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New Hammerstein Modeling and Analysis for Controlling Melt Pool Width in Powder Bed Fusion Additive Manufacturing

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Despite the advantages and emerging applications, broader adoption of powder bed fusion (PBF) additive manufacturing is challenged by insufficient reliability and in-process variations. Finite element modeling and control-oriented modeling have been shown to be effective for predicting 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 dynamics. 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 equilibrium point. Along the way, we conduct the stability and robustness analyses and formalize the Hammerstein model to facilitate the subsequent control designs.
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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 =∇ · ( kT(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<TTsol
Lf
TmTso l
+cp1(Tsol )+cp2(Tm)
2Tsol <T<Tm
cp2(T)TTm
,
(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 (1φ)4and ρpowd er =ρsol id (1φ),
where φis expressed as
φ(T)=
φ0T0<TTsol
φ0
Tsol Tm(TTm)Tsol <T<Tm
0TTm
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: kT
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:
kT
z
z=0
=Q+hc(TT0)+εσB(T4T4
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
πR2e2r2
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 ×107/(z0.5901). 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/(z0.5901), 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 eux(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(TmT0)wssi +eπ ρcp(TmT0)uxw2
ssi /8,(5)
where e is Euler’s number.
Assumptions in deriving (5) are:
1. ln(rN)
rM0, that is, r1
eN, where ris the value of rat
the width of the melt pool, M=ux
2κ, and N=2πk(TmT0)
q.
2. rM1.
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(TmT0)w0+eπρcp(TmT0)uxw2
0/8, that is,
eρcpux=8(Bq0k)/w0>0,(6)
where B=1/[π(TmT0)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.
(Bq0k)w2
ssi +kw0ws si Bw2
0q=0.(7)
Omitting the negative root, we get
wssi =pk2+4(Bq0k)Bq k
2(Bq0k)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=qq0. 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(Bq0k)B q k
2(Bq0k)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
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
cδq1
(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 in Hz,
·T(ejTs)
<1, that is, ||<1/
T(ejTs)
, 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
, we need to make sure the maximum ||is less than
1/
T(ejTs)
. 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 f1(·) 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)f1(.)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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Book
1. Introduction.- 2. Qualitative Behavior.- 3. Restrietions & Extensions.- 4. Determination of Volterra Model Parameters.- 5. Practical Considerations in Volterra Model Identification.- 6. Model-Based Controller Synthesis.- 7. Advanced Direct Synthesis Controller Design.- 8. Model Predictive Control Using Volterra Series.- 9. Application Case Studies.- 10. Summary.