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Proceedings of 2020 International Symposium on Flexible Automation

ISFA 2020

July 5-9, 2020, Chicago, Illinois, U.S.A

[ISFA2020-XXXXX]

CLOSED-LOOP SIMULATION INTEGRATING FINITE ELEMENT MODELING WITH

FEEDBACK CONTROLS IN POWDER BED FUSION 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

ABSTRACT

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.

1 Introduction

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

∗Corresponding author

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-

terials [1–3].

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 [9] have

been built to cover more complicated process dynamics. From

there, PID control [2, 11–13], sliding mode control [10], predic-

tive control [7], and iterative learning control [14] 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

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 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.

2.1 FEM

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

ρcp

dT (x,y,z,t)

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 [15].

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 [19]:

cp,e f f (T) =

cp1(T)T0<T≤Tsol

Lf

Tm−Tsol +cp1(Tsol)+cp2(Tm)

2Tsol <T<Tm

cp2(T)T≥Tm

,(2)

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

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 except

in Tsol <T<Tm[17]. Fig. 2 shows the temperature-dependent

thermal properties used in this paper.

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 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

z=h=0. The other boundary con-

dition considers surface conduction, convection, and radiation:

−k∂T

∂z

z=0

=−Q+hc(T−T0) + εσB(T4−T4

0),(3)

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:

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. 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

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)

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.

Initialization

!"# ", $ # $"

%&'( )( *( !"+ # %

"

% '( )( *( !,# % '( )( *( !-

!,# !-

(.

$ # $&!-+End if !-# !/01

FEM calculation for

!,2!-# !,3 %

4

Calculate melt pool

width 5&!-+from

%&'( )( *( !-+

Apply control

algorithms

(Baseline or RC)

Compare 5

with 51

Get control sign al

$&!6+(laser power)

COMSOL MATLAB

Retriev ing

%&'( )(*( !-+

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)

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 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 [20].

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

d(k)to y(k).

We introduce here a plug-in RC design [21] 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), 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).(4)

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-

certainties:

Q(z) = (1−αN)zm−N

1−αNz−Nq0(z−1)q0(z),(5)

where q0(z)=(1+z)n0/2n0and n0∈Z+. The closed-loop per-

formance S0(z)can be tuned by choosing different αand n0[21].

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 [22]. 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

[22] with a ﬁxed laser power of 50 W. Difference=FEM-Experiments.

Error=(FEM-Experiments)/FEM.

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%

X

200 um

Y

Rosent hal solu tion

Z

Y

Sample 1

! = #

$

Sample 2

! = 2#

$

Δ! = #

$

Sample 3

! = 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 [15]:

T(ξ,y,z)−T0=q

2πkr e−ux(r+ξ)

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.

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 [23]). 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

deviation.

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 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

10010110 2103

Frequency (Hz)

-60

-40

-20

0

Magnitude (dB)

Baseline PI control

Repetitive 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).

5 Conclusion

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.

Acknowledgment

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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