Materials Transactions, Vol. 62, No. 6 (2021) pp. 864 to 870© 2021 The Japan Institute of Metals and Materials

Modiﬁed Cellular Automaton Simulation of Metal Additive Manufacturing+1
Jun Kubo1,+2, Yuichiro Koizumi2, Takuya Ishimoto2 and Takayoshi Nakano2,+2
1

CAPCAST Inc., Shimotsuke 329-0434, Japan
Division of Materials and Manufacturing Science, Graduate School of Engineering, Osaka University, Suita 565-0871, Japan

2

Metal additive manufacturing (AM) technologies are attracting attentions not only as a fabrication process of complicated threedimensional parts but also as microstructure controlling processes. In powder bed fusion (PBF)-type AM, crystallographic texture can becontrolled by scanning strategies of energy beam. To optimize microstructures, computer simulations for predicting microstructures play very
important roles. In this work, we have developed simulation programs to explain the mechanism of the crystal orientation control. First, we
simulated the shape of melt pool by analyzing the heat transfer using apparent heat conductivity when the penetration of laser beam throughkeyholes was taken into consideration because of the evaporation and accompanying convections. It was assumed that the primary crystal
growth direction can be determined by the temperature gradient, and the crystals grow keeping the growth direction as generally recognized. The
shapes of simulated melt pools agree well with experimental observations. The modiﬁed cellular automaton simulations successfully reproduced
two typical textures with diﬀerent preferential orientations along the building directions of ©100ª and ©110ª when the bidirectional scanning withand without a rotation of 90°, respectively, was accomplished between the layers. [doi:10.2320/matertrans.MT-M2021009](Received January 19, 2021; Accepted March 12, 2021; Published April 16, 2021)
Keywords: additive manufacturing, simulation, melt pool, crystallographic texture

1.

Introduction

Recently, the application of metal-additive manufacturing(AM) has been extended to various ﬁelds. Metal AM, which
is characterized by a bottom-up additive production ofparts, has been developed as a means of producing threedimensional structures with complex shapes. Furthermore,this technique has been proven to be extremely useful forcontrolling the microstructure of metallic materials, particularly their crystallographic textures.1­5) Crystallographic
texture is a signiﬁcant metallographic feature that governs
mechanical properties such as Young’s modulus1) and yieldstress,4) and chemical properties such as oxidation resistance,5) of metals independently of shape. Its ﬂexible control
is indispensable for enhancing the functionality of metallic
parts.
Computer simulations are becoming increasingly necessary in a wide variety of areas such as shape optimizationof fabricated objects (topology optimization),6) optimizationof support shapes and fabrication patterns,7) prediction ofresidual stress,8) elucidation of the causes of pores,9) andfunctional evaluation based on the geometry of a fabricatedobject.10) However, the crystallographic texture evolution
mechanism, which is directly linked to the functionality ofmetallic materials, and computer simulations to control thismechanism have not been established yet because metal AM
is a technique based on rapid melting and solidiﬁcation inunits of minute melt pools formed by the scanning of theenergy beams. Simulations are generally performed assuming
various physical phenomena and physical quantities because
in situ observation and measurement during process are
extremely diﬃcult.
This study focuses on the crystallographic orientation
formation behavior, which depends on the laser scan strategy+1This Paper was Originally Published in Japanese in J. Japan Inst. Met.
Mater. 85 (2021) 103­109.
+2
Corresponding authors, E-mail: j-kubo@capcast.co.jp, nakano@mat.eng.
osaka-u.ac.jp(combination of laser scanning directions for each layer)using the selective laser melting (SLM) method for ¢-type
titanium alloys with a bcc structure, and aims at elucidating
the formation mechanism by implementing simulations basedon the cellular automaton (CA) method. Presently, most of
the simulations performed to study the crystallographic
texture in metal AM focus on predicting the crystal growth
direction and formation of grains such as columnar andequiaxed grains.11) In contrast, this study thoroughly
investigated the evolution process of the crystallographic
orientation by improving the conventional CA method.
2.

