) and its historical values to obtain the control
signal q(tf). After that, the FEM is updated by assigning the
iterative variables t0as tf,T(x,y,z,t0) as T(x,y,z,tf), and
the laser power as q(tf). After passing all closed-loop com-
puted information, MATLAB will call COMSOL to start the
FEM computation with the updated variables, and a new cy-
cle will begin. The closed-loop simulation will stop when tf
reaches to the whole simulation time tend (tfin general).
We provide in Fig. 4 the pseudocode of the main file
in the closed-loop simulation algorithm. After importing the
necessary classes for building the FEM, controller, and in-
3 Copyright c
by ASME
terface, the algorithm opens the FEM file and initializes the
parameters of the FEM and controller. Inside the while loop,
the main file calls the FEM software to compute the FEM
simulation and then retrieves the predicted temperature dis-
tribution T(x,y,z,tf). Based on T(x,y,z,tf), the algorithm
calculates the melt pool width w(tf) and thereafter processes
w(tf) and its historical values through the controller to get
the new laser power. At last, the algorithm updates the iter-
ative variables of the start time, the initial temperature, and
the laser power to get prepared for a new iteration. The Ap-
pendix provides the main MATLAB commands used in the
closed-loop simulation.
The proposed closed-loop simulation establishes not
only a bidirectional communication between FEM software
and programming environment, but also an interface specif-
ically for the purpose of simulating feedback controllers in
FEM. The built architecture will benefit and guide experi-
ments by validating beforehand the effectiveness of the servo
designs. Next we will present the detailed FEM and con-
troller designs in Sections 3 and 4, respectively.
3 FEM of Thermal Fields in PBF
In this section, we use the COMSOL Multiphysics 5.3a
software to build and refine the FEM of the temperature re-
sponse in PBF, which serves as an essential component of
the proposed closed-loop simulation. We verify the FEM
configuration by comparing the numerical results with the
experimental and analytical solutions. The FEM developed
in this section considers conduction, latent heat of fusion,
surface convection, and surface radiation. The principal ob-
jective of the paper is not to create a new high-fidelity FEM
at microscopic fluid flow level but to develop and test the
closed-loop simulation architecture. Therefore, we omit the
effects of evaporation, fluid flow, and Marangoni force for
easier testing in the general control community. The FEM
built in this section is intended as one application example
of the proposed architecture between FEM and controller.
However, the physics of complex melt flow (e.g., Marangoni
and surface tension effects) can be readily added to increase
the model accuracy and provide more microscopic details of
the melt pool that are beyond the focus of this paper.
3.1 FEM
We have elaborated the governing equation, initial con-
dition, and boundary conditions in Section 2. For the
temperature-dependent thermal properties in (1), we adopt
k,cp, and ρin [4, 23] for the solid and liquid materials, as
shown in Fig. 5. We generate the thermal properties of the
powder material from those of the solid material by consid-
ering the powder-bed porosity φ[24, 25]:
kpowder =ksolid (1 −φ)4and ρpowder =ρsolid (1 −φ),
Table 1: Parameters of the FEM. Note that the laser spot
diameter of 70µm is for model verification in Section 3.2.
