Abstract

Laser powder bed fusion (L-PBF) parts often require post-processing prior to use in engineering applications to improve mechanical properties and modify the as-built surface topography. The ability to tune the L-PBF process parameters to obtain designer as-built surface topography could reduce the need for post-processing. However, the relationship between the as-built surface topography and the L-PBF process parameters is currently not well-understood. In this paper, we derive data-driven models from surface topography data and L-PBF process parameters using machine learning (ML) algorithms. The prediction accuracy of the data-driven models derived from ML algorithms exceeds that of the multivariate regression benchmark because the latter does not always capture the complex relationship between the as-built surface topography parameters and the corresponding L-PBF process parameters in a single best-fit equation. Data-driven models based on decision tree (interpretable) and artificial neural network (non-interpretable) algorithms display the highest prediction accuracy. We also show experimental evidence that thermocapillary convection and melt track overlap are important drivers of the formation of as-built surface topography.

1 Introduction

Laser powder bed fusion (L-PBF) is an additive manufacturing (AM) process in which a laser selectively melts and fuses metal powder particles in a layer-by-layer fashion to create macroscale parts with free-form geometry [1,2], e.g., to manufacture complex parts for biomedical [3] and aerospace [4] applications. In recent years, research and development of the L-PBF process has expanded the portfolio of available metal powders [5], increased process speed, and improved print quality by tuning the process parameters to increase resolution, reduce porosity, or reduce the as-built surface roughness [6].

Despite these advancements, L-PBF parts still require post-processing before use in engineering applications to reduce residual stress [7], improve mechanical properties [8], and modify the as-built surface topography [9]. Common post-processing methods include [10]; heat treatment to reduce porosity [1113], chemical cleaning to remove unmelted metal powder particles from the as-built surfaces [14,15], and mechanical processing to physically modify the surface topography [16,17]. Even with these post-processing methods, tailoring the surface topography remains difficult. Hence, the ability to tune the L-PBF process parameters to obtain designer as-built surface topography is of interest, because it would reduce the need for costly post-processing. However, the relationship between the as-built surface topography and the L-PBF process parameters is currently not well-understood. Thus, we attempt to address this problem by deriving data-driven models from as-built surface topography data and the corresponding L-PBF process parameters using machine learning (ML) algorithms. This knowledge is important to contribute to transitioning L-PBF from a prototyping to a production manufacturing process.

Others have tried to predict the as-built surface topography from the L-PBF process parameters using trial-error, multivariate regression, or ML algorithms. Trial-error methodologies phenomenologically show the effect of the L-PBF process parameter on the surface topography without the use of formal statistical analysis. For instance, Eidt et al. [18] determined that increasing bulk laser power decreases the areal surface roughness (Sa, Sq, Sv, and Sp) of as-built Inconel 718 surfaces because the melt pool size increases with increasing bulk laser power, which smoothens the surface topography. Similarly, Triantaphyllou et al. [19] distinguished up-facing from down-facing as-built Ti–6Al–4V surfaces because the surface topography shows a predominance of valleys and peaks, respectively, which alters the skewness parameter Ssk.

Multivariate regression relates surface topography and L-PBF process parameters by means of a best-fit equation. Several publications documented the effect of L-PBF process parameters on the average surface roughness (Ra or Sa) of as-built surfaces and showed that Ra increases with increasing contour laser power [16], increasing hatch spacing (Calignano et al. [16]), and increasing build orientation [20] for up-facing surfaces. Alternatively, for down-facing surfaces, Ra increases with decreasing contour laser power [21] and decreasing build orientation [20]. Few publications also studied the effect of L-PBF process parameters on other R- and S- parameters. For instance, Fox et al. [22] measured the effect of build orientation on the surface peak count (Rpc), mean element profile width (Rsm), and mean element profile height (Rc) of as-built down-facing surfaces. They concluded that Rpc decreases and Rsm and Rc increase with increasing build orientation as a result of an increasing number of unmelted metal powder particles that adhere to the as-built surface. Whip et al. [23] observed that Sa, Sv, Smr2, and Svk decrease with increasing contour laser power and decreasing scan speed because of increasing melt track overlap. In our research group, Detwiler et al. [24] used multivariate regression to show that R- and S-parameters inaccurately characterize the surface topography of as-built L-PBF surfaces compared to deterministic surface topography parameters (summit density, standard deviation of summit heights, and mean summit radius).

