Figure 3. The simulated temperature distribution and single-layer multi-track isothermograms of LPBF Hastelloy X, located at the bottom of the powder bed, are presented for various laser energy densities. (a) depicts the single-point temperature distribution at the bottom of the powder bed, followed by the isothermograms corresponding to laser energy densities of (b) 31 J/mm3 , (c) 43 J/mm3 , (d) 53 J/mm3 , (e) 67 J/mm3 , and (f) 91 J/mm3 .

An integrated multiscale simulation guiding the processing optimisation for additively manufactured nickel-based superalloys

적층 가공된 니켈 기반 초합금의 가공 최적화를 안내하는 통합 멀티스케일 시뮬레이션

Xing He, Bing Yang, Decheng Kong, Kunjie Dai, Xiaoqing Ni, Zhanghua Chen
& Chaofang Dong


Microstructural defects in laser powder bed fusion (LPBF) metallic materials are correlated with processing parameters. A multi-physics model and a crystal plasticity framework are employed to predict microstructure growth in molten pools and assess the impact of manufacturing defects on plastic damage parameters. Criteria for optimising the LPBF process are identified, addressing microstructural defects and tensile properties of LPBF Hastelloy X at various volumetric energy densities (VED). The results show that higher VED levels foster a specific Goss texture {110} <001>, with irregular lack of fusion defects significantly affecting plastic damage, especially near the material surface. A critical threshold emerges between manufacturing defects and grain sizes in plastic strain accumulation. The optimal processing window for superior Hastelloy X mechanical properties ranges from 43 to 53 J/mm3 . This work accelerates the development of superior strengthductility alloys via LPBF, streamlining the trial-and-error process and reducing associated costs.

Figure 3. The simulated temperature distribution and single-layer multi-track isothermograms of LPBF Hastelloy X, located at the bottom of the powder bed, are presented for various laser energy densities. (a) depicts the single-point temperature distribution at the bottom of the powder bed, followed by the isothermograms corresponding to laser energy densities of (b) 31 J/mm3 , (c) 43 J/mm3 , (d) 53 J/mm3 , (e) 67 J/mm3 , and (f) 91 J/mm3 .
Figure 3. The simulated temperature distribution and single-layer multi-track isothermograms of LPBF Hastelloy X, located at the bottom of the powder bed, are presented for various laser energy densities. (a) depicts the single-point temperature distribution at the bottom of the powder bed, followed by the isothermograms corresponding to laser energy densities of (b) 31 J/mm3 , (c) 43 J/mm3 , (d) 53 J/mm3 , (e) 67 J/mm3 , and (f) 91 J/mm3 .

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Figure 1. Experimental setup and materials. (a) Schematic of the DED process, where three types of base materials were adopted—B1 (IN718), B2 (IN625), and B3 (SS316L), and two types of powder materials were adopted—P1 (IN718) and P2 (SS316L). (b) In situ high-speed imaging of powder flow and the SEM images of IN718 and SS316L powder particle. (c) Powder size statistics, and (d) element composition of powder IN718 (P1) and SS316L (P2).

Printability disparities in heterogeneous materialcombinations via laser directed energy deposition:a comparative stud

Jinsheng Ning1,6, Lida Zhu1,6,∗, Shuhao Wang2, Zhichao Yang1, Peihua Xu1,Pengsheng Xue3, Hao Lu1, Miao Yu1, Yunhang Zhao1, Jiachen Li4, Susmita Bose5 and Amit Bandyopadhyay5,∗


적층 제조는 바이메탈 및 다중 재료 구조의 제작 가능성을 제공합니다. 그러나 재료 호환성과 접착성은 부품의 성형성과 최종 품질에 직접적인 영향을 미칩니다. 적합한 프로세스를 기반으로 다양한 재료 조합의 기본 인쇄 가능성을 이해하는 것이 중요합니다.

여기에서는 두 가지 일반적이고 매력적인 재료 조합(니켈 및 철 기반 합금)의 인쇄 적성 차이가 레이저 지향 에너지 증착(DED)을 통해 거시적 및 미시적 수준에서 평가됩니다.

증착 프로세스는 현장 고속 이미징을 사용하여 캡처되었으며, 용융 풀 특징 및 트랙 형태의 차이점은 특정 프로세스 창 내에서 정량적으로 조사되었습니다. 더욱이, 다양한 재료 쌍으로 처리된 트랙과 블록의 미세 구조 다양성이 비교적 정교해졌고, 유익한 다중 물리 모델링을 통해 이종 재료 쌍 사이에 제시된 기계적 특성(미세 경도)의 불균일성이 합리화되었습니다.

재료 쌍의 서로 다른 열물리적 특성에 의해 유발된 용융 흐름의 차이와 응고 중 결과적인 요소 혼합 및 국부적인 재합금은 재료 조합 간의 인쇄 적성에 나타난 차이점을 지배합니다.

이 작업은 서로 다른 재료의 증착에서 현상학적 차이에 대한 심층적인 이해를 제공하고 바이메탈 부품의 보다 안정적인 DED 성형을 안내하는 것을 목표로 합니다.

Additive manufacturing provides achievability for the fabrication of bimetallic and
multi-material structures; however, the material compatibility and bondability directly affect the
parts’ formability and final quality. It is essential to understand the underlying printability of
different material combinations based on an adapted process. Here, the printability disparities of
two common and attractive material combinations (nickel- and iron-based alloys) are evaluated
at the macro and micro levels via laser directed energy deposition (DED). The deposition
processes were captured using in situ high-speed imaging, and the dissimilarities in melt pool
features and track morphology were quantitatively investigated within specific process
windows. Moreover, the microstructure diversity of the tracks and blocks processed with varied
material pairs was comparatively elaborated and, complemented with the informative
multi-physics modeling, the presented non-uniformity in mechanical properties (microhardness)
among the heterogeneous material pairs was rationalized. The differences in melt flow induced
by the unlike thermophysical properties of the material pairs and the resulting element
intermixing and localized re-alloying during solidification dominate the presented dissimilarity
in printability among the material combinations. This work provides an in-depth understanding
of the phenomenological differences in the deposition of dissimilar materials and aims to guide
more reliable DED forming of bimetallic parts.

Figure 1. Experimental setup and materials. (a) Schematic of the DED process, where three types of base materials were adopted—B1
(IN718), B2 (IN625), and B3 (SS316L), and two types of powder materials were adopted—P1 (IN718) and P2 (SS316L). (b) In situ
high-speed imaging of powder flow and the SEM images of IN718 and SS316L powder particle. (c) Powder size statistics, and (d) element
composition of powder IN718 (P1) and SS316L (P2).
Figure 1. Experimental setup and materials. (a) Schematic of the DED process, where three types of base materials were adopted—B1 (IN718), B2 (IN625), and B3 (SS316L), and two types of powder materials were adopted—P1 (IN718) and P2 (SS316L). (b) In situ high-speed imaging of powder flow and the SEM images of IN718 and SS316L powder particle. (c) Powder size statistics, and (d) element composition of powder IN718 (P1) and SS316L (P2).
Figure 2. Deposition process and the track morphology. (a)–(c) Display the in situ captured tableaux of melt propagation and some physical
features during depositing for P1B1, P1B2, and P1B3, respectively. (d) The profiles of the melt pool at a frame of (t0 + 1) ms, and the flow
streamlines in the molten pool of each case. (e) The outer surface of the formed tracks, in which the colored arrows mark the scanning
direction. (f) Cross-section of the tracks. The parameter set used for in situ imaging was P-1000 W, S-600 mm·min–1, F-18 g·min–1. All the
scale bars are 2 mm.
Figure 2. Deposition process and the track morphology. (a)–(c) Display the in situ captured tableaux of melt propagation and some physical features during depositing for P1B1, P1B2, and P1B3, respectively. (d) The profiles of the melt pool at a frame of (t0 + 1) ms, and the flow streamlines in the molten pool of each case. (e) The outer surface of the formed tracks, in which the colored arrows mark the scanning direction. (f) Cross-section of the tracks. The parameter set used for in situ imaging was P-1000 W, S-600 mm·min–1, F-18 g·min–1. All the scale bars are 2 mm.


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Fig. 3. (a–c) Snapshots of the CtFD simulation of laser-beam irradiation: (a) Top, (b) longitudinal vertical cross-sectional, and (c) transversal vertical cross-sectional views. (d) z-position of the solid/liquid interface during melting and solidification.

Solute segregation in a rapidly solidified Hastelloy-X Ni-based superalloy during laser powder bed fusion investigated by phase-field simulations and computational thermal-fluid dynamics

Masayuki Okugawa ab, Kenji Saito a, Haruki Yoshima a, Katsuhiko Sawaizumi a, Sukeharu Nomoto c, Makoto Watanabe c, Takayoshi Nakano ab, Yuichiro Koizumi abShow moreAdd to MendeleyShareCite

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Solute segregation significantly affects material properties and is a critical issue in the laser powder-bed fusion (LPBF) additive manufacturing (AM) of Ni-based superalloys. To the best of our knowledge, this is the first study to demonstrate a computational thermal-fluid dynamics (CtFD) simulation coupled multi-phase-field (MPF) simulation with a multicomponent-composition model of Ni-based superalloy to predict solute segregation under solidification conditions in LPBF. The MPF simulation of the Hastelloy-X superalloy reproduced the experimentally observed submicron-sized cell structure. Significant solute segregations were formed within interdendritic regions during solidification at high cooling rates of up to 10K s-1, a characteristic feature of LPBF. Solute segregation caused a decrease in the solidus temperature (TS), with a reduction of up to 30.4 K, which increases the risk of liquation cracks during LPBF. In addition, the segregation triggers the formation of carbide phases, which increases the susceptibility to ductility dip cracking. Conversely, we found that the decrease in TS is suppressed at the melt-pool boundary regions, where re-remelting occurs during the stacking of the layer above. Controlling the re-remelting behavior is deemed to be crucial for designing crack-free alloys. Thus, we demonstrated that solute segregation at the various interfacial regions of Ni-based multicomponent alloys can be predicted by the conventional MPF simulation. The design of crack-free Ni-based superalloys can be expedited by MPF simulations of a broad range of element combinations and their concentrations in multicomponent Ni-based superalloys.

Graphical abstract


Laser powder-bed fusion, Hastelloy-X Nickel-based superalloy, solute element segregation, computational thermal-fluid dynamics simulation, phase-field method

1. Introduction

Additive manufacturing (AM) technologies have attracted considerable attention as they allow us to easily build three-dimensional (3D) parts with complex geometries. Among the wide range of available AM techniques, laser powder-bed fusion (LPBF) has emerged as a preferred technique for metal AM [1][2][3][4][5]. In LPBF, metal products are built layer-by-layer by scanning laser, which fuse metal powder particles into bulk solids.

Significant attempts have been made to integrate LPBF techniques within the aerospace industry, with a particular focus on weldable Ni-based superalloys, such as IN718 [6][7][8], IN625 [9][10], and Hastelloy-X (HX) [11][12][13][14]. Non-weldable alloys, such as IN738LC [15][16] and CMSX-4 [1][17] are also suitable for their sufficient creep resistance under higher temperature conditions. However, non-weldable alloys are difficult to build using LPBF because of their susceptibility to cracking during the process. In general, a macro solute-segregation during solidification is suppressed by the rapid cooling conditions (up to 108 K s-1) unique to the LPBF process [18]. However, the solute segregation still occurs in the interdendritic regions that are smaller than the micrometer scale [5][19][20][21]; these regions are suggested to be related to the hot cracks in LPBF-fabricated parts. Therefore, an understanding of solute segregation is essential for the fabrication of reliable LPBF-fabricated parts while avoiding cracks.

The multiphase-field (MPF) method has gained popularity for modeling the microstructure evolution and solute segregation under rapid cooling conditions [5][20][21][22][23][24][25][26][27][28]. Moreover, quantifiable predictions have been achieved by combining the MPF method with temperature distribution analysis methods such as the finite-element method (FEM) [20] and computational thermal-fluid dynamics (CtFD) simulations [28]. These aforementioned studies have used binary-approximated multicomponent systems, such as Ni–Nb binary alloys, to simulate IN718 alloys. While MPF simulations using binary alloy systems can effectively reproduce microstructure formations and segregation behaviors, the binary approximation might be affected by the chemical interactions between the removed solute elements in the target multicomponent alloy. The limit of absolute stability predicted by the Mullins-Sekerka theory [29] is also crucial because the limit velocity is close to the solidification rate in the LPBF process and is different in multicomponent and binary-approximated systems. The difference between the solidus and liquidus temperatures, ΔT0, directly determines the absolute stability according to the Mullins-Sekerka theory. For example, the ΔT0 values of IN718 and its binary-approximated Ni–5 wt.%Nb alloy are 134 K [28] and 71 K [30], respectively. The solidification rate compared to the limit of absolute stability, i.e., the relative non-equilibrium of solidification, changes by simplification of the system. It is therefore important to use the composition of the actual multicomponent system in such simulations. However, to the best of our knowledge, there is no MPF simulation using a multicomponent model coupled with a temperature analysis simulation to predict solute segregation in a Ni-based superalloy.

In this study, we demonstrate that the conventional MPF model can reproduce experimentally observed dendritic structures by performing a phase-field simulation using the temperature distribution obtained by a CtFD simulation of a multicomponent Ni-based alloy (conventional solid-solution hardening-type HX). The MPF simulation revealed that the segregation behavior of solute elements largely depends on the regions of the melt pool, such as the cell boundary, the interior of the melt-pool boundary, and heat-affected regions. The sensitivities of the various interfaces to liquation and solidification cracks are compared based on the predicted concentration distributions. Moreover, the feasibility of using the conventional MPF model for LPBF is discussed in terms of the absolute stability limit.

2. Methods

2.1. Laser-beam irradiation experiments

Rolled and recrystallized HX ingots with dimensions of 20 × 50 × 10 mm were used as the specimens for laser-irradiation experiments. The specimens were irradiated with a laser beam scanned along straight lines of 10 mm in length using a laser AM machine (EOS 290 M, EOS) equipped with a 400 W Yb-fiber laser. Irradiation was performed with a beam power of P = 300 W and a scanning speed of V = 600 mm s-1, which are the conditions generally used in the LPBF fabrication of Ni-based superalloy [8]. The corresponding line energy was 0.5 J mm-1. The samples were cut perpendicular to the beam-scanning direction for cross-sectional observation using a field-emission scanning electron microscope (FE-SEM, JEOL JSM 6500). Crystal orientation analysis was performed by electron backscatter diffraction (EBSD). The sizes of each crystal grain and their aspect ratios were evaluated by analyzing the EBSD data.

2.2. CtFD simulation

CtFD simulations of the laser-beam irradiation of HX were performed using a 3D thermo-fluid analysis software (Flow Science FLOW-3D® with Flow-3D Weld module). A Gaussian heat source model was used, in which the irradiation intensity distribution of the beam is regarded as a symmetrical Gaussian distribution over the entire beam. The distribution of the beam irradiation intensity is expressed by the following equation.(1)q̇=2ηPπR2exp−2r2R2.

Here, P is the power, R is the effective beam radius, r is the actual beam radius, and η is the beam absorption rate of the substrate. To improve the accuracy of the model, η was calculated by assuming multiple reflections using the Fresnel equation:(2)�=1−121+1−�cos�21+1+�cos�2+�2−2�cos�+2cos2��2+2�cos�+2cos2�.

ε is the Fresnel coefficient and θ is the incident angle of the laser. A local laser melt causes the vaporization of the material and results in a high vapor pressure. This vapor pressure acts as a recoil pressure on the surface, pushing the weld pool down. The recoil pressure is reproduced using the following equation.(3)precoil=Ap0exp∆HLVRTV1−TVT.

Here, p0 is the atmospheric pressure, ∆HLV is the latent heat of vaporization, R is the gas constant, and TV is the boiling point at the saturated vapor pressure. A is a ratio coefficient that is generally assumed to be 0.54, indicating that the recoil pressure due to evaporation is 54% of the vapor pressure at equilibrium on the liquid surface.

Table 1 shows the parameters used in the simulations. Most parameters were evaluated using an alloy physical property calculation software (Sente software JMatPro v11). The values in a previously published study [31] were used for the emissivity and the Stefan–Boltzmann constant, and the values for pure Ni [32] were used for the heat of vaporization and vaporization temperatures. The Fresnel coefficient, which determines the beam absorption efficiency, was used as a fitting parameter to reproduce the morphology of the experimentally observed melt region, and a Fresnel coefficient of 0.12 was used in this study.

Table 1. Parameters used in the CtFD simulations.

Density at 298.15 Kρ8.24 g cm-3[]
Liquidus temperatureTL1628.15 K[]
Solidus temperatureTS1533.15 K[]
Viscosity at TLη6.8 g m-1 s-1[]
Specific heat at 298.15 KCP0.439 J g-1 K-1[]
Thermal conductivity at 298.15 Kλ10.3 W m-1 K-1[]
Surface tension at TLγL1.85 J m-2[]
Temperature coefficient of surface tensiondγL/dT–2.5 × 10−4 J m-2 K-1[]
Stefan–Boltzmann constantσ5.67 × 10-8 W m-2 K-4[31]
Heat of fusionΔHSL2.76 × 102 J g-1[32]
Heat of vaporizationΔHLV4.29 × 10J g-1[32]
Vaporization temperatureTV3110 K[32]

Calculated using JMatPro v11.

The dimensions of the computational domain of the numerical model were 4.0 mm in the beam-scanning direction, 0.4 mm in width, and 0.3 mm in height. A uniform mesh size of 10 μm was applied throughout the computational domain. The boundary condition of continuity was applied to all boundaries except for the top surface. The temperature was initially set to 300 K. P and V were set to their experimental values, i.e., 300 W and 600 mm s-1, respectively. Solidification conditions based on the temperature gradient, G, the solidification rate, R, and the cooling rate were evaluated, and the obtained temperature distribution was used in the MPF simulations.

2.3. MPF simulation

Two-dimensional MPF simulations weakly coupled with the CtFD simulation were performed using the Microstructure Evolution Simulation Software (MICRESS) [33][34][35][36][37] with the TQ-Interface for Thermo-Calc [38]. A simplified HX alloy composition of Ni-21.4Cr-17.6Fe-0.46Mn-8.80Mo-0.39Si-0.50W-1.10Co-0.08 C (mass %) was used in this study. The Gibbs free energy and diffusion coefficient of the system were calculated using the TCNI9 thermodynamic database [39] and the MOBNi5 mobility database [40]. Τhe equilibrium phase diagram calculated using Thermo-Calc indicates that the face-centered cubic (FCC) and σ phases appear as the equilibrium solid phases [19]. However, according to the time-temperature-transformation (TTT) diagram [41], the phases are formed after the sample is maintained for tens of hours in a temperature range of 1073 to 1173 K. Therefore, only the liquid and FCC phases were assumed to appear in the MPF simulations. The simulation domain was 5 × 100 μm, and the grid size Δx and interface width were set to 0.025 and 0.1 µm, respectively. The interfacial mobility between the solid and liquid phases was set to 1.0 × 10-8 m4 J-1 s-1. Initially, one crystalline nucleus with a [100] crystal orientation was placed at the left bottom of the simulation domain, with the liquid phase occupying the remainder of the domain. The model was solidified under the temperature field distribution obtained by the CtFD simulation. The concentration distribution and crystal orientation of the solidified model were examined. The primary dendrite arm space (PDAS) was compared to the experimental PDAS measured by the cross-sectional SEM observation.

In an actual LPBF process, solidified layers are remelted and resolidified during the stacking of the one layer above, thereby greatly affecting solute element distributions in those regions. Therefore, remelting and resolidification simulations were performed to examine the effect of remelting on solute segregation. The solidified model was remelted and resolidified by applying a time-dependent temperature field shifted by 60 μm in the height direction, assuming reheating during the stacking of the upper layer (i.e., the upper 40 μm region of the simulation box was remelted and resolidified). The changes in the composition distribution and formed microstructure were investigated.

3. Results

3.1. Experimental observation of melt pool

Fig. 1 shows a cross-sectional optical microscopy image and corresponding inverse pole figure (IPF) orientation maps obtained from the laser-melted region of HX. The dashed line indicates the fusion line. A deep melted region was formed by keyhole-mode melting due to the vaporization of the metal and resultant recoil pressure. Epitaxial growth from the unmelted region was observed. Columnar crystal grains with an average diameter of 5.46 ± 0.32 μm and an aspect ratio of 3.61 ± 0.13 appeared at the melt regions (Figs. 1b–1d). In addition, crystal grains growing in the z direction could be observed in the lower center.

Fig. 1

Fig. 2a shows a cross-sectional backscattering electron image (BEI) obtained from the laser-melted region indicated by the black square in Fig. 1a. The bright particles with a diameter of approximately 2 μm observed outside the melt pool. It is well known that M6C, M23C6, σ, and μ precipitate phases are formed in Hastelloy-X [41]. These precipitates mainly consisted of Mo, Cr, Fe, and Ni; The μ and M6C phases are rich in Mo, while the σ and M23C6 phases are rich in Cr. The SEM energy dispersive X-ray spectroscopy analysis suggested that the bright particles are the stable precipitates as shown in Fig. S2 and Table S1. Conversely, there are no carbides in the melt pool. This suggests that the cooling rate is extremely high during LPBF, which prevents the formation of a stable carbide during solidification. Figs. 2b–2f show magnified BEI images at different height positions indicated in Fig. 2a. Bright regions are observed between the cells, which become fragmentary at the center of the melt pool, as indicated by the yellow arrow heads in Figs. 2e and 2f.

Fig. 2

3.2. CtFD simulation

Figs. 3a–3c show snapshots of the CtFD simulation of HX at 2.72 ms, with the temperature indicated in color. A melt pool with an elongated teardrop shape formed and keyhole-mode melting was observed at the front of the melt region. The cooling rate, temperature gradient (G), and solidification rate (R) were evaluated from the temporal change in the temperature distribution of the CtFD simulation results. The z-position of the solid/liquid interface during the melting and solidification processes is shown in Fig. 3d. The interface goes down rapidly during melting and then rises during solidification. The MPF simulation of the microstructure formation during solidification was performed using the temperature distribution. Moreover, the microstructure formation process during the fabrication of the upper layer was investigated by remelting and resolidifying the solidified layer using the same temperature distribution with a 60 μm upward shift, corresponding to the layer thickness commonly used in the LPBF of Ni-based superalloys.

Fig. 3

Figs. 4a–4c show the changes in the cooling rate, temperature gradient, and solidification rate in the center line of the melt pool parallel to the z direction. To output the solidification conditions at the solid/liquid interface in the melt pool, only the data of the mesh where the solid phase ratio was close to 0.5 were plotted. Solidification occurred where the cooling rate was in the range of 2.1 × 105–1.6 × 10K s-1G was in the range of 3.6 × 105–1.9 × 10K m-1, and R was in the range of 8.2 × 10−2–6.3 × 10−1 m s-1. The cooling rate was the highest near the fusion line and decreased as the interface approached the center of the melt region (Fig. 4a). G also exhibited the highest value in the regions near the fusion line and decreased throughout the solid/liquid interface toward the center of the melt pool (Fig. 4b). R had the lowest value near the fusion line and increased as the interface approached the center of the melt region (Fig. 4c).

Fig. 4

3.3. MPF simulations coupled with CtFD simulation

MPF simulations of solidification, remelting, and resolidification were performed using the temperature-time distribution obtained by the CtFD simulation. Fig. 5 shows the MPF solidified models colored by phase and Mo concentration. All the computational domains show the FCC phase after the solidification (Fig. 5a). Dendrites grew parallel to the heat flow direction, and solute segregations were observed in the interdendritic regions. At the bottom of the melt pool (Fig. 5d), planar interface growth occurred before the formation of primary dendrites. The bottom of the melt pool is the turning point of the solid/liquid interface from the downward motion in melting to the upward motion in solidification. Thus, the solidification rate at the boundary is zero, and is extremely low immediately above the molt-pool boundary. Here, the lower limit of the solidification rate (R) for dendritic growth can be represented by the constitutional supercooling criterion [29]Vcs = (G × DL) / ΔT, and planar interface growth occurs at R < VcsDL and ΔT denote the diffusion coefficient in the liquid and the equilibrium freezing range, respectively. The results suggest that planar interface growth occurs at the bottom of the melt pool, resulting in a dark region with a different solute element distribution. Some of the primary dendrites were diminished by competition with other dendrites. In addition, secondary dendrite arms could be seen in the upper regions (Fig. 5c), where solidification occurred at a lower cooling rate. The fragmentation of the solute segregation near the secondary dendrite arms is similar to that observed in the experimental melt pool shown in Figs. 2e and 2f, and the secondary dendrite arms are suggested to have appeared at the center of the melt region. Fig. 6 shows the PDASs measured from the MPF simulation models, compared to the experimental PDASs measured by the cross-sectional SEM observation of the laser-melted regions (Fig. 2). The PDAS obtained by the MPF simulation become larger as the solidification progress. Ghosh et al. [21] evident by the phase-field method that the PDAS decreases as the cooling rate increases under the rapid cooling conditions obtained by the finite element analysis. In this study, the cooling rate was decreased as the interface approached the center of the melt region (Fig. 4a), and the trends in PDAS changes with respect to cooling rate is same as the reported trend [21]. The simulated trends of the PDAS with the position in the melt pool agreed well with the experimental trends. However, all PDASs in the simulation were larger than those observed in the experiment at the same positions. Ode et al. [42] reported that PDAS differences between 2D and 3D MPF simulations can be represented by PDAS2D = 1.12 × PDAS3D owing to differences in the effects of the interfacial energy and diffusivity. We also performed 2D and 3D MPF simulations under the solidification conditions of G = 1.94 × 10K m-1 and R = 0.82 m s-1 (Fig. S1), and found that the PDAS from the 2D MPF simulation was 1.26 times larger than that from the 3D simulation. Therefore, the cell structure obtained by the CtFD simulation coupled with the 2D MPF simulation agreed well with the experimental results over the entire melt pool region considering the dimensional effects.

Fig. 5
Fig. 6

Fig. 7b1 and 7c1 show the concentration profiles of the solidified model along the growth direction indicated by dashed lines in Fig. 7a. The differences in concentrations from the alloy composition are also shown in Fig. 7b2 and 7c2. Cr, Mo, C, Mn, and W were segregated to the interdendritic regions, while Si, Fe, and Co were depressed. The solute segregation behavior agrees with the experimentally observation [43] and the prediction by the Scheil-Gulliver simulation [19]. Segregation occurred to the highest degree in Mo, while the ratio of segregation to the alloy composition was remarkable in C. The concentration fluctuations correlated with the position in the melt pool and decreased at the center of the melt pool, which was suggested to correspond to the lower cooling rate in this region. Conversely, droplets that appeared between secondary dendrite arms in the upper regions of the simulation domain exhibited a locally high segregation of solute elements, with the same amount of segregation as that at the bottom of the melt pool.

Fig. 7

3.4. Remelting and resolidification simulation

The solidified model was subjected to remelting and resolidification conditions by shifting the temperature profile upward by 60 µm to reveal the effect of reheating on the solute segregation behavior. Figs. 8a and 8b shows the simulation domains of the HX model after resolidification, colored by phase and Mo concentration. The magnified MPF models during the resolidification of the regions indicated by rectangles in Figs. 8a and 8b are also shown as Figs. 8c and 8d. Dendrites grew from the bottom of the remelted region, with the segregation of solute elements occurring in the interdendritic regions. The entire domain become the FCC phase after the resolidification, as shown in Fig. 8a. The bottom of the remelted regions exhibited a different microstructure, and Mo was depressed at the remelted regions, rather than the interdendritic regions. The different solute segregation behavior [44] and the microstructure formation [45] at the melt pool boundary is also observed in LPBF manufactured 316 L stainless steel. We found that this microstructure was formed by further remelting during the resolidification process, which is shown in Fig. 9. Here, the solidified HX model was heated, and the interdendritic regions were preferentially melted while concentration fluctuations were maintained (Fig. 9a1 and 9a2). Subsequently, planer interface growth occurs near the melt pool boundary where the solidification rate is almost zero, and the dendrites outside of the boundary are grown epitaxially (Fig. 9b1 and 9b2). However, these remelted again because of the temperature rise (Fig. 9c1 and 9c2, and the temperature-time profile shown in Fig. 9e). The remelted regions then cooled and solidified with the abnormal solute segregations (Fig. 9d1 and 9d2). Then, dendrite grows from amplified fluctuations under the solidification rate larger than the criterion of constitutional supercooling (Fig. 9d1, 9d2, and Fig. 8d). It has been reported [46][47] that temperature rising owning to latent heat affects microstructure formation: phase-field simulations of a Ni–Al binary alloy suggest that the release of latent heat during solidification increases the average temperature of the system [46] and strongly influences the solidification conditions [47]. In this study, the release of latent heat during solidification is considered in CtFD simulations for calculating the temperature distribution, and the temperature increase is suggested to have also occurred due to the release of latent heat.

Fig. 8
Fig. 9

Fig. 10b1 and 10c1 show the solute element concentration line profiles of the resolidified model along the growth direction indicated by dashed lines in Fig. 10a. Fig. 10b2 and 10c2 show the corresponding differences in concentration from the alloy composition. The segregation behavior of solute elements at the interdendritic regions (Fig. 10b1 and 10b2) was the same as that in the solidified model (Figs. 7b1 and 7b2). Here, Cr, Mo, C, Mn, and W were segregated to the interdendritic regions, while Si, Fe, and Co were depressed. However, the concentration fluctuations at the interdendritic regions were larger than those in the solidified model. Moreover, the segregation of the outside of the melt pool, i.e., the heat-affected zone, was remarkable throughout remelting and resolidification. Different segregation behaviors were observed in the re-remelted region: Mo, Si, Mn, and W were segregated, while Ni, Fe, and Co were depressed. These solute segregations caused by remelting are expected to heavily influence the crack behavior.

Fig. 10

4. Discussion

4.1. Effect of segregation of solute elements on liquation cracking susceptibility

Strong solute segregation was observed between the interdendritic regions of the solidified alloy (Fig. 7). In addition, the solute segregation behavior was significantly affected by remelting and resolidification and varied across the alloy. Solute segregation can be categorized by the regions shown in Fig. 11a1–11a4, namely the cell boundary (Fig. 11a1), interior of the melt-pool boundary (Fig. 11a2), re-remelted regions (Fig. 11a3), and heat-affected regions (Fig. 11a4). The concentration profiles of these regions are shown in Fig. 11b1–11b4. Solute segregation was the highest in the cell boundary region. The solute segregation in the heat-affected region was almost the same as that in the cell boundary region, but seemed to have been attenuated by reheating during remelting and resolidification. The interior of the melt-pool boundary region also had the same tendency for solute segregation. However, the amount of Cr segregation was smaller than that of Mo. A decrease in the Cr concentration was also mitigated, and the concentration remained the same as that in the alloy composition. Fig. 11c1–11c4 show the chemical potentials of the solute elements for the FCC phase at 1073 K calculated using the compositions of those interfacial regions. All the interfacial regions showed non-constant chemical potentials for each element along the perpendicular direction, but the fluctuations of the chemical potentials differed by the type of interfaces. In particular, the fluctuation of the chemical potential of C at the cell boundary region was the largest, suggesting it can be relaxed easily by heat treatment. On the other hand, the fluctuations of the other elements in all the regions were small. The solute segregations are most likely to remain after the heat treatment and are supposed to affect the cracking susceptibilities.

Fig. 11

The solidus temperatures TS, the difference between the liquidus and solidus temperatures (i.e., the brittle temperature range (BTR)), and the fractions of the equilibrium precipitate phases at 1073 K of the interfacial regions were calculated as the liquation, solidification, and ductility dip cracking susceptibilities, respectively. At the cell boundary (Fig. 12a1), interior of the melt-pool boundary (Fig. 12a1), and heat-affected regions (Fig. 12a1), the internal and interfacial regions exhibited higher and lower TS compared to that of the alloy composition, respectively. The lowest Ts was obtained with the composition at the cell boundary region, which is the largest solute-segregated region. It has been suggested that strong segregations of solute elements in LPBF lead to liquation cracks [16]. This study also supports this suggestion, and liquation cracks are more likely to occur at the interfacial regions indicated by predicting the solute segregation behavior using the MPF model. Additionally, the BTRs of the cell boundary, interior of the melt-pool boundary, and heat-affected regions were wider at the interdendritic regions, and solidification cracks were also likely to occur in these regions. Moreover, within the solute segregation regions, the fraction of the precipitate phases in these interfacial regions was larger than that calculated using the alloy composition (Fig. 12c1, 12c2, and 12c4). This indicates that ductility dip cracking is also likely to occur at the cell boundary, interior of the melt-pool boundary, and in heat-affected regions. Contrarily, we found that the re-remelted region exhibited a higher TS and smaller BTR even in the interfacial region (Fig. 12a3 and 12b3), where the solute segregation behavior was different from that of the other regions. In addition, the re-remelting region exhibited less precipitation compared with the other segregated regions (Fig. 12c3). The re-remelting caused by the latent heat can attenuate solute segregation, prevent Ts from decreasing, decrease the BTR, and decrease the amount of precipitate phases. Alloys with a large amount of latent heat are expected to increase the re-remelting region, thereby decreasing the susceptibility to liquation and ductility dip cracks due to solute element segregation. This can be a guide for designing alloys for the LPBF process. As mentioned in Section 3.4, the microstructure [45] and the solute segregation behavior [44] at the melt pool boundary of LPBF-manufactured 316 L stainless steel are observed, and they are different from that of the interdendritic regions. Experimental observations of the solute segregation behavior in the LPBF-fabricated Ni-based alloys are currently underway.

Fig. 12

4.2. Applicability of the conventional MPF simulation to microstructure formation under LPBF

As the solidification growth rate increases, segregation coefficients approach 1, and the fluctuation of the solid/liquid interface is suppressed by the interfacial tension. The interface growth occurs in a flat fashion instead of having a cellular morphology at a velocity above the absolute stability limit, Ras, predicted by the Mullins-Sekerka theory [29]Ras = (ΔT0 DL) / (k Γ) where ΔT0DLk, and Γ are the difference between the liquidus and solidus temperatures, equilibrium segregation coefficient, the diffusivity of liquid, and the Gibbs-Thomson coefficient, respectively.

The Ras of HX was calculated using the equation and the thermodynamic parameters obtained by the TCNI9 thermodynamic database [39]. The calculated Ras of HX was 3.9 m s-1 and is ten times larger than that of the Ni–Nb alloy (approximately 0.4 m s-1[20]. The HX alloy was solidified under R values in the range of 8.2 × 10−2–6.3 × 10−1 m s-1. The theoretically calculated criterion is larger than the evaluated R, and is in agreement with the experiment in which dendritic growth is observed in the melt pool (Fig. 5). In contrast, Karayagiz et al. [20] reported that the R of the Ni–Nb binary alloy under LPBF was as high as approximately 2 m s-1, and planar interface growth was observed to be predominant under the high-growth-rate conditions. These experimentally observed microstructures agree well with the prediction by the Mullins-Sekerka theory about the relationship between the morphology and solidification rates.

In this study, the solidification microstructure formed by the laser-beam irradiation of an HX multicomponent Ni-based superalloy was reproduced by a conventional MPF simulation, in which the system was assumed to be in a quasi-equilibrium condition. Boussinot et al. [24] also suggested that the conventional phase-field model can be applied to simulate the microstructure of an IN718 multicomponent Ni-based superalloy in LPBF. In contrast, Kagayaski et al. [20] suggested that the conventional MPF simulation cannot be applied to the solidification of the Ni-Nb binary alloy system and that the finite interface dissipation model proposed by Steinbach et al. [48][49] is necessary to simulate the high solidification rates observed in LPBF. The difference in the applicability of the conventional MPF method to HX and Ni–Nb binary alloys is presumed to arise from the differences in the non-equilibrium degree of these systems under the high solidification rates of LPBF. The results suggest that Ras can be used as a simple index to apply the conventional MPF model for solidification in LPBF. Solidification becomes a non-equilibrium process as the solidification rate approaches the limit of absolute stability, Ras. In this study, the solidification of the HX multicomponent system occurred under a relatively low solidification rate compared to Ras, and the microstructure of the conventional MPF model was successfully reproduced in the physical experiment. However, note that the limit of absolute stability predicted by the Mullins-Sekerka theory was originally proposed for solidification in a binary alloy system, and further investigation is required to consider its applicability to multicomponent alloy systems. Moreover, the fast solidification, such as in the LPBF process, causes segregation coefficient approaching a value of 1 [20][21][25] corresponds to a diffusion length that is on the order of the atomic interface thickness. When the segregation coefficient approaches 1, solute undercooling disappears; hence, there is no driving force to amplify fluctuations regardless of whether interfacial tension is present. This phenomenon should be further investigated in future studies.