Texture Evolution of ¢-Type Titanium Alloy (Ti­
15Mo­5Zr­3Al) Depending on the Laser Scan
Strategy under SLM

Figure 1 depicts the crystallographic texture evolution
behavior of the Ti­15Mo­5Zr­3Al alloy (mass%) in SLM
fabrication. The observed crystallographic orientations
diﬀered signiﬁcantly simply by changing the scanning
strategy of the laser beam. In the case where the laser scans
only along the X-axis (scan strategy X), texture formation
with predominate orientation of ©100ª in the X (scan)
direction and ©110ª in Y and Z directions were observed. In
contrast, in case where the laser scan direction was alternatelychanged to X- and Y-axes (scan strategy XY), ©100ª aligned
in all X, Y, and Z directions. The factors aﬀecting theformation of crystallographic orientation caused by theselaser scanning patterns were investigated using simulation.
3.

Analysis Methods

The SLM process using the Ti­15Mo­5Zr­3Al alloy
was analyzed in this study. Computations were performed
primarily for analyzing heat transfer and crystallographic
orientation during the fabrication.
3.1 Heat transfer analysis
In the heat transfer analysis, the change in temperature

Modiﬁed Cellular Automaton Simulation of Metal Additive Manufacturing865

Fig. 1 Experimental inverse pole ﬁgure (IPF) maps1) on the vertical cross sections (y­z plane) of beta-type Ti­15Mo­5Zr­3Al alloy
fabricated by SLM with scan strategy X (without rotation) and scan strategy XY (with 90° rotation). [after T. Ishimoto et al.1)].
Table 1 Parameters used for simulation.

Fig. 2 FEM model for analysis.

distribution over time was calculated by solving the heat
conduction equation given below.12) The modiﬁcation factors
were adjusted so that the melt pool shape agreed with thatof the experimental results because probable eﬀects such as
those due to convection and evaporation cannot be expressedin eq. (1).
 2

@T
@ T
@2 T
@2 T
@fL
¼ ­ l 2 þ m 2 þ n 2  μL
cμ
; ð1Þ
@t
@x
@y
@z
@t
where c is the speciﬁc heat; μ is the density; T is thetemperature; t is the time; ­ is the thermal conductivity; L is
the latent heat; fL is the liquid phase fraction; and l, m, andn are the modiﬁcation factors considering the anisotropy ofheat transfer.
The analysis was performed based on a ﬁnite element
method (FEM). As depicted in Fig. 2, a cuboid with a base
part 3 cm in height placed below the fabrication part of600 µm © 600 µm © 600 µm dimensions was modeled witha 10 µm size mesh. Table 1 lists the parameters used in thesimulation. A value smaller than the heat transfer coeﬃcient
between metals commonly used in casting analysis andgreater than the heat transfer coeﬃcient between metal andair (4188.0 W·m¹2·K¹1) was used in the analysis13) because
the heat transfer coeﬃcient, which is the boundary condition
between the fabrication part and the surrounding metal
powder, is unknown. Moreover, the validity of the analysis
results was conﬁrmed by comparing the melt pool shapes
obtained via actual experiments because the actual measure-ment of temperature distribution changes is diﬃcult in the
SLM process with an ultrahigh cooling rate.14,15)
3.2 Crystallographic orientation analysis
As depicted in Fig. 1, crystallographic orientations diﬀer
based on the scan strategy of the laser beam. The underlying
reason was investigated by referring to the classic solidiﬁcation theory. Considering the columnar and equiaxed
grains observed in general solidiﬁcation, as depicted in theschematic in Fig. 3,16,17) the crystal growth in the initialperiod of solidiﬁcation proceeds in the direction of thethermal gradient perpendicular to the isothermal plane. It is
generally known that, even when solidiﬁcation begins withcrystals having ﬁne random orientations like chill crystals,
the crystals with the preferential growth orientation(orientation of ©100ª for the cubic crystal) aligned with thethermal gradient direction grows preferentially. However,even when the preferential growth direction is oriented in the
direction of the thermal gradient, various columnar crystals
do grow having orientations that diﬀer because of the rotation
around the growth direction. Moreover, when the directionof the thermal gradient changes with the progress ofsolidiﬁcation and the directions of the preferential growth

