Multi-Phase-Field Framework for Epitaxial Grain Growth in Selective Laser Melting Additive Manufacturing with Multi-Track and Multi-Layer

Abstract

In this study, a multi-phase-field (MPF) framework for predicting epitaxial grain growth in selective laser melting (SLM) additive manufacturing (AM) with multi-track and multi-layer scanning was developed. The spatiotemporal change in temperature was approximated using the Rosenthal equation, which provides a theoretical solution for the temperature distribution due to a moving point heat source. The powder bed was modeled as a polycrystalline layer. Large-scale MPF simulations for SLM-AM were performed using parallel computing with multiple graphics processing units. Using the MPF framework developed herein, we simulated SLM-AM with four tracks and four layers for 316L stainless steel. By observing the epitaxial grain growth process on two-dimensional cross-sections and in three dimensions, we clarified a typical growth procedure of grains with characteristic 3D shapes. The MPF framework will potentially enable a systematic estimation of the material microstructures formed during SLM-AM.

1. Introduction

Metal additive manufacturing (AM), which can fabricate parts with complex geometries, is the most promising process technology, and many studies on AM have been conducted from various viewpoints.^1–3) In metal AM, selective laser melting (SLM) is one of the most widely used AM techniques based on powder bed fusion (PBF). The mechanical and fatigue properties as well as the corrosion of fabricated parts are largely affected by material microstructures formed during SLM-AM.^4–11) Recently, not only the optimization of the complex geometry of parts but also the optimization of crystallographic texture has been studied in SLM-AM.^12–15) Systematic evaluation of three-dimensional (3D) material microstructures under various building conditions is not straightforward owing to many building parameters in SLM-AM, such as laser power, scanning speed, layer thickness, hatch distance, and scanning strategy.^16–19) In addition, the direct observation of the growth process of 3D material microstructures in SLM-AM is almost impossible. Therefore, numerical simulations to predict the evolution of material microstructure are indispensable.

To the best of our knowledge, numerical studies aimed at predicting the material microstructures in SLM-AM can be largely divided into two categories.^20–22) One is to investigate the detailed material microstructures at the scale of the cell/dendrite, where the phase-field (PF)^23,24) is the most powerful method.^25–29) However, PF simulations have been limited to only a small region in the melt pool^26,27,30) and the two-dimensional (2D) problem^{25,28,29,31–33)} due to the large computational cost of the diffuse interface model. Competitive growth between dendrites/cells is quite different between 2D and 3D problems.^34–39) In addition, melt flow in the melt pool drastically changes competitive growth.^40,41) It is almost impossible to achieve 3D full-scale PF simulations of dendrite/cell growth with melt flow in a melt pool, even when using a supercomputer, as the solidification condition in the melt pool of SLM is quite severe with a high cooling rate and temperature gradient.⁴²⁾ The other direction is a coarse-grained model in which the cell/dendrite structures are not considered. The cellular automaton^43,44) model has been widely used,^45–48) and it enables the prediction of material microstructure under melt flow.^49–51) The PF method is also used as a coarse-grained model.^52–57) Chadwick and Voorhees⁵⁵⁾ studied the development of epitaxially grown grains using a multi-phase-field (MPF) coupled with the Rosenthal equation.⁵⁸⁾ They successfully reproduced curved grains and showed small equiaxed grains around the top surface without nucleation.^59–61)

In this study, we developed an MPF framework for multi-layer and multi-track SLM-AM by extending the Chadwick and Voorhees method.⁵⁵⁾ The MPF simulations become large-scale even when using a coarse-grained model. Therefore, we implemented parallel computing using multiple graphics processing units (GPU).^62,63) Using the MPF framework, SLM-AM with four tracks and four layers was simulated, and the growth of 3D grains with characteristic shapes was investigated.

2. Model

As a numerical model, we used the MPF model with a double-obstacle potential developed by Steinbach.^64,65) Among the many existing models with multiple PF variables,⁶⁶⁾ the MPF model can introduce grain boundary anisotropy dependent on grain boundary misorientation with high accuracy⁶⁶⁾ and efficiently compute grain growth with a massive number of grains^67,68) by introducing active parameter tracking (APT).^69–72)

In the present SLM simulation, we considered polycrystals in the substrate, solidified layer, and powder bed, liquid in the melt pool, and the gas phase over the material. For each grain or phase, a PF variable ϕ_i was assigned with ϕ_i = 1 in each grain and ϕ_i = 0 in the other, and it changed smoothly and steeply at the grain boundary, solid-liquid interface, and surface of the solid and melt pool. The time evolution of ϕ_i follows:   

\begin{align} \frac{\partial \phi_{i}}{\partial t} &= -\frac{2}{n}\sum_{j=1}^{n} M_{ij}^{\phi} \Bigg[ \sum_{k=1}^{n} \left\{ \frac{1}{2}(a_{ik}^{2} - a_{jk}^{2}) + (W_{ik} - W_{jk}) \phi_{k} \right\} \\ &\quad - 30\phi_{i}{}^{2} \phi_{j}{}^{2}\Delta f_{ij} \Bigg], \end{align} (1)

where n is the number of PF variables with a non-zero value at a grid point; a_ij, W_ij, and M^ϕ_ij are the gradient coefficient, height of the double-obstacle potential, and PF mobility, respectively. These are related to the energy γ_ij, thickness δ, and mobility M_ij of the interface/surface/grain boundary as follows:⁷²⁾   

\begin{equation} a_{ij} = \frac{2}{\pi}\sqrt{2\delta \gamma_{ij}},\ W_{ij} = \frac{4\gamma_{ij}}{\delta},\ \text{and}\ M_{ij}^{\phi} = \frac{\pi^{2}}{8\delta}M_{ij}. \end{equation} (2)