Parameters Value
Size of powder bed 5mm ×10mm ×50 µm
Size of substrate 5mm ×10mm ×2 mm
Powder/substrate material Ti6Al4V
Laser power q60 W or varying [27]
Scan speed ux100mm/s [27]
Track length L5mm
Laser spot diameter 2R220µm (or 70 µm [27])
Ambient/initial temp. 293.15K
Initial porosity φ00.4 [4, 23]
Emissivity 0.35 [4]
Powder bed absorptance 0.25 [4]
Time step ts0.5ms
Melting point Tm1923.15K [28]
Solidus temperature Tsol 1873K [28]
Latent heat of fusion Lf295 kJ/kg [28]
Convection h.t. coeff.hc12.7 W/(m2·K) [4]
k,cp, and ρSee Fig. 5 [4, 23–25]
where φis expressed as
φ(T)=
φ0T0<T≤Tsol
φ0
Tsol−Tm(T−Tm)Tsol <T<Tm
0T≥Tm
,
with φ0denoting the initial porosity. Here, the heat capacity
is assumed to be the same for the powder and solid materials
[24]. Especially, we account for the latent heat of fusion
Lfby introducing the effective heat capacity to the powder
material [26]:
cp,e f f (T)=
cp1(T)T0<T≤Tsol
Lf
Tm−Tsol +cp1(Tsol)+cp2(Tm)
2Tsol <T<Tm
cp2(T)T≥Tm
,(3)
where T0is the ambient temperature, Tsol the solidus tem-
perature, Tmthe melting point, cp1the heat capacity of the
powder, and cp2the heat capacity of the liquid. Table 1 lists
the process parameters used in this study unless otherwise
specified. To justify the parameter values, we introduce here
the volumetric energy density Ev, which is a key factor in the
PBF process and directly impacts on the properties of as-built
parts. Evis defined as: Ev=P/(vth), where Pis the laser
power, vthe laser scan speed, tthe layer thickness, and hthe
hatch spacing [29]. During the in-layer multitrack printing
(Sections 4.2 and 5), P=60 W, v=100 mm/s, t=50µm, and
h=60µm, which gives Ev=200 J/mm3. This volumetric en-
ergy density is, for example, in the ranges of 71 ∼373J/mm3
as studied in [30] and 15 ∼240J/mm3as in [31].
Fig. 6a shows the bidirectional scan scheme used in this
study and the whole build with a substrate and a thin layer
4 Copyright c
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500 1000 1500 2000 2500
Temperature (K)
0
10
20
30
40
50
Thermal conductivity (W/m .K)
500 1000 1500 2000 2500
Temperature (K)
2000
2500
3000
3500
4000
4500
Density (kg/m3)
500 1000 1500 2000 2500
Temperature (K)
0
1
2
3
4
5
6
7
Heat Capacity (J/g .K)
Figure 5: Temperature-dependent thermal properties of Ti6Al4V [4, 23–25]. Solid line: solid and liquid materials. Dash-
dotted line: powder material. The two vertical dotted lines respectively indicate Tsol and Tm.
X
Z
Y
Powder bed: F ree triangular
and Swept (2mm)
Substrate: Free
tetrahedral (3.5 mm)
10 mm
5mm
2mm
50 µm
…
Laser tracks
Powder bed: F ree quad
and Swept (60 µm)
(K)
100 µm
2𝑐!
2𝑐"
(𝑥#$%&' , 𝑦#$ %&' )
𝑥#𝑥'
(a) (b)
Figure 6: (a): a thin layer of powder bed and the substrate
with selective meshing scheme. (b): surface temperature dis-
tribution at t=0.14s. The lined isotherm indicates T=Tm.
of powder bed. Here, we use a selective meshing scheme to
balance model accuracy with computation time: a fine quad-
and-swept mesh with a maximum element size of 60µm is
applied to the central powder bed region that directly inter-
acts with the energy beam, whereas less fine tetrahedral mesh
(3.5mm) and triangular-and-swept mesh (2 mm) are applied
to the substrate and the peripheral powder bed, respectively.
Fig. 6b showcases a surface temperature distribution, where
the isotherm of T=Tmindicates the melt pool geometry.
From the FEM-predicted temperature distribution, Fig.