866

J. Kubo, Y. Koizumi, T. Ishimoto and T. Nakano

Fig. 3Fig. 4

Schematic illustrations of crystal growth in casting [after Chalmers15) and Flemings16)].

Schematic illustrations of Nth layer melt pool.

thermal gradient of the crystals are diﬀerent, growth does notoccur along the preferential growth direction oriented in thethermal gradient direction as in the initial solidiﬁcation.
Under such conditions, it is widely known that the growth ofcrystals continues even under unfavorable thermal gradients
when the supercooling rate is low and new crystal nuclei are
not generated. These were considered in interpreting theexperimental results depicted in Fig. 1.
It is likely that a state as depicted in the schematic in Fig. 4
will occur when the deepest point of the melt pool caused by
the laser beam for layer N reaches layer N-2. New crystal
nuclei are generated in the interface between the C region andthe powder because columnar grains do not grow from theunmelted powder. The initial growth for such crystals orientsin the thermal gradient direction as depicted in Fig. 3. At thistime, if there is no eﬀect of convection, the thermal gradientdirection will be perpendicular to the interface between the
C region and the powder. Because convection actually occurs
as depicted in Fig. 4, it is likely that the inclination increases
further. In this study, this convection eﬀect was modeledby increasing the apparent thermal conductivity in the Z
direction by introducing the modiﬁcation factors listed inTable 1 in eq. (1). Moreover, the solid­liquid interface isnearly parallel to the scan direction because the laser scan

speed in metal AM is mostly greater than the migration rateof the solid­liquid interface. Accordingly, for both the X and
XY scan strategies, in region C in Fig. 4, the ©100ª directionaligns with the laser scan direction (X direction in Fig. 4); the©110ª direction aligns for other directions (Y and Z directionsin Fig. 4).
Figure 5(a) depicts the schematic of layer N + 1 for scanstrategy X. In this illustration, region CA is identical asregion C in Fig. 4, and the crystallographic texture where©100ª orients in the X direction and ©110ª orients in Y andZ directions is formed. Regarding the solidiﬁcation in the
C region, when no new nuclei are generated at the interface
between C and CA regions as explained earlier, thepreferential growth direction of the adjacent CA region isinherited. Similar to the CA region, columnar grains growand ©100ª orients in the X direction and ©110ª orients in Y
and Z directions (Fig. 5(a) lower row). Because the layers
are added in this manner, for all the layers in scan strategy X,
the primary orientation is always ©100ª//X direction and©110ª//Y and Z directions.
Figure 5(b) depicts the schematic of layer N + 1 for scanstrategy XY. Similar to the scan strategy X, region CA in
Fig. 5(b) represents region C (©100ª//X direction and ©110ª//Y and Z directions) in Fig. 4. A magniﬁed schematic of theinterface between C and CA regions is illustrated in the lower
row in Fig. 5(b). The crystallographic orientation in the CA
region adjacent to the melt surface diﬀers from that depicted
in Fig. 5(a) because, compared with Fig. 5(a), there is arotation of 90° around the Z-axis in Fig. 5(b). As described
previously, when no nuclei are generated in the interface
between C and CA regions, columnar grains grow byinheriting the preferential direction of growth of the adjacentunmelted region (CA region). In such a case, although oneof the ©100ª (e.g. [100]) is ﬁxed in the X directions in
Fig. 5(b), the other two ©100ª (e.g. [010] and [001]) remain to
be undetermined. As described previously, for the growth ofcolumnar grains, the preferential growth direction is more
dominant than the thermal gradient direction. Accordingly,
for the other two ©100ª directions that are not primal, thefollowing was considered assuming that they are aﬀectedby the thermal gradient. For scan strategy X where thethermal gradient direction is nearly parallel to the preferential
growth direction, it was assumed that no rotation about thepreferential growth direction occurred without being aﬀectedModiﬁed Cellular Automaton Simulation of Metal Additive Manufacturing