In this study, we did not consider the anisotropy of γ_ij and M_ij depending on the disorientation and interface direction. The last term on the right side of eq. (1) is the thermodynamic driving force term, and Δf_ij is expressed as   

\begin{equation} \Delta f_{ij} = \omega_{ij}\Delta f_{\textit{SL}}, \end{equation} (3)

where Δf_SL is the driving force for solid-liquid interface migration with   

\begin{equation} \Delta f_{\textit{SL}} = \frac{L (T_{m} - T)}{T_{m}}, \end{equation} (4)

where L is the latent heat and T_m is the melting temperature. ω_ij in eq. (3) is a matrix that applies Δf_SL only to the solid-liquid interface. In eq. (1), ϕ_i²ϕ_j² of the driving force term is introduced to solve eq. (1) more stably than using the form of ϕ_iϕ_j.⁷³⁾

The temperature T was simply approximated by the Rosenthal equation,⁵⁸⁾ which is a theoretical equation for spatiotemporal distribution in a semi-infinite plate with a moving point heat source on a surface:   

\begin{equation} T (x,y,z,t) = T_{0} + \frac{Q}{2\pi \kappa_{T}R}\exp \left[ - \frac{V_{\textit{scan}}}{2D_{T}} (R + Y)\right], \end{equation} (5)

with   

\begin{align} X &= x - x_{0},\ Y = \pm(y - y_{0} - V_{\textit{scan}} (t - t_{0})),\\ Z &= z - z_{0},\ \text{and}\ R = \sqrt{X^{2} + Y^{2} + Z^{2}}. \end{align} (6)

Here, we assumed that the point heat source moves along the y-axis on a zx-plane, where we set the coordinates as follows: the z-axis is the building direction (BD), y-axis is the scanning direction (SD), and x-axis is the transverse direction (TD). In eq. (5), T₀ is the initial temperature, Q is the power of the heat source, κ_T is the thermal conductivity, V_scan is the scanning velocity of the heat source, D_T is the thermal diffusion coefficient, (x₀, y₀, z₀) is the initial coordinate of the heat source at t_track = 0, R is the distance from the point heat source, and (X, Y, Z) are the coordinates of the point heat source at time t − t₀ for the initial time t₀ at each track. The sign of the Y-coordinate in eq. (6) is determined by the direction of heat source.

Equation (1) was discretized by a normal finite difference method as follows: the time was discretized by the first-order forward difference, and the Laplacian was discretized by the second-order central difference. The code was implemented using CUDA C with a message passing interface (MPI) for parallel computing using multiple GPUs.^62,63)

3. Computational Conditions

The simulation of SLM-AM with multi-track and multi-pass was performed for a well using 316L stainless steel with the material parameters listed in Table 1. The thermal conductivity, κ_T, and thermal diffusivity, D_T, were assumed to be constants with an average value of those of liquid and solid at the melting temperature of T_m = 1,700 K.^55,74) The solid-liquid interface energy, γ_SL, and grain boundary energy, γ_GB, and surface energy of the liquid and solid, γ_surf, were assumed to be isotropic and constant, with the same value as γ_SL = γ_GB = γ_surf = 0.2 J/m². The grain boundary mobility, M_GB, was also set to be a constant at the melting temperature T_m, while using the temperature-dependent equation M_GB = M₀ exp(−Q_GB/RT) with the pre-exponential coefficient M₀, activation energy Q_GB, and gas constant R.⁷⁵⁾

Table 1 Material parameters for 316L.

The computational conditions are listed in Table 2. The laser power, Q, and scan speed, V_scan, were set as Q = 30 W and V_scan = 500 mm/s. Under these scanning conditions, the melt pool size can be roughly estimated as an isothermal surface with T_m = 1,700 K using the Rosenthal equation, eq. (5), as shown in Fig. 1: width w = 80.0 µm, height h = 40.0 µm, and length l = 145.6 µm. In the simulation, the actual melt pool sizes were w = 80.2 µm, h = 40.2 µm, and l = 152.2 µm. Although w and h were almost the same within the error range, the length, l, was approximately 4.5% longer in the actual melt pool, with details being approximately 4% shorter in the front, l₁, and 5.9% longer in the back, l₂.

Table 2 Computational conditions.

Fig. 1

Shape and size of the melt pool, numerically computed by eq. (1) with the Rosenthal equation, eq. (5). The blue line indicates the isotherm with the melting temperature T_m = 1,700 K.

Figures 2 and 3 show the computational domain size and the scanning strategy in the TD-BD and TD-SD planes, respectively. The layer thickness was set as t_layer = 30 µm.⁷⁶⁾ The hatch, which is the distance between two neighboring tracks, was set to d_hatch = 50 µm to avoid non-overlapping areas under the conditions of t_layer = 30 µm, w = 80 µm, and h = 40 µm.⁷⁷⁾ The bi-directional scanning strategy⁷⁸⁾ was employed for all four layers. The domain size in the BD, L_BD, was set to L_BD = 160 µm by considering the four layers with t_layer = 30 µm, initial substrate thickness of 22.8 µm, and gas layer thickness of 17.2 µm in the 4th layer scanning. The domain size in the SD, L_SD, was set to L_SD = 240 µm so that the entire melt pool would fit. Because of the four tracks in each layer and the periodic boundary condition in the TD, the domain size in the TD, L_TD, was L_TD = 200 µm. The computational domain with L_TD × L_SD × L_BD = 200 × 240 × 160 µm³ was divided into n_TD × n_SD × n_BD = 640 × 768 × 512 grids with the grid size of Δx = 0.3125 µm. The scanning time in one track was set as t_track = 40,000 steps × Δt = 0.0008 s, with a time increment Δt = 2 × 10⁻⁸ s. In the time period t_track, the point heat source migrated to 400 µm with V_scan = 500 mm/s, where the heat source point started from y₀ = −65Δx (or y₀ = L_SD + 65Δx) and ended at y = L_SD + 417Δx (or y = −417Δx) to pass the melt pool over the computational domain. After four-track scanning of each layer, the next powder bed with t_layer = 30 µm was prepared on the top. The powder bed was assumed to be a polycrystalline layer with the same average diameter as the initial substrate, d_ave = 6.5 µm.⁵⁵⁾ To create a polycrystalline layer, we computed the grain growth process with 300Δt. Thus, the total time step in the simulation was 40,000 steps × 16 tracks + 300 steps × three layers = 640,900 steps, which reduces to the total time t_total = 0.012818 s. Here, the time to prepare the first polycrystalline layer and substrate was not included. In SD and BD, a zero Neumann boundary condition was set for the PF variables.