7 shows the pseudocode of how to calculate the melt pool
width. The basic principle is to search around the position of
the laser beam to find the maximum width of the melt pool
bounded by Tm. The first step is to locate the position of
the laser beam (xla ser,yla ser ), as shown in Fig. 6b. Then we
identify the points to traverse by testing and ensuring that
the rectangle 2cx×2cybe inclusive of anticipated melt pool
widths (Fig. 6b). A set Xstores the xcoordinates of these
points with an increment ∆x, and a set Ystores the ycoordi-
nates with ∆y. For each yin Y, we search in Xto find the left
boundary xland right boundary xrthat have the temperature
of Tm. A width wyfor a certain yis given by xr−xl. After
going over all the points, we identify the melt pool width as
the maximum wy. In this study, we choose ∆x= ∆y=1µm,
cx=150µm, and cy=100 µm.
3.2 Model Verification
This section verifies the developed FEM by comparing
the numerical melt pool widths with the experimental and
Algorithm Calculation of Melt Pool Width
1
Locate laser beam position: 𝑥!"#$% and 𝑦!"#$%
2
Points to traverse:
𝑋 =
{
𝑥!"#$% − 𝑐&∶ ∆𝑥 ∶* 𝑥!"#$% + 𝑐&
}
𝑌 =
.
𝑦!"#$% − 𝑐'∶ ∆𝑦 ∶* 𝑦!"#$% + 𝑐'
/
3
for
𝑦
in
𝑌
:
1
for 𝑥 in 𝑋:
1
Locate 𝑇(𝑥!, 𝑦)= 𝑇( and 𝑇(𝑥%, 𝑦)= 𝑇(
2
Width defined by 𝑇( for a certain 𝑦:
𝑤'= 𝑥%− 𝑥!
4
Melt pool width 𝑤 = max9𝑤':
• In this case study, ∆𝑥 = ∆𝑦 = 1&µm, 𝑐!=150&µm, and 𝑐"=100 µm
Figure 7: Pseudocode for calculating melt pool width.
Table 2: Melt pool widths from FEM and experimental re-
sults [27] with a fixed laser power of 50 W and different
scan speeds. Difference (Diff.)=FEM-Experiments (Exper.).
Error=|FEM-Exper.|/FEM.
Scan speed FEM(µm) Exper. (µm) Diff. (µm) Error (%)
100 mm/s 182 165.7~175.4 6.6~16.3 3.6~9.0
200 mm/s 152.6 140.7~142.9 9.7~11.9 6.4~7.8
300 mm/s 132.6 120.7~125.4 7.2~11.9 5.4~9.0
analytical solutions. Table 2 first compares the melt pool
widths obtained from the developed FEM with the experi-
mental results in [27]. The laser power is fixed to be 50W,
the laser spot diameter is 70 µm, and the scan speed is 100,
200, or 300 mm/s. The FEM gives reasonable predictions of
the melt pool widths with errors less than 10%.
Then we compare the numerical melt pool widths with
the analytical solutions. When a moving point laser source is
acting on a thick plate and the thermal properties of the plate
are constant, the analytical solution of (1) in the steady state
is the Rosenthal equation [22]:
T(ξ, y,z)−T0=q
2πkr e−ux(r+ξ)
2κ,(4)
where (ξ, y,z) is a coordinate system attached to the moving
laser source, r=pξ2+y2+z2, and κ=k/(ρcp). For compar-
ison, we adapt the FEM to accommodate the assumptions of
the Rosenthal equation, such as constant thermal properties
(k=5 W/(m·K), cp=1.1 J/(g·K), and ρ=4300 kg/m3) and
5 Copyright c
by ASME
200 um
Rosenth al solut ion Z
Y
Sample 1
𝑡 = 𝑇
!
Sample 2
𝑡 = 2𝑇
!
Δ𝑡 = 𝑇
!
Sample 3
𝑡 = 3𝑇
!
X
Y
(a)
(c)
(b)
Figure 8: Melt pool widths from the FEM and analytical so-
lutions. (b) and (c) share the same scale and legend.
point heat source. Fig. 8 compares the numerical and analyt-
ical solutions. From Fig. 8a, we can tell that the melt pool
widths obtained from the FEM and the Rosenthal equation
match well with each other under different combinations of
scan speeds and laser powers. Also, as shown in Figs. 8b
and 8c, after 27 samples, the numerical melt pool geometry
reaches to the steady state and matches with the Rosenthal
solution (the outline).
