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New Hammerstein Modeling and Analysis for

Controlling Melt Pool Width in Powder Bed

Fusion Additive Manufacturing

Dan Wang

Dept. of Mechanical Engineering

University of Washington

Seattle, Washington, 98195

Email: daw1230@uw.edu

Xinyu Zhao

Dept. of Mechanical Engineering

University of Connecticut

Storrs, Connecticut, 06269

Email: xinyu.zhao@uconn.edu

Xu Chen∗

Dept. of Mechanical Engineering

University of Washington

Seattle, Washington, 98195

Email: chx@uw.edu

ABSTRACT

Despite the advantages and emerging applications,

broader adoption of powder bed fusion (PBF) additive man-

ufacturing is challenged by insufﬁcient reliability and in-

process variations. Finite element modeling and control-

oriented modeling have been shown to be effective for pre-

dicting and engineering part qualities in PBF. This paper ﬁrst

builds a ﬁnite element model (FEM) of the thermal ﬁelds to

look into the convoluted thermal interactions during the PBF

process. Using the FEM data, we identify a novel surrogate

system model from the laser power to the melt pool width.

Linking a linear model with a memoryless nonlinear sub-

model, we develop a physics-based Hammerstein model that

captures the complex spatiotemporal thermomechanical dy-

namics. We verify the accuracy of the Hammerstein model

using the FEM and prove that the linearized model is only

a representation of the Hammerstein model around the equi-

librium point. Along the way, we conduct the stability and

robustness analyses and formalize the Hammerstein model

to facilitate the subsequent control designs.

1 INTRODUCTION

Different from conventional subtractive machining, ad-

ditive manufacturing (AM) builds a part directly from its dig-

ital model by joining materials layer by layer. In particular,

by applying high-precision lasers or electron beams as the

energy source, powder bed fusion (PBF) AM has enabled

unprecedented fabrication of complex parts from polymeric

and metallic powder materials. However, broader adoption

of the technology remains challenged by insufﬁcient relia-

∗Corresponding author

bility and in-process variations induced by, for example, un-

certain laser-material interactions, environmental vibrations,

powder recycling, imperfect interactions of mechanical com-

ponents, and recursive thermal histories of materials [1–5].

A typical part in PBF is built from many thousands of

thin layers. Within each layer, the energy beam is regu-

lated 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. Appropriate modeling of this sophis-

ticated dynamic system plays a fundamental role in under-

standing and regulating the PBF and related techniques such

as laser metal deposition (LMD). Current researches employ

the ﬁnite element model (FEM) to explore energy-deposition

mechanisms and control-oriented modeling to build mathe-

matical models for regulating in-process variations. Particu-

larly, [7,10, 11] adopt FEM to investigate the effects of var-

ious scan conﬁgurations on the thermal ﬁelds of the pow-

der bed, the geometries of the melt pool, and the mechanical

properties of the printed parts. In control-oriented model-

ing, [2, 12–14] apply low-order system models, and [13–15]

further build nonlinear submodels of LMD from laser power

and scan speed to layer height and melt pool temperature

to cover more process dynamics. Based on the attained

models, subsequent control algorithms such as PID control

[2,16–18], sliding mode control [15], predictive control [12],

and iterative learning control [19] have proved their efﬁcien-

cies in improving the dimensional accuracy of the printed

parts.

This paper establishes a new modeling and understand-

ing of PBF by taking advantage of the FEM and control-

oriented modeling. We ﬁrst develop an FEM of the thermal

ﬁelds to look into the convoluted thermal interactions dur-

1 Copyright © 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)

Fig. 1: Temperature-dependent thermal properties of Ti6Al4V [6–9]. Solid line: solid and liquid materials. Dash-dotted line: powder

material. The two vertical dotted lines respectively indicate Tsol and Tm.

ing the PBF process. The developed FEM then serves as a

simulation platform to provide data for verifying and iden-

tifying parameters of the proposed modeling schemes. In

the control-oriented modeling of PBF, stepping beyond com-

monly used low-order system models, this paper develops a

physics-based Hammerstein model that accommodates more

of the complex spatiotemporal thermomechanical dynamics.

The Hammerstein model is formulated by concatenating a

memoryless nonlinear submodel derived from the Rosenthal

equation to a linear model obtained from system identiﬁ-

cation techniques with the laser power as the input and the

melt pool width as the output. We verify the accuracy of the

Hammerstein model using the FEM and prove that the identi-

ﬁed model is only a linear representation of the Hammerstein

model around the equilibrium point. Along the way, we ana-

lyze the stability and robustness properties of the models and

present a generic control scheme of the Hammerstein model.