5. Conclusions

We simulated solute segregation in a multicomponent HX alloy under the LPBF process by an MPF simulation using the temperature distributions obtained by a CtFD simulation. We set the parameters of the CtFD simulation to match the melt pool shape formed in the laser-irradiation experiment and found that solidification occurred under high cooling rates of up to 1.6 × 10K s-1.

MPF simulations using the temperature distributions from CtFD simulation could reproduce the experimentally observed PDAS and revealed that significant solute segregation occurred at the interdendritic regions. Equilibrium thermodynamic calculations using the alloy compositions of the segregated regions when considering crack sensitivities suggested a decrease in the solidus temperature and an increase in the amount of carbide precipitation, thereby increasing the susceptibility to liquation and ductility dip cracks in these regions. Notably, these changes were suppressed at the melt-pool boundary region, where re-remelting occurred during the stacking of the layer above. This effect can be used to achieve a novel in-process segregation attenuation.

Our study revealed that a conventional MPF simulation weakly coupled with a CtFD simulation can be used to study the solidification of multicomponent alloys in LPBF, contrary to the cases of binary alloys investigated in previous studies. We discussed the applicability of the conventional MPF model to the LPBF process in terms of the limit of absolute stability, Ras, and suggested that alloys with a high limit velocity, i.e., multicomponent alloys, can be simulated using the conventional MPF model even under the high solidification velocity conditions of LPBF.

CRediT authorship contribution statement

Masayuki Okugawa: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Takayoshi Nakano: Writing – review & editing, Validation, Supervision, Funding acquisition. Yuichiro Koizumi: Writing – review & editing, Visualization, Validation, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. Sukeharu Nomoto: Writing – review & editing, Validation, Investigation. Makoto Watanabe: Writing – review & editing, Validation, Supervision, Funding acquisition. Katsuhiko Sawaizumi: Validation, Software, Investigation, Formal analysis, Data curation. Kenji Saito: Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. Haruki Yoshima: Visualization, Validation, Software, Investigation, Formal analysis, Data curation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper


This work was partly supported by the Cabinet Office, Government of Japan, Cross-ministerial Strategic Innovation Promotion Program (SIP), “Materials Integration for Revolutionary Design System of Structural Materials,” (funding agency: The Japan Science and Technology Agency), by JSPS KAKENHI Grant Numbers 21H05018 and 21H05193, and by CREST Nanomechanics: Elucidation of macroscale mechanical properties based on understanding nanoscale dynamics for innovative mechanical materials (Grant Number: JPMJCR2194) from the Japan Science and Technology Agency (JST). The authors would like to thank Mr. H. Kawabata and Mr. K. Kimura for their technical support with the sample preparations and laser beam irradiation experiments.

Appendix A. Supplementary material

Download : Download Word document (654KB)

Supplementary material.

Data availability

Data will be made available on request.


Figure 5. Simulation of the molten pool under low-speed scanning (1.06 m/s). (a) Sequential solidification of the molten pool at the end of the melt track for laser powers of 190 and 340 W, respectively. (b) Recoil pressure on the molten pool at the keyhole for laser powers of 190 and 340 W, respectively. (c) The force diagram of the melt at the back of the keyhole at t = 750 μs in case B. (d) Temperature gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case A. (e) Temperature gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case B.

Revealing formation mechanism of end of processdepression in laser powder bed fusion by multiphysics meso-scale simulation

다중물리 메조 규모 시뮬레이션을 통해 레이저 분말층 융합에서 공정 종료의 함몰 형성 메커니즘 공개

Haodong Chen a,b, Xin Lin a,b,c, Yajing Sund, Shuhao Wanga,b, Kunpeng Zhu a,b,c and Binbin Dana,b

To link to this article:


Unintended end-of-process depression (EOPD) commonly occurs in laser powder bed fusion (LPBF), leading to poor surface quality and lower fatigue strength, especially for many implants. In this study, a high-fidelity multi-physics meso-scale simulation model is developed to uncover the forming mechanism of this defect. A defect-process map of the EOPD phenomenon is obtained using this simulation model. It is found that the EOPD formation mechanisms are different under distinct regions of process parameters. At low scanning speeds in keyhole mode, the long-lasting recoil pressure and the large temperature gradient easily induce EOPD. While at high scanning speeds in keyhole mode, the shallow molten pool morphology and the large solidification rate allow the keyhole to evolve into an EOPD quickly. Nevertheless, in the conduction mode, the Marangoni effects along with a faster solidification rate induce EOPD. Finally, a ‘step’ variable power strategy is proposed to optimise the EOPD defects for the case with high volumetric energy density at low scanning speeds. This work provides a profound understanding and valuable insights into the quality control of LPBF fabrication.

의도하지 않은 공정 종료 후 함몰(EOPD)은 LPBF(레이저 분말층 융합)에서 흔히 발생하며, 특히 많은 임플란트의 경우 표면 품질이 떨어지고 피로 강도가 낮아집니다. 본 연구에서는 이 결함의 형성 메커니즘을 밝히기 위해 충실도가 높은 다중 물리학 메조 규모 시뮬레이션 모델을 개발했습니다.

이 시뮬레이션 모델을 사용하여 EOPD 현상의 결함 프로세스 맵을 얻습니다. EOPD 형성 메커니즘은 공정 매개변수의 별개 영역에서 서로 다른 것으로 밝혀졌습니다.

키홀 모드의 낮은 스캔 속도에서는 오래 지속되는 반동 압력과 큰 온도 구배로 인해 EOPD가 쉽게 유발됩니다. 키홀 모드에서 높은 스캐닝 속도를 유지하는 동안 얕은 용융 풀 형태와 큰 응고 속도로 인해 키홀이 EOPD로 빠르게 진화할 수 있습니다.

그럼에도 불구하고 전도 모드에서는 더 빠른 응고 속도와 함께 마랑고니 효과가 EOPD를 유발합니다. 마지막으로, 낮은 스캐닝 속도에서 높은 체적 에너지 밀도를 갖는 경우에 대해 EOPD 결함을 최적화하기 위한 ‘단계’ 가변 전력 전략이 제안되었습니다.

이 작업은 LPBF 제조의 품질 관리에 대한 심오한 이해와 귀중한 통찰력을 제공합니다.

Figure 5. Simulation of the molten pool under low-speed scanning (1.06 m/s). (a) Sequential solidification of the molten pool at the
end of the melt track for laser powers of 190 and 340 W, respectively. (b) Recoil pressure on the molten pool at the keyhole for laser
powers of 190 and 340 W, respectively. (c) The force diagram of the melt at the back of the keyhole at t = 750 μs in case B. (d) Temperature gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case A. (e) Temperature
gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case B.
Figure 5. Simulation of the molten pool under low-speed scanning (1.06 m/s). (a) Sequential solidification of the molten pool at the end of the melt track for laser powers of 190 and 340 W, respectively. (b) Recoil pressure on the molten pool at the keyhole for laser powers of 190 and 340 W, respectively. (c) The force diagram of the melt at the back of the keyhole at t = 750 μs in case B. (d) Temperature gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case A. (e) Temperature gradient at the solid–liquid interface of the molten pool at the moment the laser is deactivated in case B.


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Schematic diagram of HP-LPBF melting process.

Modeling and numerical studies of high-precision laser powder bed fusion

Yi Wei ;Genyu Chen;Nengru Tao;Wei Zhou

In order to comprehensively reveal the evolutionary dynamics of the molten pool and the state of motion of the fluid during the high-precision laser powder bed fusion (HP-LPBF) process, this study aims to deeply investigate the specific manifestations of the multiphase flow, solidification phenomena, and heat transfer during the process by means of numerical simulation methods. Numerical simulation models of SS316L single-layer HP-LPBF formation with single and double tracks were constructed using the discrete element method and the computational fluid dynamics method. The effects of various factors such as Marangoni convection, surface tension, vapor recoil, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool have been paid attention to during the model construction process. The results show that the molten pool exhibits a “comet” shape, in which the temperature gradient at the front end of the pool is significantly larger than that at the tail end, with the highest temperature gradient up to 1.69 × 108 K/s. It is also found that the depth of the second track is larger than that of the first one, and the process parameter window has been determined preliminarily. In addition, the application of HP-LPBF technology helps to reduce the surface roughness and minimize the forming size.


Heat transferNonequilibrium thermodynamicsSolidification processComputer simulationDiscrete element methodLasersMass transferFluid mechanicsComputational fluid dynamicsMultiphase flows


Laser powder bed fusion (LPBF) has become a research hotspot in the field of additive manufacturing of metals due to its advantages of high-dimensional accuracy, good surface quality, high density, and high material utilization.1,2 With the rapid development of electronics, medical, automotive, biotechnology, energy, communication, and optics, the demand for microfabrication technology is increasing day by day.3 High-precision laser powder bed fusion (HP-LPBF) is one of the key manufacturing technologies for tiny parts in the fields of electronics, medical, automotive, biotechnology, energy, communication, and optics because of its process characteristics such as small focal spot diameter, small powder particle size, and thin powder layup layer thickness.4–13 Compared with LPBF, HP-LPBF has the significant advantages of smaller focal spot diameter, smaller powder particle size, and thinner layer thickness. These advantages make HP-LPBF perform better in producing micro-fine parts, high surface quality, and parts with excellent mechanical properties.

HP-LPBF is in the exploratory stage, and researchers have already done some exploratory studies on the focal spot diameter, the amount of defocusing, and the powder particle size. In order to explore the influence of changing the laser focal spot diameter on the LPBF process characteristics of the law, Wildman et al.14 studied five groups of different focal spot diameter LPBF forming 316L stainless steel (SS316L) processing effect, the smallest focal spot diameter of 26 μm, and the results confirm that changing the focal spot diameter can be achieved to achieve the energy control, so as to control the quality of forming. Subsequently, Mclouth et al.15 proposed the laser out-of-focus amount (focal spot diameter) parameter, which characterizes the distance between the forming plane and the laser focal plane. The laser energy density was controlled by varying the defocusing amount while keeping the laser parameters constant. Sample preparation at different focal positions was investigated, and their microstructures were characterized. The results show that the samples at the focal plane have finer microstructure than those away from the focal plane, which is the effect of higher power density and smaller focal spot diameter. In order to explore the influence of changing the powder particle size on the characteristics of the LPBF process, Qian et al.16 carried out single-track scanning simulations on powder beds with average powder particle sizes of 70 and 40 μm, respectively, and the results showed that the melt tracks sizes were close to each other under the same process parameters for the two particle-size distributions and that the molten pool of powder beds with small particles was more elongated and the edges of the melt tracks were relatively flat. In order to explore the superiority of HP-LPBF technology, Xu et al.17 conducted a comparative analysis of HP-LPBF and conventional LPBF of SS316L. The results showed that the average surface roughness of the top surface after forming by HP-LPBF could reach 3.40 μm. Once again, it was verified that HP-LPBF had higher forming quality than conventional LPBF. On this basis, Wei et al.6 comparatively analyzed the effects of different laser focal spot diameters on different powder particle sizes formed by LPBF. The results showed that the smaller the laser focal spot diameter, the fewer the defects on the top and side surfaces. The above research results confirm that reducing the laser focal spot diameter can obtain higher energy density and thus better forming quality.

LPBF involves a variety of complex systems and mechanisms, and the final quality of the part is influenced by a large number of process parameters.18–24 Some research results have shown that there are more than 50 factors affecting the quality of the specimen. The influencing factors are mainly categorized into three main groups: (1) laser parameters, (2) powder parameters, and (3) equipment parameters, which interact with each other to determine the final specimen quality. With the continuous development of technologies such as computational materials science and computational fluid dynamics (CFD), the method of studying the influence of different factors on the forming quality of LPBF forming process has been shifted from time-consuming and laborious experimental characterization to the use of numerical simulation methods. As a result, more and more researchers are adopting this approach for their studies. Currently, numerical simulation studies on LPBF are mainly focused on the exploration of molten pool, temperature distribution, and residual stresses.

  1. Finite element simulation based on continuum mechanics and free surface fluid flow modeling based on fluid dynamics are two common approaches to study the behavior of LPBF molten pool.25–28 Finite element simulation focuses on the temperature and thermal stress fields, treats the powder bed as a continuum, and determines the molten pool size by plotting the elemental temperature above the melting point. In contrast, fluid dynamics modeling can simulate the 2D or 3D morphology of the metal powder pile and obtain the powder size and distribution by certain algorithms.29 The flow in the molten pool is mainly affected by recoil pressure and the Marangoni effect. By simulating the molten pool formation, it is possible to predict defects, molten pool shape, and flow characteristics, as well as the effect of process parameters on the molten pool geometry.30–34 In addition, other researchers have been conducted to optimize the laser processing parameters through different simulation methods and experimental data.35–46 Crystal growth during solidification is studied to further understand the effect of laser parameters on dendritic morphology and solute segregation.47–54 A multi-scale system has been developed to describe the fused deposition process during 3D printing, which is combined with the conductive heat transfer model and the dendritic solidification model.55,56
  2. Relevant scholars have adopted various different methods for simulation, such as sequential coupling theory,57 Lagrangian and Eulerian thermal models,58 birth–death element method,25 and finite element method,59 in order to reveal the physical phenomena of the laser melting process and optimize the process parameters. Luo et al.60 compared the LPBF temperature field and molten pool under double ellipsoidal and Gaussian heat sources by ANSYS APDL and found that the diffusion of the laser energy in the powder significantly affects the molten pool size and the temperature field.
  3. The thermal stresses obtained from the simulation correlate with the actual cracks,61 and local preheating can effectively reduce the residual stresses.62 A three-dimensional thermodynamic finite element model investigated the temperature and stress variations during laser-assisted fabrication and found that powder-to-solid conversion increases the temperature gradient, stresses, and warpage.63 Other scholars have predicted residual stresses and part deflection for LPBF specimens and investigated the effects of deposition pattern, heat, laser power, and scanning strategy on residual stresses, noting that high-temperature gradients lead to higher residual stresses.64–67 

In short, the process of LPBF forming SS316L is extremely complex and usually involves drastic multi-scale physicochemical changes that will only take place on a very small scale. Existing literature employs DEM-based mesoscopic-scale numerical simulations to investigate the effects of process parameters on the molten pool dynamics of LPBF-formed SS316L. However, a few studies have been reported on the key mechanisms of heating and solidification, spatter, and convective behavior of the molten pool of HP-LPBF-formed SS316L with small laser focal spot diameters. In this paper, the geometrical properties of coarse and fine powder particles under three-dimensional conditions were first calculated using DEM. Then, numerical simulation models for single-track and double-track cases in the single-layer HP-LPBF forming SS316L process were developed at mesoscopic scale using the CFD method. The flow genesis of the melt in the single-track and double-track molten pools is discussed, and their 3D morphology and dimensional characteristics are discussed. In addition, the effects of laser process parameters, powder particle size, and laser focal spot diameter on the temperature field, characterization information, and defects in the molten pool are discussed.


A. 3D powder bed modeling

HP-LPBF is an advanced processing technique for preparing target parts layer by layer stacking, the process of which involves repetitive spreading and melting of powders. In this process, both the powder spreading and the morphology of the powder bed are closely related to the results of the subsequent melting process, while the melted surface also affects the uniform distribution of the next layer of powder. For this reason, this chapter focuses on the modeling of the physical action during the powder spreading process and the theory of DEM to establish the numerical model of the powder bed, so as to lay a solid foundation for the accuracy of volume of fluid (VOF) and CFD.

1. DEM

DEM is a numerical technique for calculating the interaction of a large number of particles, which calculates the forces and motions of the spheres by considering each powder sphere as an independent unit. The motion of the powder particles follows the laws of classical Newtonian mechanics, including translational and rotational,38,68–70 which are expressed as follows:����¨=���+∑��ij,



where �� is the mass of unit particle i in kg, ��¨ is the advective acceleration in m/s2, And g is the gravitational acceleration in m/s2. �ij is the force in contact with the neighboring particle � in N. �� is the rotational inertia of the unit particle � in kg · m2. ��¨ is the unit particle � angular acceleration in rad/s2. �ij is the vector pointing from unit particle � to the contact point of neighboring particle �⁠.

Equations (1) and (2) can be used to calculate the velocity and angular velocity variations of powder particles to determine their positions and velocities. A three-dimensional powder bed model of SS316L was developed using DEM. The powder particles are assumed to be perfect spheres, and the substrate and walls are assumed to be rigid. To describe the contact between the powder particles and between the particles and the substrate, a non-slip Hertz–Mindlin nonlinear spring-damping model71 was used with the following expression:�hz=��������+��[(�����ij−�eff����)−(�����+�eff����)],


where �hz is the force calculated using the Hertzian in M. �� and �� are the radius of unit particles � and � in m, respectively. �� is the overlap size of the two powder particles in m. ��⁠, �� are the elastic constants in the normal and tangential directions, respectively. �ij is the unit vector connecting the centerlines of the two powder particles. �eff is the effective mass of the two powder particles in kg. �� and �� are the viscoelastic damping constants in the normal and tangential directions, respectively. �� and �� are the components of the relative velocities of the two powder particles. ��� is the displacement vector between two spherical particles. The schematic diagram of overlapping powder particles is shown in Fig. 1.

FIG. 1.


Schematic diagram of overlapping powder particles.

Because the particle size of the powder used for HP-LPBF is much smaller than 100 μm, the effect of van der Waals forces must be considered. Therefore, the cohesive force �jkr of the Hertz–Mindlin model was used instead of van der Waals forces,72 with the following expression:�jkr=−4��0�*�1.5+4�*3�*�3,




where �* is the equivalent Young’s modulus in GPa; �* is the equivalent particle radius in m; �0 is the surface energy of the powder particles in J/m2; α is the contact radius in m; �� and �� are the Young’s modulus of the unit particles � and �⁠, respectively, in GPa; and �� and �� are the Poisson’s ratio of the unit particles � and �⁠, respectively.

2. Model building

Figure 2 shows a 3D powder bed model generated using DEM with a coarse powder geometry of 1000 × 400 × 30 μm3. The powder layer thickness is 30 μm, and the powder bed porosity is 40%. The average particle size of this spherical powder is 31.7 μm and is normally distributed in the range of 15–53 μm. The geometry of the fine powder was 1000 × 400 × 20 μm3, with a layer thickness of 20 μm, and the powder bed porosity of 40%. The average particle size of this spherical powder is 11.5 μm and is normally distributed in the range of 5–25 μm. After the 3D powder bed model is generated, it needs to be imported into the CFD simulation software for calculation, and the imported geometric model is shown in Fig. 3. This geometric model is mainly composed of three parts: protective gas, powder bed, and substrate. Under the premise of ensuring the accuracy of the calculation, the mesh size is set to 3 μm, and the total number of coarse powder meshes is 1 704 940. The total number of fine powder meshes is 3 982 250.

FIG. 2.


Three-dimensional powder bed model: (a) coarse powder, (b) fine powder.

FIG. 3.


Geometric modeling of the powder bed computational domain: (a) coarse powder, (b) fine powder.

B. Modeling of fluid mechanics simulation

In order to solve the flow, melting, and solidification problems involved in HP-LPBF molten pool, the study must follow the three governing equations of conservation of mass, conservation of energy, and conservation of momentum.73 The VOF method, which is the most widely used in fluid dynamics, is used to solve the molten pool dynamics model.

1. VOF

VOF is a method for tracking the free interface between the gas and liquid phases on the molten pool surface. The core idea of the method is to define a volume fraction function F within each grid, indicating the proportion of the grid space occupied by the material, 0 ≤ F ≤ 1 in Fig. 4. Specifically, when F = 0, the grid is empty and belongs to the gas-phase region; when F = 1, the grid is completely filled with material and belongs to the liquid-phase region; and when 0 < F < 1, the grid contains free surfaces and belongs to the mixed region. The direction normal to the free surface is the direction of the fastest change in the volume fraction F (the direction of the gradient of the volume fraction), and the direction of the gradient of the volume fraction can be calculated from the values of the volume fractions in the neighboring grids.74 The equations controlling the VOF are expressed as follows:𝛻����+�⋅(��→)=0,


where t is the time in s and �→ is the liquid velocity in m/s.

FIG. 4.


Schematic diagram of VOF.

The material parameters of the mixing zone are altered due to the inclusion of both the gas and liquid phases. Therefore, in order to represent the density of the mixing zone, the average density �¯ is used, which is expressed as follows:72�¯=(1−�1)�gas+�1�metal,


where �1 is the proportion of liquid phase, �gas is the density of protective gas in kg/m3, and �metal is the density of metal in kg/m3.

2. Control equations and boundary conditions

Figure 5 is a schematic diagram of the HP-LPBF melting process. First, the laser light strikes a localized area of the material and rapidly heats up the area. Next, the energy absorbed in the region is diffused through a variety of pathways (heat conduction, heat convection, and surface radiation), and this process triggers complex phase transition phenomena (melting, evaporation, and solidification). In metals undergoing melting, the driving forces include surface tension and the Marangoni effect, recoil due to evaporation, and buoyancy due to gravity and uneven density. The above physical phenomena interact with each other and do not occur independently.

FIG. 5.


Schematic diagram of HP-LPBF melting process.

  1. Laser heat sourceThe Gaussian surface heat source model is used as the laser heat source model with the following expression:�=2�0����2exp(−2�12��2),(9)where � is the heat flow density in W/m2, �0 is the absorption rate of SS316L, �� is the radius of the laser focal spot in m, and �1 is the radial distance from the center of the laser focal spot in m. The laser focal spot can be used for a wide range of applications.
  2. Energy absorptionThe formula for calculating the laser absorption �0 of SS316L is as follows:�0=0.365(�0[1+�0(�−20)]/�)0.5,(10)where �0 is the direct current resistivity of SS316L at 20 °C in Ω m, �0 is the resistance temperature coefficient in ppm/°C, � is the temperature in °C, and � is the laser wavelength in m.
  3. Heat transferThe basic principle of heat transfer is conservation of energy, which is expressed as follows:𝛻𝛻𝛻�(��)��+�·(��→�)=�·(�0����)+��,(11)where � is the density of liquid phase SS316L in kg/m3, �� is the specific heat capacity of SS316L in J/(kg K), 𝛻� is the gradient operator, t is the time in s, T is the temperature in K, 𝛻�� is the temperature gradient, �→ is the velocity vector, �0 is the coefficient of thermal conduction of SS316L in W/(m K), and  �� is the thermal energy dissipation term in the molten pool.
  4. Molten pool flowThe following three conditions need to be satisfied for the molten pool to flow:
    • Conservation of mass with the following expression:𝛻�·(��→)=0.(12)
    • Conservation of momentum (Navier–Stokes equation) with the following expression:𝛻𝛻𝛻𝛻���→��+�(�→·�)�→=�·[−pI+�(��→+(��→)�)]+�,(13)where � is the pressure in Pa exerted on the liquid phase SS316L microelement, � is the unit matrix, � is the fluid viscosity in N s/m2, and � is the volumetric force (gravity, atmospheric pressure, surface tension, vapor recoil, and the Marangoni effect).
    • Conservation of energy, see Eq. (11)
  5. Surface tension and the Marangoni effectThe effect of temperature on the surface tension coefficient is considered and set as a linear relationship with the following expression:�=�0−��dT(�−��),(14)where � is the surface tension of the molten pool at temperature T in N/m, �� is the melting temperature of SS316L in K, �0 is the surface tension of the molten pool at temperature �� in Pa, and σdσ/ dT is the surface tension temperature coefficient in N/(m K).In general, surface tension decreases with increasing temperature. A temperature gradient causes a gradient in surface tension that drives the liquid to flow, known as the Marangoni effect.
  6. Metal vapor recoilAt higher input energy densities, the maximum temperature of the molten pool surface reaches the evaporation temperature of the material, and a gasification recoil pressure occurs vertically downward toward the molten pool surface, which will be the dominant driving force for the molten pool flow.75 The expression is as follows:��=0.54�� exp ���−���0���,(15)where �� is the gasification recoil pressure in Pa, �� is the ambient pressure in kPa, �� is the latent heat of evaporation in J/kg, �0 is the gas constant in J/(mol K), T is the surface temperature of the molten pool in K, and Te is the evaporation temperature in K.
  7. Solid–liquid–gas phase transitionWhen the laser hits the powder layer, the powder goes through three stages: heating, melting, and solidification. During the solidification phase, mutual transformations between solid, liquid, and gaseous states occur. At this point, the latent heat of phase transition absorbed or released during the phase transition needs to be considered.68 The phase transition is represented based on the relationship between energy and temperature with the following expression:�=�����,(�<��),�(��)+�−����−����,(��<�<��)�(��)+(�−��)����,(��<�),,(16)where �� and �� are solid and liquid phase density, respectively, of SS316L in kg/m3. �� and �� unit volume of solid and liquid phase-specific heat capacity, respectively, of SS316L in J/(kg K). �� and ��⁠, respectively, are the solidification temperature and melting temperature of SS316L in K. �� is the latent heat of the phase transition of SS316L melting in J/kg.

3. Assumptions

The CFD model was computed using the commercial software package FLOW-3D.76 In order to simplify the calculation and solution process while ensuring the accuracy of the results, the model makes the following assumptions:

  1. It is assumed that the effects of thermal stress and material solid-phase thermal expansion on the calculation results are negligible.
  2. The molten pool flow is assumed to be a Newtonian incompressible laminar flow, while the effects of liquid thermal expansion and density on the results are neglected.
  3. It is assumed that the surface tension can be simplified to an equivalent pressure acting on the free surface of the molten pool, and the effect of chemical composition on the results is negligible.
  4. Neglecting the effect of the gas flow field on the molten pool.
  5. The mass loss due to evaporation of the liquid metal is not considered.
  6. The influence of the plasma effect of the molten metal on the calculation results is neglected.

It is worth noting that the formulation of assumptions requires a trade-off between accuracy and computational efficiency. In the above models, some physical phenomena that have a small effect or high difficulty on the calculation results are simplified or ignored. Such simplifications make numerical simulations more efficient and computationally tractable, while still yielding accurate results.

4. Initial conditions

The preheating temperature of the substrate was set to 393 K, at which time all materials were in the solid state and the flow rate was zero.

5. Material parameters

The material used is SS316L and the relevant parameters required for numerical simulations are shown in Table I.46,77,78


SS316L-related parameters.

Density of solid metal (kg/m3�metal 7980 
Solid phase line temperature (K) �� 1658 
Liquid phase line temperature (K) �� 1723 
Vaporization temperature (K) �� 3090 
Latent heat of melting (⁠ J/kg⁠) �� 2.60×105 
Latent heat of evaporation (⁠ J/kg⁠) �� 7.45×106 
Surface tension of liquid phase (N /m⁠) � 1.60 
Liquid metal viscosity (kg/m s) �� 6×10−3 
Gaseous metal viscosity (kg/m s) �gas 1.85×10−5 
Temperature coefficient of surface tension (N/m K) ��/�T 0.80×10−3 
Molar mass (⁠ kg/mol⁠) 0.05 593 
Emissivity � 0.26 
Laser absorption �0 0.35 
Ambient pressure (kPa) �� 101 325 
Ambient temperature (K) �0 300 
Stefan–Boltzmann constant (W/m2 K4� 5.67×10−8 
Thermal conductivity of metals (⁠ W/m K⁠) � 24.55 
Density of protective gas (kg/m3�gas 1.25 
Coefficient of thermal expansion (/K) �� 16×10−6 
Generalized gas constant (⁠ J/mol K⁠) 8.314 


With the objective of studying in depth the evolutionary patterns of single-track and double-track molten pool development, detailed observations were made for certain specific locations in the model, as shown in Fig. 6. In this figure, P1 and P2 represent the longitudinal tangents to the centers of the two melt tracks in the XZ plane, while L1 is the transverse profile in the YZ plane. The scanning direction is positive and negative along the X axis. Points A and B are the locations of the centers of the molten pool of the first and second melt tracks, respectively (x = 1.995 × 10−4, y = 5 × 10−7, and z = −4.85 × 10−5).

FIG. 6.


Schematic diagram of observation position.

A. Single-track simulation

A series of single-track molten pool simulation experiments were carried out in order to investigate the influence law of laser power as well as scanning speed on the HP-LPBF process. Figure 7 demonstrates the evolution of the 3D morphology and temperature field of the single-track molten pool in the time period of 50–500 μs under a laser power of 100 W and a scanning speed of 800 mm/s. The powder bed is in the natural cooling state. When t = 50 μs, the powder is heated by the laser heat and rapidly melts and settles to form the initial molten pool. This process is accompanied by partial melting of the substrate and solidification together with the melted powder. The molten pool rapidly expands with increasing width, depth, length, and temperature, as shown in Fig. 7(a). When t = 150 μs, the molten pool expands more obviously, and the temperature starts to transfer to the surrounding area, forming a heat-affected zone. At this point, the width of the molten pool tends to stabilize, and the temperature in the center of the molten pool has reached its peak and remains largely stable. However, the phenomenon of molten pool spatter was also observed in this process, as shown in Fig. 7(b). As time advances, when t = 300 μs, solidification begins to occur at the tail of the molten pool, and tiny ripples are produced on the solidified surface. This is due to the fact that the melt flows toward the region with large temperature gradient under the influence of Marangoni convection and solidifies together with the melt at the end of the bath. At this point, the temperature gradient at the front of the bath is significantly larger than at the end. While the width of the molten pool was gradually reduced, the shape of the molten pool was gradually changed to a “comet” shape. In addition, a slight depression was observed at the top of the bath because the peak temperature at the surface of the bath reached the evaporation temperature, which resulted in a recoil pressure perpendicular to the surface of the bath downward, creating a depressed region. As the laser focal spot moves and is paired with the Marangoni convection of the melt, these recessed areas will be filled in as shown in Fig. 7(c). It has been shown that the depressed regions are the result of the coupled effect of Marangoni convection, recoil pressure, and surface tension.79 By t = 500 μs, the width and height of the molten pool stabilize and show a “comet” shape in Fig. 7(d).

FIG. 7.


Single-track molten pool process: (a) t = 50  ��⁠, (b) t = 150  ��⁠, (c) t = 300  ��⁠, (d) t = 500  ��⁠.

Figure 8 depicts the velocity vector diagram of the P1 profile in a single-track molten pool, the length of the arrows represents the magnitude of the velocity, and the maximum velocity is about 2.36 m/s. When t = 50 μs, the molten pool takes shape, and the velocities at the two ends of the pool are the largest. The variation of the velocities at the front end is especially more significant in Fig. 8(a). As the time advances to t = 150 μs, the molten pool expands rapidly, in which the velocity at the tail increases and changes more significantly, while the velocity at the front is relatively small. At this stage, the melt moves backward from the center of the molten pool, which in turn expands the molten pool area. The melt at the back end of the molten pool center flows backward along the edge of the molten pool surface and then converges along the edge of the molten pool to the bottom center, rising to form a closed loop. Similarly, a similar closed loop is formed at the front end of the center of the bath, but with a shorter path. However, a large portion of the melt in the center of the closed loop formed at the front end of the bath is in a nearly stationary state. The main cause of this melt flow phenomenon is the effect of temperature gradient and surface tension (the Marangoni effect), as shown in Figs. 8(b) and 8(e). This dynamic behavior of the melt tends to form an “elliptical” pool. At t = 300 μs, the tendency of the above two melt flows to close the loop is more prominent and faster in Fig. 8(c). When t = 500 μs, the velocity vector of the molten pool shows a stable trend, and the closed loop of melt flow also remains stable. With the gradual laser focal spot movement, the melt is gradually solidified at its tail, and finally, a continuous and stable single track is formed in Fig. 8(d).

FIG. 8.


Vector plot of single-track molten pool velocity in XZ longitudinal section: (a) t = 50  ��⁠, (b) t = 150  ��⁠, (c) t = 300  ��⁠, (d) t = 500  ��⁠, (e) molten pool flow.

In order to explore in depth the transient evolution of the molten pool, the evolution of the single-track temperature field and the melt flow was monitored in the YZ cross section. Figure 9(a) shows the state of the powder bed at the initial moment. When t = 250 μs, the laser focal spot acts on the powder bed and the powder starts to melt and gradually collects in the molten pool. At this time, the substrate will also start to melt, and the melt flow mainly moves in the downward and outward directions and the velocity is maximum at the edges in Fig. 9(b). When t = 300 μs, the width and depth of the molten pool increase due to the recoil pressure. At this time, the melt flows more slowly at the center, but the direction of motion is still downward in Fig. 9(c). When t = 350 μs, the width and depth of the molten pool further increase, at which time the intensity of the melt flow reaches its peak and the direction of motion remains the same in Fig. 9(d). When t = 400 μs, the melt starts to move upward, and the surrounding powder or molten material gradually fills up, causing the surface of the molten pool to begin to flatten. At this time, the maximum velocity of the melt is at the center of the bath, while the velocity at the edge is close to zero, and the edge of the melt starts to solidify in Fig. 9(e). When t = 450 μs, the melt continues to move upward, forming a convex surface of the melt track. However, the melt movement slows down, as shown in Fig. 9(f). When t = 500 μs, the melt further moves upward and its speed gradually becomes smaller. At the same time, the melt solidifies further, as shown in Fig. 9(g). When t = 550 μs, the melt track is basically formed into a single track with a similar “mountain” shape. At this stage, the velocity is close to zero only at the center of the molten pool, and the flow behavior of the melt is poor in Fig. 9(h). At t = 600 μs, the melt stops moving and solidification is rapidly completed. Up to this point, a single track is formed in Fig. 9(i). During the laser action on the powder bed, the substrate melts and combines with the molten state powder. The powder-to-powder fusion is like the convergence of water droplets, which are rapidly fused by surface tension. However, the fusion between the molten state powder and the substrate occurs driven by surface tension, and the molten powder around the molten pool is pulled toward the substrate (a wetting effect occurs), which ultimately results in the formation of a monolithic whole.38,80,81

FIG. 9.


Evolution of single-track molten pool temperature and melt flow in the YZ cross section: (a) t = 0  ��⁠, (b) t = 250  ��⁠, (c) t = 300  ��⁠, (d) t = 350  ��⁠, (e) t = 400  ��⁠, (f) t = 450  ��⁠, (g) t = 500  ��⁠, (h) t = 550  ��⁠, (i) t = 600  ��⁠.

The wetting ability between the liquid metal and the solid substrate in the molten pool directly affects the degree of balling of the melt,82,83 and the wetting ability can be measured by the contact angle of a single track in Fig. 10. A smaller value of contact angle represents better wettability. The contact angle α can be calculated by�=�1−�22,


where �1 and �2 are the contact angles of the left and right regions, respectively.

FIG. 10.


Schematic of contact angle.

Relevant studies have confirmed that the wettability is better at a contact angle α around or below 40°.84 After measurement, a single-track contact angle α of about 33° was obtained under this process parameter, which further confirms the good wettability.

B. Double-track simulation

In order to deeply investigate the influence of hatch spacing on the characteristics of the HP-LPBF process, a series of double-track molten pool simulation experiments were systematically carried out. Figure 11 shows in detail the dynamic changes of the 3D morphology and temperature field of the double-track molten pool in the time period of 2050–2500 μs under the conditions of laser power of 100 W, scanning speed of 800 mm/s, and hatch spacing of 0.06 mm. By comparing the study with Fig. 7, it is observed that the basic characteristics of the 3D morphology and temperature field of the second track are similar to those of the first track. However, there are subtle differences between them. The first track exhibits a basically symmetric shape, but the second track morphology shows a slight deviation influenced by the difference in thermal diffusion rate between the solidified metal and the powder. Otherwise, the other characteristic information is almost the same as that of the first track. Figure 12 shows the velocity vector plot of the P2 profile in the double-track molten pool, with a maximum velocity of about 2.63 m/s. The melt dynamics at both ends of the pool are more stable at t = 2050 μs, where the maximum rate of the second track is only 1/3 of that of the first one. Other than that, the rest of the information is almost no significant difference from the characteristic information of the first track. Figure 13 demonstrates a detailed observation of the double-track temperature field and melts flow in the YZ cross section, and a comparative study with Fig. 9 reveals that the width of the second track is slightly wider. In addition, after the melt direction shifts from bottom to top, the first track undergoes four time periods (50 μs) to reach full solidification, while the second track takes five time periods. This is due to the presence of significant heat buildup in the powder bed after the forming of the first track, resulting in a longer dynamic time of the melt and an increased molten pool lifetime. In conclusion, the level of specimen forming can be significantly optimized by adjusting the laser power and hatch spacing.

FIG. 11.


Double-track molten pool process: (a) t = 2050  ��⁠, (b) t = 2150  ��⁠, (c) t = 2300  ��⁠, (d) t = 2500  ��⁠.

FIG. 12.


Vector plot of double-track molten pool velocity in XZ longitudinal section: (a) t = 2050  ��⁠, (b) t = 2150  ��⁠, (c) t = 2300  ��⁠, (d) t = 2500  ��⁠.

FIG. 13.