Recently, researchers have also related the as-built surface topography to the L-PBF process parameters using data-driven models derived from ML algorithms. For instance, Khorasani et al. [25] derived a data-driven model using an artificial neural network (ANN) to predict the average surface roughness (Sa) of Ti–6Al–4V as-built surfaces as a function of L-PBF contour laser power, hatch spacing, scan speed, and pattern angle. They determined that Sa increases with increasing contour laser power and decreasing scan speed. Cao et al. [26] derived a data-driven model using a whale optimization algorithm (WOA) that enables minimizing the average surface roughness (Sa) of 316L steel as-built surfaces by tuning laser power, scan speed, and layer thickness. They determined that layer thickness shows the greatest effect on Sa. Similarly, Özel et al. [27] related the as-built surface topography parameters (Sa, Sq, Ssk, and Sku) of nickel 625 alloy to the energy density, contour laser power, and scan speed, using genetic programming and an ANN, and determined that models based on genetic programming show a higher prediction accuracy than those derived from an ANN.

Trial-error methods are time-consuming and do not necessarily lead to an optimal solution. Multivariate regression methods do not always capture the potentially complex relationship between surface topography parameters and L-PBF process parameters in a single best-fit equation that covers the entire range of the process parameters [28]. Additionally, the relationship and interaction between parameters can be challenging to visualize and interpret [29]. Publications that use ML algorithms to derive data-driven models that relate surface topography and process parameters only discuss ANN and genetic programming algorithms, without considering interpretable ML algorithms [25,27]. Furthermore, those publications focus solely on R- and S-parameters to describe the surface topography, even though it is well-known that those parameters do not uniquely describe the surface topography [30]. In contrast, deterministic surface topography parameters are based on actual surface features rather than averages or statistics and, thus, represent the surface topography more accurately than traditional R- and S-parameters [31,32].

Hence, the objective of this paper is to compare the prediction accuracy of data-driven models derived from as-built surface topography and corresponding L-PBF process parameters, using different interpretable and non-interpretable ML algorithms. We determine the deterministic surface topography parameters of as-built Inconel 718 surfaces, manufactured with different L-PBF process parameters, and we compare the prediction accuracy of different data-driven models with a multivariate regression benchmark. Additionally, using the results of the data-driven models, we explain the physical mechanisms that drive the experimental observations. The data-driven models are useful to tune L-PBF process parameters and manufacture as-built surfaces with designer surface topography, reducing the need for post-processing of as-built L-PBF parts.

2 Methods and Materials

2.1 Specimen Fabrication.

We use a 3D systems ProX SMP 320 L-PBF device and recycled Inconel 718 powder (ASTM F1877, mean particle diameter of 39.98 µm [33]) to manufacture the specimens considered in this study (ASTM E466-07, nominal geometry 101.60 mm × 19.05 mm × 3.10 mm [34]). Figure 1 schematically illustrates the location and orientation (α = 0 deg, α = 60 deg, and α = 90 deg) of the specimens on the build plate, with respect to the recoater and gas flow directions. The S1 and S2, and S3 and S4 as-built surfaces correspond to the inlet and outlet of the recoater and gas flow directions, respectively. Inert gas flow prevents oxidation and clears emissions from the process chamber to maintain a free path between the laser and the powder bed [35]. S5 is the top surface of the specimens, parallel to the build plate.