Fig. 2

Computational domain size in the TD-BD plane and bi-directional scanning strategy. The semicircle indicates the shape of melt pool in normal section to the SD. Because of periodic boundary condition in the TD, the 1st track locates on both sides of the TD. The figures from 1 to 16 are the consecutive track numbers in the simulation. Size is shown in microns.

Fig. 3

Computational domain sizes in the TD-SD plane and bi-directional scanning strategy in a layer. Size is shown in microns.

The simulation was performed in parallel on eight NVIDIA Tesla A100 GPUs, and the execution time was approximately 51 h.

4. Results

4.1 Grain structure on the outer surfaces of the computational domain

Figure 4 shows the grain structure on the outer surfaces of the computational domain at the ends of all 16 tracks, or at the end of the 4th track of the 4th layer. The color indicates the number of grains (PF variables) of 53,411. Characteristic curved grain structures were observed on all the surfaces. The top plane (TD-SD plane) does not correspond to the surface of the computational domain but that of the solid, wherein the gas layer was removed. The top surface was characterized by two types of shapes: small elongated grains along the centerline of the melt pool and tangent-shaped grains connecting the two centerlines of the melt pool. The grains located at the starting points of each track were longer and larger than the other grains because of their contact with the boundary and the absence of grains behind them. These grains probably do not appear in the actual SLM process because such sharp boundaries are impossible in real SLM-AM. The top surface became uneven because of grain growth by considering the gas phase. In the MPF model used in this study, the PF variables were not conserved. Thus, although the surface solid slightly reduced the volume, the reduction in volume was negligible because of the short period. The side SD-BD plane was characterized by long stepwise grains, in which some grains grow epitaxially over four layers from the initial substrate grains. Such long grains over some layers have been reported in experimental observations.^16,79) There were some types of grains on the TD-BD boundary. Along the centerlines of tracks 1 and 3, very long boundary grains were observed. Small elongated grains were observed along the centerlines of tracks 2 and 4. Large U-shaped grains were mainly observed between the centerlines of the melt pool, while horn-shaped grains were observed only between tracks 2 and 3. Around the top surface, thin elongated grains between the tracks and small equiaxed grains around the centerline of tracks 2 and 4 were observed.

Fig. 4

Grain structures on the outer surfaces of the computational domain at the completion of all tracks. Color indicates the number of grains and the grain boundary is indicated by the grey line.

4.2 Grain formation in each cross-section

In this section, we discuss the formation process of grain morphologies by observing the cross sections.

Figure 5 shows the grain structures on the TD-BD cross-section at y = L_SD/2 = 120 µm at the end of the polycrystalline layer preparation and each track. The color indicates the grain number, and the grain boundary is indicated by a grey line, similar to that in Fig. 4. The white solid and dashed lines represent the melt pool shapes of the track and finished tracks, respectively. The white long dashed short dashed line is the border between the polycrystalline layer and solidified layer. In the 1st layer, the powder bed and substrate were prepared as one polycrystalline substrate without distinction. For the subsequent layers, polycrystalline layers were placed on top of each layer. In the area within the solid white line immediately after passing the melt pool, a typical fan-shaped structure was observed, where thin grains extended from the outside toward the center of the melt pool, and small equiaxed grains formed around the center of the melt pool.⁵⁹⁾ For the 1st and 4th tracks in all layers, the grain structure in the melt pool was mirror symmetric with respect to the centerline of the melt pool, whereas for the 2nd and 3rd tracks, it was different in size between the left and right. Because both sides of the 1st track were small power grains, the developed grains were small and symmetric about the centerline. The 4th track was located between the solidified tracks. Thus, the developed left and right grains were larger than those of the 1st track, while maintaining the right and left symmetry. In the 2nd and 3rd tracks, the left side shows the powder grains and the right side shows the coarse solidified grains. Thus, the grains on the right were coarser than those on the left. In the present simulation, because grain nucleation was not considered, all grains grew epitaxially from the surrounding grains. New grains were introduced only from the powder grains. Observing the difference in each layer, the finest grains were formed during the scanning of the 1st layer, and the size of grains epitaxially growing from solidified grains gradually increased with the layer. In addition, the region with small equiaxed grains around the center of the melt pool decreased as the number of layers increased. The final grain structure at the end of the 4th track of the 4th layer was different from that on the TD-BD surface shown in Fig. 3. This implies that the grain structure on the TD-BD surface shown in Fig. 3 was largely affected by the boundary condition.

Fig. 5

Grain structures on the TD-BD cross-section at y = L_SD/2 = 120 µm at the end of polycrystalline layer preparation and each track.