Evolution of double-track molten pool temperature and melt flow in the YZ cross section: (a) t = 2250  ��⁠, (b) t = 2300  ��⁠, (c) t = 2350  ��⁠, (d) t = 2400  ��⁠, (e) t = 2450  ��⁠, (f) t = 2500  ��⁠, (g) t = 2550  ��⁠, (h) t = 2600  ��⁠, (i) t = 2650  ��⁠.

In order to quantitatively detect the molten pool dimensions as well as the remolten region dimensions, the molten pool characterization information in Fig. 14 is constructed by drawing the boundary on the YZ cross section based on the isothermal surface of the liquid phase line. It can be observed that the heights of the first track and second track are basically the same, but the depth of the second track increases relative to the first track. The molten pool width is mainly positively correlated with the laser power as well as the scanning speed (the laser line energy density �⁠). However, the remelted zone width is negatively correlated with the hatch spacing (the overlapping ratio). Overall, the forming quality of the specimens can be directly influenced by adjusting the laser power, scanning speed, and hatch spacing.

FIG. 14.


Double-track molten pool characterization information on YZ cross section.

In order to study the variation rule of the temperature in the center of the molten pool with time, Fig. 15 demonstrates the temperature variation curves with time for two reference points, A and B. Among them, the red dotted line indicates the liquid phase line temperature of SS316L. From the figure, it can be seen that the maximum temperature at the center of the molten pool in the first track is lower than that in the second track, which is mainly due to the heat accumulation generated after passing through the first track. The maximum temperature gradient was calculated to be 1.69 × 108 K/s. When the laser scanned the first track, the temperature in the center of the molten pool of the second track increased slightly. Similarly, when the laser scanned the second track, a similar situation existed in the first track. Since the temperature gradient in the second track is larger than that in the first track, the residence time of the liquid phase in the molten pool of the first track is longer than that of the second track.

FIG. 15.


Temperature profiles as a function of time for two reference points A and B.

C. Simulation analysis of molten pool under different process parameters

In order to deeply investigate the effects of various process parameters on the mesoscopic-scale temperature field, molten pool characteristic information and defects of HP-LPBF, numerical simulation experiments on mesoscopic-scale laser power, scanning speed, and hatch spacing of double-track molten pools were carried out.

1. Laser power

Figure 16 shows the effects of different laser power on the morphology and temperature field of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. When P = 50 W, a smaller molten pool is formed due to the lower heat generated by the Gaussian light source per unit time. This leads to a smaller track width, which results in adjacent track not lapping properly and the presence of a large number of unmelted powder particles, resulting in an increase in the number of defects, such as pores in the specimen. The surface of the track is relatively flat, and the depth is small. In addition, the temperature gradient before and after the molten pool was large, and the depression location appeared at the biased front end in Fig. 16(a). When P = 100 W, the surface of the track is flat and smooth with excellent lap. Due to the Marangoni effect, the velocity field of the molten pool is in the form of “vortex,” and the melt has good fluidity, and the maximum velocity reaches 2.15 m/s in Fig. 16(b). When P = 200 W, the heat generated by the Gaussian light source per unit time is too large, resulting in the melt rapidly reaching the evaporation temperature, generating a huge recoil pressure, forming a large molten pool, and the surface of the track is obviously raised. The melt movement is intense, especially the closed loop at the center end of the molten pool. At this time, the depth and width of the molten pool are large, leading to the expansion of the remolten region and the increased chance of the appearance of porosity defects in Fig. 16(c). The results show that at low laser power, the surface tension in the molten pool is dominant. At high laser power, recoil pressure is its main role.

FIG. 16.


Simulation results of double-track molten pool under different laser powers: (a) P = 50 W, (b) P = 100 W, (c) P = 200 W.

Table II shows the effect of different laser powers on the characteristic information of the double-track molten pool at a scanning speed of 800 mm/s and a hatch spacing of 0.06 mm. The negative overlapping ratio in the table indicates that the melt tracks are not lapped, and 26/29 indicates the melt depth of the first track/second track. It can be seen that with the increase in laser power, the melt depth, melt width, melt height, and remelted zone show a gradual increase. At the same time, the overlapping ratio also increases. Especially in the process of laser power from 50 to 200 W, the melting depth and melting width increased the most, which increased nearly 2 and 1.5 times, respectively. Meanwhile, the overlapping ratio also increases with the increase in laser power, which indicates that the melting and fusion of materials are better at high laser power. On the other hand, the dimensions of the molten pool did not change uniformly with the change of laser power. Specifically, the depth-to-width ratio of the molten pool increased from about 0.30 to 0.39 during the increase from 50 to 120 W, which further indicates that the effective heat transfer in the vertical direction is greater than that in the horizontal direction with the increase in laser power. This dimensional response to laser power is mainly affected by the recoil pressure and also by the difference in the densification degree between the powder layer and the metal substrate. In addition, according to the experimental results, the contact angle shows a tendency to increase and then decrease during the process of laser power increase, and always stays within the range of less than 33°. Therefore, in practical applications, it is necessary to select the appropriate laser power according to the specific needs in order to achieve the best processing results.


Double-track molten pool characterization information at different laser powers.

Laser power (W)Depth (μm)Width (μm)Height (μm)Remolten region (μm)Overlapping ratio (%)Contact angle (°)
50 16 54 11 −10 23 
100 26/29 74 14 18 23.33 33 
200 37/45 116 21 52 93.33 28 

2. Scanning speed

Figure 17 demonstrates the effect of different scanning speeds on the morphology and temperature field of the double-track molten pool at a laser power of 100 W and a hatch spacing of 0.06 mm. With the gradual increase in scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. When � = 200 mm/s, the slow scanning speed causes the material to absorb too much heat, which is very easy to trigger the overburning phenomenon. At this point, the molten pool is larger and the surface morphology is uneven. This situation is consistent with the previously discussed scenario with high laser power in Fig. 17(a). However, when � = 1600 mm/s, the scanning speed is too fast, resulting in the material not being able to absorb sufficient heat, which triggers the powder particles that fail to melt completely to have a direct effect on the bonding of the melt to the substrate. At this time, the molten pool volume is relatively small and the neighboring melt track cannot lap properly. This result is consistent with the previously discussed case of low laser power in Fig. 17(b). Overall, the ratio of the laser power to the scanning speed (the line energy density �⁠) has a direct effect on the temperature field and surface morphology of the molten pool.

FIG. 17.


Simulation results of double-track molten pool under different scanning speed: (a)  � = 200 mm/s, (b)  � = 1600 mm/s.

Table III shows the effects of different scanning speed on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and hatch spacing of 0.06 mm. It can be seen that the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. With the increase in scanning speed, the melt depth, melt width, melt height, remelted zone, and overlapping ratio show a gradual decreasing trend. Among them, the melt depth and melt width decreased faster, while the melt height and remolten region decreased relatively slowly. In addition, when the scanning speed was increased from 200 to 800 mm/s, the decreasing speeds of melt depth and melt width were significantly accelerated, while the decreasing speeds of overlapping ratio were relatively slow. When the scanning speed was further increased to 1600 mm/s, the decreasing speeds of melt depth and melt width were further accelerated, and the un-lapped condition of the melt channel also appeared. In addition, the contact angle increases and then decreases with the scanning speed, and both are lower than 33°. Therefore, when selecting the scanning speed, it is necessary to make reasonable trade-offs according to the specific situation, and take into account the factors of melt depth, melt width, melt height, remolten region, and overlapping ratio, in order to achieve the best processing results.


Double-track molten pool characterization information at different scanning speeds.

Scanning speed (mm/s)Depth (μm)Width (μm)Height (μm)Remolten region (μm)Overlapping ratio (%)Contact angle (°)
200 55/68 182 19/32 124 203.33 22 
1600 13 50 11 −16.67 31 

3. Hatch spacing

Figure 18 shows the effect of different hatch spacing on the morphology and temperature field of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. The surface morphology and temperature field of the first track and second track are basically the same, but slightly different. The first track shows a basically symmetric morphology along the scanning direction, while the second track shows a slight offset due to the difference in the heat transfer rate between the solidified material and the powder particles. When the hatch spacing is too small, the overlapping ratio increases and the probability of defects caused by remelting phenomenon grows. When the hatch spacing is too large, the neighboring melt track cannot overlap properly, and the powder particles are not completely melted, leading to an increase in the number of holes. In conclusion, the ratio of the line energy density � to the hatch spacing (the volume energy density E) has a significant effect on the temperature field and surface morphology of the molten pool.

FIG. 18.


Simulation results of double-track molten pool under different hatch spacings: (a) H = 0.03 mm, (b) H = 0.12 mm.

Table IV shows the effects of different hatch spacing on the characteristic information of the double-track molten pool under the condition of laser power of 100 W and scanning speed of 800 mm/s. It can be seen that the hatch spacing has little effect on the melt depth, melt width, and melt height, but has some effect on the remolten region. With the gradual expansion of hatch spacing, the remolten region shows a gradual decrease. At the same time, the overlapping ratio also decreased with the increase in hatch spacing. In addition, it is observed that the contact angle shows a tendency to increase and then remain stable when the hatch spacing increases, which has a more limited effect on it. Therefore, trade-offs and decisions need to be made on a case-by-case basis when selecting the hatch spacing.


Double-track molten pool characterization information at different hatch spacings.

Hatch spacing (mm)Depth (μm)Width (μm)Height (μm)Remolten region (μm)Overlapping ratio (%)Contact angle (°)
0.03 25/27 82 14 59 173.33 30 
0.12 26 78 14 −35 33 

In summary, the laser power, scanning speed, and hatch spacing have a significant effect on the formation of the molten pool, and the correct selection of these three process parameters is crucial to ensure the forming quality. In addition, the melt depth of the second track is slightly larger than that of the first track at higher line energy density � and volume energy density E. This is mainly due to the fact that a large amount of heat accumulation is generated after the first track, forming a larger molten pool volume, which leads to an increase in the melt depth.

D. Simulation analysis of molten pool with powder particle size and laser focal spot diameter

Figure 19 demonstrates the effect of different powder particle sizes and laser focal spot diameters on the morphology and temperature field of the double-track molten pool under a laser power of 100 W, a scanning speed of 800 mm/s, and a hatch spacing of 0.06 mm. In the process of melting coarse powder with small laser focal spot diameter, the laser energy cannot completely melt the larger powder particles, resulting in their partial melting and further generating excessive pore defects. The larger powder particles tend to generate zigzag molten pool edges, which cause an increase in the roughness of the melt track surface. In addition, the molten pool is also prone to generate the present spatter phenomenon, which can directly affect the quality of forming. The volume of the formed molten pool is relatively small, while the melt depth, melt width, and melt height are all smaller relative to the fine powder in Fig. 19(a). In the process of melting fine powders with a large laser focal spot diameter, the laser energy is able to melt the fine powder particles sufficiently, even to the point of overmelting. This results in a large number of fine spatters being generated at the edge of the molten pool, which causes porosity defects in the melt track in Fig. 19(b). In addition, the maximum velocity of the molten pool is larger for large powder particle sizes compared to small powder particle sizes, which indicates that the temperature gradient in the molten pool is larger for large powder particle sizes and the melt motion is more intense. However, the size of the laser focal spot diameter has a relatively small effect on the melt motion. However, a larger focal spot diameter induces a larger melt volume with greater depth, width, and height. In conclusion, a small powder size helps to reduce the surface roughness of the specimen, and a small laser spot diameter reduces the minimum forming size of a single track.

FIG. 19.


Simulation results of double-track molten pool with different powder particle size and laser focal spot diameter: (a) focal spot = 25 μm, coarse powder, (b) focal spot = 80 μm, fine powder.

Table V shows the maximum temperature gradient at the reference point for different powder sizes and laser focal spot diameters. As can be seen from the table, the maximum temperature gradient is lower than that of HP-LPBF for both coarse powders with a small laser spot diameter and fine powders with a large spot diameter, a phenomenon that leads to an increase in the heat transfer rate of HP-LPBF, which in turn leads to a corresponding increase in the cooling rate and, ultimately, to the formation of finer microstructures.


Maximum temperature gradient at the reference point for different powder particle sizes and laser focal spot diameters.

Laser power (W)Scanning speed (mm/s)Hatch spacing (mm)Average powder size (μm)Laser focal spot diameter (μm)Maximum temperature gradient (×107 K/s)
100 800 0.06 31.7 25 7.89 
11.5 80 7.11 


In this study, the geometrical characteristics of 3D coarse and fine powder particles were first calculated using DEM and then numerical simulations of single track and double track in the process of forming SS316L from monolayer HP-LPBF at mesoscopic scale were developed using CFD method. The effects of Marangoni convection, surface tension, recoil pressure, gravity, thermal convection, thermal radiation, and evaporative heat dissipation on the heat and mass transfer in the molten pool were considered in this model. The effects of laser power, scanning speed, and hatch spacing on the dynamics of the single-track and double-track molten pools, as well as on other characteristic information, were investigated. The effects of the powder particle size on the molten pool were investigated comparatively with the laser focal spot diameter. The main conclusions are as follows:

  1. The results show that the temperature gradient at the front of the molten pool is significantly larger than that at the tail, and the molten pool exhibits a “comet” morphology. At the top of the molten pool, there is a slightly concave region, which is the result of the coupling of Marangoni convection, recoil pressure, and surface tension. The melt flow forms two closed loops, which are mainly influenced by temperature gradients and surface tension. This special dynamic behavior of the melt tends to form an “elliptical” molten pool and an almost “mountain” shape in single-track forming.
  2. The basic characteristics of the three-dimensional morphology and temperature field of the second track are similar to those of the first track, but there are subtle differences. The first track exhibits a basically symmetrical shape; however, due to the difference in thermal diffusion rates between the solidified metal and the powder, a slight asymmetry in the molten pool morphology of the second track occurs. After forming through the first track, there is a significant heat buildup in the powder bed, resulting in a longer dynamic time of the melt, which increases the life of the molten pool. The heights of the first track and second track remained essentially the same, but the depth of the second track was greater relative to the first track. In addition, the maximum temperature gradient was 1.69 × 108 K/s during HP-LPBF forming.
  3. At low laser power, the surface tension in the molten pool plays a dominant role. At high laser power, recoil pressure becomes the main influencing factor. With the increase of laser power, the effective heat transfer in the vertical direction is superior to that in the horizontal direction. With the gradual increase of scanning speed, the surface morphology of the molten pool evolves from circular to elliptical. In addition, the scanning speed has a significant effect on the melt depth, melt width, melt height, remolten region, and overlapping ratio. Too large or too small hatch spacing will lead to remelting or non-lap phenomenon, which in turn causes the formation of defects.
  4. When using a small laser focal spot diameter, it is difficult to completely melt large powder particle sizes, resulting in partial melting and excessive porosity generation. At the same time, large powder particles produce curved edges of the molten pool, resulting in increased surface roughness of the melt track. In addition, spatter occurs, which directly affects the forming quality. At small focal spot diameters, the molten pool volume is relatively small, and the melt depth, the melt width, and the melt height are correspondingly small. Taken together, the small powder particle size helps to reduce surface roughness, while the small spot diameter reduces the forming size.


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Predicting solid-state phase transformations during metal additive manufacturing: A case study on electron-beam powder bed fusion of Inconel-738

Predicting solid-state phase transformations during metal additive manufacturing: A case study on electron-beam powder bed fusion of Inconel-738

금속 적층 제조 중 고체 상 변형 예측: Inconel-738의 전자빔 분말층 융합에 대한 사례 연구

Nana Kwabena Adomako a, Nima Haghdadi a, James F.L. Dingle bc, Ernst Kozeschnik d, Xiaozhou Liao bc, Simon P. Ringer bc, Sophie Primig a


Metal additive manufacturing (AM) has now become the perhaps most desirable technique for producing complex shaped engineering parts. However, to truly take advantage of its capabilities, advanced control of AM microstructures and properties is required, and this is often enabled via modeling. The current work presents a computational modeling approach to studying the solid-state phase transformation kinetics and the microstructural evolution during AM. Our approach combines thermal and thermo-kinetic modelling. A semi-analytical heat transfer model is employed to simulate the thermal history throughout AM builds. Thermal profiles of individual layers are then used as input for the MatCalc thermo-kinetic software. The microstructural evolution (e.g., fractions, morphology, and composition of individual phases) for any region of interest throughout the build is predicted by MatCalc. The simulation is applied to an IN738 part produced by electron beam powder bed fusion to provide insights into how γ′ precipitates evolve during thermal cycling. Our simulations show qualitative agreement with our experimental results in predicting the size distribution of γ′ along the build height, its multimodal size character, as well as the volume fraction of MC carbides. Our findings indicate that our method is suitable for a range of AM processes and alloys, to predict and engineer their microstructures and properties.

Graphical Abstract



Additive manufacturing, Simulation, Thermal cycles, γ′ phase, IN738

1. Introduction

Additive manufacturing (AM) is an advanced manufacturing method that enables engineering parts with intricate shapes to be fabricated with high efficiency and minimal materials waste. AM involves building up 3D components layer-by-layer from feedstocks such as powder [1]. Various alloys, including steel, Ti, Al, and Ni-based superalloys, have been produced using different AM techniques. These techniques include directed energy deposition (DED), electron- and laser powder bed fusion (E-PBF and L-PBF), and have found applications in a variety of industries such as aerospace and power generation [2][3][4]. Despite the growing interest, certain challenges limit broader applications of AM fabricated components in these industries and others. One of such limitations is obtaining a suitable and reproducible microstructure that offers the desired mechanical properties consistently. In fact, the AM as-built microstructure is highly complex and considerably distinctive from its conventionally processed counterparts owing to the complicated thermal cycles arising from the deposition of several layers upon each other [5][6].

Several studies have reported that the solid-state phases and solidification microstructure of AM processed alloys such as CMSX-4, CoCr [7][8], Ti-6Al-4V [9][10][11]IN738 [6]304L stainless steel [12], and IN718 [13][14] exhibit considerable variations along the build direction. For instance, references [9][10] have reported that there is a variation in the distribution of α and β phases along the build direction in Ti-alloys. Similarly, the microstructure of an L-PBF fabricated martensitic steel exhibits variations in the fraction of martensite [15]. Furthermore, some of the present authors and others [6][16][17][18][19][20] have recently reviewed and reported that there is a difference in the morphology and fraction of nanoscale precipitates as a function of build height in Ni-based superalloys. These non-uniformities in the as-built microstructure result in an undesired heterogeneity in mechanical and other important properties such as corrosion and oxidation [19][21][22][23]. To obtain the desired microstructure and properties, additional processing treatments are utilized, but this incurs extra costs and may lead to precipitation of detrimental phases and grain coarsening. Therefore, a through-process understanding of the microstructure evolution under repeated heating and cooling is now needed to further advance 3D printed microstructure and property control.

It is now commonly understood that the microstructure evolution during printing is complex, and most AM studies concentrate on the microstructure and mechanical properties of the final build only. Post-printing studies of microstructure characteristics at room temperature miss crucial information on how they evolve. In-situ measurements and modelling approaches are required to better understand the complex microstructural evolution under repeated heating and cooling. Most in-situ measurements in AM focus on monitoring the microstructural changes, such as phase transformations and melt pool dynamics during fabrication using X-ray scattering and high-speed X-ray imaging [24][25][26][27]. For example, Zhao et al. [25] measured the rate of solidification and described the α/β phase transformation during L-PBF of Ti-6Al-4V in-situ. Also, Wahlmann et al. [21] recently used an L-PBF machine coupled with X-ray scattering to investigate the changes in CMSX-4 phase during successive melting processes. Although these techniques provide significant understanding of the basic principles of AM, they are not widely accessible. This is due to the great cost of the instrument, competitive application process, and complexities in terms of the experimental set-up, data collection, and analysis [26][28].

Computational modeling techniques are promising and more widely accessible tools that enable advanced understanding, prediction, and engineering of microstructures and properties during AM. So far, the majority of computational studies have concentrated on physics based process models for metal AM, with the goal of predicting the temperature profile, heat transfer, powder dynamics, and defect formation (e.g., porosity) [29][30]. In recent times, there have been efforts in modeling of the AM microstructure evolution using approaches such as phase-field [31], Monte Carlo (MC) [32], and cellular automata (CA) [33], coupled with finite element simulations for temperature profiles. However, these techniques are often restricted to simulating the evolution of solidification microstructures (e.g., grain and dendrite structure) and defects (e.g., porosity). For example, Zinovieva et al. [33] predicted the grain structure of L-PBF Ti-6Al-4V using finite difference and cellular automata methods. However, studies on the computational modelling of the solid-state phase transformations, which largely determine the resulting properties, remain limited. This can be attributed to the multi-component and multi-phase nature of most engineering alloys in AM, along with the complex transformation kinetics during thermal cycling. This kind of research involves predictions of the thermal cycle in AM builds, and connecting it to essential thermodynamic and kinetic data as inputs for the model. Based on the information provided, the thermokinetic model predicts the history of solid-state phase microstructure evolution during deposition as output. For example, a multi-phase, multi-component mean-field model has been developed to simulate the intermetallic precipitation kinetics in IN718 [34] and IN625 [35] during AM. Also, Basoalto et al. [36] employed a computational framework to examine the contrasting distributions of process-induced microvoids and precipitates in two Ni-based superalloys, namely IN718 and CM247LC. Furthermore, McNamara et al. [37] established a computational model based on the Johnson-Mehl-Avrami model for non-isothermal conditions to predict solid-state phase transformation kinetics in L-PBF IN718 and DED Ti-6Al-4V. These models successfully predicted the size and volume fraction of individual phases and captured the repeated nucleation and dissolution of precipitates that occur during AM.

In the current study, we propose a modeling approach with appreciably short computational time to investigate the detailed microstructural evolution during metal AM. This may include obtaining more detailed information on the morphologies of phases, such as size distribution, phase fraction, dissolution and nucleation kinetics, as well as chemistry during thermal cycling and final cooling to room temperature. We utilize the combination of the MatCalc thermo-kinetic simulator and a semi-analytical heat conduction model. MatCalc is a software suite for simulation of phase transformations, microstructure evolution and certain mechanical properties in engineering alloys. It has successfully been employed to simulate solid-state phase transformations in Ni-based superalloys [38][39], steels [40], and Al alloys [41] during complex thermo-mechanical processes. MatCalc uses the classical nucleation theory as well as the so-called Svoboda-Fischer-Fratzl-Kozeschnik (SFFK) growth model as the basis for simulating precipitation kinetics [42]. Although MatCalc was originally developed for conventional thermo-mechanical processes, we will show that it is also applicable for AM if the detailed time-temperature profile of the AM build is known. The semi-analytical heat transfer code developed by Stump and Plotkowski [43] is used to simulate these profile throughout the AM build.

1.1. Application to IN738

Inconel-738 (IN738) is a precipitation hardening Ni-based superalloy mainly employed in high-temperature components, e.g. in gas turbines and aero-engines owing to its exceptional mechanical properties at temperatures up to 980 °C, coupled with high resistance to oxidation and corrosion [44]. Its superior high-temperature strength (∼1090 MPa tensile strength) is provided by the L12 ordered Ni3(Al,Ti) γ′ phase that precipitates in a face-centered cubic (FCC) γ matrix [45][46]. Despite offering great properties, IN738, like most superalloys with high γ′ fractions, is challenging to process owing to its propensity to hot cracking [47][48]. Further, machining of such alloys is challenging because of their high strength and work-hardening rates. It is therefore difficult to fabricate complex INC738 parts using traditional manufacturing techniques like casting, welding, and forging.

The emergence of AM has now made it possible to fabricate such parts from IN738 and other superalloys. Some of the current authors’ recent research successfully applied E-PBF to fabricate defect-free IN738 containing γ′ throughout the build [16][17]. The precipitated γ′ were heterogeneously distributed. In particular, Haghdadi et al. [16] studied the origin of the multimodal size distribution of γ′, while Lim et al. [17] investigated the gradient in γ′ character with build height and its correlation to mechanical properties. Based on these results, the present study aims to extend the understanding of the complex and site-specific microstructural evolution in E-PBF IN738 by using a computational modelling approach. New experimental evidence (e.g., micrographs not published previously) is presented here to support the computational results.

2. Materials and Methods

2.1. Materials preparation

IN738 Ni-based superalloy (59.61Ni-8.48Co-7.00Al-17.47Cr-3.96Ti-1.01Mo-0.81W-0.56Ta-0.49Nb-0.47C-0.09Zr-0.05B, at%) gas-atomized powder was used as feedstock. The powders, with average size of 60 ± 7 µm, were manufactured by Praxair and distributed by Astro Alloys Inc. An Arcam Q10 machine by GE Additive with an acceleration voltage of 60 kV was used to fabricate a 15 × 15 × 25 mm3 block (XYZ, Z: build direction) on a 316 stainless steel substrate. The block was 3D-printed using a ‘random’ spot melt pattern. The random spot melt pattern involves randomly selecting points in any given layer, with an equal chance of each point being melted. Each spot melt experienced a dwell time of 0.3 ms, and the layer thickness was 50 µm. Some of the current authors have previously characterized the microstructure of the very same and similar builds in more detail [16][17]. A preheat temperature of ∼1000 °C was set and kept during printing to reduce temperature gradients and, in turn, thermal stresses [49][50][51]. Following printing, the build was separated from the substrate through electrical discharge machining. It should be noted that this sample was simultaneously printed with the one used in [17] during the same build process and on the same build plate, under identical conditions.

2.2. Microstructural characterization

The printed sample was longitudinally cut in the direction of the build using a Struers Accutom-50, ground, and then polished to 0.25 µm suspension via standard techniques. The polished x-z surface was electropolished and etched using Struers A2 solution (perchloric acid in ethanol). Specimens for image analysis were polished using a 0.06 µm colloidal silica. Microstructure analyses were carried out across the height of the build using optical microscopy (OM) and scanning electron microscopy (SEM) with focus on the microstructure evolution (γ′ precipitates) in individual layers. The position of each layer being analyzed was determined by multiplying the layer number by the layer thickness (50 µm). It should be noted that the position of the first layer starts where the thermal profile is tracked (in this case, 2 mm from the bottom). SEM images were acquired using a JEOL 7001 field emission microscope. The brightness and contrast settings, acceleration voltage of 15 kV, working distance of 10 mm, and other SEM imaging parameters were all held constant for analysis of the entire build. The ImageJ software was used for automated image analysis to determine the phase fraction and size of γ′ precipitates and carbides. A 2-pixel radius Gaussian blur, following a greyscale thresholding and watershed segmentation was used [52]. Primary γ′ sizes (>50 nm), were measured using equivalent spherical diameters. The phase fractions were considered equal to the measured area fraction. Secondary γ′ particles (<50 nm) were not considered here. The γ′ size in the following refers to the diameter of a precipitate.

2.3. Hardness testing

A Struers DuraScan tester was utilized for Vickers hardness mapping on a polished x-z surface, from top to bottom under a maximum load of 100 mN and 10 s dwell time. 30 micro-indentations were performed per row. According to the ASTM standard [53], the indentations were sufficiently distant (∼500 µm) to assure that strain-hardened areas did not interfere with one another.

2.4. Computational simulation of E-PBF IN738 build

2.4.1. Thermal profile modeling

The thermal history was generated using the semi-analytical heat transfer code (also known as the 3DThesis code) developed by Stump and Plotkowski [43]. This code is an open-source C++ program which provides a way to quickly simulate the conductive heat transfer found in welding and AM. The key use case for the code is the simulation of larger domains than is practicable with Computational Fluid Dynamics/Finite Element Analysis programs like FLOW-3D AM. Although simulating conductive heat transfer will not be an appropriate simplification for some investigations (for example the modelling of keyholding or pore formation), the 3DThesis code does provide fast estimates of temperature, thermal gradient, and solidification rate which can be useful for elucidating microstructure formation across entire layers of an AM build. The mathematics involved in the code is as follows:

In transient thermal conduction during welding and AM, with uniform and constant thermophysical properties and without considering fluid convection and latent heat effects, energy conservation can be expressed as:(1)��∂�∂�=�∇2�+�̇where � is density, � specific heat, � temperature, � time, � thermal conductivity, and �̇ a volumetric heat source. By assuming a semi-infinite domain, Eq. 1 can be analytically solved. The solution for temperature at a given time (t) using a volumetric Gaussian heat source is presented as:(2)��,�,�,�−�0=33�����32∫0�1������exp−3�′�′2��+�′�′2��+�′�′2����′(3)and��=12��−�′+��2for�=�,�,�(4)and�′�′=�−���′Where � is the vector �,�,� and �� is the location of the heat source.

The numerical integration scheme used is an adaptive Gaussian quadrature method based on the following nondimensionalization:(5)�=��xy2�,�′=��xy2�′,�=��xy,�=��xy,�=��xy,�=���xy

A more detailed explanation of the mathematics can be found in reference [43].

The main source of the thermal cycling present within a powder-bed fusion process is the fusion of subsequent layers. Therefore, regions near the top of a build are expected to undergo fewer thermal cycles than those closer to the bottom. For this purpose, data from the single scan’s thermal influence on multiple layers was spliced to represent the thermal cycles experienced at a single location caused by multiple subsequent layers being fused.

The cross-sectional area simulated by this model was kept constant at 1 × 1 mm2, and the depth was dependent on the build location modelled with MatCalc. For a build location 2 mm from the bottom, the maximum number of layers to simulate is 460. Fig. 1a shows a stitched overview OM image of the entire build indicating the region where this thermal cycle is simulated and tracked. To increase similarity with the conditions of the physical build, each thermal history was constructed from the results of two simulations generated with different versions of a random scan path. The parameters used for these thermal simulations can be found in Table 1. It should be noted that the main purpose of the thermal profile modelling was to demonstrate how the conditions at different locations of the build change relative to each other. Accurately predicting the absolute temperature during the build would require validation via a temperature sensor measurement during the build process which is beyond the scope of the study. Nonetheless, to establish the viability of the heat source as a suitable approximation for this study, an additional sensitivity analysis was conducted. This analysis focused on the influence of energy input on γ′ precipitation behavior, the central aim of this paper. This was achieved by employing varying beam absorption energies (0.76, 0.82 – the values utilized in the simulation, and 0.9). The direct impact of beam absorption efficiency on energy input into the material was investigated. Specifically, the initial 20 layers of the build were simulated and subsequently compared to experimental data derived from SEM. While phase fractions were found to be consistent across all conditions, disparities emerged in the mean size of γ′ precipitates. An absorption efficiency of 0.76 yielded a mean size of approximately 70 nm. Conversely, absorption efficiencies of 0.82 and 0.9 exhibited remarkably similar mean sizes of around 130 nm, aligning closely with the outcomes of the experiments.

Fig. 1

Table 1. A list of parameters used in thermal simulation of E-PBF.

Spatial resolution5 µm
Time step0.5 s
Beam diameter200 µm
Beam penetration depth1 µm
Beam power1200 W
Beam absorption efficiency0.82
Thermal conductivity25.37 W/(m⋅K)
Chamber temperature1000 °C
Specific heat711.756 J/(kg⋅K)
Density8110 kg/m3

2.4.2. Thermo-kinetic simulation

The numerical analyses of the evolution of precipitates was performed using MatCalc version 6.04 (rel 0.011). The thermodynamic (‘mc_ni.tdb’, version 2.034) and diffusion (‘mc_ni.ddb’, version 2.007) databases were used. MatCalc’s basic principles are elaborated as follows:

The nucleation kinetics of precipitates are computed using a computational technique based on a classical nucleation theory [54] that has been modified for systems with multiple components [42][55]. Accordingly, the transient nucleation rate (�), which expresses the rate at which nuclei are formed per unit volume and time, is calculated as:(6)�=�0��*∙�xp−�*�∙�∙exp−��where �0 denotes the number of active nucleation sites, �* the rate of atomic attachment, � the Boltzmann constant, � the temperature, �* the critical energy for nucleus formation, τ the incubation time, and t the time. � (Zeldovich factor) takes into consideration that thermal excitation destabilizes the nucleus as opposed to its inactive state [54]. Z is defined as follows:(7)�=−12�kT∂2∆�∂�2�*12where ∆� is the overall change in free energy due to the formation of a nucleus and n is the nucleus’ number of atoms. ∆�’s derivative is evaluated at n* (critical nucleus size). �* accounts for the long-range diffusion of atoms required for nucleation, provided that the matrix’ and precipitates’ composition differ. Svoboda et al. [42] developed an appropriate multi-component equation for �*, which is given by:(8)�*=4��*2�4�∑�=1��ki−�0�2�0��0�−1where �* denotes the critical radius for nucleation, � represents atomic distance, and � is the molar volume. �ki and �0� represent the concentration of elements in the precipitate and matrix, respectively. The parameter �0� denotes the rate of diffusion of the ith element within the matrix. The expression for the incubation time � is expressed as [54]:(9)�=12�*�2

and �*, which represents the critical energy for nucleation:(10)�*=16�3�3∆�vol2where � is the interfacial energy, and ∆Gvol the change in the volume free energy. The critical nucleus’ composition is similar to the γ′ phase’s equilibrium composition at the same temperature. � is computed based on the precipitate and matrix compositions, using a generalized nearest neighbor broken bond model, with the assumption of interfaces being planar, sharp, and coherent [56][57][58].

In Eq. 7, it is worth noting that �* represents the fundamental variable in the nucleation theory. It contains �3/∆�vol2 and is in the exponent of the nucleation rate. Therefore, even small variations in γ and/or ∆�vol can result in notable changes in �, especially if �* is in the order of �∙�. This is demonstrated in [38] for UDIMET 720 Li during continuous cooling, where these quantities change steadily during precipitation due to their dependence on matrix’ and precipitate’s temperature and composition. In the current work, these changes will be even more significant as the system is exposed to multiple cycles of rapid cooling and heating.

Once nucleated, the growth of a precipitate is assessed using the radius and composition evolution equations developed by Svoboda et al. [42] with a mean-field method that employs the thermodynamic extremal principle. The expression for the total Gibbs free energy of a thermodynamic system G, which consists of n components and m precipitates, is given as follows:(11)�=∑���0��0�+∑�=1�4���33��+∑�=1��ki�ki+∑�=1�4���2��.

The chemical potential of component � in the matrix is denoted as �0�(�=1,…,�), while the chemical potential of component � in the precipitate is represented by �ki(�=1,…,�,�=1,…,�). These chemical potentials are defined as functions of the concentrations �ki(�=1,…,�,�=1,…,�). The interface energy density is denoted as �, and �� incorporates the effects of elastic energy and plastic work resulting from the volume change of each precipitate.

Eq. (12) establishes that the total free energy of the system in its current state relies on the independent state variables: the sizes (radii) of the precipitates �� and the concentrations of each component �ki. The remaining variables can be determined by applying the law of mass conservation to each component �. This can be represented by the equation:(12)��=�0�+∑�=1�4���33�ki,

Furthermore, the global mass conservation can be expressed by equation:(13)�=∑�=1���When a thermodynamic system transitions to a more stable state, the energy difference between the initial and final stages is dissipated. This model considers three distinct forms of dissipation effects [42]. These include dissipations caused by the movement of interfaces, diffusion within the precipitate and diffusion within the matrix.

Consequently, �̇� (growth rate) and �̇ki (chemical composition’s rate of change) of the precipitate with index � are derived from the linear system of equation system:(14)�ij��=��where �� symbolizes the rates �̇� and �̇ki [42]. Index i contains variables for precipitate radius, chemical composition, and stoichiometric boundary conditions suggested by the precipitate’s crystal structure. Eq. (10) is computed separately for every precipitate �. For a more detailed description of the formulae for the coefficients �ij and �� employed in this work please refer to [59].

The MatCalc software was used to perform the numerical time integration of �̇� and �̇ki of precipitates based on the classical numerical method by Kampmann and Wagner [60]. Detailed information on this method can be found in [61]. Using this computational method, calculations for E-PBF thermal cycles (cyclic heating and cooling) were computed and compared to experimental data. The simulation took approximately 2–4 hrs to complete on a standard laptop.

3. Results

3.1. Microstructure

Fig. 1 displays a stitched overview image and selected SEM micrographs of various γ′ morphologies and carbides after observations of the X-Z surface of the build from the top to 2 mm above the bottom. Fig. 2 depicts a graph that charts the average size and phase fraction of the primary γ′, as it changes with distance from the top to the bottom of the build. The SEM micrographs show widespread primary γ′ precipitation throughout the entire build, with the size increasing in the top to bottom direction. Particularly, at the topmost height, representing the 460th layer (Z = 22.95 mm), as seen in Fig. 1b, the average size of γ′ is 110 ± 4 nm, exhibiting spherical shapes. This is representative of the microstructure after it solidifies and cools to room temperature, without experiencing additional thermal cycles. The γ′ size slightly increases to 147 ± 6 nm below this layer and remains constant until 0.4 mm (∼453rd layer) from the top. At this position, the microstructure still closely resembles that of the 460th layer. After the 453rd layer, the γ′ size grows rapidly to ∼503 ± 19 nm until reaching the 437th layer (1.2 mm from top). The γ′ particles here have a cuboidal shape, and a small fraction is coarser than 600 nm. γ′ continue to grow steadily from this position to the bottom (23 mm from the top). A small fraction of γ′ is > 800 nm.