Fig. 5

867

Schematic illustrations of (N + 1)th layer melt pool. (a) scan strategy X, (b) scan strategy XY.

Fig. 6 Comparison between the geometries of simulated melt pools and fusion lines experimentally observed by SEM (after Ishimoto
et al.1)).

by the thermal gradient direction. In contrast, for the scanstrategy XY, where the thermal gradient direction and thepreferential growth direction are signiﬁcantly diﬀerent, it was
assumed that rotation around the preferential growth direction
occurred. In particular, the plane containing the preferential
growth direction vector and the thermal gradient direction
vector was estimated, and the second ©100ª direction was set
as the direction within the same plane perpendicular to thepreferential growth direction. Determining the two ©100ª
directions enables determining the third direction of ©100ª.
In the simulation, ﬁrst, the temperature at each timeinterval was calculated using heat transfer analysis. Whenthe temperature dropped below the solidus temperature, if
there were no elements nearby with the temperature lower
than the solidus temperature, crystal nuclei were generated,
and its growth direction corresponded to the thermal gradientdirection. In contrast, when the temperature dropped below

the solidus temperature for an element and there was anothernearby element with a temperature lower than the solidus
temperature, the growth direction of that element wasinherited. In this manner, the crystal structure was generated
over time by adopting the idea similar to the CA method,18,19)
where the change of state of an element is determinedsequentially from the state of an adjacent element. The other©100ª directions diﬀerent from the crystal growth direction
were also calculated using the thermal gradient along theplane perpendicular to the laser beam scanning direction.
4.

Results of the Simulation

4.1 Heat transfer analysis
Figure 6 depicts a comparison between the shapes of thesimulated melt pools and experimentally observed fusionlines on the cross sections of the products. For scan strategies

868

J. Kubo, Y. Koizumi, T. Ishimoto and T. Nakano

X and XY, the simulated shapes of the melt pools agree
well with the actual fusion lines observed in the products.
In the simulation, U-shaped melt pools is conﬁrmed in the
cross-section perpendicular to the laser scan direction. This
corresponds to the scale-like pattern (fusion lines) observed
across the whole cross section in the product fabricated withscan strategy X. Also in scan strategy XY, fusion lines
formed by laser scan along both X direction (scale-likeshape) and Y direction (elongated shape in Y direction) are
well reproduced.
4.2 Crystallographic orientation distribution
Figures 7 and 8 depict the simulation results for scanstrategy X. Figure 7 depicts the results of the ﬁrst few layers
of fabrication. Only the unmelted region is shown. Figure 8
depicts the comparison of the simulated orientation map(lower row) with the inverse pole ﬁgure (IPF) map obtained
from the actual product (upper row). The simulation results
are illustrated in color by calculating cos ¤, where ¤ is theangle between the ©100ª direction. Accordingly, the red(¤ = 0°) and green (¤ = 45°) indicate ©100ª and ©110ª,
respectively, as represented in the color bar. The simulation

Fig. 7 Early analysis results of scan strategy X at (a) 2.1005 s, (b) 3.1005 s,(c) 3.1307 s, (d) 3.1407 s.