Figure 6 shows the top views of the grain structure at the 22,500th step in each track during the 4th layer scanning. The melt pool shape is indicated by the white dashed line; note that in the dashed line, the grains have information in the depth direction. The grains on the top surface first solidified toward the centerline of the melt pool, or the TD, and then gradually changed the growth direction to SD. Thus, the grain shape became sickle-like, which has been observed in simulations with a single track.⁴⁵⁾ The sickle-shaped grains were observed on both sides of the melt pool in the 1st track, and on the left side of the melt pool in the 2nd and 3rd tracks. The sickle-shaped grains changed to tangent-shaped grains due to scanning of the neighboring track as shown on the right of the melt pool in the 2nd and 3rd tracks and on both sides of the melt pool in the 4th track. Behind the melt pool on the centerline, elongated thin grains were formed. Inside the melt pool, as shown in Fig. 5, the grain morphologies on the left and right were similar for the 1st and 4th tracks, whereas the grain shape and size were different for both tracks. For the 2nd and 3rd tracks, the grain morphologies were different on the left and right. These differences of grain shape in the melt pool depend on whether the scanning area touches the powder particles or solidified area, similar to the 4th layer scanning shown in Fig. 5.

Fig. 6

Top views (TD-SD plane) of grain structures at the 22,500th step in each track at the 4th layer scan.

Figure 7 shows the grain structures on the SD-BD cross-sections with x = (a) 240Δx (the middle between the 3rd and 4th tracks), (b) 320Δx (centerline of the 3rd track), and (c) 400Δx (the middle between the 3rd and 2nd tracks) at the end of the 3rd track of the 4th layer. Grains with long stepwise shapes are observed in Fig. 7(b), which are similar to those on the SD-BD surface of Fig. 4. The layers are distinguished by the white dashed line, and the bottom of the melt pool and the melt pool shape of the 4th layer scan are indicated by the black dashed line. Curved long grains suddenly changed their growth direction at the bottom of the melt pool, and grains in the upper region of the bottom of the melt pool had a sickle-like shape. Because the temperature distributions to the BD and TD from the centerline of the melt pool were the same due to the Rosenthal equation, the shapes of the grains which formed at the surface in Fig. 6 and 4th layer in the centerline section in Fig. 7(b) were similar. In Figs. 7(a) and (c), the location of the bottom of the melt pool was close to that of the 4th layer. In Fig. 7(a), while the grains located at the surface of the 3rd layer were slightly elongated in the upward direction, the upper region is filled with small equiaxed grains. From Fig. 5, it can be seen that small equiaxed grains grew from the powder grains. As shown in Fig. 7(c), columnar grains grew toward the surface, and small equiaxed grains formed around the surface. Interestingly, the columnar grains in Fig. 7(c) were slightly curved in the opposite direction to those in Fig. 7(b). The small grains at the surface grew from the small grains located at the centerline of the 2nd track, and the columnar grains grew from the columnar grains formed during the 2nd track.

Fig. 7

Grain structures on the SD-BD cross-sections with x = (a) 240Δx (the middle of the 3rd and 4th tracks), (b) 320Δx (centerline of the 3rd track), and (c) 400Δx (the middle of the 3rd and 2nd tracks) at the end of the 3rd track of the 4th layer.

As shown in Figs. 5–7, the growth of grains with characteristic shapes can be clearly observed. However, imaging a 3D structure from cross-sectional shapes is not straightforward.

4.3 Characteristic 3D grain structures

Figure 8 shows two different views of the 3D structures of grains numbered 1 to 24 in Fig. 7. The grain colors are the same as those in previous figures. Grains 13–20 with a sickle-like shape grew from the powder grains. Grains 1–4 with a tangent-like shape grew from the sickle-shaped grains that formed on the surface during the 2nd track scanning. Grains 21–24 with a waterfowl-like shape grew from the tangent-shaped grains located on the top surface of the 3rd layer. Grains 5–8 were of a plate-like shape and grew from the waterfowl-shaped grains formed during the 2nd track scanning. Finally, grains 9–12 grew over several layers.

Fig. 8

Two different views of grains 3D structures, numbered 1 to 24 (as shown in Fig. 7). The color indicates the grain number, which is same as in Fig. 7.

The 4,776 grains remained after the final 16th track (4th track in the 4th layer), as shown in Fig. 4. In all the remaining grains, the number of large grains with a volume size more than 100 times the initial average grain size was 48. Here, 12 grains were in contact with the TD-BD boundary, and the remaining 36 grains are shown in Fig. 9(d). The 35 grains had a twisted U-shape, and they bridged two neighboring tracks. Only grain A, which is also seen in Fig. 4, grew straight upward along the centerline of the melt pool. The growth procedures for these large grains are shown in Fig. 9(a)–(c). After the 1st layer scanning (Fig. 9(a)), almost all grains had a tangent-like shape on the top surface similar to grains 1–4 in Fig. 8, while grains B and C had plate-like shapes similar to the grains 5–8 in Fig. 8. At the end of the 2nd layer scanning (Fig. 9(b)), almost all grains had plate-like shapes similar to grains 5–8 in Fig. 8, and some grains were generated here and had a tangent-like shape. Grains B and C had a twisted U-shape. The two upper projections extended upward in the 3rd and 4th layers (Fig. 9(c)(d)). Figure 10 shows a close-up view of the growth process for the largest grain, D. In the 1st and 2nd layers, the morphologies at the end of the 2nd track are also shown. A typical morphological change could be clearly seen: sickle-like, tangent-like, waterfowl-like, plate-like, and twisted U-shape. These grain morphologies, except for the U-shape, are also observed in Fig. 8.