Fig. 2

Besides primary γ′, secondary γ′ with sizes ranging from 5 to 50 nm were also found. These secondary γ′ precipitates, as seen in Fig. 1f, were present only in the bottom and middle regions. A detailed analysis of the multimodal size distribution of γ′ can be found in [16]. There is no significant variation in the phase fraction of the γ′ along the build. The phase fraction is ∼ 52%, as displayed in Fig. 2. It is worth mentioning that the total phase fraction of γ′ was estimated based on the primary γ′ phase fraction because of the small size of secondary γ′. Spherical MC carbides with sizes ranging from 50 to 400 nm and a phase fraction of 0.8% were also observed throughout the build. The carbides are the light grey precipitates in Fig. 1g. The light grey shade of carbides in the SEM images is due to their composition and crystal structure [52]. These carbides are not visible in Fig. 1b-e because they were dissolved during electro-etching carried out after electropolishing. In Fig. 1g, however, the sample was examined directly after electropolishing, without electro-etching.

Table 2 shows the nominal and measured composition of γ′ precipitates throughout the build by atom probe microscopy as determined in our previous study [17]. No build height-dependent composition difference was observed in either of the γ′ precipitate populations. However, there was a slight disparity between the composition of primary and secondary γ′. Among the main γ′ forming elements, the primary γ′ has a high Ti concentration while secondary γ′ has a high Al concentration. A detailed description of the atom distribution maps and the proxigrams of the constituent elements of γ′ throughout the build can be found in [17].

Table 2. Bulk IN738 composition determined using inductively coupled plasma atomic emission spectroscopy (ICP-AES). Compositions of γ, primary γ′, and secondary γ′ at various locations in the build measured by APT. This information is reproduced from data in Ref. [17] with permission.

γ matrix
Primary γ′
Secondary γ′

3.2. Hardness

Fig. 3a shows the Vickers hardness mapping performed along the entire X-Z surface, while Fig. 3b shows the plot of average hardness at different build heights. This hardness distribution is consistent with the γ′ precipitate size gradient across the build direction in Fig. 1Fig. 2. The maximum hardness of ∼530 HV1 is found at ∼0.5 mm away from the top surface (Z = 22.5), where γ′ particles exhibit the smallest observed size in Fig. 2b. Further down the build (∼ 2 mm from the top), the hardness drops to the 440–490 HV1 range. This represents the region where γ′ begins to coarsen. The hardness drops further to 380–430 HV1 at the bottom of the build.

Fig. 3

3.3. Modeling of the microstructural evolution during E-PBF

3.3.1. Thermal profile modeling

Fig. 4 shows the simulated thermal profile of the E-PBF build at a location of 23 mm from the top of the build, using a semi-analytical heat conduction model. This profile consists of the time taken to deposit 460 layers until final cooling, as shown in Fig. 4a. Fig. 4b-d show the magnified regions of Fig. 4a and reveal the first 20 layers from the top, a single layer (first layer from the top), and the time taken for the build to cool after the last layer deposition, respectively.

Fig. 4

The peak temperatures experienced by previous layers decrease progressively as the number of layers increases but never fall below the build preheat temperature (1000 °C). Our simulated thermal cycle may not completely capture the complexity of the actual thermal cycle utilized in the E-PBF build. For instance, the top layer (Fig. 4c), also representing the first deposit’s thermal profile without additional cycles (from powder heating, melting, to solidification), recorded the highest peak temperature of 1390 °C. Although this temperature is above the melting range of the alloy (1230–1360 °C) [62], we believe a much higher temperature was produced by the electron beam to melt the powder. Nevertheless, the solidification temperature and dynamics are outside the scope of this study as our focus is on the solid-state phase transformations during deposition. It takes ∼25 s for each layer to be deposited and cooled to the build temperature. The interlayer dwell time is 125 s. The time taken for the build to cool to room temperature (RT) after final layer deposition is ∼4.7 hrs (17,000 s).

3.3.2. MatCalc simulation

During the MatCalc simulation, the matrix phase is defined as γ. γ′, and MC carbide are included as possible precipitates. The domain of these precipitates is set to be the matrix (γ), and nucleation is assumed to be homogenous. In homogeneous nucleation, all atoms of the unit volume are assumed to be potential nucleation sitesTable 3 shows the computational parameters used in the simulation. All other parameters were set at default values as recommended in the version 6.04.0011 of MatCalc. The values for the interfacial energies are automatically calculated according to the generalized nearest neighbor broken bond model and is one of the most outstanding features in MatCalc [56][57][58]. It should be noted that the elastic misfit strain was not included in the calculation. The output of MatCalc includes phase fraction, size, nucleation rate, and composition of the precipitates. The phase fraction in MatCalc is the volume fraction. Although the experimental phase fraction is the measured area fraction, it is relatively similar to the volume fraction. This is because of the generally larger precipitate size and similar morphology at the various locations along the build [63]. A reliable phase fraction comparison between experiment and simulation can therefore be made.

Table 3. Computational parameters used in the simulation.

Precipitation domainγ
Nucleation site γ′Bulk (homogenous)
Nucleation site MC carbideBulk (Homogenous)
Precipitates class size250
Regular solution critical temperature γ′2500 K[64]
Calculated interfacial energyγ′ = 0.080–0.140 J/m2 and MC carbide = 0.410–0.430 J/m2 Precipitate phase fraction

Fig. 5a shows the simulated phase fraction of γ′ and MC carbide during thermal cycling. Fig. 5b is a magnified view of 5a showing the simulated phase fraction at the center points of the top 70 layers, whereas Fig. 5c corresponds to the first two layers from the top. As mentioned earlier, the top layer (460th layer) represents the microstructure after solidification. The microstructure of the layers below is determined by the number of thermal cycles, which increases with distance to the top. For example, layers 459, 458, 457, up to layer 1 (region of interest) experience 1, 2, 3 and 459 thermal cycles, respectively. In the top layer in Fig. 5c, the volume fraction of γ′ and carbides increases with temperature. For γ′, it decreases to zero when the temperature is above the solvus temperature after a few seconds. Carbides, however, remain constant in their volume fraction reaching equilibrium (phase fraction ∼ 0.9%) in a short time. The topmost layer can be compared to the first deposit, and the peak in temperature symbolizes the stage where the electron beam heats the powder until melting. This means γ′ and carbide precipitation might have started in the powder particles during heating from the build temperature and electron beam until the onset of melting, where γ′ dissolves, but carbides remain stable [28].

Fig. 5

During cooling after deposition, γ′ reprecipitates at a temperature of 1085 °C, which is below its solvus temperature. As cooling progresses, the phase fraction increases steadily to ∼27% and remains constant at 1000 °C (elevated build temperature). The calculated equilibrium fraction of phases by MatCalc is used to show the complex precipitation characteristics in this alloy. Fig. 6 shows that MC carbides form during solidification at 1320 °C, followed by γ′, which precipitate when the solidified layer cools to 1140 °C. This indicates that all deposited layers might contain a negligible amount of these precipitates before subsequent layer deposition, while being at the 1000 °C build temperature or during cooling to RT. The phase diagram also shows that the equilibrium fraction of the γ′ increases as temperature decreases. For instance, at 1000, 900, and 800 °C, the phase fractions are ∼30%, 38%, and 42%, respectively.

Fig. 6

Deposition of subsequent layers causes previous layers to undergo phase transformations as they are exposed to several thermal cycles with different peak temperatures. In Fig. 5c, as the subsequent layer is being deposited, γ′ in the previous layer (459th layer) begins to dissolve as the temperature crosses the solvus temperature. This is witnessed by the reduction of the γ′ phase fraction. This graph also shows how this phase dissolves during heating. However, the phase fraction of MC carbide remains stable at high temperatures and no dissolution is seen during thermal cycling. Upon cooling, the γ′ that was dissolved during heating reprecipitates with a surge in the phase fraction until 1000 °C, after which it remains constant. This microstructure is similar to the solidification microstructure (layer 460), with a similar γ′ phase fraction (∼27%).

The complete dissolution and reprecipitation of γ′ continue for several cycles until the 50th layer from the top (layer 411), where the phase fraction does not reach zero during heating to the peak temperature (see Fig. 5d). This indicates the ‘partial’ dissolution of γ′, which continues progressively with additional layers. It should be noted that the peak temperatures for layers that underwent complete dissolution were much higher (1170–1300 °C) than the γ′ solvus.

The dissolution and reprecipitation of γ′ during thermal cycling are further confirmed in Fig. 7, which summarizes the nucleation rate, phase fraction, and concentration of major elements that form γ′ in the matrix. Fig. 7b magnifies a single layer (3rd layer from top) within the full dissolution region in Fig. 7a to help identify the nucleation and growth mechanisms. From Fig. 7b, γ′ nucleation begins during cooling whereby the nucleation rate increases to reach a maximum value of approximately 1 × 1020 m−3s−1. This fast kinetics implies that some rearrangement of atoms is required for γ′ precipitates to form in the matrix [65][66]. The matrix at this stage is in a non-equilibrium condition. Its composition is similar to the nominal composition and remains unchanged. The phase fraction remains insignificant at this stage although nucleation has started. The nucleation rate starts declining upon reaching the peak value. Simultaneously, diffusion-controlled growth of existing nuclei occurs, depleting the matrix of γ′ forming elements (Al and Ti). Thus, from (7)(11), ∆�vol continuously decreases until nucleation ceases. The growth of nuclei is witnessed by the increase in phase fraction until a constant level is reached at 27% upon cooling to and holding at build temperature. This nucleation event is repeated several times.

Fig. 7

At the onset of partial dissolution, the nucleation rate jumps to 1 × 1021 m−3s−1, and then reduces sharply at the middle stage of partial dissolution. The nucleation rate reaches 0 at a later stage. Supplementary Fig. S1 shows a magnified view of the nucleation rate, phase fraction, and thermal profile, underpinning this trend. The jump in nucleation rate at the onset is followed by a progressive reduction in the solute content of the matrix. The peak temperatures (∼1130–1160 °C) are lower than those in complete dissolution regions but still above or close to the γ′ solvus. The maximum phase fraction (∼27%) is similar to that of the complete dissolution regions. At the middle stage, the reduction in nucleation rate is accompanied by a sharp drop in the matrix composition. The γ′ fraction drops to ∼24%, where the peak temperatures of the layers are just below or at γ′ solvus. The phase fraction then increases progressively through the later stage of partial dissolution to ∼30% towards the end of thermal cycling. The matrix solute content continues to drop although no nucleation event is seen. The peak temperatures are then far below the γ′ solvus. It should be noted that the matrix concentration after complete dissolution remains constant. Upon cooling to RT after final layer deposition, the nucleation rate increases again, indicating new nucleation events. The phase fraction reaches ∼40%, with a further depletion of the matrix in major γ′ forming elements. γ′ size distribution

Fig. 8 shows histograms of the γ′ precipitate size distributions (PSD) along the build height during deposition. These PSDs are predicted at the end of each layer of interest just before final cooling to room temperature, to separate the role of thermal cycles from final cooling on the evolution of γ′. The PSD for the top layer (layer 460) is shown in Fig. 8a (last solidified region with solidification microstructure). The γ′ size ranges from 120 to 230 nm and is similar to the 44 layers below (2.2 mm from the top).

Fig. 8

Further down the build, γ′ begins to coarsen after layer 417 (44th layer from top). Fig. 8c shows the PSD after the 44th layer, where the γ′ size exhibits two peaks at ∼120–230 and ∼300 nm, with most of the population being in the former range. This is the onset of partial dissolution where simultaneously with the reprecipitation and growth of fresh γ′, the undissolved γ′ grows rapidly through diffusive transport of atoms to the precipitates. This is shown in Fig. 8c, where the precipitate class sizes between 250 and 350 represent the growth of undissolved γ′. Although this continues in the 416th layer, the phase fractions plot indicates that the onset of partial dissolution begins after the 411th layer. This implies that partial dissolution started early, but the fraction of undissolved γ′ was too low to impact the phase fraction. The reprecipitated γ′ are mostly in the 100–220 nm class range and similar to those observed during full dissolution.

As the number of layers increases, coarsening intensifies with continued growth of more undissolved γ′, and reprecipitation and growth of partially dissolved ones. Fig. 8d, e, and f show this sequence. Further down the build, coarsening progresses rapidly, as shown in Figs. 8d, 8e, and 8f. The γ′ size ranges from 120 to 1100 nm, with the peaks at 160, 180, and 220 nm in Figs. 8d, 8e, and 8f, respectively. Coarsening continues until nucleation ends during dissolution, where only the already formed γ′ precipitates continue to grow during further thermal cycling. The γ′ size at this point is much larger, as observed in layers 361 and 261, and continues to increase steadily towards the bottom (layer 1). Two populations in the ranges of ∼380–700 and ∼750–1100 nm, respectively, can be seen. The steady growth of γ′ towards the bottom is confirmed by the gradual decrease in the concentration of solute elements in the matrix (Fig. 7a). It should be noted that for each layer, the γ′ class with the largest size originates from continuous growth of the earliest set of the undissolved precipitates.

Fig. 9Fig. 10 and supplementary Figs. S2 and S3 show the γ′ size evolution during heating and cooling of a single layer in the full dissolution region, and early, middle stages, and later stages of partial dissolution, respectively. In all, the size of γ′ reduces during layer heating. Depending on the peak temperature of the layer which varies with build height, γ′ are either fully or partially dissolved as mentioned earlier. Upon cooling, the dissolved γ′ reprecipitate.

Fig. 9
Fig. 10

In Fig. 9, those layers that underwent complete dissolution (top layers) were held above γ′ solvus temperature for longer. In Fig. 10, layers at the early stage of partial dissolution spend less time in the γ′ solvus temperature region during heating, leading to incomplete dissolution. In such conditions, smaller precipitates are fully dissolved while larger ones shrink [67]. Layers in the middle stages of partial dissolution have peak temperatures just below or at γ′ solvus, not sufficient to achieve significant γ′ dissolution. As seen in supplementary Fig. S2, only a few smaller γ′ are dissolved back into the matrix during heating, i.e., growth of precipitates is more significant than dissolution. This explains the sharp decrease in concentration of Al and Ti in the matrix in this layer.

The previous sections indicate various phenomena such as an increase in phase fraction, further depletion of matrix composition, and new nucleation bursts during cooling. Analysis of the PSD after the final cooling of the build to room temperature allows a direct comparison to post-printing microstructural characterization. Fig. 11 shows the γ′ size distribution of layer 1 (460th layer from the top) after final cooling to room temperature. Precipitation of secondary γ′ is observed, leading to the multimodal size distribution of secondary and primary γ′. The secondary γ′ size falls within the 10–80 nm range. As expected, a further growth of the existing primary γ′ is also observed during cooling.

Fig. 11 γ′ chemistry after deposition

Fig. 12 shows the concentration of the major elements that form γ′ (Al, Ti, and Ni) in the primary and secondary γ′ at the bottom of the build, as calculated by MatCalc. The secondary γ′ has a higher Al content (13.5–14.5 at% Al), compared to 13 at% Al in the primary γ′. Additionally, within the secondary γ′, the smallest particles (∼10 nm) have higher Al contents than larger ones (∼70 nm). In contrast, for the primary γ′, there is no significant variation in the Al content as a function of their size. The Ni concentration in secondary γ′ (71.1–72 at%) is also higher in comparison to the primary γ′ (70 at%). The smallest secondary γ′ (∼10 nm) have higher Ni contents than larger ones (∼70 nm), whereas there is no substantial change in the Ni content of primary γ′, based on their size. As expected, Ti shows an opposite size-dependent variation. It ranges from ∼ 7.7–8.7 at% Ti in secondary γ′ to ∼9.2 at% in primary γ′. Similarly, within the secondary γ′, the smallest (∼10 nm) have lower Al contents than the larger ones (∼70 nm). No significant variation is observed for Ti content in primary γ′.

Fig. 12

4. Discussion

A combined modelling method is utilized to study the microstructural evolution during E-PBF of IN738. The presented results are discussed by examining the precipitation and dissolution mechanism of γ′ during thermal cycling. This is followed by a discussion on the phase fraction and size evolution of γ′ during thermal cycling and after final cooling. A brief discussion on carbide morphology is also made. Finally, a comparison is made between the simulation and experimental results to assess their agreement.

4.1. γ′ morphology as a function of build height

4.1.1. Nucleation of γ′

The fast precipitation kinetics of the γ′ phase enables formation of γ′ upon quenching from higher temperatures (above solvus) during thermal cycling [66]. In Fig. 7b, for a single layer in the full dissolution region, during cooling, the initial increase in nucleation rate signifies the first formation of nuclei. The slight increase in nucleation rate during partial dissolution, despite a decrease in the concentration of γ′ forming elements, may be explained by the nucleation kinetics. During partial dissolution and as the precipitates shrink, it is assumed that the regions at the vicinity of partially dissolved precipitates are enriched in γ′ forming elements [68][69]. This differs from the full dissolution region, in which case the chemical composition is evenly distributed in the matrix. Several authors have attributed the solute supersaturation of the matrix around primary γ′ to partial dissolution during isothermal ageing [69][70][71][72]. The enhanced supersaturation in the regions close to the precipitates results in a much higher driving force for nucleation, leading to a higher nucleation rate upon cooling. This phenomenon can be closely related to the several nucleation bursts upon continuous cooling of Ni-based superalloys, where second nucleation bursts exhibit higher nucleation rates [38][68][73][74].

At middle stages of partial dissolution, the reduction in the nucleation rate indicates that the existing composition and low supersaturation did not trigger nucleation as the matrix was closer to the equilibrium state. The end of a nucleation burst means that the supersaturation of Al and Ti has reached a low level, incapable of providing sufficient driving force during cooling to or holding at 1000 °C for further nucleation [73]. Earlier studies on Ni-based superalloys have reported the same phenomenon during ageing or continuous cooling from the solvus temperature to RT [38][73][74].

4.1.2. Dissolution of γ′ during thermal cycling

γ′ dissolution kinetics during heating are fast when compared to nucleation due to exponential increase in phase transformation and diffusion activities with temperature [65]. As shown in Fig. 9Fig. 10, and supplementary Figs. S2 and S3, the reduction in γ′ phase fraction and size during heating indicates γ′ dissolution. This is also revealed in Fig. 5 where phase fraction decreases upon heating. The extent of γ′ dissolution mostly depends on the temperature, time spent above γ′ solvus, and precipitate size [75][76][77]. Smaller γ′ precipitates are first to be dissolved [67][77][78]. This is mainly because more solute elements need to be transported away from large γ′ precipitates than from smaller ones [79]. Also, a high temperature above γ′ solvus temperature leads to a faster dissolution rate [80]. The equilibrium solvus temperature of γ′ in IN738 in our MatCalc simulation (Fig. 6) and as reported by Ojo et al. [47] is 1140 °C and 1130–1180 °C, respectively. This means the peak temperature experienced by previous layers decreases progressively from γ′ supersolvus to subsolvus, near-solvus, and far from solvus as the number of subsequent layers increases. Based on the above, it can be inferred that the degree of dissolution of γ′ contributes to the gradient in precipitate distribution.

Although the peak temperatures during later stages of partial dissolution are much lower than the equilibrium γ′ solvus, γ′ dissolution still occurs but at a significantly lower rate (supplementary Fig. S3). Wahlmann et al. [28] also reported a similar case where they observed the rapid dissolution of γ′ in CMSX-4 during fast heating and cooling cycles at temperatures below the γ′ solvus. They attributed this to the γ′ phase transformation process taking place in conditions far from the equilibrium. While the same reasoning may be valid for our study, we further believe that the greater surface area to volume ratio of the small γ′ precipitates contributed to this. This ratio means a larger area is available for solute atoms to diffuse into the matrix even at temperatures much below the solvus [81].

4.2. γ′ phase fraction and size evolution

4.2.1. During thermal cycling

In the first layer, the steep increase in γ′ phase fraction during heating (Fig. 5), which also represents γ′ precipitation in the powder before melting, has qualitatively been validated in [28]. The maximum phase fraction of 27% during the first few layers of thermal cycling indicates that IN738 theoretically could reach the equilibrium state (∼30%), but the short interlayer time at the build temperature counteracts this. The drop in phase fraction at middle stages of partial dissolution is due to the low number of γ′ nucleation sites [73]. It has been reported that a reduction of γ′ nucleation sites leads to a delay in obtaining the final volume fraction as more time is required for γ′ precipitates to grow and reach equilibrium [82]. This explains why even upon holding for 150 s before subsequent layer deposition, the phase fraction does not increase to those values that were observed in the previous full γ′ dissolution regions. Towards the end of deposition, the increase in phase fraction to the equilibrium value of 30% is as a result of the longer holding at build temperature or close to it [83].

During thermal cycling, γ′ particles begin to grow immediately after they first precipitate upon cooling. This is reflected in the rapid increase in phase fraction and size during cooling in Fig. 5 and supplementary Fig. S2, respectively. The rapid growth is due to the fast diffusion of solute elements at high temperatures [84]. The similar size of γ′ for the first 44 layers from the top can be attributed to the fact that all layers underwent complete dissolution and hence, experienced the same nucleation event and growth during deposition. This corresponds with the findings by Balikci et al. [85], who reported that the degree of γ′ precipitation in IN738LC does not change when a solution heat treatment is conducted above a certain critical temperature.

The increase in coarsening rate (Fig. 8) during thermal cycling can first be ascribed to the high peak temperature of the layers [86]. The coarsening rate of γ′ is known to increase rapidly with temperature due to the exponential growth of diffusion activity. Also, the simultaneous dissolution with coarsening could be another reason for the high coarsening rate, as γ′ coarsening is a diffusion-driven process where large particles grow by consuming smaller ones [78][84][86][87]. The steady growth of γ′ towards the bottom of the build is due to the much lower layer peak temperature, which is almost close to the build temperature, and reduced dissolution activity, as is seen in the much lower solute concentration in γ′ compared to those in the full and partial dissolution regions.

4.2.2. During cooling

The much higher phase fraction of ∼40% upon cooling signifies the tendency of γ′ to reach equilibrium at lower temperatures (Fig. 4). This is due to the precipitation of secondary γ′ and a further increase in the size of existing primary γ′, which leads to a multimodal size distribution of γ′ after cooling [38][73][88][89][90]. The reason for secondary γ′ formation during cooling is as follows: As cooling progresses, it becomes increasingly challenging to redistribute solute elements in the matrix owing to their lower mobility [38][73]. A higher supersaturation level in regions away from or free of the existing γ′ precipitates is achieved, making them suitable sites for additional nucleation bursts. More cooling leads to the growth of these secondary γ′ precipitates, but as the temperature and in turn, the solute diffusivity is low, growth remains slow.

4.3. Carbides

MC carbides in IN738 are known to have a significant impact on the high-temperature strength. They can also act as effective hardening particles and improve the creep resistance [91]. Precipitation of MC carbides in IN738 and several other superalloys is known to occur during solidification or thermal treatments (e.g., hot isostatic pressing) [92]. In our case, this means that the MC carbides within the E-PBF build formed because of the thermal exposure from the E-PBF thermal cycle in addition to initial solidification. Our simulation confirms this as MC carbides appear during layer heating (Fig. 5). The constant and stable phase fraction of MC carbides during thermal cycling can be attributed to their high melting point (∼1360 °C) and the short holding time at peak temperatures [75][93][94]. The solvus temperature for most MC carbides exceeds most of the peak temperatures observed in our simulation, and carbide dissolution kinetics at temperatures above the solvus are known to be comparably slow [95]. The stable phase fraction and random distribution of MC carbides signifies the slight influence on the gradient in hardness.

4.4. Comparison of simulations and experiments

4.4.1. Precipitate phase fraction and morphology as a function of build height

A qualitative agreement is observed for the phase fraction of carbides, i.e. ∼0.8% in the experiment and ∼0.9% in the simulation. The phase fraction of γ′ differs, with the experiment reporting a value of ∼51% and the simulation, 40%. Despite this, the size distribution of primary γ′ along the build shows remarkable consistency between experimental and computational analyses. It is worth noting that the primary γ′ morphology in the experimental analysis is observed in the as-fabricated state, whereas the simulation (Fig. 8) captures it during deposition process. The primary γ′ size in the experiment is expected to experience additional growth during the cooling phase. Regardless, both show similar trends in primary γ′ size increments from the top to the bottom of the build. The larger primary γ’ size in the simulation versus the experiment can be attributed to the fact that experimental and simulation results are based on 2D and 3D data, respectively. The absence of stereological considerations [96] in our analysis could have led to an underestimation of the precipitate sizes from SEM measurements. The early starts of coarsening (8th layer) in the experiment compared to the simulation (45th layer) can be attributed to a higher actual γ′ solvus temperature than considered in our simulation [47]. The solvus temperature of γ′ in a Ni-based superalloy is mainly determined by the detailed composition. A high amount of Cr and Co are known to reduce the solvus temperature, whereas Ta and Mo will increase it [97][98][99]. The elemental composition from our experimental work was used for the simulation except for Ta. It should be noted that Ta is not included in the thermodynamic database in MatCalc used, and this may have reduced the solvus temperature. This could also explain the relatively higher γ′ phase fraction in the experiment than in simulation, as a higher γ′ solvus temperature will cause more γ′ to precipitate and grow early during cooling [99][100].

Another possible cause of this deviation can be attributed to the extent of γ′ dissolution, which is mainly determined by the peak temperature. It can be speculated that individual peak temperatures at different layers in the simulation may have been over-predicted. However, one needs to consider that the true thermal profile is likely more complicated in the actual E-PBF process [101]. For example, the current model assumes that the thermophysical properties of the material are temperature-independent, which is not realistic. Many materials, including IN738, exhibit temperature-dependent properties such as thermal conductivityspecific heat capacity, and density [102]. This means that heat transfer simulations may underestimate or overestimate the temperature gradients and cooling rates within the powder bed and the solidified part. Additionally, the model does not account for the reduced thermal diffusivity through unmelted powder, where gas separating the powder acts as insulation, impeding the heat flow [1]. In E-PBF, the unmelted powder regions with trapped gas have lower thermal diffusivity compared to the fully melted regions, leading to localized temperature variations, and altered solidification behavior. These limitations can impact the predictions, particularly in relation to the carbide dissolution, as the peak temperatures may be underestimated.

While acknowledging these limitations, it is worth emphasizing that achieving a detailed and accurate representation of each layer’s heat source would impose tough computational challenges. Given the substantial layer count in E-PBF, our decision to employ a semi-analytical approximation strikes a balance between computational feasibility and the capture of essential trends in thermal profiles across diverse build layers. In future work, a dual-calibration strategy is proposed to further reduce simulation-experiment disparities. By refining temperature-independent thermophysical property approximations and absorptivity in the heat source model, and by optimizing interfacial energy descriptions in the kinetic model, the predictive precision could be enhanced. Further refining the simulation controls, such as adjusting the precipitate class size may enhance quantitative comparisons between modeling outcomes and experimental data in future work.

4.4.2. Multimodal size distribution of γ′ and concentration

Another interesting feature that sees qualitative agreement between the simulation and the experiment is the multimodal size distribution of γ′. The formation of secondary γ′ particles in the experiment and most E-PBF Ni-based superalloys is suggested to occur at low temperatures, during final cooling to RT [16][73][90]. However, so far, this conclusion has been based on findings from various continuous cooling experiments, as the study of the evolution during AM would require an in-situ approach. Our simulation unambiguously confirms this in an AM context by providing evidence for secondary γ′ precipitation during slow cooling to RT. Additionally, it is possible to speculate that the chemical segregation occurring during solidification, due to the preferential partitioning of certain elements between the solid and liquid phases, can contribute to the multimodal size distribution during deposition [51]. This is because chemical segregation can result in variations in the local composition of superalloys, which subsequently affects the nucleation and growth of γ′. Regions with higher concentrations of alloying elements will encourage the formation of larger γ′ particles, while regions with lower concentrations may favor the nucleation of smaller precipitates. However, it is important to acknowledge that the elevated temperature during the E-PBF process will largely homogenize these compositional differences [103][104].

A good correlation is also shown in the composition of major γ′ forming elements (Al and Ti) in primary and secondary γ′. Both experiment and simulation show an increasing trend for Al content and a decreasing trend for Ti content from primary to secondary γ′. The slight composition differences between primary and secondary γ′ particles are due to the different diffusivity of γ′ stabilizers at different thermal conditions [105][106]. As the formation of multimodal γ′ particles with different sizes occurs over a broad temperature range, the phase chemistry of γ′ will be highly size dependent. The changes in the chemistry of various γ′ (primary, secondary, and tertiary) have received significant attention since they have a direct influence on the performance [68][105][107][108][109]. Chen et al. [108][109], reported a high Al content in the smallest γ′ precipitates compared to the largest, while Ti showed an opposite trend during continuous cooling in a RR1000 Ni-based superalloy. This was attributed to the temperature and cooling rate at which the γ′ precipitates were formed. The smallest precipitates formed last, at the lowest temperature and cooling rate. A comparable observation is evident in the present investigation, where the secondary γ′ forms at a low temperature and cooling rate in comparison to the primary. The temperature dependence of γ′ chemical composition is further evidenced in supplementary Fig. S4, which shows the equilibrium chemical composition of γ′ as a function of temperature.

5. Conclusions

A correlative modelling approach capable of predicting solid-state phase transformations kinetics in metal AM was developed. This approach involves computational simulations with a semi-analytical heat transfer model and the MatCalc thermo-kinetic software. The method was used to predict the phase transformation kinetics and detailed morphology and chemistry of γ′ and MC during E-PBF of IN738 Ni-based superalloy. The main conclusions are:

  • 1.The computational simulations are in qualitative agreement with the experimental observations. This is particularly true for the γ′ size distribution along the build height, the multimodal size distribution of particles, and the phase fraction of MC carbides.
  • 2.The deviations between simulation and experiment in terms of γ′ phase fraction and location in the build are most likely attributed to a higher γ′ solvus temperature during the experiment than in the simulation, which is argued to be related to the absence of Ta in the MatCalc database.
  • 3.The dissolution and precipitation of γ′ occur fast and under non-equilibrium conditions. The level of γ′ dissolution determines the gradient in γ′ size distribution along the build. After thermal cycling, the final cooling to room temperature has further significant impacts on the final γ′ size, morphology, and distribution.
  • 4.A negligible amount of γ′ forms in the first deposited layer before subsequent layer deposition, and a small amount of γ′ may also form in the powder induced by the 1000 °C elevated build temperature before melting.

Our findings confirm the suitability of MatCalc to predict the microstructural evolution at various positions throughout a build in a Ni-based superalloy during E-PBF. It also showcases the suitability of a tool which was originally developed for traditional thermo-mechanical processing of alloys to the new additive manufacturing context. Our simulation capabilities are likely extendable to other alloy systems that undergo solid-state phase transformations implemented in MatCalc (various steels, Ni-based superalloys, and Al-alloys amongst others) as well as other AM processes such as L-DED and L-PBF which have different thermal cycle characteristics. New tools to predict the microstructural evolution and properties during metal AM are important as they provide new insights into the complexities of AM. This will enable control and design of AM microstructures towards advanced materials properties and performances.

CRediT authorship contribution statement

Primig Sophie: Writing – review & editing, Supervision, Resources, Project administration, Funding acquisition, Conceptualization. Adomako Nana Kwabena: Writing – original draft, Writing – review & editing, Visualization, Software, Investigation, Formal analysis, Conceptualization. Haghdadi Nima: Writing – review & editing, Supervision, Project administration, Methodology, Conceptualization. Dingle James F.L.: Methodology, Conceptualization, Software, Writing – review & editing, Visualization. Kozeschnik Ernst: Writing – review & editing, Software, Methodology. Liao Xiaozhou: Writing – review & editing, Project administration, Funding acquisition. Ringer Simon P: Writing – review & editing, Project administration, Funding acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


This research was sponsored by the Department of Industry, Innovation, and Science under the auspices of the AUSMURI program – which is a part of the Commonwealth’s Next Generation Technologies Fund. The authors acknowledge the facilities and the scientific and technical assistance at the Electron Microscope Unit (EMU) within the Mark Wainwright Analytical Centre (MWAC) at UNSW Sydney and Microscopy Australia. Nana Adomako is supported by a UNSW Scientia PhD scholarship. Michael Haines’ (UNSW Sydney) contribution to the revised version of the original manuscript is thankfully acknowledged.

Appendix A. Supplementary material

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Supplementary material.

Data Availability

Data will be made available on request.


Thermo-fluid modeling of influence of attenuated laser beam intensity profile on melt pool behavior in laser-assisted powder-based direct energy deposition

레이저 보조 분말 기반 직접 에너지 증착에서 용융 풀 거동에 대한 감쇠 레이저 빔 강도 프로파일의 영향에 대한 열유체 모델링

Thermo-fluid modeling of influence of attenuated laser beam intensity profile on melt pool behavior in laser-assisted powder-based direct energy deposition

Mohammad Sattari, Amin Ebrahimi, Martin Luckabauer, Gert-willem R.B.E. Römer

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Professional

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A numerical framework based on computational fluid dynamics (CFD), using the finite volume method (FVM) and volume of fluid (VOF) technique is presented to investigate the effect of the laser beam intensity profile on melt pool behavior in laser-assisted powder-based directed energy deposition (L-DED). L-DED is an additive manufacturing (AM) process that utilizes a laser beam to fuse metal powder particles. To assure high-fidelity modeling, it was found that it is crucial to accurately model the interaction between the powder stream and the laser beam in the gas region above the substrate. The proposed model considers various phenomena including laser energy attenuation and absorption, multiple reflections of the laser rays, powder particle stream, particle-fluid interaction, temperature-dependent properties, buoyancy effects, thermal expansion, solidification shrinkage and drag, and Marangoni flow. The latter is induced by temperature and element-dependent surface tension. The model is validated using experimental results and highlights the importance of considering laser energy attenuation. Furthermore, the study investigates how the laser beam intensity profile affects melt pool size and shape, influencing the solidification microstructure and mechanical properties of the deposited material. The proposed model has the potential to optimize the L-DED process for a variety of materials and provides insights into the capability of numerical modeling for additive manufacturing optimization.

Original languageEnglish
Title of host publicationFlow-3D World Users Conference
Publication statusPublished – 2023
EventFlow-3D World User Conference – Strasbourg, France
Duration: 5 Jun 2023 → 7 Jun 2023


ConferenceFlow-3D World User Conference
Period5/06/23 → 7/06/23
Numerical simulation on molten pool behavior of narrow gap gas tungsten arc welding

좁은 간격 가스 텅스텐 아크 용접의 용융 풀 거동에 대한 수치 시뮬레이션

Numerical simulation on molten pool behavior of narrow gap gas tungsten arc welding

The International Journal of Advanced Manufacturing Technology (2023)Cite this article


As a highly efficient thick plate welding resolution, narrow gap gas tungsten arc welding (NG-GTAW) is in the face of a series of problems like inter-layer defects like pores, lack of fusion, inclusion of impurity, and the sensitivity to poor sidewall fusion, which is hard to be repaired after the welding process. This study employs numerical simulation to investigate the molten pool behavior in NG-GTAW root welding. A 3D numerical model was established, where a body-fitted coordinate system was applied to simulate the electromagnetic force, and a bridge transition model was developed to investigate the wire–feed root welding. The simulated results were validated experimentally. Results show that the molten pool behavior is dominated by electromagnetic force when the welding current is relatively high, and the dynamic change of the vortex actually determines the molten pool morphology. For self-fusion welding, there are two symmetric inward vortices in the cross-section and one clockwise vortex in the longitudinal section. With the increasing welding current, the vortices in the cross-section gradually move to the arc center with a decreasing range, while the vortex in the longitudinal section moves backward. With the increasing traveling speed, the vortices in the cross-section move toward the surface of the molten pool with a decreasing range, and the horizontal component of liquid metal velocity changes in the longitudinal section. For wire–feed welding, the filling metal strengthens the downward velocity component; as a result, the vortex formation is blocked in the cross-section and is strengthened in the longitudinal section.

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Data availability

The raw/processed data required cannot be shared at this time as the data also forms part of an ongoing study.