4 Repetitive Controller Design
This section looks into another fundamental component
of the closed-loop simulation, namely, the controller design.
We first adopt the FEM in Section 3 to investigate the pe-
riodic PBF thermal cycles and then design a repetitive con-
troller to regulate these cycles.
4.1 Periodic Thermal Interactions: Hatch Spacing
We first examine how hatch spacing affects the melt pool
variation, especially during the transition from the end of one
track (named P1) to the start of the adjacent track to be sin-
tered (P2). Here, hatch spacing is defined as the distance
between two adjacent scan vectors and denoted as ∆xin Fig.
1. When the hatch spacing (e.g., 47 µm in (a1)-(a4) of Fig.
9) is much less than half of the melt pool width (around the
laser spot radius 110µm), the laser spot at P2 will be centered
inside the melt pool region of P1 and thus can take advan-
tage of the accumulated heat, yielding a well-developed melt
pool at 50.5ms. When the hatch spacing (e.g., 100 µm in
(b1)-(b4) of Fig. 9) is close to or larger than half of the melt
pool width, the laser spot at P2 will be centered out of the
melt pool region of P1, yielding a lower initial temperature
at P2. Besides, since the laser is turned offduring the tran-
sition, a larger hatch spacing (100µm) gives a longer dwell
time (1ms) with the same scan speed (100 mm/s). Hence,
previously fused tracks would have more time to cool down,
which also yields a lower initial temperature at P2. There-
fore, the melt pool evolves slower at P2, and two imma-
50.5 ms 51 ms 51.5 ms 52 ms
(a1) (a2) (a3) (a4)
550.5 ms 551 ms 551.5 ms 552 ms
(b1) (b2) (b3) (b4)
(K)
Figure 9: Melt pool variations at the start of the 2nd track
with hatch spacing of 47µm (i.e., dwell time of 0.47 ms) in
(a1)-(a4) and the start of the 12th track with hatch spacing of
100µm (i.e., dwell time of 1 ms) in (b1)-(b4).
0 100 200 300 400 500 600 700 800 900 1000
Samples
2.3
2.4
2.5
2.6
2.7
2.8
Melt pool width (m)
10-4
Track
No. 1 2 3 4 5 6 7 8 9 10
(a) Evolution of melt pool width (time-domain)
0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)
0
1
2
3
Magnitude (dB)
10-6
(b) FFT of melt-pool-width evolution (frequency-domain)
Figure 10: In-layer thermal disturbance with constant laser
power.
ture states show up at 550.5ms and 551 ms, which will cause
more porosity in the printed part.
To get consistent part quality in PBF, a stable melt pool
is desired during the transition. The immature melt pool
states can be eliminated by decreasing the hatch spacing.
However, there is a trade-offbetween melt pool stability and
printing efficiency since simply shortening the hatch spac-
ing increases printing time. Next we will study periodic melt
pool variations that are intrinsic in the PBF process and de-
mand much more involved solutions, such as advanced con-
trol algorithms.
6 Copyright c
by ASME
Preheated part
(K)
(b)(a)
13.5 ms 25 ms 50 ms
(c1)(c2) (c3)
Figure 11: (a): mesh with added preheated part. (The thin
layer of powder bed is hidden to unveil the added part.) (b):
surface temperature distribution at t=49ms. (c1)-(c3): sur-
face temperature distributions (top view) during the printing
of the first track.
4.2 Periodic Thermal Interactions: In-layer Effects
To investigate the in-layer thermal cycles, we bidirec-
tionally print 10 tracks in the first layer with a hatch spacing
of 60µm (Fig. 6a). Fig. 6b illustrates the simulated surface
temperature profile at 0.14s. From Fig. 10a, we observe that
the melt pool width changes over time and structurally de-
viates from the steady-state value 246µm as extracted from
the first track. Most importantly, the start of each track has
larger melt pool widths than the rest of the track. This is
because in bidirectional scanning, when the energy beam ap-
proaches the end of one track, the large latent heat does not
have enough time to dissipate out before the next track starts.