Fig. 9

3D structure of the 35 largest grains without contact with the TD-BD boundary at end of each layer.

Fig. 10

Growth process of the largest grain D (as shown in Fig. 9). In the 1st and 2nd layers, the morphologies at the end of the 2nd track are also shown in addition to those at the end of each layer. The sickle-like shape at the end of the 2nd track and tangent-like shape in (a) the 1st layer, waterfowl-like shape at the end of the 2nd track and plate-like shape in (b) the 2nd layer, and twisted U-shape in (c) the 3rd and (d) 4th layers.

Experimental observation of the 3D morphological changes is challenging, and imaging the 3D structure of the twisted U-shaped large grains from the 2D cross-sections is difficult. As demonstrated above, the MPF framework developed herein can express the detailed 3D morphological change during multi-track and multi-layer SLB-AM.

5. Discussion

We showed that the developed MPF framework can express the growth process of complicated grains in SLM-AM with multi-track and multi-layer scanning. Here, we discuss some issues with the proposed model which can be addressed in future work.

The modeling of the powder bed is an issue in the present model. Because the liquid flow was not considered in this model, we could not directly use the powder particles. However, we must express epitaxial growth from unmelted particles.⁶⁰⁾ The polycrystalline layer model introduced in this study is considered to be a reasonable idea. However, we need to discuss the grain size and thickness of the polycrystalline layer to express the epitaxial growth from the unmelted powder particle.⁸⁰⁾ In addition, it is necessary to model the surface mound shape after passing the laser.^59,61,81,82)

The shape of the melt pool is important for determining grain morphologies.^77,83) In the present model, we employed the Rosenthal equation, which provides the spatiotemporal distribution of temperature for the moving point heat source. However, the actual melt pool shape deviates from the one expressed by the Rosenthal equation.⁷⁶⁾ When a keyhole is generated in the melt pool, the shape of the melt pool changes significantly.⁸⁴⁾ Thus, it is important to summarize the relationship between laser power, scanning rate, material, and melt pool shape.⁸⁵⁾

In the present model, we do not consider the nucleation of new grains. Gaussian distribution of the nucleation density is often used to model the nucleation as a function of the undercooling.^49,86,87) Moreover, we need to consider anisotropy in the interface/surface/grain energy and mobility.^68,88)

6. Conclusions

A multi-phase-field (MPF) framework was developed to predict material microstructure evolutions during multi-track and multi-layer selective laser melting (SLM) in additive manufacturing (AM). The spatiotemporal distribution of the temperature was approximated by the Rosenthal equation, and the powder bed was expressed as a polycrystalline layer. To enable large-scale MPF simulations during multi-track and multi-layer SLM-AM, we employed parallel computing using multiple graphics processing units. Through the SLM-AM simulation with four layers and four tracks and a bi-directional scanning strategy for 316L, we observed the material microstructural evolutions on the 2D cross-sections and in the 3D, and clarified the growth process of grains with a characteristic 3D shape. Finally, we discussed issues associated with the developed MPF framework. We believe that the MPF framework can predict the material microstructure evolution under various building conditions and scanning strategies in SLM-AM.⁸⁹⁾

Acknowledgments

This work was supported by KAKENHI Grant-in-Aid for Scientific Research 22H05282. This work was also supported by “Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures (JHPCN)” and “High Performance Computing Infrastructure” in Japan (Project ID: jh220023).