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Figure 5. Schematic view of flap and support structure [32]

Design Optimization of Ocean Renewable Energy Converter Using a Combined Bi-level Metaheuristic Approach

결합된 Bi-level 메타휴리스틱 접근법을 사용한 해양 재생 에너지 변환기의 설계 최적화

Erfan Amini a1, Mahdieh Nasiri b1, Navid Salami Pargoo a, Zahra Mozhgani c, Danial Golbaz d, Mehrdad Baniesmaeil e, Meysam Majidi Nezhad f, Mehdi Neshat gj, Davide Astiaso Garcia h, Georgios Sylaios i


In recent years, there has been an increasing interest in renewable energies in view of the fact that fossil fuels are the leading cause of catastrophic environmental consequences. Ocean wave energy is a renewable energy source that is particularly prevalent in coastal areas. Since many countries have tremendous potential to extract this type of energy, a number of researchers have sought to determine certain effective factors on wave converters’ performance, with a primary emphasis on ambient factors. In this study, we used metaheuristic optimization methods to investigate the effects of geometric factors on the performance of an Oscillating Surge Wave Energy Converter (OSWEC), in addition to the effects of hydrodynamic parameters. To do so, we used CATIA software to model different geometries which were then inserted into a numerical model developed in Flow3D software. A Ribed-surface design of the converter’s flap is also introduced in this study to maximize wave-converter interaction. Besides, a Bi-level Hill Climbing Multi-Verse Optimization (HCMVO) method was also developed for this application. The results showed that the converter performs better with greater wave heights, flap freeboard heights, and shorter wave periods. Additionally, the added ribs led to more wave-converter interaction and better performance, while the distance between the flap and flume bed negatively impacted the performance. Finally, tracking the changes in the five-dimensional objective function revealed the optimum value for each parameter in all scenarios. This is achieved by the newly developed optimization algorithm, which is much faster than other existing cutting-edge metaheuristic approaches.


Wave Energy Converter


Hydrodynamic Effects

Geometric Design

Metaheuristic Optimization

Multi-Verse Optimizer


The increase in energy demand, the limitations of fossil fuels, as well as environmental crises, such as air pollution and global warming, are the leading causes of calling more attention to harvesting renewable energy recently [1][2][3]. While still in its infancy, ocean wave energy has neither reached commercial maturity nor technological convergence. In recent decades, remarkable progress has been made in the marine energy domain, which is still in the early stage of development, to improve the technology performance level (TPL) [4][5]and technology readiness level (TRL) of wave energy converters (WECs). This has been achieved using novel modeling techniques [6][7][8][9][10][11][12][13][14] to gain the following advantages [15]: (i) As a source of sustainable energy, it contributes to the mix of energy resources that leads to greater diversity and attractiveness for coastal cities and suppliers. [16] (ii) Since wave energy can be exploited offshore and does not require any land, in-land site selection would be less expensive and undesirable visual effects would be reduced. [17] (iii) When the best layout and location of offshore site are taken into account, permanent generation of energy will be feasible (as opposed to using solar energy, for example, which is time-dependent) [18].

In general, the energy conversion process can be divided into three stages in a WEC device, including primary, secondary, and tertiary stages [19][20]. In the first stage of energy conversion, which is the subject of this study, the wave power is converted to mechanical power by wave-structure interaction (WSI) between ocean waves and structures. Moreover, the mechanical power is transferred into electricity in the second stage, in which mechanical structures are coupled with power take-off systems (PTO). At this stage, optimal control strategies are useful to tune the system dynamics to maximize power output [10][13][12]. Furthermore, the tertiary energy conversion stage revolves around transferring the non-standard AC power into direct current (DC) power for energy storage or standard AC power for grid integration [21][22]. We discuss only the first stage regardless of the secondary and tertiary stages. While Page 1 of 16 WECs include several categories and technologies such as terminators, point absorbers, and attenuators [15][23], we focus on oscillating surge wave energy converters (OSWECs) in this paper due to its high capacity for industrialization [24].

Over the past two decades, a number of studies have been conducted to understand how OSWECs’ structures and interactions between ocean waves and flaps affect converters performance. Henry et al.’s experiment on oscillating surge wave energy converters is considered as one of the most influential pieces of research [25], which demonstrated how the performance of oscillating surge wave energy converters (OSWECs) is affected by seven different factors, including wave period, wave power, flap’s relative density, water depth, free-board of the flap, the gap between the tubes, gap underneath the flap, and flap width. These parameters were assessed in their two models in order to estimate the absorbed energy from incoming waves [26][27]. In addition, Folly et al. investigated the impact of water depth on the OSWECs performance analytically, numerically, and experimentally. According to this and further similar studies, the average annual incident wave power is significantly reduced by water depth. Based on the experimental results, both the surge wave force and the power capture of OSWECs increase in shallow water [28][29]. Following this, Sarkar et al. found that under such circumstances, the device that is located near the coast performs much better than those in the open ocean [30]. On the other hand, other studies are showing that the size of the converter, including height and width, is relatively independent of the location (within similar depth) [31]. Subsequently, Schmitt et al. studied OSWECs numerically and experimentally. In fact, for the simulation of OSWEC, OpenFOAM was used to test the applicability of Reynolds-averaged Navier-Stokes (RANS) solvers. Then, the experimental model reproduced the numerical results with satisfying accuracy [32]. In another influential study, Wang et al. numerically assessed the effect of OSWEC’s width on their performance. According to their findings, as converter width increases, its efficiency decreases in short wave periods while increases in long wave periods [33]. One of the main challenges in the analysis of the OSWEC is the coupled effect of hydrodynamic and geometric variables. As a result, numerous cutting-edge geometry studies have been performed in recent years in order to find the optimal structure that maximizes power output and minimizes costs. Garcia et al. reviewed hull geometry optimization studies in the literature in [19]. In addition, Guo and Ringwood surveyed geometric optimization methods to improve the hydrodynamic performance of OSWECs at the primary stage [14]. Besides, they classified the hull geometry of OSWECs based on Figure 1. Subsequently, Whittaker et al. proposed a different design of OSWEC called Oyster2. There have been three examples of different geometries of oysters with different water depths. Based on its water depth, they determined the width and height of the converter. They also found that in the constant wave period the less the converter’s width, the less power captures the converter has [34]. Afterward, O’Boyle et al. investigated a type of OSWEC called Oyster 800. They compared the experimental and numerical models with the prototype model. In order to precisely reproduce the shape, mass distribution, and buoyancy properties of the prototype, a 40th-scale experimental model has been designed. Overall, all the models were fairly accurate according to the results [35].

Inclusive analysis of recent research avenues in the area of flap geometry has revealed that the interaction-based designs of such converters are emerging as a novel approach. An initiative workflow is designed in the current study to maximizing the wave energy extrication by such systems. To begin with, a sensitivity analysis plays its role of determining the best hydrodynamic values for installing the converter’s flap. Then, all flap dimensions and characteristics come into play to finalize the primary model. Following, interactive designs is proposed to increase the influence of incident waves on the body by adding ribs on both sides of the flap as a novel design. Finally, a new bi-level metaheuristic method is proposed to consider the effects of simultaneous changes in ribs properties and other design parameters. We hope this novel approach will be utilized to make big-scale projects less costly and justifiable. The efficiency of the method is also compared with four well known metaheuristic algorithms and out weight them for this application.

This paper is organized as follows. First, the research methodology is introduced by providing details about the numerical model implementation. To that end, we first introduced the primary model’s geometry and software details. That primary model is later verified with a benchmark study with regard to the flap angle of rotation and water surface elevation. Then, governing equations and performance criteria are presented. In the third part of the paper, we discuss the model’s sensitivity to lower and upper parts width (we proposed a two cross-sectional design for the flap), bottom elevation, and freeboard. Finally, the novel optimization approach is introduced in the final part and compared with four recent metaheuristic algorithms.

2. Numerical Methods

In this section, after a brief introduction of the numerical software, Flow3D, boundary conditions are defined. Afterwards, the numerical model implementation, along with primary model properties are described. Finally, governing equations, as part of numerical process, are discussed.

2.1Model Setup

FLOW-3D is a powerful and comprehensive CFD simulation platform for studying fluid dynamics. This software has several modules to solve many complex engineering problems. In addition, modeling complex flows is simple and effective using FLOW-3D’s robust meshing capabilities [36]. Interaction between fluid and moving objects might alter the computational range. Dynamic meshes are used in our modeling to take these changes into account. At each time step, the computational node positions change in order to adapt the meshing area to the moving object. In addition, to choose mesh dimensions, some factors are taken into account such as computational accuracy, computational time, and stability. The final grid size is selected based on the detailed procedure provided in [37]. To that end, we performed grid-independence testing on a CFD model using three different mesh grid sizes of 0.01, 0.015, and 0.02 meters. The problem geometry and boundary conditions were defined the same, and simulations were run on all three grids under the same conditions. The predicted values of the relevant variable, such as velocity, was compared between the grids. The convergence behavior of the numerical solution was analyzed by calculating the relative L2 norm error between two consecutive grids. Based on the results obtained, it was found that the grid size of 0.02 meters showed the least error, indicating that it provided the most accurate and reliable solution among the three grids. Therefore, the grid size of 0.02 meters was selected as the optimal spatial resolution for the mesh grid.

In this work, the flume dimensions are 10 meters long, 0.1 meters wide, and 2.2 meters high, which are shown in figure2. In addition, input waves with linear characteristics have a height of 0.1 meters and a period of 1.4 seconds. Among the linear wave methods included in this software, RNGk-ε and k- ε are appropriate for turbulence model. The research of Lopez et al. shows that RNGk- ε provides the most accurate simulation of turbulence in OSWECs [21]. We use CATIA software to create the flap primary model and other innovative designs for this project. The flap measures 0.1 m x 0.65 m x 0.360 m in x, y and z directions, respectively. In Figure 3, the primary model of flap and its dimensions are shown. In this simulation, five boundaries have been defined, including 1. Inlet, 2. Outlet, 3. Converter flap, 4. Bed flume, and 5. Water surface, which are shown in figure 2. Besides, to avoid wave reflection in inlet and outlet zones, Flow3D is capable of defining some areas as damping zones, the length of which has to be one to one and a half times the wavelength. Therefore, in the model, this length is considered equal to 2 meters. Furthermore, there is no slip in all the boundaries. In other words, at every single time step, the fluid velocity is zero on the bed flume, while it is equal to the flap velocity on the converter flap. According to the wave theory defined in the software, at the inlet boundary, the water velocity is called from the wave speed to be fed into the model.


In the current study, we utilize the Schmitt experimental model as a benchmark for verification, which was developed at the Queen’s University of Belfast. The experiments were conducted on the flap of the converter, its rotation, and its interaction with the water surface. Thus, the details of the experiments are presented below based up on the experimental setup’s description [38]. In the experiment, the laboratory flume has a length of 20m and a width of 4.58m. Besides, in order to avoid incident wave reflection, a wave absorption source is devised at the end of the left flume. The flume bed, also, includes two parts with different slops. The flap position and dimensions of the flume can be seen in Figure4. In addition, a wave-maker with 6 paddles is installed at one end. At the opposite end, there is a beach with wire meshes. Additionally, there are 6 indicators to extract the water level elevation. In the flap model, there are three components: the fixed support structure, the hinge, and the flap. The flap measures 0.1m x 0.65m x 0.341m in x, y and z directions, respectively. In Figure5, the details are given [32]. The support structure consists of a 15 mm thick stainless steel base plate measuring 1m by 1.4m, which is screwed onto the bottom of the tank. The hinge is supported by three bearing blocks. There is a foam centerpiece on the front and back of the flap which is sandwiched between two PVC plates. Enabling changes of the flap, three metal fittings link the flap to the hinge. Moreover, in this experiment, the selected wave is generated based on sea wave data at scale 1:40. The wave height and the wave period are equal to 0.038 (m) and 2.0625 (s), respectively, which are tantamount to a wave with a period of 13 (s) and a height of 1.5 (m).

Two distinct graphs illustrate the numerical and experi-mental study results. Figure6 and Figure7 are denoting the angle of rotation of flap and surface elevation in computational and experimental models, respectively. The two figures roughly represent that the numerical and experimental models are a good match. However, for the purpose of verifying the match, we calculated the correlation coefficient (C) and root mean square error (RMSE). According to Figure6, correlation coefficient and RMSE are 0.998 and 0.003, respectively, and in Figure7 correlation coefficient and RMSE are respectively 0.999 and 0.001. Accordingly, there is a good match between the numerical and empirical models. It is worth mentioning that the small differences between the numerical and experimental outputs may be due to the error of the measuring devices and the calibration of the data collection devices.

Including continuity equation and momentum conserva- tion for incompressible fluid are given as [32][39]:(1)

where P represents the pressure, g denotes gravitational acceleration, u represents fluid velocity, and Di is damping coefficient. Likewise, the model uses the same equation. to calculate the fluid velocity in other directions as well. Considering the turbulence, we use the two-equation model of RNGK- ε. These equations are:

(3)��t(��)+����(����)=����[�eff�������]+��-��and(4)���(��)+����(����)=����[�eff�������]+�1�∗����-��2��2�Where �2� and �1� are constants. In addition, �� and �� represent the turbulent Prandtl number of � and k, respectively.

�� also denote the production of turbulent kinetic energy of k under the effect of velocity gradient, which is calculated as follows:(5)��=�eff[�����+�����]�����(6)�eff=�+��(7)�eff=�+��where � is molecular viscosity,�� represents turbulence viscosity, k denotes kinetic energy, and ∊∊ is energy dissipation rate. The values of constant coefficients in the two-equation RNGK ∊-∊ model is as shown in the Table 1 [40].Table 2.

Table 1. Constant coefficients in RNGK- model


Table 2. Flap properties

Joint height (m)0.476
Height of the center of mass (m)0.53
Weight (Kg)10.77

It is worth mentioning that the volume of fluid method is used to separate water and air phases in this software [41]. Below is the equation of this method [40].(8)����+����(���)=0where α and 1 − α are portion of water phase and air phase, respectively. As a weighting factor, each fluid phase portion is used to determine the mixture properties. Finally, using the following equations, we calculate the efficiency of converters [42][34][43]:(9)�=14|�|2�+�2+(�+�a)2(�n2-�2)2where �� represents natural frequency, I denotes the inertia of OSWEC, Ia is the added inertia, F is the complex wave force, and B denotes the hydrodynamic damping coefficient. Afterward, the capture factor of the converter is calculated by [44]:(10)��=�1/2��2����gw where �� represents the capture factor, which is the total efficiency of device per unit length of the wave crest at each time step [15], �� represent the dimensional amplitude of the incident wave, w is the flap’s width, and Cg is the group velocity of the incident wave, as below:(11)��=��0·121+2�0ℎsinh2�0ℎwhere �0 denotes the wave number, h is water depth, and H is the height of incident waves.

According to previous sections ∊,����-∊ modeling is used for all models simulated in this section. For this purpose, the empty boundary condition is used for flume walls. In order to preventing wave reflection at the inlet and outlet of the flume, the length of wave absorption is set to be at least one incident wavelength. In addition, the structured mesh is chosen, and the mesh dimensions are selected in two distinct directions. In each model, all grids have a length of 2 (cm) and a height of 1 (cm). Afterwards, as an input of the software for all of the models, we define the time step as 0.001 (s). Moreover, the run time of every simulation is 30 (s). As mentioned before, our primary model is Schmitt model, and the flap properties is given in table2. For all simulations, the flume measures 15 meters in length and 0.65 meters in width, and water depth is equal to 0.335 (m). The flap is also located 7 meters from the flume’s inlet.

Finally, in order to compare the results, the capture factor is calculated for each simulation and compared to the primary model. It is worth mentioning that capture factor refers to the ratio of absorbed wave energy to the input wave energy.

According to primary model simulation and due to the decreasing horizontal velocity with depth, the wave crest has the highest velocity. Considering the fact that the wave’s orbital velocity causes the flap to move, the contact between the upper edge of the flap and the incident wave can enhance its performance. Additionally, the numerical model shows that the dynamic pressure decreases as depth increases, and the hydrostatic pressure increases as depth increases.

To determine the OSWEC design, it is imperative to understand the correlation between the capture factor, wave period, and wave height. Therefore, as it is shown in Figure8, we plot the change in capture factor over the variations in wave period and wave height in 3D and 2D. In this diagram, the first axis features changes in wave period, the second axis displays changes in wave height, and the third axis depicts changes in capture factor. According to our wave properties in the numerical model, the wave period and wave height range from 2 to 14 seconds and 2 to 8 meters, respectively. This is due to the fact that the flap does not oscillate if the wave height is less than 2 (m), and it does not reverse if the wave height is more than 8 (m). In addition, with wave periods more than 14 (s), the wavelength would be so long that it would violate the deep-water conditions, and with wave periods less than 2 (s), the flap would not oscillate properly due to the shortness of wavelength. The results of simulation are shown in Figure 8. As it can be perceived from Figure 8, in a constant wave period, the capture factor is in direct proportion to the wave height. It is because of the fact that waves with more height have more energy to rotate the flap. Besides, in a constant wave height, the capture factor increases when the wave period increases, until a given wave period value. However, the capture factor falls after this point. These results are expected since the flap’s angular displacement is not high in lower wave periods, while the oscillating motion of that is not fast enough to activate the power take-off system in very high wave periods.

As is shown in Figure 9, we plot the change in capture factor over the variations in wave period (s) and water depth (m) in 3D. As it can be seen in this diagram, the first axis features changes in water depth (m), the second axis depicts the wave period (s), and the third axis displays OSWEC’s capture factor. The wave period ranges from 0 to 10 seconds based on our wave properties, which have been adopted from Schmitt’s model, while water depth ranges from 0 to 0.5 meters according to the flume and flap dimensions and laboratory limitations. According to Figure9, for any specific water depth, the capture factor increases in a varying rate when the wave period increases, until a given wave period value. However, the capture factor falls steadily after this point. In fact, the maximum capture factor occurs when the wave period is around 6 seconds. This trend is expected since, in a specific water depth, the flap cannot oscillate properly when the wavelength is too short. As the wave period increases, the flap can oscillate more easily, and consequently its capture factor increases. However, the capture factor drops in higher wave periods because the wavelength is too large to move the flap. Furthermore, in a constant wave period, by changing the water depth, the capture factor does not alter. In other words, the capture factor does not depend on the water depth when it is around its maximum value.

3Sensitivity Analysis

Based on previous studies, in addition to the flap design, the location of the flap relative to the water surface (freeboard) and its elevation relative to the flume bed (flap bottom elevation) play a significant role in extracting energy from the wave energy converter. This study measures the sensitivity of the model to various parameters related to the flap design including upper part width of the flap, lower part width of the flap, the freeboard, and the flap bottom elevation. Moreover, as a novel idea, we propose that the flap widths differ in the lower and upper parts. In Figure10, as an example, a flap with an upper thickness of 100 (mm) and a lower thickness of 50 (mm) and a flap with an upper thickness of 50 (mm) and a lower thickness of 100 (mm) are shown. The influence of such discrepancy between the widths of the upper and lower parts on the interaction between the wave and the flap, or in other words on the capture factor, is evaluated. To do so, other parameters are remained constant, such as the freeboard, the distance between the flap and the flume bed, and the wave properties.

In Figure11, models are simulated with distinct upper and lower widths. As it is clear in this figure, the first axis depicts the lower part width of the flap, the second axis indicates the upper part width of the flap, and the colors represent the capture factor values. Additionally, in order to consider a sufficient range of change, the flap thickness varies from half to double the value of the primary model for each part.

According to this study, the greater the discrepancy in these two parts, the lower the capture factor. It is on account of the fact that when the lower part of the flap is thicker than the upper part, and this thickness difference in these two parts is extremely conspicuous, the inertia against the motion is significant at zero degrees of rotation. Consequently, it is difficult to move the flap, which results in a low capture factor. Similarly, when the upper part of the flap is thicker than the lower part, and this thickness difference in these two parts is exceedingly noticeable, the inertia is so great that the flap can not reverse at the maximum degree of rotation. As the results indicate, the discrepancy can enhance the performance of the converter if the difference between these two parts is around 20%. As it is depicted in the Figure11, the capture factor reaches its own maximum amount, when the lower part thickness is from 5 to 6 (cm), and the upper part thickness is between 6 and 7 (cm). Consequently, as a result of this discrepancy, less material will be used, and therefore there will be less cost.

As illustrated in Figure12, this study examines the effects of freeboard (level difference between the flap top and water surface) and the flap bottom elevation (the distance between the flume bed and flap bottom) on the converter performance. In this diagram, the first axis demonstrates the freeboard and the second axis on the left side displays the flap bottom elevation, while the colors indicate the capture factor. In addition, the feasible range of freeboard is between -15 to 15 (cm) due to the limitation of the numerical model, so that we can take the wave slamming and the overtopping into consideration. Additionally, based on the Schmitt model and its scaled model of 1:40 of the base height, the flap bottom should be at least 9 (cm) high. Since the effect of surface waves is distributed over the depth of the flume, it is imperative to maintain a reasonable flap height exposed to incoming waves. Thus, the maximum flap bottom elevation is limited to 19 (cm). As the Figure12 pictures, at constant negative values of the freeboard, the capture factor is in inverse proportion with the flap bottom elevation, although slightly.

Furthermore, at constant positive values of the freeboard, the capture factor fluctuates as the flap bottom elevation decreases while it maintains an overall increasing trend. This is on account of the fact that increasing the flap bottom elevation creates turbulence flow behind the flap, which encumbers its rotation, as well as the fact that the flap surface has less interaction with the incoming waves. Furthermore, while keeping the flap bottom elevation constant, the capture factor increases by raising the freeboard. This is due to the fact that there is overtopping with adverse impacts on the converter performance when the freeboard is negative and the flap is under the water surface. Besides, increasing the freeboard makes the wave slam more vigorously, which improves the converter performance.

Adding ribs to the flap surface, as shown in Figure13, is a novel idea that is investigated in the next section. To achieve an optimized design for the proposed geometry of the flap, we determine the optimal number and dimensions of ribs based on the flap properties as our decision variables in the optimization process. As an example, Figure13 illustrates a flap with 3 ribs on each side with specific dimensions.

Figure14 shows the flow velocity field around the flap jointed to the flume bed. During the oscillation of the flap, the pressure on the upper and lower surfaces of the flap changes dynamically due to the changing angle of attack and the resulting change in the direction of fluid flow. As the flap moves upwards, the pressure on the upper surface decreases, and the pressure on the lower surface increases. Conversely, as the flap moves downwards, the pressure on the upper surface increases, and the pressure on the lower surface decreases. This results in a cyclic pressure variation around the flap. Under certain conditions, the pressure field around the flap can exhibit significant variations in magnitude and direction, forming vortices and other flow structures. These flow structures can affect the performance of the OSWEC by altering the lift and drag forces acting on the flap.

4Design Optimization

We consider optimizing the design parameters of the flap of converter using a nature-based swarm optimization method, that fall in the category of metaheuristic algorithms [45]. Accordingly, we choose four state-of-the-art algorithms to perform an optimization study. Then, based on their performances to achieve the highest capture factor, one of them will be chosen to be combined with the Hill Climb algorithm to carry out a local search. Therefore, in the remainder of this section, we discuss the search process of each algorithm and visualize their performance and convergence curve as they try to find the best values for decision variables.

4.1. Metaheuristic Approaches

As the first considered algorithm, the Gray Wolf Optimizer (GWO) algorithm simulates the natural leadership and hunting performance of gray wolves which tend to live in colonies. Hunters must obey the alpha wolf, the leader, who is responsible for hunting. Then, the beta wolf is at the second level of the gray wolf hierarchy. A subordinate of alpha wolf, beta stands under the command of the alpha. At the next level in this hierarchy, there are the delta wolves. They are subordinate to the alpha and beta wolves. This category of wolves includes scouts, sentinels, elders, hunters, and caretakers. In this ranking, omega wolves are at the bottom, having the lowest level and obeying all other wolves. They are also allowed to eat the prey just after others have eaten. Despite the fact that they seem less important than others, they are really central to the pack survival. Since, it has been shown that without omega wolves, the entire pack would experience some problems like fighting, violence, and frustration. In this simulation, there are three primary steps of hunting including searching, surrounding, and finally attacking the prey. Mathematically model of gray wolves’ hunting technique and their social hierarchy are applied in determined by optimization. this study. As mentioned before, gray wolves can locate their prey and surround them. The alpha wolf also leads the hunt. Assuming that the alpha, beta, and delta have more knowledge about prey locations, we can mathematically simulate gray wolf hunting behavior. Hence, in addition to saving the top three best solutions obtained so far, we compel the rest of the search agents (also the omegas) to adjust their positions based on the best search agent. Encircling behavior can be mathematically modeled by the following equations: [46].(12)�→=|�→·��→(�)-�→(�)|(13)�→(�+1)=��→(�)-�→·�→(14)�→=2.�2→(15)�→=2�→·�1→-�→Where �→indicates the position vector of gray wolf, ��→ defines the vector of prey, t indicates the current iteration, and �→and �→are coefficient vectors. To force the search agent to diverge from the prey, we use �→ with random values greater than 1 or less than -1. In addition, C→ contains random values in the range [0,2], and �→ 1 and �2→ are random vectors in [0,1]. The second considered technique is the Moth Flame Optimizer (MFO) algorithm. This method revolves around the moths’ navigation mechanism, which is realized by positioning themselves and maintaining a fixed angle relative to the moon while flying. This effective mechanism helps moths to fly in a straight path. However, when the source of light is artificial, maintaining an angle with the light leads to a spiral flying path towards the source that causes the moth’s death [47]. In MFO algorithm, moths and flames are both solutions. The moths are actual search agents that fly in hyper-dimensional space by changing their position vectors, and the flames are considered pins that moths drop when searching the search space [48]. The problem’s variables are the position of moths in the space. Each moth searches around a flame and updates it in case of finding a better solution. The fitness value is the return value of each moth’s fitness (objective) function. The position vector of each moth is passed to the fitness function, and the output of the fitness function is assigned to the corresponding moth. With this mechanism, a moth never loses its best solution [49]. Some attributes of this algorithm are as follows:

  • •It takes different values to converge moth in any point around the flame.
  • •Distance to the flame is lowered to be eventually minimized.
  • •When the position gets closer to the flame, the updated positions around the flame become more frequent.

As another method, the Multi-Verse Optimizer is based on a multiverse theory which proposes there are other universes besides the one in which we all live. According to this theory, there are more than one big bang in the universe, and each big bang leads to the birth of a new universe [50]. Multi-Verse Optimizer (MVO) is mainly inspired by three phenomena in cosmology: white holes, black holes, and wormholes. A white hole has never been observed in our universe, but physicists believe the big bang could be considered a white hole [51]. Black holes, which behave completely in contrast to white holes, attract everything including light beams with their extremely high gravitational force [52]. In the multiverse theory, wormholes are time and space tunnels that allow objects to move instantly between any two corners of a universe (or even simultaneously from one universe to another) [53]. Based on these three concepts, mathematical models are designed to perform exploration, exploitation, and local search, respectively. The concept of white and black holes is implied as an exploration phase, while the concept of wormholes is considered as an exploitation phase by MVO. Additionally, each solution is analogous to a universe, and each variable in the solution represents an object in that universe. Furthermore, each solution is assigned an inflation rate, and the time is used instead of iterations. Following are the universe rules in MVO:

  • •The possibility of having white hole increases with the inflation rate.
  • •The possibility of having black hole decreases with the inflation rate.
  • •Objects tend to pass through black holes more frequently in universes with lower inflation rates.
  • •Regardless of inflation rate, wormholes may cause objects in universes to move randomly towards the best universe. [54]

Modeling the white/black hole tunnels and exchanging objects of universes mathematically was accomplished by using the roulette wheel mechanism. With every iteration, the universes are sorted according to their inflation rates, then, based on the roulette wheel, the one with the white hole is selected as the local extremum solution. This is accomplished through the following steps:

Assume that


Where ��� represents the jth parameter of the ith universe, Ui indicates the ith universe, NI(Ui) is normalized inflation rate of the ith universe, r1 is a random number in [0,1], and j xk shows the jth parameter of the kth universe selected by a roulette wheel selection mechanism [54]. It is assumed that wormhole tunnels always exist between a universe and the best universe formed so far. This mechanism is as follows:(17)���=if�2<���:��+���×((���-���)×�4+���)�3<0.5��-���×((���-���)×�4+���)�3≥0.5����:���where Xj indicates the jth parameter of the best universe formed so far, TDR and WEP are coefficients, where Xj indicates the jth parameter of the best universelbjshows the lower bound of the jth variable, ubj is the upper bound of the jth variable, and r2, r3, and r4 are random numbers in [1][54].

Finally, one of the newest optimization algorithms is WOA. The WOA algorithm simulates the movement of prey and the whale’s discipline when looking for their prey. Among several species, Humpback whales have a specific method of hunting [55]. Humpback whales can recognize the location of prey and encircle it before hunting. The optimal design position in the search space is not known a priori, and the WOA algorithm assumes that the best candidate solution is either the target prey or close to the optimum. This foraging behavior is called the bubble-net feeding method. Two maneuvers are associated with bubbles: upward spirals and double loops. A unique behavior exhibited only by humpback whales is bubble-net feeding. In fact, The WOA algorithm starts with a set of random solutions. At each iteration, search agents update their positions for either a randomly chosen search agent or the best solution obtained so far [56][55]. When the best search agent is determined, the other search agents will attempt to update their positions toward that agent. It is important to note that humpback whales swim around their prey simultaneously in a circular, shrinking circle and along a spiral-shaped path. By using a mathematical model, the spiral bubble-net feeding maneuver is optimized. The following equation represents this behavior:(18)�→(�+1)=�′→·�bl·cos(2��)+�∗→(�)


X→(t+ 1) indicates the distance of the it h whale to the prey (best solution obtained so far),� is a constant for defining the shape of the logarithmic spiral, l is a random number in [−1, 1], and dot (.) is an element-by-element multiplication [55].

Comparing the four above-mentioned methods, simulations are run with 10 search agents for 400 iterations. In Figure 15, there are 20 plots the optimal values of different parameters in optimization algorithms. The five parameters of this study are freeboard, bottom elevations, number of ribs on the converter, rib thickness, and rib Height. The optimal value for each was found by optimization algorithms, naming WOA, MVO, MFO, and GWO. By looking through the first row, the freeboard parameter converges to its maximum possible value in the optimization process of GWO after 300 iterations. Similarly, MFO finds the same result as GWO. In contrast, the freeboard converges to its minimum possible value in MVO optimizing process, which indicates positioning the converter under the water. Furthermore, WOA found the optimal value of freeboard as around 0.02 after almost 200 iterations. In the second row, the bottom elevation is found at almost 0.11 (m) in all algorithms; however, the curves follow different trends in each algorithm. The third row shows the number of ribs, where results immediately reveal that it should be over 4. All algorithms coincide at 5 ribs as the optimal number in this process. The fourth row displays the trends of algorithms to find optimal rib thickness. MFO finds the optimal value early and sets it to around 0.022, while others find the same value in higher iterations. Finally, regarding the rib height, MVO, MFO, and GWO state that the optimal value is 0.06 meters, but WOA did not find a higher value than 0.039.

4.2. HCMVO Bi-level Approach

Despite several strong search characteristics of MVO and its high performance in various optimization problems, it suffers from a few deficiencies in local and global search mechanisms. For instance, it is trapped in the local optimum when wormholes stochastically generate many solutions near the best universe achieved throughout iterations, especially in solving complex multimodal problems with high dimensions [57]. Furthermore, MVO needs to be modified by an escaping strategy from the local optima to enhance the global search abilities. To address these shortages, we propose a fast and effective meta-algorithm (HCMVO) to combine MVO with a Random-restart hill-climbing local search. This meta-algorithm uses MVO on the upper level to develop global tracking and provide a range of feasible and proper solutions. The hill-climbing algorithm is designed to develop a comprehensive neighborhood search around the best-found solution proposed by the upper-level (MVO) when MVO is faced with a stagnation issue or falling into a local optimum. The performance threshold is formulated as follows.(20)Δ����THD=∑�=1�����TH��-����TH��-1�where BestTHDis the best-found solution per generation, andM is related to the domain of iterations to compute the average performance of MVO. If the proposed best solution by the local search is better than the initial one, the global best of MVO will be updated. HCMVO iteratively runs hill climbing when the performance of MVO goes down, each time with an initial condition to prepare for escaping such undesirable situations. In order to get a better balance between exploration and exploitation, the search step size linearly decreases as follows:(21)��=��-����Ma�iter��+1where iter and Maxiter are the current iteration and maximum number of evaluation, respectively. �� stands for the step size of the neighborhood search. Meanwhile, this strategy can improve the convergence rate of MVO compared with other algorithms.

Algorithm 1 shows the technical details of the proposed optimization method (HCMVO). The initial solution includes freeboard (�), bottom elevation (�), number of ribs (Nr), rib thickness (�), and rib height(�).

5. Conclusion

The high trend of diminishing worldwide energy resources has entailed a great crisis upon vulnerable societies. To withstand this effect, developing renewable energy technologies can open doors to a more reliable means, among which the wave energy converters will help the coastal residents and infrastructure. This paper set out to determine the optimized design for such devices that leads to the highest possible power output. The main goal of this research was to demonstrate the best design for an oscillating surge wave energy converter using a novel metaheuristic optimization algorithm. In this regard, the methodology was devised such that it argued the effects of influential parameters, including wave characteristics, WEC design, and interaction criteria.

To begin with, a numerical model was developed in Flow 3D software to simulate the response of the flap of a wave energy converter to incoming waves, followed by a validation study based upon a well-reputed experimental study to verify the accuracy of the model. Secondly, the hydrodynamics of the flap was investigated by incorporating the turbulence. The effect of depth, wave height, and wave period are also investigated in this part. The influence of two novel ideas on increasing the wave-converter interaction was then assessed: i) designing a flap with different widths in the upper and lower part, and ii) adding ribs on the surface of the flap. Finally, four trending single-objective metaheuristic optimization methods

Empty CellAlgorithm 1: Hill Climb Multiverse Optimization
01:procedure HCMVO
04:Initialize parameters�ER,�DR,�EP,Best�,���ite��▹Wormhole existence probability (WEP)
06:��=Normalize the inflation rate��
07:for iter in[1,⋯,���iter]do
13:White HoleIndex=Roulette�heelSelection(-��)
14:�(Black HoleIndex,�)=��(White HoleIndex,�)
15:end if
24:end if
25:end if
26:end for
27:end for
31:ifΔBestTHD<��then▹Perform hill climbing local search
33:end if
34:end for
35:return�,BestTHD▹Final configuration
36:end procedure

The implementation details of the hill-climbing algorithm applied in HCMPA can be seen in Algorithm 2. One of the critical parameters isg, which denotes the resolution of the neighborhood search around the proposed global best by MVO. If we set a small step size for hill-climbing, the convergence speed will be decreased. On the other hand, a large step size reinforces the exploration ability. Still, it may reduce the exploitation ability and in return increase the act of jumping from a global optimum or surfaces with high-potential solutions. Per each decision variable, the neighborhood search evaluates two different direct searches, incremental or decremental. After assessing the generated solutions, the best candidate will be selected to iterate the search algorithm. It is noted that the hill-climbing algorithm should not be applied in the initial iteration of the optimization process due to the immense tendency for converging to local optima. Meanwhile, for optimizing largescale problems, hill-climbing is not an appropriate selection. In order to improve understanding of the proposed hybrid optimization algorithm’s steps, the flowchart of HCMVO is designed and can be seen in Figure 16.

Figure 17 shows the observed capture factor (which is the absorbed energy with respect to the available energy) by each optimization algorithm from iterations 1 to 400. The algorithms use ten search agents in their modified codes to find the optimal solutions. While GWO and MFO remain roughly constant after iterations 54 and 40, the other three algorithms keep improving the capture factor. In this case, HCMVO and MVO worked very well in the optimizing process with a capture factor obtained by the former as 0.594 and by the latter as 0.593. MFO almost found its highest value before the iteration 50, which means the exploration part of the algorithm works out well. Similarly, HCMVO does the same. However, it keeps finding the better solution during the optimization process until the last iteration, indicating the strong exploitation part of the algorithm. GWO reveals a weakness in exploration and exploitation because not only does it evoke the least capture factor value, but also the curve remains almost unchanged throughout 350 iterations.

Figure 18 illustrates complex interactions between the five optimization parameters and the capture factor for HCMVO (a), MPA (b), and MFO (c) algorithms. The first interesting observation is that there is a high level of nonlinear relationships among the setting parameters that can make a multi-modal search space. The dark blue lines represent the best-found configuration throughout the optimisation process. Based on both HCMVO (a) and MVO (b), we can infer that the dark blue lines concentrate in a specific range, showing the high convergence ability of both HCMVO and MVO. However, MFO (c) could not find the exact optimal range of the decision variables, and the best-found solutions per generation distribute mostly all around the search space.

Empty CellAlgorithm 1: Hill Climb Multiverse Optimization
01:procedure HCMVO
03:Initialize the constraints��1�,��1�
04:�1�=Mi�1�+���1�/�▹Compute the step size,�is search resolution
07:Main loop
08:for iter≤���ita=do
13:t = t+1
14:end while
18:end for
20:end procedure

were utilized to illuminate the optimum values of the design parameters, and the best method was chosen to develop a new algorithm that performs both local and global search methods.

