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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

by ASME

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

by ASME

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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