REFERENCES

• 1)   W.E.  Frazier: J. Mater. Eng. Perform. 23 (2014) 1917–1928. doi:10.1007/s11665-014-0958-z
• 2)   T.  DebRoy,  H.L.  Wei,  J.S.  Zuback,  T.  Mukherjee,  J.W.  Elmer,  J.O.  Milewski,  A.M.  Beese,  A.  Wilson-Heid,  A.  De and  W.  Zhang: Prog. Mater. Sci. 92 (2018) 112–224. doi:10.1016/j.pmatsci.2017.10.001
• 3)   D.  Herzog,  V.  Seyda,  E.  Wycisk and  C.  Emmelmann: Acta Mater. 117 (2016) 371–392. doi:10.1016/j.actamat.2016.07.019
• 4)   Y.  Tsutsumi,  T.  Ishimoto,  T.  Oishi,  T.  Manaka,  P.  Chen,  M.  Ashida,  K.  Doi,  H.  Katayama,  T.  Hanawa and  T.  Nakano: Addit. Manuf. 45 (2021) 102066. doi:10.1016/j.addma.2021.102066
• 5)   Z.  Zhang,  S.  Wang,  H.  Liu,  L.  Wang and  X.  Xiao: J. Mater. Eng. Perform. (2022). doi:10.1007/s11665-022-07416-w
• 6)   E.  Liverani,  S.  Toschi,  L.  Ceschini and  A.  Fortunato: J. Mater. Process. Technol. 249 (2017) 255–263. doi:10.1016/j.jmatprotec.2017.05.042
• 7)   A.  Yadollahi,  N.  Shamsaei,  S.M.  Thompson and  D.W.  Seely: Mater. Sci. Eng. A 644 (2015) 171–183. doi:10.1016/j.msea.2015.07.056
• 8)   A.B.  Spierings,  T.L.  Starr and  K.  Wegener: Rapid Prototyping J. 19 (2013) 88–94. doi:10.1108/13552541311302932
• 9)   V.B.  Vukkum and  R.K.  Gupta: Mater. Des. 221 (2022) 110874. doi:10.1016/j.matdes.2022.110874
• 10)   D.  Kong,  C.  Dong,  X.  Ni,  L.  Zhang,  J.  Yao,  C.  Man,  X.  Cheng,  K.  Xiao and  X.  Li: J. Mater. Sci. Technol. 35 (2019) 1499–1507. doi:10.1016/j.jmst.2019.03.003
• 11)   T.  Morita,  Y.  Oka,  S.  Tsutsumi,  S.  Takesue,  N.  Higuchi and  H.  Sakai: Mater. Trans. 63 (2022) 854–863. doi:10.2320/matertrans.MT-M2022018
• 12)   S.-H.  Sun,  K.  Hagihara and  T.  Nakano: Mater. Des. 140 (2018) 307–316. doi:10.1016/j.matdes.2017.11.060
• 13)   T.  Ishimoto,  S.  Wu,  Y.  Ito,  S.-H.  Sun,  H.  Amano and  T.  Nakano: ISIJ Int. 60 (2020) 1758–1764. doi:10.2355/isijinternational.ISIJINT-2019-744
• 14)   T.  Todo,  T.  Ishimoto,  O.  Gokcekaya,  J.  Oh and  T.  Nakano: Scr. Mater. 206 (2022) 114252. doi:10.1016/j.scriptamat.2021.114252
• 15)   B.  Rankouhi,  K.M.  Bertsch,  G.  Meric de Bellefon,  M.  Thevamaran,  D.J.  Thoma and  K.  Suresh: Mater. Sci. Eng. A 776 (2020) 139050. doi:10.1016/j.msea.2020.139050
• 16)   D.J.  Rowenhorst,  L.  Nguyen,  A.D.  Murphy-Leonard and  R.W.  Fonda: Curr. Opin. Solid State Mater. Sci. 24 (2020) 100819. doi:10.1016/j.cossms.2020.100819
• 17)   A.T.  Polonsky,  C.A.  Lang,  K.G.  Kvilekval,  M.I.  Latypov,  M.P.  Echlin,  B.S.  Manjunath and  T.M.  Pollock: Integr. Mater. Manuf. Innov. 8 (2019) 37–51. doi:10.1007/s40192-019-00126-7
• 18)   A.T.  Polonsky,  N.  Raghavan,  M.P.  Echlin,  M.M.  Kirka,  R.R.  Dehoff and  T.M.  Pollock: Mater. Charact. 190 (2022) 112043. doi:10.1016/j.matchar.2022.112043
• 19)   M.P.  Echlin,  T.L.  Burnett,  A.T.  Polonsky,  T.M.  Pollock and  P.J.  Withers: Curr. Opin. Solid State Mater. Sci. 24 (2020) 100817. doi:10.1016/j.cossms.2020.100817
• 20)   J.H.K.  Tan,  S.L.  Sing and  W.Y.  Yeong: Virtual Phys. Prototyp. 15 (2020) 87–105. doi:10.1080/17452759.2019.1677345
• 21)   C.  Körner,  M.  Markl and  J.A.  Koepf: Metall. Mater. Trans. A 51 (2020) 4970–4983. doi:10.1007/s11661-020-05946-3
• 22)   M.M.  Francois et al.: Curr. Opin. Solid State Mater. Sci. 21 (2017) 198–206. doi:10.1016/j.cossms.2016.12.001
• 23)   T.  Takaki: ISIJ Int. 54 (2014) 437–444. doi:10.2355/isijinternational.54.437
• 24)   M.  Ohno: ISIJ Int. 60 (2020) 2745–2754. doi:10.2355/isijinternational.ISIJINT-2020-174
• 25)   F.  Yu,  Y.  Wei and  X.  Liu: Int. J. Heat Mass Transfer 142 (2019) 118450. doi:10.1016/j.ijheatmasstransfer.2019.118450
• 26)   K.  Karayagiz,  L.  Johnson,  R.  Seede,  V.  Attari,  B.  Zhang,  X.  Huang,  S.  Ghosh,  T.  Duong,  I.  Karaman,  A.  Elwany and  R.  Arróyave: Acta Mater. 185 (2020) 320–339. doi:10.1016/j.actamat.2019.11.057
• 27)   T.  Keller,  G.  Lindwall,  S.  Ghosh,  L.  Ma,  B.M.  Lane,  F.  Zhang,  U.R.  Kattner,  E.A.  Lass,  J.C.  Heigel,  Y.  Idell,  M.E.  Williams,  A.J.  Allen,  J.E.  Guyer and  L.E.  Levine: Acta Mater. 139 (2017) 244–253. doi:10.1016/j.actamat.2017.05.003