The correlation between hydrodynamic parameters and the capture factor of the converter was supported by the results. For any given water depth, the capture factor increases as the wave period increases, until a certain wave period value (6 seconds) is reached, after which the capture factor gradually decreases. It is expected since the flap cannot oscillate effectively when the wavelength is too short for a certain water depth. Conversely, when the wavelength is too long, the capture factor decreases. Furthermore, under a constant wave period, increasing the water depth does not affect the capture factor. Regarding the sensitivity analysis, the study found that increasing the flap bottom elevation causes turbulence flow behind the flap and limitation of rotation, which leads to less interaction with the incoming waves. Furthermore, while keeping the flap bottom elevation constant, increasing the freeboard improves the capture factor. Overtopping happens when the freeboard is negative and the flap is below the water surface, which has a detrimental influence on converter performance. Furthermore, raising the freeboard causes the wave impact to become more violent, which increases converter performance.

In the last part, we discussed the search process of each algorithm and visualized their performance and convergence curves as they try to find the best values for decision variables. Among the four selected metaheuristic algorithms, the Multi-verse Optimizer proved to be the most effective in achieving the best answer in terms of the WEC capture factor. However, the MVO needed modifications regarding its escape approach from the local optima in order to improve its global search capabilities. To overcome these constraints, we presented a fast and efficient meta-algorithm (HCMVO) that combines MVO with a Random-restart hill-climbing local search. On a higher level, this meta-algorithm employed MVO to generate global tracking and present a range of possible and appropriate solutions. Taken together, the results demonstrated that there is a significant degree of nonlinearity among the setup parameters that might result in a multimodal search space. Since MVO was faced with a stagnation issue or fell into a local optimum, we constructed a complete neighborhood search around the best-found solution offered by the upper level. In sum, the newly-developed algorithm proved to be highly effective for the problem compared to other similar optimization methods. The strength of the current findings may encourage future investigation on design optimization of wave energy converters using developed geometry as well as the novel approach.

CRediT authorship contribution statement

Erfan Amini: Conceptualization, Methodology, Validation, Data curation, Writing – original draft, Writing – review & editing, Visualization. Mahdieh Nasiri: Conceptualization, Methodology, Validation, Data curation, Writing – original draft, Writing – review & editing, Visualization. Navid Salami Pargoo: Writing – original draft, Writing – review & editing. Zahra Mozhgani: Conceptualization, Methodology. Danial Golbaz: Writing – original draft. Mehrdad Baniesmaeil: Writing – original draft. Meysam Majidi Nezhad: . Mehdi Neshat: Supervision, Conceptualization, Writing – original draft, Writing – review & editing, Visualization. Davide Astiaso Garcia: Supervision. Georgios Sylaios: Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


This research has been carried out within ILIAD (Inte-grated Digital Framework for Comprehensive Maritime Data and Information Services) project that received funding from the European Union’s H2020 programme.

Data availability

Data will be made available on request.


Figure 14. Defects: (a) Unmelt defects(Scheme NO.4);(b) Pores defects(Scheme NO.1); (c); Spattering defect (Scheme NO.3); (d) Low overlapping rate defects(Scheme NO.5).

Molten pool structure, temperature and velocity
flow in selective laser melting AlCu5MnCdVA alloy

용융 풀 구조, 선택적 온도 및 속도 흐름 레이저 용융 AlCu5MnCdVA 합금

Pan Lu1 , Zhang Cheng-Lin2,6,Wang Liang3, Liu Tong4 and Liu Jiang-lin5
1 Aviation and Materials College, Anhui Technical College of Mechanical and Electrical Engineering, Wuhu Anhui 241000, People’s
Republic of China 2 School of Engineering Science, University of Science and Technology of China, Hefei Anhui 230026, People’s Republic of China 3 Anhui Top Additive Manufacturing Technology Co., Ltd., Wuhu Anhui 241300, People’s Republic of China 4 Anhui Chungu 3D Printing Institute of Intelligent Equipment and Industrial Technology, Anhui 241300, People’s Republic of China 5 School of Mechanical and Transportation Engineering, Taiyuan University of Technology, Taiyuan Shanxi 030024, People’s Republic of
China 6 Author to whom any correspondence should be addressed.
E-mail:,,, and


SLM, molten pool, AlCu5MnCdVA alloy, heat flow, velocity flow, numerical simulation


선택적 레이저 용융(SLM)은 열 전달, 용융, 상전이, 기화 및 물질 전달을 포함하는 복잡한 동적 비평형 프로세스인 금속 적층 제조(MAM)에서 가장 유망한 기술 중 하나가 되었습니다. 용융 풀의 특성(구조, 온도 흐름 및 속도 흐름)은 SLM의 최종 성형 품질에 결정적인 영향을 미칩니다. 이 연구에서는 선택적 레이저 용융 AlCu5MnCdVA 합금의 용융 풀 구조, 온도 흐름 및 속도장을 연구하기 위해 수치 시뮬레이션과 실험을 모두 사용했습니다.

그 결과 용융풀의 구조는 다양한 형태(깊은 오목 구조, 이중 오목 구조, 평면 구조, 돌출 구조 및 이상적인 평면 구조)를 나타냈으며, 용융 풀의 크기는 약 132 μm × 107 μm × 50 μm였습니다. : 용융풀은 초기에는 여러 구동력에 의해 깊이 15μm의 깊은 오목형상이었으나, 성형 후기에는 장력구배에 의해 높이 10μm의 돌출형상이 되었다. 용융 풀 내부의 금속 흐름은 주로 레이저 충격력, 금속 액체 중력, 표면 장력 및 반동 압력에 의해 구동되었습니다.

AlCu5MnCdVA 합금의 경우, 금속 액체 응고 속도가 매우 빠르며(3.5 × 10-4 S), 가열 속도 및 냉각 속도는 각각 6.5 × 107 K S-1 및 1.6 × 106 K S-1 에 도달했습니다. 시각적 표준으로 표면 거칠기를 선택하고, 낮은 레이저 에너지 AlCu5MnCdVA 합금 최적 공정 매개변수 창을 수치 시뮬레이션으로 얻었습니다: 레이저 출력 250W, 부화 공간 0.11mm, 층 두께 0.03mm, 레이저 스캔 속도 1.5m s-1 .

또한, 실험 프린팅과 수치 시뮬레이션과 비교할 때, 용융 풀의 폭은 각각 약 205um 및 약 210um이었고, 인접한 두 용융 트랙 사이의 중첩은 모두 약 65um이었다. 결과는 수치 시뮬레이션 결과가 실험 인쇄 결과와 기본적으로 일치함을 보여 수치 시뮬레이션 모델의 정확성을 입증했습니다.

Selective Laser Melting (SLM) has become one of the most promising technologies in Metal Additive Manufacturing (MAM), which is a complex dynamic non-equilibrium process involving heat transfer, melting, phase transition, vaporization and mass transfer. The characteristics of the molten pool (structure, temperature flow and velocity flow) have a decisive influence on the final forming quality of SLM. In this study, both numerical simulation and experiments were employed to study molten pool structure, temperature flow and velocity field in Selective Laser Melting AlCu5MnCdVA alloy. The results showed the structure of molten pool showed different forms(deep-concave structure, double-concave structure, plane structure, protruding structure and ideal planar structure), and the size of the molten pool was approximately 132 μm × 107 μm × 50 μm: in the early stage, molten pool was in a state of deep-concave shape with a depth of 15 μm due to multiple driving forces, while a protruding shape with a height of 10 μm duo to tension gradient in the later stages of forming. The metal flow inside the molten pool was mainly driven by laser impact force, metal liquid gravity, surface tension and recoil pressure. For AlCu5MnCdVA alloy, metal liquid solidification speed was extremely fast(3.5 × 10−4 S), the heating rate and cooling rate reached 6.5 × 107 K S−1 and 1.6 × 106 K S−1 , respectively. Choosing surface roughness as a visual standard, low-laser energy AlCu5MnCdVA alloy optimum process parameters window was obtained by numerical simulation: laser power 250 W, hatching space 0.11 mm, layer thickness 0.03 mm, laser scanning velocity 1.5 m s−1 . In addition, compared with experimental printing and numerical simulation, the width of the molten pool was about 205 um and about 210 um, respectively, and overlapping between two adjacent molten tracks was all about 65 um. The results showed that the numerical simulation results were basically consistent with the experimental print results, which proved the correctness of the numerical simulation model.

Figure 1. AlCu5MnCdVA powder particle size distribution.
Figure 1. AlCu5MnCdVA powder particle size distribution.
Figure 2. AlCu5MnCdVA powder
Figure 2. AlCu5MnCdVA powder
Figure 3. Finite element model and calculation domains of SLM.
Figure 3. Finite element model and calculation domains of SLM.
Figure 4. SLM heat transfer process.
Figure 4. SLM heat transfer process.
Figure 14. Defects: (a) Unmelt defects(Scheme NO.4);(b) Pores defects(Scheme NO.1); (c); Spattering defect (Scheme NO.3); (d) Low
overlapping rate defects(Scheme NO.5).
Figure 17. Two-pass molten tracks overlapping for Scheme NO.2.
Figure 17. Two-pass molten tracks overlapping for Scheme NO.2.


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Development of macro-defect-free PBF-EB-processed Ti–6Al–4V alloys with superior plasticity using PREP-synthesized powder and machine learning-assisted process optimization

Development of macro-defect-free PBF-EB-processed Ti–6Al–4V alloys with superior plasticity using PREP-synthesized powder and machine learning-assisted process optimization

Yunwei GuiabKenta Aoyagib Akihiko Chibab
aDepartment of Materials Processing, Graduate School of Engineering, Tohoku University, 6-6 Aramaki Aza Aoba, Aoba-ku, Sendai, 980-8579, Japan
bInstitute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan

Received 14 October 2022, Revised 23 December 2022, Accepted 3 January 2023, Available online 5 January 2023.Show lessAdd to MendeleyShareCite rights and content


The elimination of internal macro-defects is a key issue in Ti–6Al–4V alloys fabricated via powder bed fusion using electron beams (PBF-EB), wherein internal macro-defects mainly originate from the virgin powder and inappropriate printing parameters. This study compares different types powders by combining support vector machine techniques to determine the most suitable powder for PBF-EB and to predict the processing window for the printing parameters without internal macro-defects. The results show that powders fabricated via plasma rotating electrode process have the best sphericity, flowability, and minimal porosity and are most suitable for printing. Surface roughness criterion was also applied to determine the quality of the even surfaces, and support vector machine was used to construct processing maps capable of predicting a wide range of four-dimensional printing parameters to obtain macro-defect-free samples, offering the possibility of subsequent development of Ti–6Al–4V alloys with excellent properties. The macro-defect-free samples exhibited good elongation, with the best overall mechanical properties being the ultimate tensile strength and elongation of 934.7 MPa and 24.3%, respectively. The elongation of the three macro-defect-free samples was much higher than that previously reported for additively manufactured Ti–6Al–4V alloys. The high elongation of the samples in this work is mainly attributed to the elimination of internal macro-defects.


Additive manufacturing (AM) technologies can rapidly manufacture complex or custom parts, reducing process steps and saving manufacturing time [[1], [2], [3], [4]], and are widely used in the aerospace, automotive, and other precision industries [5,6]. Powder bed fusion using an electron beam (PBF-EB) is an additive manufacturing method that uses a high-energy electron beam to melt metal powders layer by layer to produce parts. In contrast to selective laser melting, PBF-EB involves the preparation of samples in a high vacuum environment, which effectively prevents the introduction of impurities such as O and N. It also involves a preheating process for the print substrate and powder, which reduces residual thermal stress on the sample and subsequent heat treatment processes [[2], [3], [4],7]. Due to these features and advantages, PBF-EB technology is a very important AM technology with great potential in metallic materials. Moreover, PBF-EB is the ideal AM technology for the manufacture of complex components made of many alloys, such as titanium alloys, nickel-based superalloys, aluminum alloys and stainless steels [[2], [3], [4],8].

Ti–6Al–4V alloy is one of the prevalent commercial titanium alloys possessing high specific strength, excellent mechanical properties, excellent corrosion resistance, and good biocompatibility [9,10]. It is widely used in applications requiring low density and excellent corrosion resistance, such as the aerospace industry and biomechanical applications [11,12]. The mechanical properties of PBF-EB-processed Ti–6Al–4V alloys are superior to those fabricated by casting or forging, because the rapid cooling rate in PBF-EB results in finer grains [[12], [13], [14], [15], [16], [17], [18]]. However, the PBF-EB-fabricated parts often include internal macro-defects, which compromises their mechanical properties [[19], [20], [21], [22]]. This study focused on the elimination of macro-defects, such as porosity, lack of fusion, incomplete penetration and unmelted powders, which distinguishes them from micro-defects such as vacancies, dislocations, grain boundaries and secondary phases, etc. Large-sized fusion defects cause a severe reduction in mechanical strength. Smaller defects, such as pores and cracks, lead to the initiation of fatigue cracking and rapidly accelerate the cracking process [23]. The issue of internal macro-defects must be addressed to expand the application of the PBF-EB technology. The main studies for controlling internal macro-defects are online monitoring of defects, remelting and hot isostatic pressing (HIP). The literatures [24,25] report the use of infrared imaging or other imaging techniques to identify defects, but the monitoring of smaller sized defects is still not adequate. And in some cases remelting does not reduce the internal macro-defects of the part, but instead causes coarsening of the macrostructure and volatilization of some metal elements [23]. The HIP treatment does not completely eliminate the internal macro-defects, the original defect location may still act as a point of origin of the crack, and the subsequent treatment will consume more time and economic costs [23]. Therefore, optimizing suitable printing parameters to avoid internal macro-defects in printed parts at source is of great industrial value and research significance, and is an urgent issue in PBF-EB related technology.

There are two causes of internal macro-defects in the AM process: gas pores trapped in the virgin powder and the inappropriate printing parameters [7,23]. Gui et al. [26] classify internal macro-defects during PBF-EB process according to their shape, such as spherical defects, elongated shape defects, flat shape defects and other irregular shape defects. Of these, spherical defects mainly originate from raw material powders. Other shape defects mainly originate from lack of fusion or unmelted powders caused by unsuitable printing parameters, etc. The PBF-EB process requires powders with good flowability, and spherical powders are typically chosen as raw materials. The prevalent techniques for the fabrication of pre-alloyed powders are gas atomization (GA), plasma atomization (PA), and the plasma rotating electrode process (PREP) [27,28]. These methods yield powders with different characteristics that affect the subsequent fabrication. The selection of a suitable powder for PBF-EB is particularly important to produce Ti–6Al–4V alloys without internal macro-defects. The need to optimize several printing parameters such as beam current, scan speed, line offset, and focus offset make it difficult to eliminate internal macro-defects that occur during printing [23]. Most of the studies [11,12,22,[29], [30], [31], [32], [33]] on the optimization of AM processes for Ti–6Al–4V alloys have focused on samples with a limited set of parameters (e.g., power–scan speed) and do not allow for the guidance and development of unknown process windows for macro-defect-free samples. In addition, process optimization remains a time-consuming problem, with the traditional ‘trial and error’ method demanding considerable time and economic costs. The development of a simple and efficient method to predict the processing window for alloys without internal macro-defects is a key issue. In recent years, machine learning techniques have increasingly been used in the field of additive manufacturing and materials development [[34], [35], [36], [37]]. Aoyagi et al. [38] recently proposed a novel and efficient method based on a support vector machine (SVM) to optimize the two-dimensional process parameters (current and scan speed) and obtain PBF-EB-processed CoCr alloys without internal macro-defects. The method is one of the potential approaches toward effective optimization of more than two process parameters and makes it possible for the machine learning techniques to accelerate the development of alloys without internal macro-defects.

Herein, we focus on the elimination of internal macro-defects, such as pores, lack of fusion, etc., caused by raw powders and printing parameters. The Ti–6Al–4V powders produced by three different methods were compared, and the powder with the best sphericity, flowability, and minimal porosity was selected as the feedstock for subsequent printing. The relationship between the surface roughness and internal macro-defects in the Ti–6Al–4V components was also investigated. The combination of SVM and surface roughness indices (Sdr) predicted a wider four-dimensional processing window for obtaining Ti–6Al–4V alloys without internal macro-defects. Finally, we investigated the tensile properties of Ti–6Al–4V alloys at room temperature with different printing parameters, as well as the corresponding microstructures and fracture types.

Section snippets

Starting materials

Three types of Ti–6Al–4V alloy powders, produced by GA, PA, and PREP, were compared. The particle size distribution of the powders was determined using a laser particle size analyzer (LS230, Beckman Coulter, USA), and the flowability was measured using a Hall flowmeter (JIS-Z2502, Tsutsui Scientific Instruments Co., Ltd., Japan), according to the ASTM B213 standard. The powder morphology and internal macro-defects were determined using scanning electron microscopy (SEM, JEOL JCM-6000) and X-ray 

Comparison of the characteristics of GA, PA, and PREP Ti–6Al–4V powders

The particle size distributions (PSDs) and flowability of the three types of Ti–6Al–4V alloy powders produced by GA, PA, and PREP are shown in Fig. 2. Although the average particle sizes are similar (89.4 μm for GA, 82.5 μm for PA, and 86.1μm for PREP), the particle size range is different for the three types of powder (6.2–174.8 μm for GA, 27.3–139.2 μm for PA, and 39.4–133.9 μm for PREP). The flowability of the GA, PA, and PREP powders was 30.25 ± 0.98, 26.54 ± 0.37, and 25.03 ± 0.22 (s/50


The characteristics of the three types of Ti–6Al–4V alloy powders produced via GA, PA, and PREP were compared. The PREP powder with the best sphericity, flowability, and low porosity was found to be the most favorable powder for subsequent printing of Ti–6Al–4V alloys without internal macro-defects. The quantitative criterion of Sdr <0.015 for even surfaces was also found to be applicable to Ti–6Al–4V alloys. The process maps of Ti–6Al–4V alloys include two regions, high beam current/scan speed 

Uncited references

[55]; [56]; [57]; [58]; [59]; [60]; [61]; [62]; [63]; [64]; [65].

CRediT authorship contribution statement

Yunwei Gui: Writing – original draft, Visualization, Validation, Investigation. Kenta Aoyagi: Writing – review & editing, Supervision, Resources, Methodology, Funding acquisition, Conceptualization. Akihiko Chiba: Supervision, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


This study was based on the results obtained from project JPNP19007, commissioned by the New Energy and Industrial Technology Development Organization (NEDO). This work was also supported by JSPS KAKENHI (Proposal No. 21K03801) and the Inter-University Cooperative Research Program (Proposal nos. 18G0418, 19G0411, and 20G0418) of the Cooperative Research and Development Center for Advanced Materials, Institute for Materials Research, Tohoku University. It was also supported by the Council for

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Fig. 8. Comparison of the wave pattern for : (a) Ship wave only; (b) Ship wave in the presence of a following current.

균일한 해류가 존재하는 선박 파도의 수치 시뮬레이션

Numerical simulation of ship waves in the presence of a uniform current

CongfangAiYuxiangMaLeiSunGuohaiDongState Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China


• Ship waves in the presence of a uniform current are studied by a non-hydrostatic model.

• Effects of a following current on characteristic wave parameters are investigated.

• Effects of an opposing current on characteristic wave parameters are investigated.

• The response of the maximum water level elevation to the ship draft is discussed.


이 논문은 균일한 해류가 존재할 때 선박파의 생성 및 전파를 시뮬레이션하기 위한 비정역학적 모델을 제시합니다. 선박 선체의 움직임을 표현하기 위해 움직이는 압력장 방법이 모델에 통합되었습니다.

뒤따르거나 반대 방향의 균일한 흐름이 있는 경우의 선박 파도의 수치 결과를 흐름이 없는 선박 파도의 수치 결과와 비교합니다. 추종 또는 반대 균일 전류가 존재할 때 계산된 첨단선 각도는 분석 솔루션과 잘 일치합니다. 추종 균일 전류와 반대 균일 전류가 특성파 매개변수에 미치는 영향을 제시하고 논의합니다.

선박 흘수에 대한 최대 수위 상승의 응답은 추종 또는 반대의 균일한 흐름이 있는 경우에도 표시되며 흐름이 없는 선박 파도의 응답과 비교됩니다. 선박 선체 측면의 최대 수위 상승은 Froude 수 Fr’=Us/gh의 특정 범위에 대해 다음과 같은 균일한 흐름의 존재에 의해 증가될 수 있음이 밝혀졌습니다.

여기서 Us는 선박 속도이고 h는 물입니다. 깊이. 균일한 해류를 무시하면 추종류나 반대류가 존재할 때 선박 흘수에 대한 최대 수위 상승의 응답이 과소평가될 수 있습니다.

본 연구는 선박파의 해석에 있어 균일한 해류의 영향을 고려해야 함을 시사합니다.

This paper presents a non-hydrostatic model to simulate the generation and propagation of ship waves in the presence of a uniform current. A moving pressure field method is incorporated into the model to represent the movement of a ship hull. Numerical results of ship waves in the presence of a following or an opposing uniform current are compared with those of ship waves without current. The calculated cusp-line angles in the presence of a following or opposing uniform current agree well with analytical solutions. The effects of a following uniform current and an opposing uniform current on the characteristic wave parameters are presented and discussed. The response of the maximum water level elevation to the ship draft is also presented in the presence of a following or an opposing uniform current and is compared with that for ship waves without current. It is found that the maximum water level elevation lateral to the ship hull can be increased by the presence of a following uniform current for a certain range of Froude numbers Fr′=Us/gh, where Us is the ship speed and h is the water depth. If the uniform current is neglected, the response of the maximum water level elevation to the ship draft in the presence of a following or an opposing current can be underestimated. The present study indicates that the effect of a uniform current should be considered in the analysis of ship waves.


Ship waves, Non-hydrostatic model, Following current, Opposing current, Wave parameters

1. Introduction

Similar to wind waves, ships sailing across the sea can also create free-surface undulations ranging from ripples to waves of large size (Grue, 20172020). Ship waves can cause sediment suspension and engineering structures damage and even pose a threat to flora and fauna living near the embankments of waterways (Dempwolff et al., 2022). It is quite important to understand ship waves in various environments. The study of ship waves has been conducted over a century. A large amount of research (Almström et al., 2021Bayraktar and Beji, 2013David et al., 2017Ertekin et al., 1986Gourlay, 2001Havelock, 1908Lee and Lee, 2019Samaras and Karambas, 2021Shi et al., 2018) focused on the generation and propagation of ship waves without current. When a ship navigates in the sea or in a river where tidal flows or river flows always exist, the effect of currents should be taken into account. However, the effect of currents on the characteristic parameters of ship waves is still unclear, because very few publications have been presented on this topic.

Over the past two decades, many two-dimensional (2D) Boussinesq-type models (Bayraktar and Beji, 2013Dam et al., 2008David et al., 2017Samaras and Karambas, 2021Shi et al., 2018) were developed to examine ship waves. For example, Bayraktar and Beji (2013) solved Boussinesq equations with improved dispersion characteristics to simulate ship waves due to a moving pressure field. David et al. (2017) employed a Boussinesq-type model to investigate the effects of the pressure field and its propagation speed on characteristic wave parameters. All of these Boussinesq-type models aimed to simulate ship waves without current except for that of Dam et al. (2008), who investigated the effect of currents on the maximum wave height of ship waves in a narrow channel.

In addition to Boussinesq-type models, numerical models based on the Navier-Stokes equations (NSE) or Euler equations are also capable of resolving ship waves. Lee and Lee (20192021) employed the FLOW-3D model to simulate ship waves without current and ship waves in the presence of a uniform current to confirm their equations for ship wave crests. FLOW-3D is a computational fluid dynamics (CFD) software based on the NSE, and the volume of fluid (VOF) method is used to capture the moving free surface. However, VOF-based NSE models are computationally expensive due to the treatment of the free surface. To efficiently track the free surface, non-hydrostatic models employ the so-called free surface equation and can be solved efficiently. One pioneering application for the simulation of ship waves by the non-hydrostatic model was initiated by Ma (2012) and named XBeach. Recently, Almström et al. (2021) validated XBeach with improved dispersive behavior by comparison with field measurements. XBeach employed in Almström et al. (2021) is a 2-layer non-hydrostatic model and is accurate up to Kh=4 for the linear dispersion relation (de Ridder et al., 2020), where K=2π/L is the wavenumber. L is the wavelength, and h is the still water depth. However, no applications of non-hydrostatic models on the simulation of ship waves in the presence of a uniform current have been published. For more advances in the numerical modelling of ship waves, the reader is referred to Dempwolff et al. (2022).

This paper investigates ship waves in the presence of a uniform current by using a non-hydrostatic model (Ai et al., 2019), in which a moving pressure field method is incorporated to represent the movement of a ship hull. The model solves the incompressible Euler equations by using a semi-implicit algorithm and is associated with iterating to solve the Poisson equation. The model with two, three and five layers is accurate up to Kh= 7, 15 and 40, respectively (Ai et al., 2019) in resolving the linear dispersion relation. To the best of our knowledge, ship waves in the presence of currents have been studied theoretically (Benjamin et al., 2017Ellingsen, 2014Li and Ellingsen, 2016Li et al., 2019.) and numerically (Dam et al., 2008Lee and Lee, 20192021). However, no publications have presented the effects of a uniform current on characteristic wave parameters except for Dam et al. (2008), who investigated only the effect of currents on the maximum wave height in a narrow channel for the narrow relative Froude number Fr=(Us−Uc)/gh ranging from 0.47 to 0.76, where Us is the ship speed and Uc is the current velocity. To reveal the effect of currents on the characteristic parameters of ship waves, the main objectives of this paper are (1) to validate the capability of the proposed model to resolve ship waves in the presence of a uniform current, (2) to investigate the effects of a following or an opposing current on characteristic wave parameters including the maximum water level elevation and the leading wave period in the ship wave train, (3) to show the differences in characteristic wave parameters between ship waves in the presence of a uniform current and those without current when the same relative Froude number Fr is specified, and (4) to examine the response of the maximum water level elevation to the ship draft in the presence of a uniform current.

The remainder of this paper is organized as follows. The non-hydrostatic model for ship waves is described in Section 2. Section 3 presents numerical validations for ship waves. Numerical results and discussions about the effects of a uniform current on characteristic wave parameters are provided in Section 4, and a conclusion is presented in Section 5.

2. Non-hydrostatic model for ship waves

2.1. Governing equations

The 3D incompressible Euler equations are expressed in the following form:(1)∂u∂x+∂v∂y+∂w∂z=0(2)∂u∂t+∂u2∂x+∂uv∂y+∂uw∂z=−∂p∂x(3)∂v∂t+∂uv∂x+∂v2∂y+∂vw∂z=−∂p∂y(4)∂w∂t+∂uw∂x+∂vw∂y+∂w2∂z=−∂p∂z−gwhere t is the time; u(x,y,z,t), v(x,y,z,t) and w(x,y,z,t) are the velocity components in the horizontal x, y and vertical z directions, respectively; p(x,y,z,t) is the pressure divided by a constant reference density; and g is the gravitational acceleration.

The pressure p(x,y,z,t) can be expressed as(5)p=ps+g(η−z)+qwhere ps(x,y,t) is the pressure at the free surface, η(x,y,t) is the free surface elevation, and q(x,y,z,t) is the non-hydrostatic pressure.

η(x,y,t) is calculated by the following free-surface equation:(6)∂η∂t+∂∂x∫−hηudz+∂∂y∫−hηvdz=0where z=−h(x,y) is the bottom surface.

To generate ship waves, ps(x,y,t) is determined by the following slender-body type pressure field (Bayraktar and Beji, 2013David et al., 2017Samaras and Karambas, 2021):

For −L/2≤x’≤L/2,−B/2≤y’≤B/2(7)ps(x,y,t)|t=0=pm[1−cL(x′/L)4][1−cB(y′/B)2]exp⁡[−a(y′/B)2]where x′=x−x0 and y′=y−y0. (x0,y0) is the center of the pressure field, pm is the peak pressure defined at (x0,y0), and L and B are the lengthwise and breadthwise parameters, respectively. cL, cB and a are set to 16, 2 and 16, respectively.

2.2. Numerical algorithms

In this study, the generation of ship waves is incorporated into the semi-implicit non-hydrostatic model developed by Ai et al. (2019). The 3D grid system used in the model is built from horizontal rectangular grids by adding horizontal layers. The horizontal layers are distributed uniformly along the water depth, which means the layer thickness is defined by Δz=(η+h)/Nz, where Nz is the number of horizontal layers.

In the solution procedure, the first step is to generate ship waves by implementing Eq. (7) together with the prescribed ship track. In the second step, Eqs. (1)(2)(3)(4) are solved by the pressure correction method, which can be subdivided into three stages. The first stage is to compute intermediate velocities un+1/2, vn+1/2, and wn+1/2 by solving Eqs. (2)(3)(4), which contain the non-hydrostatic pressure at the preceding time level. In the second stage, the Poisson equation for the non-hydrostatic pressure correction term is solved on the graphics processing unit (GPU) in conjunction with the conjugate gradient method. The third stage is to compute the new velocities un+1, vn+1, and wn+1 by correcting the intermediate values after including the non-hydrostatic pressure correction term. In the discretization of Eqs. (2)(3), the gradient terms of the water surface ∂η/∂x and ∂η/∂y are discretized by means of the semi-implicit method (Vitousek and Fringer, 2013), in which the implicitness factor θ=0.5 is used. The model is second-order accurate in time for free-surface flows. More details about the model can be found in Ai et al. (2019).

3. Model validation

In this section, we validate the proposed model in resolving ship waves. The numerical experimental conditions are provided in Table 1 and Table 2. In Table 2, Case A with the current velocity of Uc = 0.0 m/s represents ship waves without current. Both Case B and Case C correspond to the cases in the presence of a following current, while Case D and Case E represent the cases in the presence of an opposing current. The current velocities are chosen based on the observed currents at 40.886° N, 121.812° E, which is in the Liaohe Estuary. The measured data were collected from 14:00 on September 18 (GMT + 08:00) to 19:00 on September 19 in 2021. The maximum flood velocity is 1.457 m/s, and the maximum ebb velocity is −1.478 m/s. The chosen current velocities are between the maximum flood velocity and the maximum ebb velocity.

Table 1. Summary of ship speeds.

CaseWater depth h (m)Ship speed Us (m/s)Froude number Fr′=Us/gh

Table 2. Summary of current velocities.

Current velocity
Uc (m/s)−0.5−1.0

Notably, the Froude number Fr′=Us/gh presented in Table 1 is defined by the ship speed Us only and is different from the relative Froude number Fr when a uniform current is presented. According to the theory of Lee and Lee (2021), with the same relative Froude number, the cusp-line angles in the presence of a following or an opposing uniform current are identical to those without current. As a result, for the test cases presented in Table 1Table 2, all calculated cusp-line angles follow the analytical solution of Havelock (1908), when the relative Froude number Fr is introduced.

As shown in Fig. 1, the dimensions of the computational domain are −420≤x≤420 m and −200≤y≤200 m, which are similar to those of David et al. (2017). The ship track follows the x axis and ranges from −384 m to 384 m. The ship hull is represented by Eq. (7), in which the length L and the beam B are set to 14.0 m and 7.0 m, respectively, and the peak pressure value is pm= 5000 Pa. In the numerical simulations, grid convergence tests reveal that the horizontal grid spacing of Δx=Δy= 1.0 m and two horizontal layers are adequate. The numerical results with different numbers of horizontal layers are shown in the Appendix.

Fig. 1

Fig. 2Fig. 3 compare the calculated cusp-line angles θc with the analytical solutions of Havelock (1908) for ship waves in the presence of a following uniform current and an opposing uniform current, respectively. The calculated cusp-line angles without current are also depicted in Fig. 2Fig. 3. All calculated cusp-line angles are in good agreement with the analytical solutions, except that the model tends to underpredict the cusp-line angle for 0.9<Fr<1.0. Notably, a similar underprediction of the cusp-line angle can also be found in David et al. (2017).

Fig. 2
Fig. 3

4. Results and discussions

This section presents the effects of a following current and opposing current on the maximum water level elevation and the leading wave period in the wave train based on the test cases presented in Table 1Table 2. Moreover, the response of the maximum water level elevation to the ship draft in the presence of a uniform current is examined.

4.1. Effects of a following current on characteristic wave parameters

To present the effect of a following current on the maximum wave height, the variations of the maximum water level elevation ηmax with the Froude number Fr′ at gauge points G1 and G2 are depicted in Fig. 4. The positions of gauge points G1 and G2 are shown in Fig. 1. The maximum water level elevation is an analogue to the maximum wave height and is presented in this study, because maximum wave heights at different positions away from the ship track vary throughout the wave train (David et al., 2017). In general, the variations of ηmax with the Froude number Fr′ in the three cases show a similar behavior, in which with the increase in Fr′, ηmax increases and then decreases. The presence of the following currents decreases ηmax for Fr′≤0.8 and Fr′≥1.2. Specifically, the following currents have a significant effect on ηmax for Fr′≤0.8. Notably, ηmax can be increased by the presence of the following currents for 0.9≤Fr′≤1.1. Compared with Case A, at location G1 ηmax is amplified 1.25 times at Fr′=0.925 in Case B and 1.31 times at Fr′=1.025 in Case C. Similarly, at location G2 ηmax is amplified 1.15 times at Fr′=1.025 in Case B and 1.11 times at Fr′=1.075 in Case C. The fact that ηmax can be increased by the presence of a following current for 0.9≤Fr′≤1.1 implies that if a following uniform current is neglected, then ηmax may be underestimated.

Fig. 4

To show the effect of a following current on the wave period, Fig. 5 depicts the variation of the leading wave period Tp in the wave train at gauge point G2 with the Froude number Fr′. Similar to David et al. (2017), Tp is defined by the wave period of the first wave with a leading trough in the wave train. The leading wave periods for Fr′= 0.6 and 0.7 were not given in Case B and Case C, because the leading wave heights for Fr′= 0.6 and 0.7 are too small to discern the leading wave periods. Compared with Case A, the presence of a following current leads to a larger Tp for 0.925≤Fr′≤1.1 and a smaller Tp for Fr′≥1.3. For Fr′= 0.8 and 0.9, Tp in Case B is larger than that in Case A and Tp in Case C is smaller than that in Case A. In all three cases, Tp decreases with increasing Fr′ for Fr′>1.0. However, this decreasing trend becomes very gentle after Fr′≥1.4. Notably, as shown in Fig. 5, Fr′=1.2 tends to be a transition point at which the following currents have a very limited effect on Tp. Moreover, before the transition point, Tp in Case B and Case C are larger than that in Case A (only for 0.925≤Fr′≤1.2), but after the transition point the reverse is true.

Fig. 5

As mentioned previously, the cusp-line angles for ship waves in the presence of a following or an opposing current are identical to those for ship waves only with the same relative Froude number Fr. However, with the same Fr, the characteristic parameters of ship waves in the presence of a following or an opposing current are quite different from those of ship waves without current. Fig. 6 shows the variations of the maximum water level elevation ηmax with Fr at gauge points G1 and G2 for ship waves in the presence of a following uniform current. Overall, the relationship curves between ηmax and Fr in Case B and Case C are lower than those in Case A. It is inferred that with the same Fr, ηmax in the presence of a following current is smaller than that without current. Fig. 7 shows the variation of the leading wave period Tp in the wave train at gauge point G2 with Fr for ship waves in the presence of a following uniform current. The overall relationship curves between Tp and Fr in Case B and Case C are also lower than those in Case A for 0.9≤Fr≤2.0. It can be inferred that with the same Fr, Tp in the presence of a following current is smaller than that without current for Fr≥0.9.

Fig. 6
Fig. 7

To compare the numerical results between the case of ship waves only and the case of ship waves in the presence of a following current with the same Fr, Fig. 8 shows the wave patterns for Fr=1.2. To obtain the case of ship waves in the presence of a following current with Fr=1.2, the ship speed Us=9.7 m/s and the current velocity Uc=0.5 m/s are adopted. Fig. 8 indicates that both the calculated cusp-line angles for the case of Us=9.2 m/s and Uc=0.0 m/s and the case of Us=9.7 m/s and Uc=0.5 m/s are equal to 56.5°, which follows the theory of Lee and Lee (2021)Fig. 9 depicts the comparison of the time histories of the free surface elevation at gauge point G2 for Fr=1.2 between the case of ship waves only and the case of ship waves in the presence of a following current. The time when the ship wave just arrived at gauge point G2 is defined as t′=0. Both the maximum water level elevation and the leading wave period in the case of Us=9.2 m/s and Uc=0.0 m/s are larger than those in the case of Us=9.7 m/s and Uc=0.5 m/s, which is consistent with the inferences based on Fig. 6Fig. 7.

Fig. 8
Fig. 8. Comparison of the wave pattern for Fr=1.2: (a) Ship wave only; (b) Ship wave in the presence of a following current.
Fig. 9
Fig. 9. Comparison of the time histories of the free surface elevation at gauge point G2 for between case of ship waves only and case of ship waves in the presence of a following current.