The resulting increased melt pool widths at the beginning of
each track form a periodic disturbance with a repetitive spec-
trum in the frequency domain (Fig. 10b). The fundamental
frequency f0of the disturbance is determined by the duration
of scanning one track t0, that is, f0=1/t0=ux/L, where ux
is the scan speed and Lis the track length. In this example,
f0=100/5=20Hz, and frequency spikes at n f0(n∈Z+, the
set of positive integers) appear in the fast Fourier transform
(FFT) of the disturbance.
The disturbance periodicity is closely related to the re-
curring laser scanning trajectories and the repetitive in-layer
thermomechanical interactions. Besides the bidirectional
scan, other scan patterns yield similar repetitive disturbances
(see, e.g., experimental results in [32]). To deal with these
undesired repetitive spectra, we develop the closed-loop sim-
ulation in Section 2 to bring automatic control algorithms
[1, 33] into FEM. More results and analyses will be elabo-
rated in Section 5.
4.3 Periodic Thermal Interactions: Combined In- and
Cross-layer Effect
This section demonstrates the combined effect of pe-
riodic in- and cross-layer thermal interactions. As shown
in Fig. 11, we put under the powder bed a Ti6Al4V part
(4.45 ×1×1mm3) that is preheated to 1200 K [34]. Due to
the high initial temperature of the added part, the powder
on top of the part has a higher initial temperature than the
powder elsewhere. The scan strategy is the same as that in
Fig. 6a. Eight tracks are scanned bidirectionally with the
hatch spacing of 50µm. The length of the laser track (5 mm)
is greater than that of the added part (1mm). This configu-
ration imitates the printing process of parts with overhang
structures, where the preheated Ti6Al4V part corresponds
to the previously fused layers. As in Fig. 6, we also use
the selective mesh scheme (Fig. 11a): triangular-and-swept
(72.6µm) for the central powder bed, triangular-and-swept
(1.5mm) for the peripheral powder bed, and free tetrahedra
(2mm) for the substrate and the added part.
In Fig. 11, plots (c1)-(c3) illustrate the top views of
the surface temperature profiles during the first-track print-
ing from 0 to 50ms. When the laser is passing over the
preheated powder at 25ms, we get a larger melt pool width,
compared to when the laser is approaching (t=13.5ms) or
leaving (t=50ms) the preheated region. The larger melt pool
width is generated due to the higher initial temperature of
powder on top of the preheated part.
During the evolution of the melt pool width in Fig. 12a,
at the beginning of each track, there is a large increase of
the melt pool width caused by the in-layer thermal interac-
tion, as explained in Section 4.2. Besides, as indicated by
the arrows in Fig. 12a, larger melt pool widths appear every
time the laser passes over the powder on top of the preheated
part. These arrowed peaks caused by the cross-layer thermal
interaction get smaller as the heat accumulated by the pre-
heated part dissipates out (Tracks 7 and 8 in Fig. 12a). This
phenomenon can also be seen from the blurrier border of the
preheated region at t=50ms in Fig. 11 (c3).
We have demonstrated that the periodic evolution of the
melt pool width is a lumped output of the repetitive in- and
cross-layer heat transfer dynamics. When comparing the fre-
quency spectra in Figs. 10b and 12b, we can tell that the
cross-layer thermal interaction changes the magnitudes of
the spectral peaks but not the harmonic frequency values.
These variations can thus be attenuated by the same feed-
back control algorithms, such as the RC algorithm to be in-
troduced next.