• 28)   S.M.  Elahi,  R.  Tavakoli,  I.  Romero and  D.  Tourret: Comput. Mater. Sci. 216 (2023) 111882. doi:10.1016/j.commatsci.2022.111882
• 29)   R.  Acharya,  J.A.  Sharon and  A.  Staroselsky: Acta Mater. 124 (2017) 360–371. doi:10.1016/j.actamat.2016.11.018
• 30)   L.  Wu and  J.  Zhang: JOM 70 (2018) 2392–2399. doi:10.1007/s11837-018-3057-z
• 31)   C.  Tang and  H.  Du: JOM 74 (2022) 2996–3009. doi:10.1007/s11837-022-05310-3
• 32)   S.M.  Elahi,  R.  Tavakoli,  A.K.  Boukellal,  T.  Isensee,  I.  Romero and  D.  Tourret: Comput. Mater. Sci. 209 (2022) 111383. doi:10.1016/j.commatsci.2022.111383
• 33)   X.  Lingda,  Z.  Guoli,  M.  Gaoyang,  W.  Chunming and  J.  Ping: J. Alloy. Compd. 858 (2021) 157669. doi:10.1016/j.jallcom.2020.157669
• 34)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: Acta Mater. 118 (2016) 230–243. doi:10.1016/j.actamat.2016.07.049
• 35)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: ISIJ Int. 56 (2016) 1427–1435. doi:10.2355/isijinternational.ISIJINT-2016-156
• 36)   T.  Takaki,  M.  Ohno,  Y.  Shibuta,  S.  Sakane,  T.  Shimokawabe and  T.  Aoki: J. Cryst. Growth 442 (2016) 14–24. doi:10.1016/j.jcrysgro.2016.01.036
• 37)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Aoki and  C.-A.  Gandin: Materialia 1 (2018) 104–113. doi:10.1016/j.mtla.2018.05.002
• 38)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta and  T.  Aoki: Comput. Mater. Sci. 203 (2022) 111143. doi:10.1016/j.commatsci.2021.111143
• 39)   D.  Tourret,  Y.  Song,  A.J.  Clarke and  A.  Karma: Acta Mater. 122 (2017) 220–235. doi:10.1016/j.actamat.2016.09.055
• 40)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta and  T.  Aoki: Comput. Mater. Sci. 171 (2020) 109209. doi:10.1016/j.commatsci.2019.109209
• 41)   T.  Takaki,  S.  Sakane,  M.  Ohno and  Y.  Shibuta: Model. Simul. Mater. Sci. Eng. 27 (2019) 054001. doi:10.1088/1361-651X/ab1a17
• 42)   P.A.  Hooper: Addit. Manuf. 22 (2018) 548–559. doi:10.1016/j.addma.2018.05.032
• 43)   C.A.  Gandin and  M.  Rappaz: Acta Metall. Mater. 42 (1994) 2233–2246. doi:10.1016/0956-7151(94)90302-6
• 44)   C.A.  Gandin and  M.  Rappaz: Acta Mater. 45 (1997) 2187–2195. doi:10.1016/S1359-6454(96)00303-5
• 45)   K.  Teferra and  D.J.  Rowenhorst: Acta Mater. 213 (2021) 116930. doi:10.1016/j.actamat.2021.116930
• 46)   O.  Zinovieva,  A.  Zinoviev,  V.  Romanova and  R.  Balokhonov: Addit. Manuf. 36 (2020) 101521. doi:10.1016/j.addma.2020.101521
• 47)   M.S.  Mohebbi and  V.  Ploshikhin: Addit. Manuf. 36 (2020) 101726. doi:10.1016/j.addma.2020.101726
• 48)   Y.  Lian,  Z.  Gan,  C.  Yu,  D.  Kats,  W.K.  Liu and  G.J.  Wagner: Mater. Des. 169 (2019) 107672. doi:10.1016/j.matdes.2019.107672
• 49)   Z.  Yang,  H.  Fang,  K.  Jin,  J.  He,  W.  Ge and  W.  Yan: Int. J. Heat Mass Transfer 189 (2022) 122735. doi:10.1016/j.ijheatmasstransfer.2022.122735
• 50)   F.  Xu,  F.  Xiong,  M.-J.  Li and  Y.  Lian: Materials 15 (2022) 6800. doi:10.3390/ma15196800
• 51)   F.  Xiong,  Z.  Gan,  J.  Chen and  Y.  Lian: J. Mater. Process. Technol. 303 (2022) 117538. doi:10.1016/j.jmatprotec.2022.117538
• 52)   T.  Xue,  Z.  Gan,  S.  Liao and  J.  Cao: npj Comput. Mater. 8 (2022) 201. doi:10.1038/s41524-022-00890-9
• 53)   A.F.  Chadwick and  P.W.  Voorhees: Int. J. Numer. Methods Eng. 123 (2022) 3093–3110. doi:10.1002/nme.6966
• 54)   M.  Yang,  L.  Wang and  W.  Yan: npj Comput. Mater. 7 (2021) 56. doi:10.1038/s41524-021-00524-6
• 55)   A.F.  Chadwick and  P.W.  Voorhees: Acta Mater. 211 (2021) 116862. doi:10.1016/j.actamat.2021.116862
• 56)   Y.  Yang,  P.  Kühn,  M.  Yi,  H.  Egger and  B.-X.  Xu: JOM 72 (2020) 1719–1733. doi:10.1007/s11837-019-03982-y
• 57)   M.  Ohno,  Y.  Shibuta and  T.  Takaki: Mater. Trans. 60 (2019) 170–179. doi:10.2320/matertrans.ME201711
• 58)   D.  Rosenthal: Trans. ASME 68 (1946) 849–865. doi:10.1115/1.4018624
• 59)   Y.  Wang,  C.  Yu,  L.  Xing,  K.  Li,  J.  Chen,  W.  Liu,  J.  Ma and  Z.  Shen: J. Mater. Process. Technol. 281 (2020) 116591. doi:10.1016/j.jmatprotec.2020.116591
• 60)   A.A.  Antonysamy,  J.  Meyer and  P.B.  Prangnell: Mater. Charact. 84 (2013) 153–168. doi:10.1016/j.matchar.2013.07.012
• 61)   Y.  Balit,  E.  Charkaluk and  A.  Constantinescu: Addit. Manuf. 31 (2020) 100862. doi:10.1016/j.addma.2019.100862
• 62)   Y.  Shibuta,  M.  Ohno and  T.  Takaki: Adv. Theory Simul. 1 (2018) 1800065. doi:10.1002/adts.201800065
• 63)   Y.  Shibuta,  M.  Ohno and  T.  Takaki: JOM 67 (2015) 1793–1804. doi:10.1007/s11837-015-1452-2
• 64)   I.  Steinbach and  F.  Pezzolla: Phys. D 134 (1999) 385–393. doi:10.1016/S0167-2789(99)00129-3
• 65)   T.  Takaki,  T.  Hirouchi,  Y.  Hisakuni,  A.  Yamanaka and  Y.  Tomita: Mater. Trans. 49 (2008) 2559–2565. doi:10.2320/matertrans.MB200805
• 66)   E.  Miyoshi,  T.  Takaki,  M.  Ohno and  Y.  Shibuta: ISIJ Int. 60 (2020) 160–167. doi:10.2355/isijinternational.ISIJINT-2019-305
• 67)   E.  Miyoshi,  T.  Takaki,  M.  Ohno,  Y.  Shibuta,  S.  Sakane,  T.  Shimokawabe and  T.  Aoki: npj Comput. Mater. 3 (2017) 25. doi:10.1038/s41524-017-0029-8
• 68)   E.  Miyoshi,  T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta and  T.  Aoki: Comput. Mater. Sci. 186 (2021) 109992. doi:10.1016/j.commatsci.2020.109992
• 69)   S.G.  Kim,  D.I.  Kim,  W.T.  Kim and  Y.B.  Park: Phys. Rev. E 74 (2006) 061605. doi:10.1103/PhysRevE.74.061605
• 70)   Y.  Suwa,  Y.  Saito and  H.  Onodera: Acta Mater. 55 (2007) 6881–6894. doi:10.1016/j.actamat.2007.08.045
• 71)   M.  Ohno,  S.  Tsuchiya and  K.  Matsuura: Acta Mater. 59 (2011) 5700–5709. doi:10.1016/j.actamat.2011.05.045
• 72)   T.  Takaki,  Y.  Hisakuni,  T.  Hirouchi,  A.  Yamanaka and  Y.  Tomita: Comput. Mater. Sci. 45 (2009) 881–888. doi:10.1016/j.commatsci.2008.12.009
• 73)   M.  Ohno,  T.  Takaki and  Y.  Shibuta: J. Comput. Phys. 335 (2017) 621–636. doi:10.1016/j.jcp.2017.01.053
• 74)  C.S. Kim: United States, (1975) p. Medium: ED; Size: Pages: 28.
• 75)   M.  Yang,  L.  Wang and  W.  Yan: Addit. Manuf. 47 (2021) 102286. doi:10.1016/j.addma.2021.102286
• 76)   O.  Andreau,  I.  Koutiri,  P.  Peyre,  J.-D.  Penot,  N.  Saintier,  E.  Pessard,  T.  De Terris,  C.  Dupuy and  T.  Baudin: J. Mater. Process. Technol. 264 (2019) 21–31. doi:10.1016/j.jmatprotec.2018.08.049
• 77)   A.K.  Agrawal,  B.  Rankouhi and  D.J.  Thoma: Curr. Opin. Solid State Mater. Sci. 26 (2022) 101024. doi:10.1016/j.cossms.2022.101024
• 78)   H.  Jia,  H.  Sun,  H.  Wang,  Y.  Wu and  H.  Wang: Int. J. Adv. Manuf. Technol. 113 (2021) 2413–2435. doi:10.1007/s00170-021-06810-3
• 79)   L.L.  Parimi,  R.G.  A,  D.  Clark and  M.M.  Attallah: Mater. Charact. 89 (2014) 102–111. doi:10.1016/j.matchar.2013.12.012
• 80)   M.J.  Heiden,  L.A.  Deibler,  J.M.  Rodelas,  J.R.  Koepke,  D.J.  Tung,  D.J.  Saiz and  B.H.  Jared: Addit. Manuf. 25 (2019) 84–103. doi:10.1016/j.addma.2018.10.019
• 81)   A.J.  Birnbaum,  J.C.  Steuben,  E.J.  Barrick,  A.P.  Iliopoulos and  J.G.  Michopoulos: Addit. Manuf. 29 (2019) 100784. doi:10.1016/j.addma.2019.100784
• 82)   J.  Shao,  G.  Yu,  S.  Li,  X.  He,  C.  Tian and  B.  Dong: J. Alloy. Compd. 898 (2022) 162976. doi:10.1016/j.jallcom.2021.162976
• 83)   A.K.  Agrawal,  G.  Meric de Bellefon and  D.  Thoma: Mater. Sci. Eng. A 793 (2020) 139841. doi:10.1016/j.msea.2020.139841
• 84)   M.  Wei,  W.J.  Ding,  G.  Vastola and  Y.-W.  Zhang: Addit. Manuf. 54 (2022) 102779. doi:10.1016/j.addma.2022.102779
• 85)   K.  Guo,  L.  Qiao,  Z.  Huang,  N.  Anwer and  Y.  Cao: Int. J. Adv. Manuf. Technol. 121 (2022) 1651–1673. doi:10.1007/s00170-022-09366-y
• 86)   M.  Rappaz and  P.H.  Thévoz: Acta Metall. 35 (1987) 2929–2933. doi:10.1016/0001-6160(87)90292-6
• 87)   A.S.  Sabau,  L.  Yuan,  J.-L.  Fattebert and  J.A.  Turner: Comput. Phys. Commun. 284 (2023) 108605. doi:10.1016/j.cpc.2022.108605
• 88)   N.  Yamanaka,  S.  Sakane and  T.  Takaki: Comput. Mater. Sci. 197 (2021) 110658. doi:10.1016/j.commatsci.2021.110658
• 89)   M.J.  Heiden,  S.C.  Jensen,  J.R.  Koepke,  D.J.  Saiz,  S.M.  Dickens and  B.H.  Jared: Materialia 21 (2022) 101356. doi:10.1016/j.mtla.2022.101356