Fig. 10 shows the response of the maximum water level elevation ηmax to the ship draft at gauge point G2 for Fr′= 1.2 in the presence of a following uniform current. pm ranges from 2500 Pa to 40,000 Pa with an interval of Δp= 2500 Pa pm0= 2500 Pa represents a reference case. ηmax0 denotes the maximum water level elevation corresponding to the case of pm0= 2500 Pa. The best-fit linear trend lines obtained by linear regression analysis for the three responses are also depicted in Fig. 10. In general, all responses of ηmax to the ship draft show a linear relationship. The coefficients of determination for the three linear trend lines are R2= 0.9901, 0.9941 and 0.9991 for Case A, Case B and Case C, respectively. R2 is used to measure how close the numerical results are to the linear trend lines. The closer R2 is to 1.0, the more linear the numerical results tend to be. As a result, the relationship curve between ηmax and the ship draft in the presence of a following uniform current tends to be more linear than that without current. Notably, with the increase in pmpm0, ηmax increases faster in Case B and Case C than Case A. This implies that neglecting the following currents can lead to the underestimation of the response of ηmax to the ship draft.

Fig. 10

4.2. Effects of an opposing current on characteristic wave parameters

Fig. 11 shows the variations of the maximum water level elevation ηmax with the Froude number Fr′ at gauge points G1 and G2 for ship waves in the presence of an opposing uniform current. The presence of opposing uniform currents leads to a significant reduction in ηmax at the two gauge points for 0.6≤Fr′≤2.0. Especially for Fr′=0.6, the decrease in ηmax is up to 73.8% in Case D and 78.4% in Case E at location G1 and up to 93.8% in Case D and 95.3% in Case E at location G2 when compared with Case A. Fig. 12 shows the variations of the leading wave period Tp at gauge point G2 with the Froude number Fr′ for ship waves in the presence of an opposing uniform current. The leading wave periods for Fr′= 0.6 and 0.7 were also not provided in Case D and Case E due to the small leading wave heights. In general, Tp decreases with increasing Fr′ in Case D and Case E for 0.8≤Fr′≤2.0. Tp in Case D and Case E are larger than that in Case A for Fr′≥1.0.

Fig. 11
Fig. 12

Fig. 13 depicts the variations of the maximum water level elevation ηmax with the relative Froude number Fr at gauge points G1 and G2 for ship waves in the presence of an opposing uniform current. Similar to Case B and Case C shown in Fig. 6, the overall relationship curves between ηmax and Fr in Case D and Case E are lower than those in Case A. This implies that with the same Fr, ηmax in the presence of an opposing current is also smaller than that without current. Fig. 14 depicts the variations of the leading wave period Tp in the wave train at gauge point G2 with Fr for ship waves in the presence of an opposing uniform current. Similar to Case B and Case C shown in Fig. 7, the overall relationship curves between Tp and Fr in Case D and Case E are lower than those in Case A for 0.9≤Fr≤2.0. This also implies that with the same Fr, Tp in the presence of an opposing current is smaller than that without current.

Fig. 13
Fig. 14

Fig. 15 shows a comparison of the wave pattern for Fr=1.2 between the case of ship waves only and the case of ship waves in the presence of an opposing current. The case of the ship wave in the presence of an opposing current with Fr=1.2 is obtained by setting the ship speed Us=8.7 m/s and the current velocity Uc=−0.5 m/s. As expected (Lee and Lee, 2021), both calculated cusp-line angles are identical. Fig. 16 depicts the comparison of the time histories of the free surface elevation at gauge point G2 for Fr=1.2 between the case of ship waves only and the case of ship waves in the presence of an opposing current. The maximum water level elevation in the case of Us=9.2 m/s and Uc=0.0 m/s is larger than that in the case of Us=8.7 m/s and Uc=−0.5 m/s, while the reverse is true for the leading wave period. Fig. 16 is consistent with the inferences based on Fig. 13Fig. 14.

Fig. 15
Fig. 16

Fig. 17 depicts the response of the maximum water level elevation ηmax to the ship draft at gauge point G2 for Fr′= 1.2 in the presence of an opposing uniform current. Similarly, the response of ηmax to the ship draft in the presence of an opposing uniform current shows a linear relationship. The coefficients of determination for the three linear trend lines are R2= 0.9901, 0.9955 and 0.9987 for Case A, Case D and Case E, respectively. This indicates that the relationship curve between ηmax and the ship draft in the presence of an opposing uniform current also tends to be more linear than that without current. In addition, ηmax increases faster with increasing pmpm0 in Case D and Case E than Case A, implying that the response of ηmax to the ship draft can also be underestimated by neglecting opposing currents.

Fig. 17

5. Conclusions

A non-hydrostatic model incorporating a moving pressure field method was used to investigate characteristic wave parameters for ship waves in the presence of a uniform current. The calculated cusp-line angles for ship waves in the presence of a following or an opposing uniform current were in good agreement with analytical solutions, demonstrating that the proposed model can accurately resolve ship waves in the presence of a uniform current.

The model results showed that the presence of a following current can result in an increase in the maximum water level elevation ηmax for 0.9≤Fr′≤1.1, while the presence of an opposing current leads to a significant reduction in ηmax for 0.6≤Fr′≤2.0. The leading wave period Tp can be increased for 0.925≤Fr′≤1.2 and reduced for Fr′≥1.3 due to the presence of a following current. However, the presence of an opposing current leads to an increase in Tp for Fr′≥1.0.

Although with the same relative Froude number Fr, the cusp-line angles for ship waves in the presence of a following or an opposing current are identical to those for ship waves without current, the maximum water level elevation ηmax and leading wave period Tp in the presence of a following or an opposing current are quite different from those without current. The present model results imply that with the same Fr, ηmax in the presence of a following or an opposing current is smaller than that without current for Fr≥0.6, and Tp in the presence of a following or an opposing current is smaller than that without current for Fr≥0.9.

The response of ηmax to the ship draft in the presence of a following current or an opposing current is similar to that without current and shows a linear relationship. However, the presence of a following or an opposing uniform current results in more linear responses of ηmax to the ship draft. Moreover, more rapid responses of ηmax to the ship draft are obtained when a following current or an opposing current is presented. This implies that the response of ηmax to the ship draft in the presence of a following current or an opposing current can be underestimated if the uniform current is neglected.

The present results have implications for ships sailing across estuarine and coastal environments, where river flows or tidal flows are significant. In these environments, ship waves can be larger than expected and the response of the maximum water level elevation to the ship draft may be more remarkable. The effect of a uniform current should be considered in the analysis of ship waves.

The present study considered only slender-body type ships. For different hull shapes, the effects of a uniform current on characteristic wave parameters need to be further investigated. Moreover, the effects of an oblique uniform current on ship waves need to be examined in future work.

CRediT authorship contribution statement

Congfang Ai: Conceptualization, Methodology, Software, Validation, Writing – original draft, Funding acquisition. Yuxiang Ma: Conceptualization, Methodology, Funding acquisition, Writing – review & editing. Lei Sun: Conceptualization, Methodology. Guohai Dong: Supervision, Funding acquisition.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


This research is financially supported by the National Natural Science Foundation of China (Grant No. 521712485172010501051979029), LiaoNing Revitalization Talents Program (Grant No. XLYC1807010) and the Fundamental Research Funds for the Central Universities (Grant No. DUT21LK01).

Appendix. Numerical results with different numbers of horizontal layers

Fig. 18 shows comparisons of the time histories of the free surface elevation at gauge point G1 for Case B and Fr′= 1.2 between the three sets of numerical results with different numbers of horizontal layers. The maximum water level elevations ηmax obtained by Nz= 3 and 4 are 0.24% and 0.35% larger than ηmax with Nz= 2, respectively. Correspondingly, the leading wave periods Tp obtained by Nz= 3 and 4 are 0.45% and 0.55% larger than Tp with Nz= 2, respectively. In general, the three sets of numerical results are very close. To reduce the computational cost, two horizontal layers Nz= 2 were chosen for this study.

Fig. 18

Data availability

Data will be made available on request.


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이종 금속 인터커넥트의 펄스 레이저 용접을 위한 가공 매개변수 최적화

Optimization of processing parameters for pulsed laser welding of dissimilar metal interconnects

본 논문은 독자의 편의를 위해 기계번역된 내용이어서 자세한 내용은 원문을 참고하시기 바랍니다.

NguyenThi TienaYu-LungLoabM.Mohsin RazaaCheng-YenChencChi-PinChiuc

aNational Cheng Kung University, Department of Mechanical Engineering, Tainan, Taiwan

bNational Cheng Kung University, Academy of Innovative Semiconductor and Sustainable Manufacturing, Tainan, Taiwan

cJum-bo Co., Ltd, Xinshi District, Tainan, Taiwan


워블 전략이 포함된 펄스 레이저 용접(PLW) 방법을 사용하여 알루미늄 및 구리 이종 랩 조인트의 제조를 위한 최적의 가공 매개변수에 대해 실험 및 수치 조사가 수행됩니다. 피크 레이저 출력과 접선 용접 속도의 대표적인 조합 43개를 선택하기 위해 원형 패킹 설계 알고리즘이 먼저 사용됩니다.

선택한 매개변수는 PLW 프로세스의 전산유체역학(CFD) 모델에 제공되어 용융 풀 형상(즉, 인터페이스 폭 및 침투 깊이) 및 구리 농도를 예측합니다. 시뮬레이션 결과는 설계 공간 내에서 PLW 매개변수의 모든 조합에 대한 용융 풀 형상 및 구리 농도를 예측하기 위해 3개의 대리 모델을 교육하는 데 사용됩니다.

마지막으로, 대체 모델을 사용하여 구성된 처리 맵은 용융 영역에 균열이나 기공이 없고 향상된 기계적 및 전기적 특성이 있는 이종 조인트를 생성하는 PLW 매개변수를 결정하기 위해 세 가지 품질 기준에 따라 필터링됩니다.

제안된 최적화 접근법의 타당성은 최적의 용접 매개변수를 사용하여 생성된 실험 샘플의 전단 강도, 금속간 화합물(IMC) 형성 및 전기 접촉 저항을 평가하여 입증됩니다.

결과는 최적의 매개변수가 1209N의 높은 전단 강도와 86µΩ의 낮은 전기 접촉 저항을 생성함을 확인합니다. 또한 용융 영역에는 균열 및 기공과 같은 결함이 없습니다.

An experimental and numerical investigation is performed into the optimal processing parameters for the fabrication of aluminum and copper dissimilar lap joints using a pulsed laser welding (PLW) method with a wobble strategy. A circle packing design algorithm is first employed to select 43 representative combinations of the peak laser power and tangential welding speed. The selected parameters are then supplied to a computational fluidic dynamics (CFD) model of the PLW process to predict the melt pool geometry (i.e., interface width and penetration depth) and copper concentration. The simulation results are used to train three surrogate models to predict the melt pool geometry and copper concentration for any combination of the PLW parameters within the design space. Finally, the processing maps constructed using the surrogate models are filtered in accordance with three quality criteria to determine the PLW parameters that produce dissimilar joints with no cracks or pores in the fusion zone and enhanced mechanical and electrical properties. The validity of the proposed optimization approach is demonstrated by evaluating the shear strength, intermetallic compound (IMC) formation, and electrical contact resistance of experimental samples produced using the optimal welding parameters. The results confirm that the optimal parameters yield a high shear strength of 1209 N and a low electrical contact resistance of 86 µΩ. Moreover, the fusion zone is free of defects, such as cracks and pores.

Fig. 1. Schematic illustration of Al-Cu lap-joint arrangement
Fig. 1. Schematic illustration of Al-Cu lap-joint arrangement
Fig. 2. Machine setup (MFQS-150W_1500W
Fig. 2. Machine setup (MFQS-150W_1500W
Fig. 5. Lap-shear mechanical tests: (a) experimental setup and specimen dimensions, and (b) two different failures of lap-joint welding.
N. Thi Tien et al.
Fig. 5. Lap-shear mechanical tests: (a) experimental setup and specimen dimensions, and (b) two different failures of lap-joint welding. N. Thi Tien et al.
Fig. 9. Simulation and experimental results for melt pool profile. (a) Simulation results for melt pool cross-section, and (b) OM image of melt pool cross-section.
(Note that laser processing parameter of 830 W and 565 mm/s is chosen.).
Fig. 9. Simulation and experimental results for melt pool profile. (a) Simulation results for melt pool cross-section, and (b) OM image of melt pool cross-section. (Note that laser processing parameter of 830 W and 565 mm/s is chosen.).



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Fig. 8 Distribution of solidification properties on the yz cross section at the maximum width of the melt pool.(a) thermal gradient G, (b) solidification velocity vT, (c) cooling rate G×vT, and (d) morphology factor G/vT. These profiles are calculated with a laser power 300 W and velocity 400 mm/s using (a1 through d1) analytical Rosenthal simulation and (a2 through d2) high-fidelity CFD simulation. The laser is moving out of the page from the upper left corner of each color map (Color figure online)

Quantifying Equiaxed vs Epitaxial Solidification in Laser Melting of CMSX-4 Single Crystal Superalloy

CMSX -4 단결정 초합금의 레이저 용융에서 등축 응고와 에피택셜 응고 정량화

본 논문은 독자의 편의를 위해 기계번역된 내용이어서 자세한 내용은 원문을 참고하시기 바랍니다.


에피택셜 과 등축 응고 사이의 경쟁은 적층 제조에서 실행되는 레이저 용융 동안 CMSX-4 단결정 초합금에서 조사되었습니다. 단일 트랙 레이저 스캔은 레이저 출력과 스캐닝 속도의 여러 조합으로 방향성 응고된 CMSX-4 합금의 분말 없는 표면에서 수행되었습니다. EBSD(Electron Backscattered Diffraction) 매핑은 새로운 방향의 식별을 용이하게 합니다. 영역 분율 및 공간 분포와 함께 융합 영역 내에서 핵을 형성한 “스트레이 그레인”은 충실도가 높은 전산 유체 역학 시뮬레이션을 사용하여 용융 풀 내의 온도 및 유체 속도 필드를 모두 추정했습니다. 이 정보를 핵 생성 모델과 결합하여 용융 풀에서 핵 생성이 발생할 확률이 가장 높은 위치를 결정했습니다. 금속 적층 가공의 일반적인 경험에 따라 레이저 용융 트랙의 응고된 미세 구조는 에피택셜 입자 성장에 의해 지배됩니다. 더 높은 레이저 스캐닝 속도와 더 낮은 출력이 일반적으로 흩어진 입자 감소에 도움이 되지만,그럼에도 불구하고 길쭉한 용융 풀에서 흩어진 입자가 분명했습니다.

The competition between epitaxial vs. equiaxed solidification has been investigated in CMSX-4 single crystal superalloy during laser melting as practiced in additive manufacturing. Single-track laser scans were performed on a powder-free surface of directionally solidified CMSX-4 alloy with several combinations of laser power and scanning velocity. Electron backscattered diffraction (EBSD) mapping facilitated identification of new orientations, i.e., “stray grains” that nucleated within the fusion zone along with their area fraction and spatial distribution. Using high-fidelity computational fluid dynamics simulations, both the temperature and fluid velocity fields within the melt pool were estimated. This information was combined with a nucleation model to determine locations where nucleation has the highest probability to occur in melt pools. In conformance with general experience in metals additive manufacturing, the as-solidified microstructure of the laser-melted tracks is dominated by epitaxial grain growth; nevertheless, stray grains were evident in elongated melt pools. It was found that, though a higher laser scanning velocity and lower power are generally helpful in the reduction of stray grains, the combination of a stable keyhole and minimal fluid velocity further mitigates stray grains in laser single tracks.


니켈 기반 초합금은 고온에서 긴 노출 시간 동안 높은 인장 강도, 낮은 산화 및 우수한 크리프 저항성을 포함하는 우수한 특성의 고유한 조합으로 인해 가스 터빈 엔진 응용 분야에서 광범위하게 사용됩니다. CMSX-4는 특히 장기 크리프 거동과 관련하여 초고강도의 2세대 레늄 함유 니켈 기반 단결정 초합금입니다. 1 , 2 ]입계의 존재가 크리프를 가속화한다는 인식은 가스 터빈 엔진의 고온 단계를 위한 단결정 블레이드를 개발하게 하여 작동 온도를 높이고 효율을 높이는 데 기여했습니다. 이러한 구성 요소는 사용 중 마모될 수 있습니다. 즉, 구성 요소의 무결성을 복원하고 단결정 미세 구조를 유지하는 수리 방법을 개발하기 위한 지속적인 작업이 있었습니다. 3 , 4 , 5 ]

적층 제조(AM)가 등장하기 전에는 다양한 용접 공정을 통해 단결정 초합금에 대한 수리 시도가 수행되었습니다. 균열 [ 6 , 7 ] 및 흩어진 입자 8 , 9 ] 와 같은 심각한 결함 이 이 수리 중에 자주 발생합니다. 일반적으로 “스트레이 그레인”이라고 하는 응고 중 모재의 방향과 다른 결정학적 방향을 가진 새로운 그레인의 형성은 니켈 기반 단결정 초합금의 수리 중 유해한 영향으로 인해 중요한 관심 대상입니다. 3 , 10 ]결과적으로 재료의 단결정 구조가 손실되고 원래 구성 요소에 비해 기계적 특성이 손상됩니다. 이러한 흩어진 입자는 특정 조건에서 에피택셜 성장을 대체하는 등축 응고의 시작에 해당합니다.

떠돌이 결정립 형성을 완화하기 위해 이전 작업은 용융 영역(FZ) 내에서 응고하는 동안 떠돌이 결정립 형성에 영향을 미치는 수지상 응고 거동 및 처리 조건을 이해하는 데 중점을 두었습니다. 11 , 12 , 13 , 14 ] 연구원들은 단결정 합금의 용접 중에 표류 결정립 형성에 대한 몇 가지 가능한 메커니즘을 제안했습니다. 12 , 13 , 14 , 15 ]응고 전단에 앞서 국부적인 구성 과냉각은 이질적인 핵 생성 및 등축 결정립의 성장을 유발할 수 있습니다. 또한 용융 풀에서 활발한 유체 흐름으로 인해 발생하는 덴드라이트 조각화는 용융 풀 경계 근처에서 새로운 결정립을 형성할 수도 있습니다. 두 메커니즘 모두에서, 표류 결정립 형성은 핵 생성 위치에 의존하며, 차이점은 수상 돌기 조각화는 수상 돌기 조각이 핵 생성 위치로 작용한다는 것을 의미하는 반면 다른 메커니즘은 재료,  를 들어 산화물 입자에서 발견되는 다른 유형의 핵 생성 위치를 사용한다는 것을 의미합니다. 잘 알려진 바와 같이, 많은 주물에 대한 반대 접근법은 TiB와 같은 핵제의 도입을 통해 등축 응고를 촉진하는 것입니다.22알루미늄 합금에서.

헌법적 과냉 메커니즘에서 Hunt 11 ] 는 정상 상태 조건에서 기둥에서 등축으로의 전이(CET)를 설명하는 모델을 개발했습니다. Gaumann과 Kurz는 Hunt의 모델을 수정하여 단결정이 응고되는 동안 떠돌이 결정립이 핵을 생성하고 성장할 수 있는 정도를 설명했습니다. 12 , 14 ] 이후 연구에서 Vitek은 Gaumann의 모델을 개선하고 출력 및 스캐닝 속도와 같은 용접 조건의 영향에 대한 보다 자세한 분석을 포함했습니다. Vitek은 또한 실험 및 모델링 기술을 통해 표류 입자 형성에 대한 기판 방향의 영향을 포함했습니다. 3 , 10 ]일반적으로 높은 용접 속도와 낮은 출력은 표류 입자의 양을 최소화하고 레이저 용접 공정 중 에피택셜 단결정 성장을 최대화하는 것으로 나타났습니다. 3,10 ] 그러나 Vitek은 덴드라이트 조각화를 고려하지 않았으며 그의 연구는 불균질 핵형성이 레이저 용접된 CMSX -4 단결정 합금에서 표류 결정립 형성을 이끄는 주요 메커니즘임을 나타냅니다. 현재 작업에서 Vitek의 수치적 방법이 채택되고 금속 AM의 급속한 특성의 더 높은 속도와 더 낮은 전력 특성으로 확장됩니다.

AM을 통한 금속 부품 제조 는 지난 10년 동안 급격한 인기 증가를 목격했습니다. 16 ] EBM(Electron Beam Melting)에 의한 CMSX-4의 제작 가능성은 자주 조사되었으나 17 , 18 , 19 , 20 , 21 ] CMSX의 제조 및 수리에 대한 조사는 매우 제한적이었다. – 4개의 단결정 구성요소는 레이저 분말 베드 융합(LPBF)을 사용하며, AM의 인기 있는 하위 집합으로, 특히 표류 입자 형성을 완화하는 메커니즘과 관련이 있습니다. 22 ]이러한 조사 부족은 주로 이러한 합금 시스템과 관련된 처리 문제로 인해 발생합니다. 2 , 19 , 22 , 23 , 24 ] 공정 매개변수( 예: 열원 전력, 스캐닝 속도, 스폿 크기, 예열 온도 및 스캔 전략)의 엄격한 제어는 완전히 조밀한 부품을 만들고 유지 관리할 수 있도록 하는 데 필수적입니다. 단결정 미세구조. 25 ] EBM을 사용하여 단결정 합금의 균열 없는 수리가 현재 가능하지만 19 , 24 ] 표류 입자를 생성하지 않는 수리는 쉽게 달성할 수 없습니다.23 , 26 ]

이 작업에서 LPBF를 대표하는 조건으로 레이저 용융을 사용하여 단결정 CMSX-4에서 표류 입자 완화를 조사했습니다. LPBF는 스캐닝 레이저 빔을 사용하여 금속 분말의 얇은 층을 기판에 녹이고 융합합니다. 층별 증착에서 레이저 빔의 사용은 급격한 온도 구배, 빠른 가열/냉각 주기 및 격렬한 유체 흐름을 경험하는 용융 풀을 생성 합니다 이것은 일반적으로 부품에 결함을 일으킬 수 있는 매우 동적인 물리적 현상으로 이어집니다. 28 , 29 , 30 ] 레이저 유도 키홀의 동역학( 예:, 기화 유발 반동 압력으로 인한 위상 함몰) 및 열유체 흐름은 AM 공정에서 응고 결함과 강하게 결합되고 관련됩니다. 31 , 32 , 33 , 34 ] 기하 구조의 급격한 변화가 발생하기 쉬운 불안정한 키홀은 다공성, 볼링, 스패터 형성 및 흔하지 않은 미세 구조 상을 포함하는 유해한 물리적 결함을 유발할 수 있습니다. 그러나 키홀 진화와 유체 흐름은 자연적으로 다음을 통해 포착 하기 어렵 습니다 .전통적인 사후 특성화 기술. 고충실도 수치 모델링을 활용하기 위해 이 연구에서는 전산유체역학(CFD)을 적용하여 표면 아래의 레이저-물질 상호 작용을 명확히 했습니다. 36 ] 이것은 응고된 용융물 풀의 단면에 대한 오랫동안 확립된 사후 특성화와 비교하여 키홀 및 용융물 풀 유체 흐름 정량화를 실행합니다.

CMSX-4 구성 요소의 레이저 기반 AM 수리 및 제조를 위한 적절한 절차를 개발하기 위해 적절한 공정 창을 설정하고 응고 중 표류 입자 형성 경향에 대한 예측 기능을 개발하는 것부터 시작합니다. 다중 합금에 대한 단일 트랙 증착은 분말 층이 있거나 없는 AM 공정에서 용융 풀 형상 및 미세 구조의 정확한 분석을 제공하는 것으로 나타났습니다. 37 , 38 , 39 ]따라서 본 연구에서는 CMSX-4의 응고 거동을 알아보기 위해 분말을 사용하지 않는 단일 트랙 레이저 스캔 실험을 사용하였다. 이는 CMSX-4 단결정의 LPBF 제조를 위한 예비 실험 지침을 제공합니다. 또한 응고 모델링은 기존 용접에서 LPBF와 관련된 급속 용접으로 확장되어 표류 입자 감소를 위한 최적의 레이저 용융 조건을 식별했습니다. 가공 매개변수 최적화를 위한 추가 지침을 제공하기 위해 용융물 풀의 매우 동적인 유체 흐름을 모델링했습니다.

재료 및 방법

단일 트랙 실험

방전 가공(EDM)을 사용하여 CMSX-4 방향성 응고 단결정 잉곳으로부터 샘플을 제작했습니다. 샘플의 최종 기하학은 치수 20의 직육면체 형태였습니다.××20××6mm. 6개 중 하나⟨ 001 ⟩⟨001⟩잉곳의 결정학적 방향은 레이저 트랙이 이 바람직한 성장 방향을 따라 스캔되도록 절단 표면에 수직으로 위치했습니다. 단일 레이저 용융 트랙은 EOS M290 기계를 사용하여 분말이 없는 샘플 표면에 만들어졌습니다. 이 기계는 최대 출력 400W, 가우시안 빔 직경 100의 이터븀 파이버 레이저가 장착된 LPBF 시스템입니다. μμ초점에서 m. 실험 중에 직사각형 샘플을 LPBF 기계용 맞춤형 샘플 홀더의 포켓에 끼워 표면을 동일한 높이로 유지했습니다. 이 맞춤형 샘플 홀더에 대한 자세한 내용은 다른 곳에서 설명합니다. 실험 은 아르곤 퍼지 분위기에서 수행되었으며 예열은 적용되지 않았습니다 단일 트랙 레이저 용융 실험은 다양한 레이저 출력(200~370W)과 스캔 속도(0.4~1.4m/s)에서 수행되었습니다.

성격 묘사

레이저 스캐닝 후, 레이저 빔 스캐닝 방향에 수직인 평면에서 FZ를 통해 다이아몬드 톱을 사용하여 샘플을 절단했습니다. 그 후, 샘플을 장착하고 220 그릿 SiC 페이퍼로 시작하여 콜로이드 실리카 현탁액 광택제로 마무리하여 자동 연마했습니다. 결정학적 특성화는 20kV의 가속 전압에서 TESCAN MIRA 3XMH 전계 방출 주사 전자 현미경(SEM)에서 수행되었습니다. EBSD 지도는0.4μm _0.4μ미디엄단계 크기. Bruker 시스템을 사용하여 EBSD 데이터를 정리하고 분석했습니다. EBSD 클린업은 그레인을 접촉시키기 위한 그레인 확장 루틴으로 시작한 다음 인덱스되지 않은 회절 패턴과 관련된 검은색 픽셀을 해결하기 위해 이웃 방향 클린업 루틴으로 이어졌습니다. 용융 풀 형태를 분석하기 위해 단면을 광학 현미경으로 분석했습니다. 광학 특성화의 대비를 향상시키기 위해 10g CuSO로 구성된 Marbles 시약의 변형으로 샘플을 에칭했습니다.44, 50mL HCl 및 70mL H22영형.

응고 모델링

구조적 과냉 기준에 기반한 응고 모델링을 수행하여 표유 입자의 성향 및 분포에 대한 가공 매개변수의 영향을 평가했습니다. 이 분석 모델링 접근 방식에 대한 자세한 내용은 이전 작업에서 제공됩니다. 3 , 10 ] 참고문헌 3 에 기술된 바와 같이 , 기본 재료의 결정학적 배향을 가진 용융 풀에서 총 표유 입자 면적 분율의 변화는 최소이므로 기본 재료 배향의 영향은 이 작업에서 고려되지 않았습니다. 우리의 LPBF 결과를 이전 작업과 비교하기 위해 Vitek의 작업에서 사용된 수학적으로 간단한 Rosenthal 방정식 3 ]또한 레이저 매개변수의 함수로 용융 풀의 모양과 FZ의 열 조건을 계산하기 위한 기준으로 여기에서 채택되었습니다. Rosenthal 솔루션은 열이 일정한 재료 특성을 가진 반무한 판의 정상 상태 점원을 통해서만 전도를 통해 전달된다고 가정하며 일반적으로 다음과 같이 표현 됩니다 40 , 41 ] .

티=티0+η피2 파이케이엑스2+와이2+지2———-√경험치[- 브이(엑스2+와이2+지2———-√− 엑스 )2α _] ,티=티0+η피2파이케이엑스2+와이2+지2경험치⁡[-V(엑스2+와이2+지2-엑스)2α],(1)

여기서 T 는 온도,티0티0본 연구에서 313K(  , EOS 기계 챔버 온도)로 설정된 주변 온도, P 는 레이저 빔 파워, V 는 레이저 빔 스캐닝 속도,ηη는 레이저 흡수율, k 는 열전도율,αα베이스 합금의 열확산율입니다. x , y , z 는 각각 레이저 스캐닝 방향, 가로 방향 및 세로 방향의 반대 방향과 정렬된 방향입니다 . 이 직교 좌표는 참조 3 의 그림 1에 있는 시스템을 따랐습니다 . CMSX-4에 대한 고상선 온도(1603K)와 액상선 온도(1669K)의 등온선 평균으로 응고 프런트( 즉 , 고체-액체 계면)를 정의했습니다. 42 , 43 , 44 ] 시뮬레이션에 사용된 열물리적 특성은 표 I 에 나열되어 있습니다.표 I CMSX-4의 응고 모델링에 사용된 열물리적 특성

풀 사이즈 테이블

열 구배는 외부 열 흐름에 의해 결정되었습니다.∇ 티∇티45 ] 에 의해 주어진 바와 같이 :

지 = | ∇ 티| =∣∣∣∂티∂엑스나^^+∂티∂와이제이^^+∂티∂지케이^^∣∣∣=(∂티∂엑스)2+(∂티∂와이)2+(∂티∂지)2————————√,G=|∇티|=|∂티∂엑스나^^+∂티∂와이제이^^+∂티∂지케이^^|=(∂티∂엑스)2+(∂티∂와이)2+(∂티∂지)2,(2)

어디나^^나^^,제이^^제이^^, 그리고케이^^케이^^는 각각 x , y 및 z 방향 을 따른 단위 벡터 입니다. 응고 등온선 속도,V티V티는 다음 관계에 의해 레이저 빔 스캐닝 속도 V 와 기하학적으로 관련됩니다.

V티= V코사인θ =V∂티∂엑스(∂티∂엑스)2+(∂티∂와이)2+(∂티∂지)2——————-√,V티=V코사인⁡θ=V∂티∂엑스(∂티∂엑스)2+(∂티∂와이)2+(∂티∂지)2,(삼)

어디θθ는 스캔 방향과 응고 전면의 법선 방향(  , 최대 열 흐름 방향) 사이의 각도입니다. 이 연구의 용접 조건과 같은 제한된 성장에서 수지상 응고 전면은 고체-액체 등온선의 속도로 성장하도록 강제됩니다.V티V티. 46 ]

응고 전선이 진행되기 전에 새로 핵 생성된 입자의 국지적 비율ΦΦ, 액체 온도 구배 G 에 의해 결정 , 응고 선단 속도V티V티및 핵 밀도N0N0. 고정된 임계 과냉각에서 모든 입자가 핵형성된다고 가정함으로써△티N△티N, 등축 결정립의 반경은 결정립이 핵 생성을 시작하는 시점부터 주상 전선이 결정립에 도달하는 시간까지의 성장 속도를 통합하여 얻습니다. 과냉각으로 대체 시간d (ΔT_) / dt = – _V티G디(△티)/디티=-V티G, 열 구배 G 사이의 다음 관계 , 등축 입자의 국부적 부피 분율ΦΦ, 수상 돌기 팁 과냉각ΔT _△티, 핵 밀도N0N0, 재료 매개변수 n 및 핵생성 과냉각△티N△티N, Gäumann 외 여러분 에 의해 파생되었습니다 . 12 , 14 ] Hunt의 모델 11 ] 의 수정에 기반함 :

지 =1엔 + 1- 4π _N03 인치( 1 − Φ )———√삼ΔT _( 1 -△티엔 + 1N△티엔 + 1) .G=1N+1-4파이N0삼인⁡(1-Φ)삼△티(1-△티NN+1△티N+1).(4)

계산을 단순화하기 위해 덴드라이트 팁 과냉각을 전적으로 구성 과냉각의 것으로 추정합니다.△티씨△티씨, 멱법칙 형식으로 근사화할 수 있습니다.△티씨= ( _V티)1 / 엔△티씨=(ㅏV티)1/N, 여기서 a 와 n 은 재료 종속 상수입니다. CMSX-4의 경우 이 값은a = 1.25 ×106ㅏ=1.25×106 s K 3.4m− 1-1,엔 = 3.4N=3.4, 그리고N0= 2 ×1015N0=2×1015미디엄− 3,-삼,참고문헌 3 에 의해 보고된 바와 같이 .△티N△티N2.5K이며 보다 큰 냉각 속도에서 응고에 대해 무시할 수 있습니다.106106 K/s. 에 대한 표현ΦΦ위의 방정식을 재배열하여 해결됩니다.

Φ= 1 -이자형에스\ 여기서\  S=- 4π _N0삼(1( 엔 + 1 ) (GN/ 아V티)1 / 엔)삼=−2.356×1019(vTG3.4)33.4.Φ=1−eS\ where\ S=−4πN03(1(n+1)(Gn/avT)1/n)3=−2.356×1019(vTG3.4)33.4.


As proposed by Hunt,[11] a value of Φ≤0.66Φ≤0.66 pct represents fully columnar epitaxial growth condition, and, conversely, a value of Φ≥49Φ≥49 pct indicates that the initial single crystal microstructure is fully replaced by an equiaxed microstructure. To calculate the overall stray grain area fraction, we followed Vitek’s method by dividing the FZ into roughly 19 to 28 discrete parts (depending on the length of the melt pool) of equal length from the point of maximum width to the end of melt pool along the x direction. The values of G and vTvT were determined at the center on the melt pool boundary of each section and these values were used to represent the entire section. The area-weighted average of ΦΦ over these discrete sections along the length of melt pool is designated as Φ¯¯¯¯Φ¯, and is given by:



where k is the index for each subsection, and AkAk and ΦkΦk are the areas and ΦΦ values for each subsection. The summation is taken over all the sections along the melt pool. Vitek’s improved model allows the calculation of stray grain area fraction by considering the melt pool geometry and variations of G and vTvT around the tail end of the pool.

수년에 걸쳐 용융 풀 현상 모델링의 정확도를 개선하기 위해 많은 고급 수치 방법이 개발되었습니다. 우리는 FLOW-3D와 함께 고충실도 CFD를 사용했습니다. FLOW-3D는 여러 물리 모델을 통합하는 상용 FVM(Finite Volume Method)입니다. 47 , 48 ] CFD는 유체 운동과 열 전달을 수치적으로 시뮬레이션하며 여기서 사용된 기본 물리 모델은 레이저 및 표면력 모델이었습니다. 레이저 모델에서는 레이 트레이싱 기법을 통해 다중 반사와 프레넬 흡수를 구현합니다. 36 ]먼저, 레이저 빔은 레이저 빔에 의해 조명되는 각 그리드 셀을 기준으로 여러 개의 광선으로 이산화됩니다. 그런 다음 각 입사 광선에 대해 입사 벡터가 입사 위치에서 금속 표면의 법선 벡터와 정렬될 때 에너지의 일부가 금속에 의해 흡수됩니다. 흡수율은 Fresnel 방정식을 사용하여 추정됩니다. 나머지 에너지는 반사광선 에 의해 유지되며 , 반사광선은 재료 표면에 부딪히면 새로운 입사광선으로 처리됩니다. 두 가지 주요 힘이 액체 금속 표면에 작용하여 자유 표면을 변형시킵니다. 금속의 증발에 의해 생성된 반동 압력은 증기 억제를 일으키는 주요 힘입니다. 본 연구에서 사용된 반동 압력 모델은피아르 자형= 특급 _{ B ( 1- _티V/ 티) }피아르 자형=ㅏ경험치⁡{비(1-티V/티)}, 어디피아르 자형피아르 자형는 반동압력, A 와 B 는 재료의 물성에 관련된 계수로 각각 75와 15이다.티V티V는 포화 온도이고 T 는 키홀 벽의 온도입니다. 표면 흐름 및 키홀 형성의 다른 원동력은 표면 장력입니다. 표면 장력 계수는 Marangoni 흐름을 포함하기 위해 온도의 선형 함수로 추정되며,σ =1.79-9.90⋅10− 4( 티− 1654케이 )σ=1.79-9.90⋅10-4(티-1654년케이)엔엠− 1-1. 49 ] 계산 영역은 베어 플레이트의 절반입니다(2300 μμ미디엄××250 μμ미디엄××500 μμm) xz 평면 에 적용된 대칭 경계 조건 . 메쉬 크기는 8입니다. μμm이고 시간 단계는 0.15입니다. μμs는 계산 효율성과 정확성 간의 균형을 제공합니다.

결과 및 논의

용융 풀 형태

이 작업에 사용된 5개의 레이저 파워( P )와 6개의 스캐닝 속도( V )는 서로 다른 29개의 용융 풀을 생성했습니다.피- 브이피-V조합. P 와 V 값이 가장 높은 것은 그림 1 을 기준으로 과도한 볼링과 관련이 있기 때문에 본 연구에서는 분석하지 않았다  .