4.4 RC
RC is designed for tracking/rejecting periodic exoge-
nous references/disturbances in applications with repetitive
tasks [35]. By learning from previous iterations, RC can
greatly enhance current control performance in the struc-
tured task space. Digital RC incorporates an internal model
1/(1 −z−N) in the controller, where zis the complex inde-
terminate in the z-transform. N=fs/f0is the period of the
signal, where fs=1/tsis the sampling frequency and f0is
7 Copyright c
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0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time (s)
2.1
2.2
2.3
2.4
2.5
2.6
Melt pool width (m)
10-4
Track 7
Track 6
Track 5
Track 4
Track 3
Track 2
Track 1
Track 8
(a)
(a) Evolution of melt pool width (time-domain). An interval between two
adjacent dashed lines indicates the printing of one track, whereas inside one
track, the interval between two adjacent dotted lines denotes when the laser is
passing over the preheated part.
0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)
0
1
2
3
Magnitude (dB)
10-6
(b)
(b) FFT of melt-pool-width evolution (frequency-domain)
Figure 12: Combined in- and cross-layer thermal distur-
bance.
C(z)FEM or P(z)
Q(z)
z−mˆ
P−1(z)z−m
d(k)
+
r(k)+e(k)
+u(k)+
y(k)
−
+
ωc(k)
+
+
Plug-in compensator
Figure 13: Block diagram of a plug-in RC design.
the fundamental disturbance frequency. Consider a baseline
feedback system consisting of P(z) and C(z) (Fig. 2). Here,
C(z) can be designed by conventional servo algorithms, such
as PID, H∞, and lead-lag compensation. The sensitivity
function S(z)=1
1+P(z)C(z)is the transfer function from d(k)
to y(k).
We introduce here a plug-in RC design [33] that uses
the internal signals e(k) and u(k) to generate a compensation
signal ωc(k) (Fig. 13). Let mdenote the relative degree of
ˆ
P(z), where ˆ
P(z) is the nominal model of P(z). The transfer
function of the overall controller from e(k) to u(k) is
Call(z)=C(z)+z−mˆ
P−1(z)Q(z)
1−z−mQ(z).(5)
10010110 2103
-60
-40
-20
0
Magnitude (dB)
1-z -mQ(z)
10010110 2103
Frequency (Hz)
-100
-80
-60
-40
-20
0
Magnitude (dB)
Q(z)
=0.999
=0.99
=0.9
Figure 14: Magnitude responses of 1 −z−mQ(z) and Q(z) with
different values of α(and n0=1) in an example of Section 5.
The internal model is integrated in Call by designing
the Qfilter as Q(z)=(1 −αN)zm−N/(1 −αNz−N), which
gives 1 −z−mQ(z)=(1 −z−N)/(1 −αNz−N). Here, α∈[0,1)
is a tuning factor that determines the attenuation width of
1−z−mQ(z). At the harmonic frequencies ωk=k2πf0ts
(k∈Z+), with z=ejωk, we have 1−z−N=0, 1 −z−mQ(z)=0,
Call(z)→ ∞ from (5), and hence the new sensitivity func-
tion S0(z)=1
1+P(z)Call(z)=0. At the intermediate frequencies
ω,k2πf0ts, with z=ejωand αbeing close to 1, Q(z)≈0,
1−z−mQ(z)≈1, Call(z)≈C(z) from (5), and thereby the orig-
inal loop shape is maintained. A smaller αcan yield a wider
attenuation width at the cost of deviating from the baseline
loop shape, as shown in Fig. 14.
During implementation, a zero-phase low-pass filter
q0(z−1)q0(z) is attached to Q(z) for robustness against high-
frequency plant uncertainties:
Q(z)=(1 −αN)zm−N
1−αNz−Nq0(z−1)q0(z),(6)
where q0(z)=(1+z)n0/2n0and n0∈Z+(Fig. 15). The closed-
loop performance S0(z) can be tuned by choosing different α
and n0[33]. The plug-in RC and the baseline control can
be merged into the closed-loop simulation by setting u(k) as
q(tf) and y(k) as w(tf) (see Fig. 3).