The remainder of this paper is structured as follows.

Section 2 builds the FEM of the thermal ﬁelds in PBF. Sec-

tion 3 identiﬁes the linear plant model from the FEM. Sec-

tion 4 derives the closed-form expressions of the steady-state

melt pool width and furthermore develops and analyzes the

main Hammerstein model. Section 5 concludes the paper.

2 FEM OF THERMAL FIELDS IN PBF

In this section, we build and reﬁne an FEM of the ther-

mal ﬁelds in PBF. The model considers surface convection,

surface radiation, conduction, and latent heat of fusion. For

brevity and without loss of generality, the effects of evapo-

ration, ﬂuid ﬂow, and Marangoni force are neglected when

implementing the FEM in COMSOL Multiphysics 5.3a soft-

ware. The governing equation for the 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 ca-

pacity, ρthe effective density, tthe time, Tthe temperature,

and qsthe rate of local internal energy generated per unit

volume [20]. When no confusion would arise in the context,

T(x,y,z,t)is abbreviated to Tin the remainder of this paper.

2.1 Nonlinear Phase Change and Temperature-

dependent Thermal Properties

Fig. 1 shows the temperature-dependent thermal prop-

erties used in this paper. We account for the latent heat of

fusion Lfby introducing the effective heat capacity [21]:

cp,ef f (T)=

cp1(T)T0<T≤Tsol

Lf

Tm−Tso l

+cp1(Tsol )+cp2(Tm)

2Tsol <T<Tm

cp2(T)T≥Tm

,

(2)

where T0is the ambient temperature, Tsol the solidus temper-

ature, Tmthe melting point, cp1the heat capacity of the solid

or powder, and cp2the heat capacity of the liquid.

For the thermal properties, we adopt k,cp, and ρin [6,7]

for the solid and liquid materials. We generate the thermal

properties of the powder material from those of the solid ma-

terial by considering the porosity φ[8, 9]:

kpow der =ks oli d (1−φ)4and ρpowd er =ρsol id (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

[8].

2.2 Initial Condition, Boundary Conditions, and Laser

Beam Proﬁle

The initial condition can take any physical conﬁgura-

tion. In this study, we assume a uniform distribution of the

initial temperature T(x,y,z,0)=T0. When the substrate (left

plot of Fig. 2) is designed to be large enough compared to

the heat affected zone, one boundary condition is established

by assuming the bottom (z=hsu b ) of the substrate has no

heat loss in the rapid laser scanning: −k∂T

∂z

z=hsu b

=0. The

2 Copyright © by ASME

X

Z

Y

Powder bed: F ree triangular

and Swept (0.83 mm)

Substrate: Free

tetrahedral (1.1 mm)

10 mm

5mm

2mm

50 µm

…

Laser tracks

Powder bed: F ree quad

and Swept (60 µm)

(K)

100 µm

Fig. 2: Left: powder bed and substrate with selective meshing

scheme. Right: surface temperature distribution at t=0.14s. The

lined isotherm indicates T=Tm.

other boundary condition considers surface conduction, con-

vection, 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

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. The Appendix has listed

the process parameters used in this study.

2.3 Meshing and Scanning Schemes

The left plot of Fig. 2 shows the built FEM with a

substrate and a thin layer of powder bed. For melt pools

in the scale of around 248 µm in diameter, we use a selec-

tive meshing scheme to balance model accuracy with com-

putation time: a ﬁne quad-and-swept mesh with a maxi-

mum element size of 60 µm is applied to the central pow-

der bed region that directly interacts with the energy beam,

whereas less ﬁner tetrahedral mesh (1.1mm) and triangular-

and-swept mesh (0.83 mm) are applied to the substrate and

the peripheral powder bed, respectively. We adopt the FEM

practice to use coarser element sizes for the substrate and the

peripheral powder bed that undergoes less signiﬁcant heat

transfer than the central melt pool. We consider both single

and multiple scans of the laser beam on the powder bed. The

left plot of Fig. 2 illustrates the bidirectional scan scheme

used in this study.

The developed FEM has been validated experimentally

and analytically in [22] and serves as a simulation platform

in this paper. Later on the data generated from the FEM such

as melt pool width will be used to identify and verify the ac-

curacies of the proposed models. In FEM, melt pool widths

are generated from the temperature distribution T(x,y,z,t)

by searching around the position of the laser beam to ﬁnd the

maximum width of the melt pool geometry bounded by Tm.