단일 트랙 용융 풀은 그림  1 과 같이 형상에 따라 네 가지 유형으로 분류할 수 있습니다 39 ] : (1) 전도 모드(파란색 상자), (2) 키홀 모드(빨간색), (3) 전환 모드(마젠타), (4) 볼링 모드(녹색). 높은 레이저 출력과 낮은 스캐닝 속도의 일반적인 조합인 키홀 모드에서 용융물 풀은 일반적으로 너비/깊이( W / D ) 비율이 0.5보다 훨씬 큰 깊고 가느다란 모양을 나타냅니다 . 스캐닝 속도가 증가함에 따라 용융 풀이 얕아져 W / D 가 약 0.5인 반원형 전도 모드 용융 풀을 나타냅니다. W / D _전환 모드 용융 풀의 경우 1에서 0.5 사이입니다. 스캐닝 속도를 1200 및 1400mm/s로 더 높이면 충분히 큰 캡 높이와 볼링 모드 용융 풀의 특징인 과도한 언더컷이 발생할 수 있습니다.

힘과 속도의 함수로서의 용융 풀 깊이와 너비는 각각 그림  2 (a)와 (b)에 표시되어 있습니다. 용융 풀 폭은 기판 표면에서 측정되었습니다. 그림  2 (a)는 깊이가 레이저 출력과 매우 선형적인 관계를 따른다는 것을 보여줍니다. 속도가 증가함에 따라 깊이  파워 곡선의 기울기는 꾸준히 감소하지만 더 높은 속도 곡선에는 약간의 겹침이 있습니다. 이러한 예상치 못한 중첩은 종종 용융 풀 형태의 동적 변화를 유발하는 유체 흐름의 영향과 레이저 스캔당 하나의 이미지만 추출되었다는 사실 때문일 수 있습니다. 이러한 선형 동작은 그림 2 (b) 의 너비에 대해 명확하지 않습니다  . 그림  2(c)는 선형 에너지 밀도 P / V 의 함수로서 용융 깊이와 폭을 보여줍니다 . 선형 에너지 밀도는 퇴적물의 단위 길이당 에너지 투입량을 측정한 것입니다. 50 ] 용융 풀 깊이는 에너지 밀도에 따라 달라지며 너비는 더 많은 분산을 나타냅니다. 동일한 에너지 밀도가 준공 부품의 용융 풀, 미세 구조 또는 속성에서 반드시 동일한 유체 역학을 초래하지는 않는다는 점에 유의하는 것이 중요합니다. 50 ]

그림 1
그림 1
그림 2
그림 2

레이저 흡수율 평가

레이저 흡수율은 LPBF 조건에서 재료 및 가공 매개변수에 따라 크게 달라진다는 것은 잘 알려져 있습니다. 31 , 51 , 52 ] 적분구를 이용한 전통적인 흡수율의 직접 측정은 일반적으로 높은 비용과 구현의 어려움으로 인해 쉽게 접근할 수 없습니다. 51 ] 그  . 39 ] 전도 모드 용융 풀에 대한 Rosenthal 방정식을 기반으로 경험적 레이저 흡수율 모델을 개발했지만 기본 가정으로 인해 키홀 용융 풀에 대한 정확한 예측을 제공하지 못했습니다. 40 ] 최근 간 . 53 ] Ti–6Al–4V에 대한 30개의 고충실도 다중 물리 시뮬레이션 사례를 사용하여 레이저 흡수에 대한 스케일링 법칙을 확인했습니다. 그러나 연구 중인 특정 재료에 대한 최소 흡수(평평한 용융 표면의 흡수율)에 대한 지식이 필요하며 이는 CMSX-4에 대해 알려지지 않았습니다. 다양한 키홀 모양의 용융 풀에 대한 레이저 흡수의 정확한 추정치를 얻기가 어렵기 때문에 상한 및 하한 흡수율로 분석 시뮬레이션을 실행하기로 결정했습니다. 깊은 키홀 모양의 용융 풀의 경우 대부분의 빛을 가두는 키홀 내 다중 반사로 인해 레이저 흡수율이 0.8만큼 높을 수 있습니다. 이것은 기하학적 현상이며 기본 재료에 민감하지 않습니다. 5152 , 54 ] 따라서 본 연구에서는 흡수율의 상한을 0.8로 설정하였다. 참고 문헌 51 에 나타낸 바와 같이 , 전도 용융 풀에 해당하는 최저 흡수율은 약 0.3이었으며, 이는 이 연구에서 합리적인 하한 값입니다. 따라서 레이저 흡수율이 스트레이 그레인 형성에 미치는 영향을 보여주기 위해 흡수율 값을 0.55 ± 0.25로 설정했습니다. Vitek의 작업에서는 1.0의 고정 흡수율 값이 사용되었습니다. 3 ]

퓨전 존 미세구조

그림  3 은 200~300W 및 600~300W 및 600~300W 범위의 레이저 출력 및 속도로 9가지 다른 처리 매개변수에 의해 생성된 CMSX-4 레이저 트랙의 yz 단면 에서 취한 EBSD 역극점도와 해당 역극점도를 보여 줍니다. 각각 1400mm/s. EBSD 맵에서 여러 기능을 쉽게 관찰할 수 있습니다. 스트레이 그레인은 EBSD 맵에서 그 방향에 해당하는 다른 RGB 색상으로 나타나고 그레인 경계를 묘사하기 위해 5도의 잘못된 방향이 사용되었습니다. 여기, 그림  3 에서 스트레이 그레인은 대부분 용융 풀의 상단 중심선에 집중되어 있으며, 이는 용접된 단결정 CMSX-4의 이전 보고서와 일치합니다. 10 ]역 극점도에서, 점 근처에 집중된 클러스터⟨ 001 ⟩⟨001⟩융합 경계에서 유사한 방향을 유지하는 단결정 기반 및 에피택셜로 응고된 덴드라이트를 나타냅니다. 그러나 흩어진 곡물은 식별할 수 있는 질감이 없는 흩어져 있는 점으로 나타납니다. 단결정 기본 재료의 결정학적 방향은 주로⟨ 001 ⟩⟨001⟩비록 샘플을 절단하는 동안 식별할 수 없는 기울기 각도로 인해 또는 단결정 성장 과정에서 약간의 잘못된 방향이 있었기 때문에 약간의 편차가 있지만. 용융 풀 내부의 응고된 수상 돌기의 기본 방향은 다시 한 번⟨ 001 ⟩⟨001⟩주상 결정립 구조와 유사한 에피택셜 성장의 결과. 그림 3 과 같이 용융 풀에서 수상돌기의 성장 방향은 하단의 수직 방향에서 상단의 수평 방향으로 변경되었습니다  . 이 전이는 주로 온도 구배 방향의 변화로 인한 것입니다. 두 번째 전환은 CET입니다. FZ의 상단 중심선 주변에서 다양한 방향의 흩어진 입자가 관찰되며, 여기서 안쪽으로 성장하는 수상돌기가 서로 충돌하여 용융 풀에서 응고되는 마지막 위치가 됩니다.

더 깊은 키홀 모양을 특징으로 하는 샘플에서 용융 풀의 경계 근처에 침전된 흩어진 입자가 분명합니다. 이러한 새로운 입자는 나중에 모델링 섹션에서 논의되는 수상돌기 조각화 메커니즘에 의해 잠재적으로 발생합니다. 결정립이 강한 열 구배에서 핵을 생성하고 성장한 결과, 대부분의 흩어진 결정립은 모든 방향에서 동일한 크기를 갖기보다는 장축이 열 구배 방향과 정렬된 길쭉한 모양을 갖습니다. 그림 3 의 전도 모드 용융 풀 흩어진 입자가 없는 것으로 입증되는 더 나은 단결정 품질을 나타냅니다. 상대적으로 낮은 출력과 높은 속도의 스캐닝 레이저에 의해 생성된 이러한 더 얕은 용융 풀에서 최소한의 결정립 핵형성이 발생한다는 것은 명백합니다. 더 큰 면적 분율을 가진 스트레이 그레인은 고출력 및 저속으로 생성된 깊은 용융 풀에서 더 자주 관찰됩니다. 국부 응고 조건에 대한 동력 및 속도의 영향은 후속 모델링 섹션에서 조사할 것입니다.

그림 3
그림 3

응고 모델링

서론에서 언급한 바와 같이 연구자들은 단결정 용접 중에 표류 결정립 형성의 가능한 메커니즘을 평가했습니다. 12 , 13 , 14 , 15 , 55 ]논의된 가장 인기 있는 두 가지 메커니즘은 (1) 응고 전단에 앞서 구성적 과냉각에 의해 도움을 받는 이종 핵형성 및 (2) 용융물 풀의 유체 흐름으로 인한 덴드라이트 조각화입니다. 첫 번째 메커니즘은 광범위하게 연구되었습니다. 이원 합금을 예로 들면, 고체는 액체만큼 많은 용질을 수용할 수 없으므로 응고 중에 용질을 액체로 거부합니다. 결과적으로, 성장하는 수상돌기 앞에서 용질 분할은 실제 온도가 국부 평형 액상선보다 낮은 과냉각 액체를 생성합니다. 충분히 광범위한 체질적으로 과냉각된 구역의 존재는 새로운 결정립의 핵형성 및 성장을 촉진합니다. 56 ]전체 과냉각은 응고 전면에서의 구성, 동역학 및 곡률 과냉각을 포함한 여러 기여의 합입니다. 일반적인 가정은 동역학 및 곡률 과냉각이 합금에 대한 용질 과냉각의 더 큰 기여와 관련하여 무시될 수 있다는 것입니다. 57 ]

서로 다른 기본 메커니즘을 더 잘 이해하려면피- 브이피-V조건에서 응고 모델링이 수행됩니다. 첫 번째 목적은 스트레이 그레인의 전체 범위를 평가하는 것입니다(Φ¯¯¯¯Φ¯) 처리 매개 변수의 함수로 국부적 표류 입자 비율의 변화를 조사하기 위해 (ΦΦ) 용융 풀의 위치 함수로. 두 번째 목적은 금속 AM의 빠른 응고 동안 응고 미세 구조와 표류 입자 형성 메커니즘 사이의 관계를 이해하는 것입니다.

그림 4
그림 4

그림  4 는 해석적으로 시뮬레이션된 표류 입자 비율을 보여줍니다.Φ¯¯¯¯Φ¯세 가지 레이저 흡수율 값에서 다양한 레이저 스캐닝 속도 및 레이저 출력에 대해. 결과는 스트레이 그레인 면적 비율이 흡수된 에너지에 민감하다는 것을 보여줍니다. 흡수율을 0.30에서 0.80으로 증가시키면Φ¯¯¯¯Φ¯약 3배이며, 이 효과는 저속 및 고출력 영역에서 더욱 두드러집니다. 다른 모든 조건이 같다면, 흡수된 전력의 큰 영향은 평균 열 구배 크기의 일반적인 감소와 용융 풀 내 평균 응고율의 증가에 기인합니다. 스캐닝 속도가 증가하고 전력이 감소함에 따라 평균 스트레이 그레인 비율이 감소합니다. 이러한 일반적인 경향은 Vitek의 작업에서 채택된 그림 5 의 파란색 영역에서 시뮬레이션된 용접 결과와 일치합니다  . 3 ] 더 큰 과냉각 구역( 즉, 지 /V티G/V티영역)은 용접 풀의 표유 입자의 면적 비율이 분홍색 영역에 해당하는 LPBF 조건의 면적 비율보다 훨씬 더 크다는 것을 의미합니다. 그럼에도 불구하고 두 데이터 세트의 일반적인 경향은 유사합니다.  , 레이저 출력이 감소하고 레이저 속도가 증가함에 따라 표류 입자의 비율이 감소합니다. 또한 그림  5 에서 스캐닝 속도가 LPBF 영역으로 증가함에 따라 표유 입자 면적 분율에 대한 레이저 매개변수의 변화 효과가 감소한다는 것을 추론할 수 있습니다. 그림  6 (a)는 그림 3 의 EBSD 분석에서 나온 실험적 표류 결정립 면적 분율  과 그림 4 의 해석 시뮬레이션 결과를  비교합니다.. 열쇠 구멍 모양의 FZ에서 정확한 값이 다르지만 추세는 시뮬레이션과 실험 데이터 모두에서 일관되었습니다. 키홀 모양의 용융 풀, 특히 전력이 300W인 2개는 분석 시뮬레이션 예측보다 훨씬 더 많은 양의 흩어진 입자를 가지고 있습니다. Rosenthal 방정식은 일반적으로 열 전달이 순전히 전도에 의해 좌우된다는 가정으로 인해 열쇠 구멍 체제의 열 흐름을 적절하게 반영하지 못하기 때문에 이러한 불일치가 실제로 예상됩니다. 39 , 40 ] 그것은 또한 그림  4 의 발견 , 즉 키홀 모드 동안 흡수된 전력의 증가가 표류 입자 형성에 더 이상적인 조건을 초래한다는 것을 검증합니다. 그림  6 (b)는 실험을 비교Φ¯¯¯¯Φ¯수치 CFD 시뮬레이션Φ¯¯¯¯Φ¯. CFD 모델이 약간 초과 예측하지만Φ¯¯¯¯Φ¯전체적으로피- 브이피-V조건에서 열쇠 구멍 조건에서의 예측은 분석 모델보다 정확합니다. 전도 모드 용융 풀의 경우 실험 값이 분석 시뮬레이션 값과 더 가깝게 정렬됩니다.

그림 5
그림 5

모의 온도 구배 G 분포 및 응고율 검사V티V티분석 모델링의 쌍은 그림  7 (a)의 CMSX-4 미세 구조 선택 맵에 표시됩니다. 제공지 /V티G/V티(  , 형태 인자)는 형태를 제어하고지 ×V티G×V티(  , 냉각 속도)는 응고된 미세 구조의 규모를 제어하고 , 58 , 59 ]지 -V티G-V티플롯은 전통적인 제조 공정과 AM 공정 모두에서 미세 구조 제어를 지원합니다. 이 플롯의 몇 가지 분명한 특징은 등축, 주상, 평면 전면 및 이러한 경계 근처의 전이 영역을 구분하는 경계입니다. 그림  7 (a)는 몇 가지 선택된 분석 열 시뮬레이션에 대한 미세 구조 선택 맵을 나타내는 반면 그림  7 (b)는 수치 열 모델의 결과와 동일한 맵을 보여줍니다. 등축 미세구조의 형성은 낮은 G 이상 에서 명확하게 선호됩니다.V티V티정황. 이 플롯에서 각 곡선의 평면 전면에 가장 가까운 지점은 용융 풀의 최대 너비 위치에 해당하는 반면 등축 영역에 가까운 지점의 끝은 용융 풀의 후면 꼬리에 해당합니다. 그림  7 (a)에서 대부분의지 -V티G-V티응고 전면의 쌍은 원주형 영역에 속하고 점차 CET 영역으로 위쪽으로 이동하지만 용융 풀의 꼬리는 다음에 따라 완전히 등축 영역에 도달하거나 도달하지 않을 수 있습니다.피- 브이피-V조합. 그림 7 (a) 의 곡선 중 어느 것도  평면 전면 영역을 통과하지 않지만 더 높은 전력의 경우에 가까워집니다. 저속 레이저 용융 공정을 사용하는 이전 작업에서는 곡선이 평면 영역을 통과할 수 있습니다. 레이저 속도가 증가함에 따라 용융 풀 꼬리는 여전히 CET 영역에 있지만 완전히 등축 영역에서 멀어집니다. CET 영역으로 떨어지는 섹션의 수도 감소합니다.Φ¯¯¯¯Φ¯응고된 물질에서.

그림 6
그림 6

그만큼지 -V티G-V티CFD 모델을 사용하여 시뮬레이션된 응고 전면의 쌍이 그림  7 (b)에 나와 있습니다. 세 방향 모두에서 각 점 사이의 일정한 간격으로 미리 정의된 좌표에서 수행된 해석 시뮬레이션과 달리, 고충실도 CFD 모델의 출력은 불규칙한 사면체 좌표계에 있었고 G 를 추출하기 전에 일반 3D 그리드에 선형 보간되었습니다. 그리고V티V티그런 다음 미세 구조 선택 맵에 플롯됩니다. 일반적인 경향은 그림  7 (a)의 것과 일치하지만 이 방법으로 모델링된 매우 동적인 유체 흐름으로 인해 결과에 더 많은 분산이 있었습니다. 그만큼지 -V티G-V티분석 열 모델의 쌍 경로는 더 연속적인 반면 수치 시뮬레이션의 경로는 용융 풀 꼬리 모양의 차이를 나타내는 날카로운 굴곡이 있습니다(이는 G 및V티V티) 두 모델에 의해 시뮬레이션됩니다.

그림 7
그림 7
그림 8
그림 8

유체 흐름을 통합한 응고 모델링

수치 CFD 모델을 사용하여 유동 입자 형성 정도에 대한 유체 흐름의 영향을 이해하고 시뮬레이션 결과를 분석 Rosenthal 솔루션과 비교했습니다. 그림  8 은 응고 매개변수 G 의 분포를 보여줍니다.V티V티,지 /V티G/V티, 그리고지 ×V티G×V티yz 단면에서 x  FLOW-3D에서 (a1–d1) 분석 열 모델링 및 (a2–d2) FVM 방법을 사용하여 시뮬레이션된 용융 풀의 최대 폭입니다. 그림  8 의 값은 응고 전선이 특정 위치에 도달할 때 정확한 값일 수도 있고 아닐 수도 있지만 일반적인 추세를 반영한다는 의미의 임시 가상 값입니다. 이 프로파일은 출력 300W 및 속도 400mm/s의 레이저 빔에서 시뮬레이션됩니다. 용융 풀 경계는 흰색 곡선으로 표시됩니다. (a2–d2)의 CFD 시뮬레이션 용융 풀 깊이는 342입니다. μμm, 측정 깊이 352와 잘 일치 μμ일치하는 길쭉한 열쇠 구멍 모양과 함께 그림 1 에 표시된 실험 FZ의 m  . 그러나 분석 모델은 반원 모양의 용융 풀을 출력하고 용융 풀 깊이는 264에 불과합니다. μμ열쇠 구멍의 경우 현실과는 거리가 멀다. CFD 시뮬레이션 결과에서 열 구배는 레이저 반사 증가와 불안정한 액체-증기 상호 작용이 발생하는 증기 함몰의 동적 부분 근처에 있기 때문에 FZ 하단에서 더 높습니다. 대조적으로 해석 결과의 열 구배 크기는 경계를 따라 균일합니다. 두 시뮬레이션 결과 모두 그림 8 (a1) 및 (a2) 에서 응고가 용융 풀의 상단 중심선을 향해 진행됨에 따라 열 구배가 점차 감소합니다  . 응고율은 그림 8 과 같이 경계 근처에서 거의 0입니다. (b1) 및 (b2). 이는 경계 영역이 응고되기 시작할 때 국부 응고 전면의 법선 방향이 레이저 스캐닝 방향에 수직이기 때문입니다. 이것은 드라이브θ → π/ 2θ→파이/2그리고V티→ 0V티→0식에서 [ 3 ]. 대조적으로 용융 풀의 상단 중심선 근처 영역에서 응고 전면의 법선 방향은 레이저 스캐닝 방향과 잘 정렬되어 있습니다.θ → 0θ→0그리고V티→ 브이V티→V, 빔 스캐닝 속도. G 와 _V티V티값이 얻어지면 냉각 속도지 ×V티G×V티및 형태 인자지 /V티G/V티계산할 수 있습니다. 그림 8 (c2)는 용융 풀 바닥 근처의 온도 구배가 매우 높고 상단에서 더 빠른 성장 속도로  인해 냉각 속도가 용융 풀의 바닥 및 상단 중심선 근처에서 더 높다는 것을 보여줍니다. 지역. 그러나 이러한 추세는 그림  8 (c1)에 캡처되지 않았습니다. 그림 8 의 형태 요인 (d1) 및 (d2)는 중심선에 접근함에 따라 눈에 띄게 감소합니다. 경계에서 큰 값은 열 구배를 거의 0인 성장 속도로 나누기 때문에 발생합니다. 이 높은 형태 인자는 주상 미세구조 형성 가능성이 높음을 시사하는 반면, 중앙 영역의 값이 낮을수록 등축 미세구조의 가능성이 더 크다는 것을 나타냅니다. Tanet al. 또한 키홀 모양의 용접 풀 59 ] 에서 이러한 응고 매개변수의 분포 를 비슷한 일반적인 경향으로 보여주었습니다. 그림  3 에서 볼 수 있듯이 용융 풀의 상단 중심선에 있는 흩어진 입자는 낮은 특징을 나타내는 영역과 일치합니다.지 /V티G/V티그림  8 (d1) 및 (d2)의 값. 시뮬레이션과 실험 간의 이러한 일치는 용융 풀의 상단 중심선에 축적된 흩어진 입자의 핵 생성 및 성장이 등온선 속도의 증가와 온도 구배의 감소에 의해 촉진됨을 보여줍니다.

그림 9
그림 9

그림  9 는 유체 속도 및 국부적 핵형성 성향을 보여줍니다.ΦΦ300W의 일정한 레이저 출력과 400, 800 및 1200mm/s의 세 가지 다른 레이저 속도에 의해 생성된 3D 용융 풀 전체에 걸쳐. 그림  9 (d)~(f)는 로컬ΦΦ해당 3D 보기에서 밝은 회색 평면으로 표시된 특정 yz 단면의 분포. 이 yz 섹션은 가장 높기 때문에 선택되었습니다.Φ¯¯¯¯Φ¯용융 풀 내의 값은 각각 23.40, 11.85 및 2.45pct입니다. 이들은 그림  3 의 실험 데이터와 비교하기에 적절하지 않을 수 있는 액체 용융 풀의 과도 값이며Φ¯¯¯¯Φ¯그림  6 의 값은 이 값이 고체-액체 계면에 가깝지 않고 용융 풀의 중간에서 취해졌기 때문입니다. 온도가 훨씬 낮아서 핵이 생존하고 성장할 수 있기 때문에 핵 형성은 용융 풀의 중간이 아닌 고체-액체 계면에 더 가깝게 발생할 가능성이 있습니다.

그림  3 (a), (d), (g), (h)에서 위쪽 중심선에서 멀리 떨어져 있는 흩어진 결정립이 있었습니다. 그들은 훨씬 더 높은 열 구배와 더 낮은 응고 속도 필드에 위치하기 때문에 과냉각 이론은 이러한 영역에서 표류 입자의 형성에 대한 만족스러운 설명이 아닙니다. 이것은 떠돌이 결정립의 형성을 야기할 수 있는 두 번째 메커니즘,  수상돌기의 팁을 가로지르는 유체 흐름에 의해 유발되는 수상돌기 조각화를 고려하도록 동기를 부여합니다. 유체 흐름이 열 구배를 따라 속도 성분을 갖고 고체-액체 계면 속도보다 클 때, 주상 수상돌기의 국지적 재용융은 용질이 풍부한 액체가 흐물흐물한 구역의 깊은 곳에서 액상선 등온선까지 이동함으로써 발생할 수 있습니다. . 55] 분리된 수상돌기는 대류에 의해 열린 액체로 운반될 수 있습니다. 풀이 과냉각 상태이기 때문에 이러한 파편은 고온 조건에서 충분히 오래 생존하여 길 잃은 입자의 핵 생성 사이트로 작용할 수 있습니다. 결과적으로 수상 돌기 조각화 과정은 활성 핵의 수를 효과적으로 증가시킬 수 있습니다.N0N0) 용융 풀 15 , 60 , 61 ] 에서 생성된 미세 구조에서 표류 입자의 면적을 증가시킵니다.

그림  9 (a) 및 (b)에서 반동 압력은 용융 유체를 아래쪽으로 흐르게 하여 결과 흐름을 지배합니다. 유체 속도의 역방향 요소는 V = 400 및 800mm/s에 대해 각각 최대값 1.0 및 1.6m/s로 더 느려집니다 . 그림  9 (c)에서 레이저 속도가 더 증가함에 따라 증기 침하가 더 얕고 넓어지고 반동 압력이 더 고르게 분포되어 증기 침강에서 주변 영역으로 유체를 밀어냅니다. 역류는 최대값 3.5m/s로 더 빨라집니다. 용융 풀의 최대 너비에서 yz 단면  의 키홀 아래 평균 유체 속도는 그림에 표시된 경우에 대해 0.46, 0.45 및 1.44m/s입니다.9 (a), (b) 및 (c). 키홀 깊이의 변동은 각 경우의 최대 깊이와 최소 깊이의 차이로 정의되는 크기로 정량화됩니다. 240 범위의 강한 증기 내림 변동 μμm은 그림 9 (a)의 V = 400mm/s 경우에서  발견 되지만 이 변동은 그림  9 (c)에서 16의 범위로  크게 감소합니다.μμ미디엄. V = 400mm/s인 경우 의 유체장과 높은 변동 범위는 이전 키홀 동역학 시뮬레이션과 일치합니다. 34 ]

따라서 V = 400mm/s 키홀 케이스의 무질서한 변동 흐름이 용융 풀 경계를 따라 응고된 주상 수상돌기에서 분리된 조각을 구동할 가능성이 있습니다. V = 1200mm/s의 경우 강한 역류 는 그림 3 에서 관찰되지 않았지만 동일한 효과를 가질 수 있습니다. . 덴드라이트 조각화에 대한 유체 유동장의 영향에 대한 이 경험적 설명은 용융 풀 경계 근처에 떠돌이 입자의 존재에 대한 그럴듯한 설명을 제공합니다. 분명히 하기 위해, 우리는 이 가설을 검증하기 위해 이 현상에 대한 직접적인 실험적 관찰을 하지 않았습니다. 이 작업에서 표유 입자 면적 분율을 계산할 때 단순화를 위해 핵 생성 모델링에 일정한 핵 생성 수 밀도가 적용되었습니다. 이는 그림  9 의 표류 입자 영역 비율 이 수지상정 조각화가 발생하는 경우 이러한 높은 유체 흐름 용융 풀에서 발생할 수 있는 것,  강화된 핵 생성 밀도를 반영하지 않는다는 것을 의미합니다.

위의 이유로 핵 형성에 대한 수상 돌기 조각화의 영향을 아직 배제할 수 없습니다. 그러나 단편화 이론은 용접 문헌 [ 62 ] 에서 검증될 만큼 충분히 개발되지 않았 으므로 부차적인 중요성만 고려된다는 점에 유의해야 합니다. 1200mm/s를 초과하는 레이저 스캐닝 속도는 최소한의 표류 결정립 면적 분율을 가지고 있음에도 불구하고 분명한 볼링을 나타내기 때문에 단결정 수리 및 AM 처리에 적합하지 않습니다. 따라서 낮은 P 및 높은 V 에 의해 생성된 응고 전면 근처에서 키홀 변동이 최소화되고 유체 속도가 완만해진 용융 풀이 생성된다는 결론을 내릴 수 있습니다., 처리 창의 극한은 아니지만 흩어진 입자를 나타낼 가능성이 가장 적습니다.

마지막으로 단일 레이저 트랙의 응고 거동을 조사하면 에피택셜 성장 동안 표류 입자 형성을 더 잘 이해할 수 있다는 점에 주목하는 것이 중요합니다. 우리의 현재 결과는 최적의 레이저 매개변수에 대한 일반적인 지침을 제공하여 최소 스트레이 그레인을 달성하고 단결정 구조를 유지합니다. 이 가이드라인은 250W 정도의 전력과 600~800mm/s의 스캔 속도로 최소 흩어진 입자에 적합한 공정 창을 제공합니다. 각 처리 매개변수를 신중하게 선택하면 과거에 스테인리스강에 대한 거의 단결정 미세 구조를 인쇄하는 데 성공했으며 이는 CMSX-4 AM 빌드에 대한 가능성을 보여줍니다. 63 ]신뢰성을 보장하기 위해 AM 수리 프로세스를 시작하기 전에 보다 엄격한 실험 테스트 및 시뮬레이션이 여전히 필요합니다. 둘 이상의 레이저 트랙 사이의 상호 작용도 고려해야 합니다. 또한 레이저, CMSX-4 분말 및 벌크 재료 간의 상호 작용이 중요하며, 수리 중에 여러 층의 CMSX-4 재료를 축적해야 하는 경우 다른 스캔 전략의 효과도 중요한 역할을 할 수 있습니다. 분말이 포함된 경우 Lopez-Galilea 등 의 연구에서 제안한 바와 같이 분말이 주로 완전히 녹지 않았을 때 추가 핵 생성 사이트를 도입하기 때문에 단순히 레이저 분말과 속도를 조작하여 흩어진 입자 형성을 완화하기 어려울 수 있습니다 . 22 ]결과적으로 CMSX-4 단결정을 수리하기 위한 레이저 AM의 가능성을 다루기 위해서는 기판 재료, 레이저 출력, 속도, 해치 간격 및 층 두께의 조합을 모두 고려해야 하며 향후 연구에서 다루어야 합니다. CFD 모델링은 2개 이상의 레이저 트랙 사이의 상호작용과 열장에 미치는 영향을 통합할 수 있으며, 이는 AM 빌드 시나리오 동안 핵 생성 조건으로 단일 비드 연구의 지식 격차를 해소할 것입니다.


LPBF 제조의 특징적인 조건 하에서 CMSX-4 단결정 의 에피택셜(기둥형)  등축 응고 사이의 경쟁을 실험적 및 이론적으로 모두 조사했습니다. 이 연구는 고전적인 응고 개념을 도입하여 빠른 레이저 용융의 미세 구조 특징을 설명하고 응고 조건과 표유 결정 성향을 예측하기 위해 해석적 및 수치적 고충실도 CFD 열 모델 간의 비교를 설명했습니다. 본 연구로부터 다음과 같은 주요 결론을 도출할 수 있다.

  • 단일 레이저 트랙의 레이저 가공 조건은 용융 풀 형상, 레이저 흡수율, 유체 흐름 및 키홀 요동, 입자 구조 및 표류 입자 형성 민감성에 강한 영향을 미치는 것으로 밝혀졌습니다.
  • 레이저 용접을 위해 개발된 이론적인 표유 결정립 핵형성 분석이 레이저 용융 AM 조건으로 확장되었습니다. 분석 모델링 결과와 단일 레이저 트랙의 미세구조 특성화를 비교하면 예측이 전도 및 볼링 조건에서 실험적 관찰과 잘 일치하는 반면 키홀 조건에서는 예측이 약간 과소하다는 것을 알 수 있습니다. 이러한 불일치는 레이저 트랙의 대표성이 없는 섹션이나 유체 속도 필드의 변화로 인해 발생할 수 있습니다. CFD 모델에서 추출한 열장에 동일한 표유 입자 계산 파이프라인을 적용하면 연구된 모든 사례에서 과대평가가 발생하지만 분석 모델보다 연장된 용융 풀의 실험 데이터와 더 정확하게 일치합니다.
  • 이 연구에서 두 가지 표류 결정립 형성 메커니즘인 불균일 핵형성 및 수상돌기 조각화가 평가되었습니다. 우리의 결과는 불균일 핵형성이 용융 풀의 상단 중심선에서 새로운 결정립의 형성으로 이어지는 주요 메커니즘임을 시사합니다.지 /V티G/V티정권.
  • 용융 풀 경계 근처의 흩어진 입자는 깊은 키홀 모양의 용융 풀에서 독점적으로 관찰되며, 이는 강한 유체 흐름으로 인한 수상 돌기 조각화의 영향이 이러한 유형의 용융 풀에서 고려하기에 충분히 강력할 수 있음을 시사합니다.
  • 일반적으로 더 높은 레이저 스캐닝 속도와 더 낮은 전력 외에도 안정적인 키홀과 최소 유체 속도는 또한 흩어진 입자 형성을 완화하고 레이저 단일 트랙에서 에피택셜 성장을 보존합니다.


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리켄 RIBF 조감도. 3 개의 입사 (RILAC, RILAC2, AVF), 4 개의 링 사이클로트론 (RRC, fRC, IRC, SRC), RI 빔 분리 장치 BigRIPS 및 다양한 실험 장치로 구성

Riken RIBF의 He 가스 스트리퍼 및 하전 변환 링 계획

国立研究開発法人理化学研究所 〒 351-0198 埼玉県和光市広沢 2-1
令和 4 年 9 月 16 日原稿受付


리켄 RI 빔 팩토리(RIBF)는 지속적으로 우라늄 빔의 대강도화에 임하고 있으며, 지난 10년간 200배 이상의 강도 증강에 성공하고 있다. He 가스를 이용한 하전 스트리퍼 (He 가스 스트리퍼)의 실현은 그 고강도화의 큰 터닝 포인트였다. 또한, 하전 변환 효율을 비약적으로 올리기 위해 현재 제안하고 있는 하전 변환 링(CSR)은 더욱 큰 강도화가 큰 열쇠가 되는 장치이다. He 스트리퍼와 CSR 계획에 관한 관련 물리 화제와 문제를 섞어 소개한다.


리켄 RI 빔 팩토리 (RIBF [1])와 같이 여러 가속기를 사용하여 중이온의 다단계 가속에서 가속가수의 선택성은 특징적인 자유도 중 하나이다. 가속기의 시작점이되는 이온 소스로부터 생성 된 이온의 원자가의 선택과 가속 도중의 원자가는 “하전 스트리퍼”라고 불리는 장치에 의해 제어 선택된다.

가능한 한 다가가 가속기에서의 가속이나 편향은 효율적이지만, 이온원으로 다가 이온을 대강도로 얻는 것은 일반적으로 어렵고, 스트리퍼로 다가로 하기 위해서는 충분히 가속되어 있어야 한다 있다. 가수를 어느 단계에서 어디까지 올리는지, 그 가속 전략의 최적화는 중이온 가속기 설계의 간이다.

특히 스트리퍼의 성능(얻어지는 가수·변환 효율·내구성·균일성 등)은 가속기 전체의 성능(가속 가능 빔 강도·가속 효율·안정성 등)을 결정하는 가장 중요한 인자라고 할 수 있다. 스트리퍼에는 다양한 기술적인 어려움이 있지만, 이온 원자 충돌의 물리 그 자체를 구현한 장치이며, 축축, 중이온 가속기의 성능은 원자 충돌 과정에 지배되고 있다고 해도 과언이 아니다 .

본 논문에서는 제가 중심으로 개발을 하고 있는 리켄 RIBF 에 있어서의 He 가스 스트리퍼[2–4]와 장래 계획의 하나 하전 변환 링(CSR[5–7])에 대해서, 관련하는 물리의 화제 와 문제를 섞어 소개한다. 모두 가장 가속하기 어려운 우라늄 빔에의 적용을 주안으로 한 것으로, 우선 RIBF에서의 우라늄 빔 가속에 대해 개관한다.

理研 RI ビームファクトリー(RIBF[1])のように複 数の加速器を用いた重イオンの多段階加速にお いて,加速価数の選択性は特徴的な自由度の一 つである.加速器の始点となるイオン源からの生 成イオンの価数の選択,そして加速途中の価数も 「荷電ストリッパー」と呼ばれる装置によって制御 選択される.なるべく多価の方が加速器での加速 や偏向は効率的あるが,イオン源で多価イオンを 大強度で得るのは一般に難しく,ストリッパーで多 価にするためには十分加速されている必要がある. 価数をどの段階でどこまで上げるのか,その加速 ストラテジーの最適化は重イオン加速器設計の肝 である.特にストリッパーの性能(得られる価数・変 換効率・耐久性・均一性など)は加速器全体の性 能(加速可能ビーム強度・加速効率・安定性など) を決める最重要因子といえる.ストリッパーには 様々な技術的な難しさはあるが,イオン原子衝突 の物理そのものを体現した装置であり,畢竟,重 イオン加速器の性能は原子衝突過程に支配され ているといっても過言ではない. 本稿では私が中心となって開発を行っている 理研 RIBF における He ガスストリッパー[2–4]と将 来計画の一つ荷電変換リング(CSR[5–7])につい て,関連する物理の話題や問題を織り交ぜながら 紹介する.いずれも最も加速の難しいウランビームへの適用を主眼としたものであり,先ず RIBF に おけるウランビーム加速について概観する.

리켄 RIBF 조감도. 3 개의 입사 (RILAC, RILAC2, AVF), 4 개의 링 사이클로트론 (RRC, fRC, IRC, SRC), RI 빔 분리 장치 BigRIPS 및 다양한 실험 장치로 구성
리켄 RIBF 조감도. 3 개의 입사 (RILAC, RILAC2, AVF), 4 개의 링 사이클로트론 (RRC, fRC, IRC, SRC), RI 빔 분리 장치 BigRIPS 및 다양한 실험 장치로 구성
그림 7 : He 가스 스트리퍼의 단면도와 실제 사진 (왼쪽 아래) 및 빔 통과시의 발광 모습 (오른쪽 아래).
그림 7 : He 가스 스트리퍼의 단면도와 실제 사진 (왼쪽 아래) 및 빔 통과시의 발광 모습 (오른쪽 아래).


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