Fig. 8results mostly reproduce the actual crystallographic texture
where the ©100ª preferentially orients in the X direction and©110ª orients in the Y and Z directions. As depicted in Fig. 8(upper row), near ©111ª orientation depicted in blue or purple
partly appears in the X and Z directions. The likely reasonof this is that the solid­liquid interface was not completely
parallel to the direction of laser scanning; however, theunderlying factors need to be studied further. In thesimulation, because the laser scan speed was fast enough toresult in the solid­liquid interface being nearly parallel tothe scanning direction, no orientation in the ©111ª direction
was observed. Moreover, grains with ©100ª orientation in the
Y and Z directions have grown (the red vertical lines in Fig. 8lower row). These grains extend directly from the melt pool
bottom and grow in the build direction (Z direction). As aresult, grain boundaries parallel to the Z direction generate.
It is likely that, because the melt pool depicted in Fig. 6
has a large curvature radius at the bottom, crystals havegrown upward from the melt pool center (bottom). When thecurvature radius at the melt pool bottom is small, even if
grains oriented in the Z direction grow there, they are
eliminated by the grains grew from the melt pool sides with
©110ª//Z orientation.4,20) Such grains are also observed in
the actual product as shown in the upper row of Fig. 8. In thismanner, grains grow from both sides of the melt pool
generating grain boundaries in the central region of themelt pool.
Figures 9 and 10 depict the simulation results for scanstrategy XY. Figure 9 depicts the results of the ﬁrst few layers
of fabrication. Only the solid phase was colored to indicate
the crystal orientation, the liquid phase was hidden, and thesolid-liquid interface was shown in brown. Since it is a threedimensional map, the solid-liquid interface behind the hidden
liquid phase is visible in brown.
For scan strategy XY, Fig. 10 depicts the comparison of
the simulated ﬁnal orientation map with the IPF map obtained
from the actual product. As depicted in Fig. 9, similar to scanstrategy X, initially, grains with ©110ª//Z direction (green)
appear, which remelt thereafter to produce new grains with
©100ª//Z direction. As a result, the most regions represent the
crystallographic orientation where ©100ª preferentially orients
in X, Y, and Z directions. The orientation with ©110ª//Z is
also observed at the topmost part; it is probably owing to thenonoccurrence of remelting and resolidiﬁcation. Moreover,Comparison of analysis results (lower side) with experimental results (upper side: after Ishimoto et al.1)) of scan strategy X.

Modiﬁed Cellular Automaton Simulation of Metal Additive Manufacturing869

theory, and casting solidiﬁcation simulation technique. The
model was then solved using the cellular automaton method
to analytically reproduce the characteristic crystallographic
orientation distributions. Since it is extremely diﬃcult toobserve the real phenomenon because metal AM involves
ultrahigh speed and superﬁne regions, the simulation was
eﬀective to understand how the crystallographic orientation
evolves. However, further studies are required to improve
the accuracy of the calculations by improving the accuracy ofparameters such as interface conditions and material properties. In addition, further studies are required to compare theexperimental results with analytical (simulation) results fora wide variety of alloy compositions and fabrication
conditions.
Acknowledgments
This work was supported by Grants-in-Aid for ScientiﬁcResearch from the Japan Society for the Promotion of
Science (JSPS) [grant number JP18H05254].
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Fig. 9 Early analysis results of scan strategy XY at (a) 3.1306 s, (b)3.1307 s, (c) 3.1318 s, (d) 3.1328 s, (e) 3.1339 s.

for the Y and Z directions, ©110ª orientation is seen, thoughless frequently. This would be because the values used in thesimulation were diﬀerent from the actual ones. The physicalproperties of the powder material and the interface conditions(heat transfer coeﬃcients) for the melt pool are poorly
understood. To improve accuracy of simulation of metal AM,
measurement of the real phenomenon is needed, although it ischallenging.
5.

Conclusion

In this study, the SLM fabrication of metals with cubic
crystal structures was investigated to elucidate the underlying
mechanism of crystallographic texture evolution, where
distinct crystallographic orientations occur depending on
scan strategies. The process of crystal growth in the meltpool created by laser scanning was modeled based on the
published experimental observations, classic solidiﬁcation

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