101102103104

-160

-150

-140

-130

-120

Magnitude (dB)

101102103104

Frequency (Hz)

-200

-100

0

100

200

Phase (degree)

Measured system using sinusoidal and PRBS signals

identified system P = 0.001671/(s+1055)

Fig. 3: Measured and identiﬁed system responses.

3 LINEAR MODEL AROUND QUASI-STEADY

EQUILIBRIUM

From the developed FEM, we identify the linear plant

model as P(s)=0.001671/(s+1055)from laser power

changes δqto melt pool width changes δwaround the equi-

librium point at (q0=60 W,w0=248.41µm). Here, q=

q0+δqand w=w0+δware the actual laser power fed to the

FEM and the melt pool width generated from the FEM, re-

spectively. The input signals fed to the FEM include a pseu-

dorandom binary sequence (PRBS) signal and multiple sinu-

soidal signals (10~300 Hz), with a magnitude of 20 W and

an add-on DC component of q0=60 W. As shown in Fig.

3, the frequency responses of the measured and identiﬁed

systems match well with each other. From a physics view-

point, the low-pass dynamics is attributed to the high-density

energy deposition of the laser and the ﬁrst-order temporal

dynamics of the temperature evolution in (1). Under the

sampling time tsof 0.5ms, the zero-order-hold equivalent of

the plant model is Pd(z)=6.493 ×10−7/(z−0.5901). For a

more general and uniﬁed analysis (Fig. 4), Pd(z)can be fur-

ther normalized to have a unit DC gain by: P(z)=Pd(z)/c=

0.4099/(z−0.5901), where cis the DC gain of Pd(z).

4 HAMMERSTEIN MODEL IN PBF

In this section, we show the limit of the linear model

subject to the complicated nonlinear thermomechanical dy-

namics of PBF and build a new physics-based Hammerstein

Model to address the limitations. After that, we analyze the

stability and robustness of the models. The Hammerstein

model is conventionally employed in system identiﬁcation

for nonlinear systems, consisting of a nonlinear static ele-

ment followed by a linear dynamic element. Recent studies

of the Hammerstein model aim at parameter estimation and

neural network based solutions [23–25]. Here, we repurpose

the method for identifying a nonlinear model for PBF.

4.1 Core Physics of the Melt Pool at Quasi-Steady State

When a moving point laser source is acting on a large

thick plate, the analytical solution of (1) in the steady state is

the Rosenthal equation [20]:

T(ξ, y,z) − T0=q

2πkr e−ux(r+ξ)

2κ,(4)

3 Copyright © by ASME

where (ξ, y,z)is a coordinate system attached to the moving

source, uxis the laser scanning speed, r=pξ2+y2+z2, and

κ=k/(ρcp).

Some assumptions and simpliﬁcations in deriving the

Rosenthal equation are:

1. The material’s physical coefﬁcients such as k,ρ, and

cpare independent of temperature. Using an average

value provides a reasonable approximation and enables

a closed-form solution to be obtained.

2. The internal heat generation is neglected, i.e., qs=0.

3. The workpiece material is homogeneous and isotropic.

4. When the powder bed is processed long enough, a Qua-

sistationary state is reached, that is, the temperature un-

dergoes no change with time with respect to the coordi-

nate system attached to the heat source, i.e., (ξ, y,z).

5. A point heat source rather than Gaussian distribution is

used.

6. The effect of latent heat of fusion is negligible since the

absorbed latent heat evolves later on.

From the Rosenthal equation in (4), the closed-form equation

relating the steady-state melt pool width (wssi or wss ) with

the laser power qis [26]:

q=πk(Tm−T0)wssi +eπ ρcp(Tm−T0)uxw2

ssi /8,(5)

where e is Euler’s number.

Assumptions in deriving (5) are:

1. −ln(r∗N)

r∗M≈0, that is, r∗≈1

eN, where r∗is the value of rat

the width of the melt pool, M=ux

2κ, and N=2πk(Tm−T0)

q.

2. r∗M1.

3. The approximation of qis found to be improved by ac-

counting for the zero-speed power in (4), that is, the ﬁrst

term on the right hand side of (5).

Under typical PBF conﬁgurations, the ﬁrst two assump-

tions are reasonably valid for all alloys except for the alloy

AlSi10Mg.

4.2 Structure of the Hammerstein Model

We start to build the Hammerstein model by lumping

the memoryless nonlinear submodel in (5) with the identiﬁed

linear dynamics P(z)that has unit DC gain (see Section 3).

As shown in Fig. 4, the Hammerstein model upgrades Pd(z)

by replacing the constant cwith the nonlinear closed-form

expression of the steady-state melt pool width f(·). In (5),

the values of parameters k,ρ, and cpare to be determined.

Substituting the equilibrium point (q0,w0)to (5) gives q0=

πk(Tm−T0)w0+eπρcp(Tm−T0)uxw2

0/8, that is,

eρcpux=8(Bq0−k)/w0>0,(6)

where B=1/[π(Tm−T0)w0]is a constant. In (5) and (6), ρ

and cpare multiplied together and related to k. Based on the

ﬁrst assumption in Section 4.1, we choose k=40 W/(m ·k)

from Fig. 1. Substituting (6) to (5) yields

f(.)P(z)

δqδwss(i)δw

Hammerstein model H

cP(z)

δqδw

Pd(z)

Fig. 4: Block diagrams of the Hammerstein model and the identiﬁed

linear model.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s)

-1

-0.5

0

0.5

1

w (m)

10-4

FEM result

Linear model Pd(z)

Hammerstein model w/o compensation

Hammerstein model w/ compensation

Fig. 5: Parameter identiﬁcation of the Hammerstein model with in-

put signal of 10 Hz.

(Bq0−k)w2

ssi +kw0ws si −Bw2

0q=0.(7)

Omitting the negative root, we get

wssi =pk2+4(Bq0−k)Bq −k

2(Bq0−k)w0.(8)

With all parameters determined, the Hammerstein

model in Fig. 4 is thus formalized around the equilib-

rium point by connecting (8) with P(z)and letting δwssi =

wssi −w0and δq=q−q0. Certainly, due to simpliﬁca-

tions in deriving (5), the direct solution (8) only works at

speciﬁc input conditions. Under the input signal of δq10 =

20Wsin(2πf tsn), where f=10 Hz, nis the discrete-time in-

teger time index, and tsis the sampling time, we can tell from

Fig. 5 that the output of the Hammerstein model (dashed

line) deviates from the FEM result (solid line).

To generalize the nonlinear block, we multiply wss i in

(8) with a compensation factor α(q):

wss =pk2+4(Bq0−k)B q −k

2(Bq0−k)w0α(q),(9)

where α(q)is a quadratic function that passes through three

points (60 W,1)(i.e., no compensation at the equilibrium

point), (80 W, α1)(i.e., the maximum laser power), and

(40 W, α2)(i.e., the minimum laser power). We identify the

parameters α1and α2, respectively, as 0.8507 and 1.1973 us-

ing the Parameter Estimation tool in MATLAB. The nonlin-

ear least square regression is used to minimize the sum of

squared errors between the FEM data and the output of the

4 Copyright © by ASME

0 20 40 60 80 100 120 140

Frequency of input signal (Hz)

4

6

8

10

12

RMS error (m)

10-6

Hammerstein model w/ compensation

Linear model Pd(z)

Fig. 6: Root mean squared (RMS) errors with respect to different

input frequencies.

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

Time (s)

-4

-2

0

2

w (m)

10-5

FEM result Hammerstein model w/ compensation

Fig. 7: Melt pool width changes with input signal of 60 Hz.

-30 -25 -20 -15 -10 -5 0 5 10 15 20

q (W)

-10

-5

0

5

wss (m)

10-5

Hammerstein model w/ compensation

Hammerstein model w/o compensation

Linear model Pd(z)

Fig. 8: Steady-state melt pool width changes of different models.

updated Hammerstein model with compensation (solid and

dash-dotted lines in Fig. 5).

The compensated Hammerstein model is achieved by

using (9) instead of (8) and letting δwss =ws s −w0. From

Figs. 5 and 6, we can tell that this Hammerstein model

(dash-dotted line) gives a better approximation (41% more

accurate in terms of root mean square errors) of the system

dynamics than the identiﬁed linear model Pd(z)at 10 Hz.

More generally, as shown in Fig. 6, under different input

frequencies, the compensated Hammerstein model consis-

tently yields smaller root mean squared errors with respect

to the FEM result than the linear model. Fig. 7 illustrates

the results of the compensated Hammerstein model and the

FEM under the input frequency of 60 Hz. Following the pro-

cedures from (6) to (9), we can adaptively build the Ham-

merstein model for each speciﬁc equilibrium point and struc-

turally draw the complete model map for the entire task space

of PBF.

We compare in Fig. 8 how δws s changes with respect to

δqunder different modeling schemes. In the identiﬁed linear

model Pd(z), the gradient of the dash-dotted line that links

δwss with δqis the constant c(Fig. 4). From Fig. 8, we

can tell that Pd(z)is only a linear representation of the non-

linear Hammerstein model (solid line) near the equilibrium

point. It is remarkable how Pd(z)identiﬁed from the FEM

data coincides tangentially with the Hammerstein model de-

rived from the governing equation. Next we will conduct the

stability and robustness analyses to investigate when Pd(z)

would fail in representing the Hammerstein model.

4.3 Stability and Robustness

Based on the Hammerstein model, we evaluate the ro-

bustness and stability properties of the linear models Pd(z)

that are commonly used in practice. Let H=Pd(z)(1+∆),

where His the Hammerstein model and ∆is the bounded

model uncertainty. From Fig. 4, we have δwss ·P(z)=

δq·cP(z)(1+∆), which gives

|∆|=

δwss

cδq−1

(10)

that is speciﬁed by the distance between the dash-dotted and

solid lines in Fig. 8.

Standard robust-stability analysis indicates that a

closed-loop system containing Pd(z)is stable if and only if

both of the following conditions hold:

1. Nominal stability condition is satisﬁed, that is, the

closed loop is stable when ∆=0.

2. Robust stability requirement is met by applying the

small gain theorem [27]: for any frequency Ωin Hz,

∆·T(ejΩTs)

<1, that is, |∆|<1/

T(ejΩTs)

, where T(z)

is the complementary sensitivity function [1].

Note that |∆|in (10) is positively correlated to the control

signal δq, that is, more laser power deviation from the equi-

librium point yields a larger |∆|. Under a certain frequency

Ω, we need to make sure the maximum |∆|is less than

1/

T(ejΩTs)

. When the condition is violated, Pd(z)will no

longer be a valid representation of the Hammerstein model.

A sufﬁcient condition for the stability of the Hammer-

stein model is the BIBO stability of the linear model P(z)

(Fig. 4) [24]. In practice, the linear model is typically a

rational transfer function, whose stability can be easily ex-

amined.

4.4 Control Implementation

Although the focus of this paper is on the modeling

of the complex physics in PBF, we have additionally vali-

dated the proposed model in closed-loop controls. Limited

in space, we will brieﬂy discuss the key concept. Fig. 9

presents a typical feedback loop when applying the Hammer-

stein model. An f−1(·) block is added and connected with

the block of the Hammerstein model. Combining these two

blocks together yields the linear model P(z). It is thereafter

standard practice to design the control algorithms for P(z).

To ﬁnd the inverse of the nonlinear element, one approach is

to use a high-order polynomial to approximate f(solid line

in Fig. 8). Besides, [23] proposes an approximate method

for the cases when fis not invertible.

5 Copyright © by ASME

C(z)f−1(.)H

d(k)

+

r(k)+e(k)

+

y(k)

−

Fig. 9: Block diagram of feedback control for a Hammerstein

model.

5 CONCLUSION

In this paper, we ﬁrst build a ﬁnite element model (FEM)

to simulate the thermal ﬁelds during the powder bed fusion

(PBF) process. Using the FEM data, we identify around

the equilibrium point the linear system model from the laser

power to the melt pool width. In addition, deriving from the

Rosenthal equation, we reach a nonlinear closed-form ex-

pression of the steady-state melt pool width. Concatenat-

ing the nonlinear expression to the identiﬁed linear model,

we develop the main Hammerstein model that captures more

of the convoluted thermomechanical dynamics of PBF. We

prove that the Hammerstein model gives a better approxi-

mation (e.g., 41% increasing at 10 Hz) of the FEM result

than the linear model. From there, we analyze the stability

and robustness properties of the models and present a generic

control scheme for the Hammerstein Model.

ACKNOWLEDGEMENT

This material is based upon work supported in part by

the National Science Foundation under Award No. 1953155.

APPENDIX

Deﬁned parameters of the FEM

Parameters Value Parameters Value

Powder bed size 15mm ×15 mm ×50 µm Material Ti6Al4V

Substrate size 15 mm ×15 mm ×5mm Track length L5mm

R220 µm/2 Time step Ts0.5ms

Absorptance 0.25 Emissivity 0.35

Tso l 1873 K Scan speed ux100 mm/s

Lf295 kJ/kg Laser power P60 W

T0/Tm293.15K/1923.15 Kφ00.4

hc12.7W/(m2·K) k,cp, and ρsee Fig. 1

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