Fig. 1
Schematic of the location and orientation of the specimens on the build plate: α = 0 deg, α = 60 deg, and α = 90 deg. The recoater travels in the x-direction (S1–S2), the inert gas flows in the y-direction (S3–S4), and the specimen is built layer-upon-layer in the z-direction (S5).
Fig. 1
Schematic of the location and orientation of the specimens on the build plate: α = 0 deg, α = 60 deg, and α = 90 deg. The recoater travels in the x-direction (S1–S2), the inert gas flows in the y-direction (S3–S4), and the specimen is built layer-upon-layer in the z-direction (S5).
Close modal

Each specimen is manufactured with different L-PBF process parameters: bulk laser power (115 W ≤ P ≤ 465 W), laser scan speed (620 mm/s ≤ v ≤ 1770 mm/s), layer thickness (t = 30 µm or t = 60 µm), and build orientation (0 deg ≤ α ≤ 90 deg). The contour laser power varies depending on the layer thickness (P = 115 W for t = 30 µm and P = 165 W for t = 60 µm) and contour scan speed (v = 625 mm/s), hatch spacing (h = 100 µm), and laser spot size (50 µm) remain constant for all specimens. The parameter ranges are chosen to maintain the laser energy density Eρ = P/vht within the recommended range for Inconel 718 (30 J/mm3Eρ ≤ 90 J/mm3). Table 1 lists the L-PBF process parameters for each of the 24 specimens used in this work. We note that these 24 specimens are a subset of 75 specimens used in previous publications related to fatigue testing [36,37] and also for multivariate regression analysis of the surface topography [24]. The as-built surfaces remained unaltered during earlier experiments.

Table 1

L-PBF process parameters of the 24 specimens considered in this work

Specimen numberContour laser power Pc (W)Bulk laser power Pb (W)Bulk laser scan speed vb (mm/s)Layer thickness t (μm)Build orientation α (deg)
111522011803060
211522011803060
31153301770300
41151156203090
51151156203090
61151681475300
71151681475300
81152751200300
91151159153060
1011533014753060
1111516811803090
121152008003090
1311527517703060
1416531510506060
15165165850600
1616539010506060
1716546514006090
1816524012506060
1916539014506090
2011522011803090
211653151050600
2216531510506090
2316531510506090
2416520010006090
Specimen numberContour laser power Pc (W)Bulk laser power Pb (W)Bulk laser scan speed vb (mm/s)Layer thickness t (μm)Build orientation α (deg)
111522011803060
211522011803060
31153301770300
41151156203090
51151156203090
61151681475300
71151681475300
81152751200300
91151159153060
1011533014753060
1111516811803090
121152008003090
1311527517703060
1416531510506060
15165165850600
1616539010506060
1716546514006090
1816524012506060
1916539014506090
2011522011803090
211653151050600
2216531510506090
2316531510506090
2416520010006090

2.2 Surface Topography Measurement and Post-processing.

Optical profilometry, confocal laser scanning microscopy (CLSM), and scanning electron microscopy have been used to measure the surface topography of L-PBF specimens [9,38]. We selected CLSM (Olympus LEXT OLS5000) with a 1.8 mm × 1.8 mm field-of-view, 0.625 µm lateral (20× optical zoom), and 0.006 µm vertical resolution, based on the size of surface topography features, resolution, and accuracy [9,38], and after performing a convergence analysis.

Figure 2(a) schematically shows a specimen, relative to the original fatigue specimen, and Fig. 2(b) identifies the measurement locations on the as-built surfaces (squares with numbers). We measure the surface topography in three different locations on each of the three (specimen 3, 6, 7, 8, 15, 21) or four (all other specimens) as-built surfaces of each specimen; i.e., we perform 9 or 12 measurements per specimen, or (6 × 3 × 3) + (18 × 4 × 3) = 270 measurements for all 24 specimens. The fracture and sectioned surfaces of the specimen are not as-built and, thus, are excluded from the analysis. Furthermore, the six specimens that are oriented under 0 deg with respect to the build plate (specimen 3, 6, 7, 8, 15, and 21) only have three as-built surfaces because the bottom surface is oriented toward the build plate and consists of support material. Figures 2(c) and 2(d) show typical as-built surface topography measurements z = f(x,y) of specimen 2. We post-process the surface topography data by removing tilt and by digitally high-pass filtering the roughness from the entire surface topography following ISO 25178-2 [39].

Fig. 2
(a) Location of the specimen with respect to the original fatigue specimen, (b) surface topography measurement locations on the four as-built surfaces, and (c) and (d) typical as-built surface topography measurements (specimen 2)
Fig. 2
(a) Location of the specimen with respect to the original fatigue specimen, (b) surface topography measurement locations on the four as-built surfaces, and (c) and (d) typical as-built surface topography measurements (specimen 2)
Close modal

2.3 Deterministic Surface Topography Parameters.

We quantify the deterministic surface topography parameters from each post-processed surface topography measurement. Deterministic surface topography parameters are based on the entire surface instead of one or multiple traces for R-, S-parameters [39] and statistical methods [40]. A nine-point peak (9PP) identification scheme [31,32] first identifies the N summits of the surface topography as the local maxima. Then, we calculate the standard deviation of summit heights σs from the N summit heights, and the summit density η = N/An, with An the surface topography measurement area (1.8 mm × 1.8 mm). The curvature of each summit i in two orthogonal directions x and y is κx,i = d2z/dx2 and κy,i = d2z/dy2, and the radius of curvature ρi of that summit is the inverse of the average of its κx and κy, i.e., ρi = −[(κx,i + κy,i)/2]−1. The mean summit radius Rs is the arithmetic mean of all individual summit radii. Finally, β = ηRsσs is a dimensionless roughness parameter.

2.4 Data-Driven Models, Machine Learning Algorithms, and Prediction Error Metrics.

We derive data-driven models that relate the deterministic surface topography parameters to the L-PBF process parameters (forward model) and vice versa (inverse model). Different physical phenomena dominate the surface topography of each as-built surface of a specimen, depending on its orientation with respect to the inert gas flow (S3 to S4) and the recoater motion (S1–S2). Thus, we derive data-driven models using different ML algorithms for each as-built surface (S1–S5) and each deterministic surface topography parameter (η, Rs, σs).

We consider three interpretable ML algorithms, including k-nearest neighbors (KNN), decision trees, and random forests, and two non-interpretable ML algorithms, including support vector machine (SVM) and artificial neural networks (ANN) [41], and we use the scikit-learn toolbox in python to implement these algorithms. We compare the prediction error of the data-driven models derived using these different ML algorithms to that of the multivariate regression model, previously described in reference [24], using the coefficient of determination (R2), root-mean-square error (RMSE), and mean absolute error (MAE) [29]. The coefficient of determination R2 is given as
R2=1i=1n(yiy^i)2i=1n(yiy¯)2
(1)
where yi and ŷi are the actual and predicted values of the ith data point, and y¯ is the arithmetic mean of n data points. RMSE is given as
RMSE=i=1n(yiy^i)2n
(2)
and MAE is defined as
MAE=i=1n|yiy^i|n
(3)

We use the hold-out method to determine the prediction error of each model and randomly split the data set 80/20 in training and validation data, which is commonly used by others [41]. Furthermore, we used a group shuffle to avoid separating the three repeat measurements of any as-built surface from each other. We note that the data sets for each as-built surface vary between 18 (S5) and 72 (S3 and S4) data points. Hence, because we have small data sets we use each surface topography measurement as an individual datapoint, rather than averaging the three repeat measurements per as-built surface, which would render the dataset even smaller.

However, we previously concluded based on a one-way ANOVA and a Tukey honestly significant difference (HSD) test [24] that only the deterministic surface topography parameters of S5 showed statistically significant differences with those of S1–S4. Therefore, when explaining the physical phenomena that drive the formation of the as-built surface topography (Sec. 3.2), we consider the surface topography parameters of S1–S4 and those of S5 as two separate data sets.

3 Results and Discussion

3.1 Prediction Accuracy of Data-Driven Models Compared to the Multivariate Regression Benchmark.

Table 2 shows an overview of the prediction error metrics (R2, RMSE, and MAE, as defined in Eqs. (1)(3)), for each deterministic surface topography parameter (η, