Proceedings of 2020 International Symposium on Flexible Automation
July 5-9, 2020, Chicago, Illinois, U.S.A
CLOSED-LOOP SIMULATION INTEGRATING FINITE ELEMENT MODELING WITH
FEEDBACK CONTROLS IN POWDER BED FUSION ADDITIVE MANUFACTURING
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Dept. of Mechanical Engineering
University of Washington
Seattle, Washington, 98195
Powder bed fusion (PBF) additive manufacturing has en-
abled unmatched agile manufacturing of a wide range of prod-
ucts from engine components to medical implants. While high-
ﬁdelity ﬁnite element modeling and feedback 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. Inte-
grating both realms, this paper builds a ﬁrst-instance closed-loop
simulation framework by utilizing the output signals retrieved
from the ﬁnite element model (FEM) to directly update the con-
trol signals sent to the model. The proposed closed-loop simu-
lation enables testing the limits of advanced 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 solutions and then use the FEM to understand the
melt-pool evolution induced by the in-layer thermomechanical
interactions. From there, we build a repetitive control algorithm
to greatly attenuate variations of the melt pool width.
Additive manufacturing (AM) builds a part directly from its
digital model by joining materials layer by layer, which is dif-
ferent from conventional subtractive machining. In particular,
powder bed fusion (PBF) AM, applying high-precision lasers or
electron beams as the energy source, has enabled unprecedented
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 in-
duced by, for example, uncertain laser-material interactions, en-
vironmental vibrations, powder recycling, imperfect interactions
of mechanical components, and complex thermal histories of ma-
Current researches employ ﬁnite element modeling and
feedback controls to understand the energy-deposition mecha-
nisms and to regulate the in-process variations in PBF and other
AM technologies such as laser metal deposition (LMD). Partic-
ularly, [4–6] adopt ﬁnite element modeling to investigate the ef-
fects of various scan patterns, scan speeds, number of lasers, and
overhanging structures on the thermal ﬁelds of the powder bed,
the geometries of the melt pool, and the mechanical properties
of the printed parts. Existing strategies on feedback controls of-
ten implement low-order system models obtained using system
identiﬁcation techniques [2, 7–9]. A nonlinear memoryless sub-
model [8, 10] and a spatial-domain Hammerstein model  have
been built to cover more complicated process dynamics. From
there, PID control [2, 11–13], sliding mode control , predic-
tive control , and iterative learning control  have proved
their efﬁciencies in improving the dimensional accuracy of the
printed parts in PBF and LMD.
Although ﬁnite element models (FEMs) and feedback con-
trols have been identiﬁed key for predicting and engineering part
qualities in PBF, existing results in each realm are developed in
separate computational architectures due to their different time
1 2 345678 9 10
FIGURE 1. Schematic of in-layer sintering process in PBF.
scales. To be more speciﬁc, feedback controls are implemented
in real time, while it can take hours or even days for FEMs to
simulate the sintering of a few layers that ﬁnishes in seconds
in reality. If we can integrate FEMs with feedback controls di-
rectly in a closed loop, however, we will be able to 1) combine
aforementioned knowledges from each realm, 2) test the limits
of advanced controls in PBF, 3) survey the parameter space fully
to generate more predictable part qualities, and 4) quickly design
controllers and update parameters for novel materials and printer
settings. These beneﬁts are more prominent when the experi-
ments 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 sim-
ulation framework that leverages modern architectures of ﬁnite-
element-modeling tools and the power of data processing and ad-
vanced controls. Speciﬁcally, we build a bidirectional communi-
cation so that the output signals (e.g., melt pool width) retrieved
from the FEM can be utilized to directly update the FEM process
parameters (e.g., laser power) in external control toolboxes (e.g.,
MATLAB). Along the course of formulating the framework, we
validate the FEM by comparing its results with experimental and
analytical solutions and furthermore apply the FEM to investi-
gate the periodic in-layer thermal interactions. Under the frame-
work of the closed-loop simulation, we then verify the effective-
ness of the repetitive control (RC) in attenuating the repetitive
variations of the melt pool width.
The remainder of this paper is structured as follows. Sec-
tion 2 builds the main closed-loop simulation framework taking
an FEM and a plug-in RC design for example. Section 3 veriﬁes
the FEM and justiﬁes the existence of the periodic in-layer ther-
mal interactions. Section 4 implement the proposed closed-loop
simulation to evaluate the performance of RC in attenuating the
periodic in-layer disturbances. Section 5 concludes the paper.
2 Proposed high-ﬁdelity closed-loop simulation
A typical part in PBF is built from many thousands of thin
layers. Within each layer (Fig. 1), the energy beam is regulated
to follow trajectories predeﬁned by the part geometry in a slicing
process. After one layer is ﬁnished printing, a new thin layer of
powder will be spread on top, and then another cycle begins. This
section frames the main high-ﬁdelity closed-loop simulation. We
ﬁrst design an FEM to simulate the thermal ﬁelds during the PBF
process. After that, a sample RC algorithm is designed and in-
troduced to the closed-loop simulation.
We use the COMSOL Multiphysics 5.3a software to build
and reﬁne the FEM of the thermal ﬁelds in PBF. The model con-
siders surface convection, surface radiation, conduction, and la-
tent heat of fusion. For brevity and without loss of generality,
the effects of evaporation, ﬂuid ﬂow, and Marangoni force are
neglected. The governing equation for conduction heat ﬂow is
dt =∇·(k∇T(x,y,z,t)) + qs,(1)
where kis the thermal conductivity, cpthe speciﬁc heat capac-
ity, ρthe effective density, tthe time, Tthe temperature, and qs
the rate of local internal energy generated per unit volume .
When no confusion would arise in the context, T(x,y,z,t)is ab-
breviated to Tin the remaining of this paper.
2.1.1 Phase change and temperature-dependent
thermal properties We account for the latent heat of fusion
Lfby introducing the effective heat capacity :
cp,e f f (T) =
where T0is the ambient temperature, Tsol the solidus temperature,
Tmthe melting point, cp1the heat capacity of the powder, and cp2
the heat capacity of the liquid.
For the thermal properties, we adopt k,cp, and ρin [4, 16]
for the solid and liquid materials. For the powder material,
we use the thermal properties generated from the solid mate-
rial by considering the porosity φ[17, 18]: kpowder =ksolid (1−
φ)4and ρpowder =ρsolid (1−φ), where φis expressed as
with φ0denoting the initial porosity. Here, the heat capacity is
assumed to be the same for the powder and solid materials except
in Tsol <T<Tm. Fig. 2 shows the temperature-dependent
thermal properties used in this paper.
500 1000 1500 2000 2500
Thermal conductivity (W/m .K)
500 1000 1500 2000 2500
500 1000 1500 2000 2500
Heat Capacity (J/g .K)
FIGURE 2. Temperature-dependent thermal properties of Ti6Al4V [4, 16–18]. Solid line: solid and liquid materials. Dash-dotted line: powder
material. The two vertical dotted lines respectively indicate Tsol and Tm.
2.1.2 Initial condition, boundary conditions, and
laser beam proﬁle The initial condition is T(x,y,z,0) = T0.
One boundary condition is established by assuming the bottom
(z=h) has no heat loss: −k∂T
z=h=0. The other boundary con-
dition considers surface conduction, convection, and radiation:
=−Q+hc(T−T0) + εσB(T4−T4
where Qis the input heat ﬂux, hcthe convection heat transfer
coefﬁcient, εthe emissivity, and σBthe Stefan-Boltzmann con-
stant. Here, we assume Qhas a Gaussian laser beam proﬁle:
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. In Appendix, we list the process param-
eters used in this study unless otherwise speciﬁed.
2.1.3 Meshing and scanning schemes The left
plot of Fig. 3 shows the built FEM with a substrate and a thin
layer of powder bed. In this FEM, 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 60 µm
is applied to the central powder bed region that directly inter-
acts with the energy beam, whereas less ﬁner tetrahedral mesh
(3.5mm) and triangular-and-swept mesh (2 mm) are applied to
the substrate and the peripheral powder bed, respectively. The
left plot of Fig. 3 also illustrates the bidirectional scan scheme
used in this study with a hatch spacing (∆xin Fig. 1) of 60 µm.
The developed FEM will be veriﬁed in Section 3.
2.2 Closed-loop simulation framework
We propose here the main closed-loop simulation frame-
work that integrates feedback controls with FEM (e.g., the FEM
in Section 2.1) and enables updating directly the control signals
of the FEM. This closed-loop framework is designed using the
Powder bed: F ree triangular
and Swept (2mm)
tetrahedral (3.5 mm)
Powder bed: F ree quad
and Swept (60 µm)
FIGURE 3. Left: powder bed and substrate with selective meshing
scheme. Right: surface temperature distribution at t=0.14s. The lined
isotherm stands for T=Tm.
!"# ", $ # $"
%&'( )( *( !"+ # %
% '( )( *( !,# % '( )( *( !-
$ # $&!-+End if !-# !/01
FEM calculation for
!,2!-# !,3 %
Calculate melt pool
%&'( )( *( !-+
(Baseline or RC)
Get control sign al
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FIGURE 4. Schematic of proposed closed-loop simulation.
software LiveLink™ for MATLAB and mainly composed of two
parts: FEM developed using COMSOL and feedback control al-
gorithms designed using MATLAB. The key idea of this closed-
loop framework is to use the output signals retrieved from the
FEM to update in MATLAB the control signals sent back to the
model step by step. As a case study, we use melt pool width as
the output signal and laser power as the control signal.
Fig. 4 illustrates the procedures of the proposed closed-loop
simulation. First of all, we initialize the FEM in Section 2.1 by
setting the start time t0as 0, the laser power qas the initial one
q0, and T(x,y,z,t0)as the ambient temperature T0. Note that
the computation time of the FEM is set as one time step from t0
to tf=t0+Ts, and afterwards, MATLAB will call the FEM re-
cursively to ﬁnish the whole simulation with a longer time tend.
The design and initialization of the FEM is completed in COM-
C(z)FEM or P(z)
FIGURE 5. Block diagram of a plug-in RC design.
SOL, while the main ﬁle of the closed-loop framework is writ-
ten in MATLAB. When the main ﬁle starts running, the com-
mand model.study(’std1’).run ﬁrst calls COMSOL to compute
the FEM (named std1) for one time step, and then the function
mphinterp retrieves the temperature distribution T(x,y,z,tf)at
t=tffrom COMSOL. Thereafter, the main ﬁle calculates the
melt pool width wat t=tffrom T(x,y,z,tf)and, based on the
control algorithms, processes w(tf)and obtains the control sig-
nal q(tf). At the ﬁnal step, the iterative variables in the FEM are
updated by assigning tfto t0,T(x,y,z,tf)to T(x,y,z,t0), and
q(tf)to the laser power. After this iteration, MATLAB will call
COMSOL again to start a new FEM computation with the up-
dated variables, and then another cycle begins. The closed-loop
simulation will stop when tfreaches to tend .
The proposed closed-loop simulation achieves updating in a
closed loop the control signals of FEM. This simulation frame-
work will beneﬁt and guide experiments by validating before-
hand the effectiveness of the servo designs. Next we will bring a
RC algorithm into the proposed closed-loop simulation.
2.3 Repetitive controller design
RC is designed for tracking/rejecting periodic exogenous
references/disturbances in applications with repetitive tasks .
By learning from previous iterations, RC can greatly enhance
current control performance in the structured task space. In digi-
tal RC, an internal model 1/(1−z−N)is incorporated in the con-
troller, where zis the complex indeterminate in the z-transform.
N=fs/f0is the period of the signal, where fs=1/Tsis the sam-
pling frequency and f0is the fundamental disturbance frequency.
Consider a baseline feedback system consisting a plant P(z)and
a baseline controller C(z)(Fig. 5 with the dotted box removed).
Here, C(z)can be designed by conventional servo algorithms,
such as PID, H∞, and lead-lag compensation. The signals r(k),
d(k),u(k), and y(k)respectively represent the reference, the in-
put disturbance, the control signal, and the system output. The
sensitivity function S(z) = 1
1+P(z)C(z)is the transfer function from
We introduce here a plug-in RC design  that uses the
internal signals e(k)and u(k)to generate a compensation signal
ωc(k)(Fig. 5). Let mdenote the relative degree of 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
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), where α∈[0,1)is
a tuning factor. 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 (4), and hence the new sensitivity func-
tion S0(z) = 1
1+P(z)Call(z)=0. At the intermediate frequencies
ω6=k2πf0Ts, with z=ejωand αbeing close to 1, Q(z)≈0,
1−z−mQ(z)≈1, Call (z)≈C(z)from (4), and thereby the origi-
nal loop shape is maintained.
During implementation, zero-phase pairs q0(z−1)q0(z)are
attached to Q(z)for robustness against high-frequency plant un-
Q(z) = (1−αN)zm−N
where q0(z)=(1+z)n0/2n0and n0∈Z+. The closed-loop per-
formance S0(z)can be tuned by choosing different αand n0.
The plug-in RC and the baseline control can be easily incorpo-
rated into the closed-loop simulation by setting u(k)as q(tf)and
y(k)as w(tf). Under the framework of the closed-loop simula-
tion, we will prove in Section 4 the effectiveness of RC in PBF.
3 Model veriﬁcation and thermal interactions
In this section, we verify the FEM in Section 2.1 and then
apply it to understand the periodic in-layer thermal cycles.
3.1 Model veriﬁcation
We compare the melt pool widths obtained from the FEM
ﬁrst with the experimental results and then with the analytical
solutions. Throughout this paper, melt pool widths are derived
from the temperature distribution (e.g., T(x,y,z,t)in the FEM)
by searching around the position of the laser beam to ﬁnd the
maximum width of the melt pool geometry bounded by Tm.
We compare in Table 1 the numerical melt pool widths with
the experimental results in . The laser power is ﬁxed to 50 W,
and the scan speed is 100, 200, or 300 mm/s. Overall, the FEM
gives reasonable predictions of the melt pool widths with errors
of 3.61%, 6.41%, and 5.44%, respectively. The main reason
that the numerical melt pool widths are slightly (less than 10 µm)
larger than the experimental results is that evaporation is ignored
TABLE 1. Melt pool widths from FEM and experimental results
 with a ﬁxed laser power of 50 W. Difference=FEM-Experiments.
Scan speed (mm/s) 100 200 300
FEM (µm) 182 152.63 132.56
Experiments (µm) Min/Max 165.71/175.43 140.71/142.85 120.71/125.35
Difference (µm) 6.57 9.78 7.21
Error 3.61% 6.41% 5.44%
Rosent hal solu tion
! = #
! = 2#
Δ! = #
! = 3#
FIGURE 6. Melt pool widths from the FEM and analytical solution.
Right and bottom left plots share the same scale and legend.
in the FEM so the overheated material and the heat within are
condensed in the melt pool.
Then we compare the FEM results with the analytical solu-
tions. 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 :
2κ, where (ξ,y,z)is a coordinate
system attached to the moving source, r=pξ2+y2+z2, and
κ=k/(ρcp). For comparison, the FEM is adapted to accom-
modate the assumptions of the Rosenthal equation, such as con-
stant thermal properties (k=5 W/(m·K), cp=1.1 J/(g·K), and
ρ=4300 kg/m3) and point heat source. Fig. 6 compares the
numerical and analytical solutions. As shown in the right plot
and the bottom left plot, after 27 samples, the numerical melt
pool geometry reaches to the steady state and matches with the
Rosenthal solution (the outline). Also, from the top left plot of
Fig. 6, we can tell that the melt pool widths obtained from the
FEM and the Rosenthal equation match well with each other un-
der different combinations of scan speeds and laser powers.
3.2 Periodic thermal interactions
After having validated the FEM, next we will adopt it to in-
vestigate the periodic in-layer thermal cycles in PBF. Here, we
bidirectionally sinter 10 tracks within one layer (Figs. 1 and 3).
The right plot in Fig. 3 illustrates the simulated surface temper-
ature distribution at t=0.14 s, where the isotherm of T=Tm
indicates the melt pool geometry. From the solid line in the top
plot of Fig. 7, we observe that the melt pool width changes over
time and structurally deviates from the steady-state value 246µm
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 re-
sulting increased melt pool widths at the beginning of each track
form a periodic disturbance with a repetitive spectrum in the fre-
quency domain (the solid line in the middle plot of Fig. 7). The
fundamental frequency f0of the disturbance is determined by the
duration of scanning one track t0, that is, f0=1/t0=ux/L, where
uxis 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 periodicity is closely related to the recur-
ring laser scanning trajectories and the repetitive in-layer ther-
momechanical interactions. Besides the bidirectional scan used
in this study, other scan patterns yield similar repetitive distur-
bances (see, e.g., experimental results in ). To deal with these
undesired repetitive spectra, we implement the closed-loop simu-
lation by bringing automatic control algorithms [1,21] into ﬁnite
element modeling, as will be discussed in Section 4.
4 Results and Analyses
This section employs the proposed closed-loop simulation
to evaluate the performances of the baseline control and RC in
attenuating the variations of the melt pool width (Section 3.2).
First, we identify the plant model of the FEM from the laser
power to the melt pool width as P(s) = 0.001671/(s+1055).
The input signals used for system identiﬁcation include a pseudo-
random 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 systems match well with each other (see Fig. 8).
After that, we design a PI controller as C(s) = Kp+Ki/s
with Kp=9.38 ×105and Ki=1.66 ×109. Under the sam-
pling time Tsof 0.5ms (i.e., fs=2 kHz), the zero-order-hold
equivalents 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. 9 shows the magnitude re-
sponse of the sensitivity function S(z)in the baseline feedback
loop composed of P(z)and C(z). Such a design provides a band-
FIGURE 7. In-layer thermal disturbance. Top: time-domain. Middle:
frequency-domain (FFT). Bottom: laser power (control signals u(k)in
Fig. 5). The three plots share the same legend. σdenotes the standard
Measured system using sinusoidal and PRBS signals
identified system P = 0.001671/(s+1055)
FIGURE 8. Measured and identiﬁed system responses.
width 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 accord-
ing to Section 2.2 integrating FEM with baseline control and RC,
respectively. Here, in Fig. 5, r(k) = 0, and d(k)comes from the
in-layer thermal interactions. From the frequency-domain results
in Fig. 7, we can tell that the baseline PI control can attenuate
Baseline PI control
FIGURE 9. Magnitude responses of sensitivity functions S(z)in base-
line control and S0(z)in RC.
to some extent the frequency spikes below the closed-loop band-
width but not the other high-frequency spikes. Compared to the
case without control, the baseline feedback loop decreases the
3σvalue of the variations of the melt pool width (y(k)in Fig. 5)
by 21.57%, where σdenotes the standard deviation.
To enhance the disturbance-attenuation performance, we
bring the plug-in RC compensator in Section 2.3 into the closed-
loop simulation. In the Q-ﬁlter design in (5), the relative de-
gree mof ˆ
P(z)(=P(z)in this example) is 1, the disturbance pe-
riod N=fs/f0=2000/20 =100, and we choose α=0.99 and
n0=1. With the plug-in RC introduced, high-gain control ef-
forts are generated exactly at 20Hz and its harmonics (see S0(z)
in the solid line of Fig. 9). The bottom plot of Fig. 7 illustrates
the control signals u(k)of the baseline control, the RC, and the
case without control. As shown in the middle plot of Fig. 7,
compared with the baseline control, RC further lowers the peri-
odic 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)
(the top plot of Fig. 7).
In this paper, we ﬁrst build a ﬁnite element model (FEM) to
simulate the temperature response in powder bed fusion (PBF)
additive manufacturing. Then we validate the FEM by compar-
ing the numerical results with the experimental and analytical so-
lutions. Employing the FEM, we justify the existence of the pe-
riodic disturbances in the evolution of the melt pool width. From
there, we develop a ﬁrst-instance closed-loop simulation frame-
work by integrating FEM with feedback controls (e.g., baseline
PI control and repetitive control) to reduce the in-process vari-
ations and advance the part quality in PBF. Implementing this
closed-loop frameworks, we validate that the repetitive control
algorithm attenuates the periodic disturbances more substantially
by 35.97% compared to the PI control.
This material is based upon work supported in part by the
National Science Foundation under Grant No. 1953155.
Appendix: deﬁned parameters of the FEM
Parameters Value Parameters Value
Powder bed size 5mm ×10 mm ×50 µm Material Ti6Al4V
Substrate size 5 mm ×10mm ×2 mm Track length L5 mm
Laser spot diameter 2R220µmm Time step Ts0.5ms
Powder bed absorptance 0.25 Emissivity 0.35
Solidus temperature Tsol 1873K Scan speed ux100mm/s
Latent heat of fusion Lf295kJ/kg Laser power q60 W
T0/Tm293.15 K/1923.15 K φ00.4
hc12.7W/(m2·K) k,cp, and ρFig. 2
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