5 Results and Analyses
Under the infrastructure of the closed-loop simulation,
we evaluate the performance of RC in attenuating the peri-
odic in-layer variations of the melt pool width.
First, we identify the plant model of the FEM in Sec-
tion 3 from the laser power to the melt pool width as P(s)=
0.001671/(s+1055). The input signals used for system iden-
tification include a pseudorandom binary sequence (PRBS)
signal and multiple sinusoidal signals (10~300 Hz), with
magnitudes of 20 W and add-on DC components of 60 W.
The frequency responses of the measured and identified sys-
tems match well with each other, as shown in Fig. 16.
8 Copyright c
by ASME
10010110 2103
-60
-40
-20
0
Magnitude (dB)
1-z -mQ(z)
10010110 2103
-100
-80
-60
-40
-20
0
Magnitude (dB)
Q(z)
n0=0
n0=1
n0=2
Figure 15: Magnitude responses of 1 −z−mQ(z) and Q(z) with
different n0(and α=0.99) in an example of Section 5.
10110210 3104
-160
-150
-140
-130
-120
Magnitude (dB)
10110210 3104
Frequency (Hz)
-200
-100
0
100
200
Phase (degree)
Measured system using sinusoidal and PRBS signals
identified system P = 0.001671/(s+1055)
Figure 16: Measured and identified system responses.
10010110 2103
Frequency (Hz)
-60
-40
-20
0
Magnitude (dB)
Baseline PI control
Repetitive control
Figure 17: Magnitude responses of sensitivity functions S(z)
in baseline control and S0(z) in RC.
After that, we design a PI controller as C(s)=Kp+Ki/s
with Kp=9.38×105and Ki=1.66×109. Under the sampling
time tsof 0.5ms (i.e., fs=2kHz), the zero-order-hold equiv-
alents of the plant and controller models respectively are
P(z)=6.493 ×10−7/(z−0.5901) and C(z)=(9.38z−1.08) ×
105/(z−1). The dashed line in Fig. 17 shows the magnitude
response of the sensitivity function S(z) in the baseline feed-
back loop that comprises P(z) and C(z). Such a design pro-
vides a bandwidth at 197Hz, which approximates the limit
of 20% of the Nyquist frequency (1000Hz) and indicates that
the PI controller is well tuned. The closed-loop simulations
are designed according to Section 3 integrating FEM with
baseline control and RC, respectively. In this disturbance-
rejection example, r(k)=0 in Fig. 13, and d(k) comes from
(a) Evolution of melt pool width (time-domain)
0 100 200 300 400 500 600 700 800 900 1000
Frequency (Hz)
0
1
2
3
Magnitude (dB)
10-6
Constant laser power (w/o control)
Baseline PI control
Repetitive control w/ q 0(z-1)q0(z)
(b) FFT of melt-pool-width evolution (frequency-domain)
(c) Laser power (control signals u(k) in Fig. 13)
Figure 18: In-layer thermal disturbance. The three plots
share the same legend. The 3σ-values of the melt pool
width respectively are 15.219µm for constant laser power,
11.937µm (21.6% decrease) for baseline PI control, and
9.744µm (35.97% decrease) for RC. σdenotes the standard
deviation. The earlier results in Fig. 10 are superimposed in
Figs. 18a and 18b for clarity.
the in-layer melt-pool-width variations (Section 4.2). From
Fig. 18b, we can tell that the baseline PI control can atten-
uate the frequency spikes below the closed-loop bandwidth
but not the other high-frequency spikes. Compared to the
case without control, the baseline feedback loop decreases
the 3σvalue of the melt-pool-width changes (y(k) in Fig. 13
with mean removed) by 21.57%, where σdenotes the stan-
dard deviation.
To enhance the disturbance-attenuation performance,
we bring the plug-in RC compensator in Section 4.4 into
the closed-loop high-fidelity simulation. In the Q-filter de-
sign in (6), the relative degree mof ˆ
P(z) is 1 with ˆ
P(z) being
equal to P(z) in this simulation example; the disturbance pe-
riod N=fs/f0=2000/20 =100; we choose α=0.99. Also,
we attach a low-pass filter with n0=1 to Q(z) as in (6). As
shown in the solid line of Fig. 17, the plug-in RC with the Q-
filter generates high-gain control efforts exactly at 20Hz and
its harmonics. Fig. 18c illustrates the control signals u(k) of
9 Copyright c
by ASME
the baseline control, the RC, and the case without control.
As shown in Fig. 18b, compared with the baseline control,
RC further lowers the periodic frequency spikes, especially
at high frequencies beyond the closed-loop bandwidth, and
decreases the 3σvalue by 35.97%. Similarly, in the time
domain, the increased control efforts of RC at the harmonic
frequencies yield a further-attenuated output y(k), as shown
in Fig. 18a.
Remark 1: Practical feedback design has an effective
servo bandwidth, above which the control efforts are required
to be small for robustness. The baseline PI control and the
Q-filter with q0(z−1)q0(z) are designed to account for this
constraint. As shown in Fig. 17, above approximately 650
Hz, the magnitudes of the sensitivity functions of baseline PI
control and RC are close to zero, leaving the corresponding
frequency components unchanged.
Remark 2: To compute the closed-loop simulation with
the FEM in Section 3, it takes approximately 19 seconds for
one time step of 0.5 ms, 32 minutes for a track of length 5
mm, and 4.5 hours for ten tracks of length 5 mm using two
Intel Xeon Gold 6146 CPUs at 3.20 GHz with 24 cores in
total.
6 Conclusions and Future Work
In this paper, we develop a first-instance closed-loop
high-fidelity simulation architecture by integrating finite el-
ement model (FEM) with feedback controls to reduce the
in-process variations and advance the part quality in pow-
der bed fusion (PBF) additive manufacturing. We build an
FEM to simulate the temperature response in PBF and then
validate the FEM by comparing its results with the exper-
imental and analytical solutions. Employing the FEM, we
justify the existence of the periodic disturbances in the evo-
lution of the melt pool width. From there, implementing the
proposed closed-loop simulation, we validate that the repet-
itive control algorithm attenuates the periodic disturbances
more substantially by 35.97% than the PI control.
One avenue for future work is to expand the proposed
closed-loop simulation to be used in multi-layer printing
when macroscopic features begin to form. The proposed ar-
chitecture enables simulating in the FEM the development of
the macroscopic temperature history and the melt pool ge-
ometries under the influence of feedback controls. To relieve
the computation burden incurred by the high-fidelity simu-
lation of multiple layers, we can start with a thin-wall build
with small numbers of tracks (say one or two) within one
layer.
Acknowledgements
This material is based upon work supported in part by
the National Science Foundation under Grant No. 1953155.
We would like to thank all the reviewers and editors for their
insightful comments and great efforts towards improving this
manuscript.
Appendix: main MATLAB commands in the proposed
closed-loop simulation
% Importing the COMSOL classes
import com.comsol.model.*
import com.comsol.model.util.*
% Open COMSOL mph file
model = mphopen('FEM_file_with_one_time_step');
ht = model.physics('ht');
% Compute FEM
model.study('std1').run;
% Get the whole temperature distribution
[x0, y0, T, tt] = mphinterp(model, {'x',…
'y', 'T', 't'}, 'coord', coord, 't', t);
% Set tf to t0
time = mphglobal(model,'t','solnum','end');
model.param.set('t0', time);
% Set T(tf) to T(t0)
ht.feature('init1').set('T', 'T');
v1 = model.sol('sol1').feature('v1');
v1.set('initsol', 'sol1');
% Set q(tf) to q
model.param.set('laserpower', q);
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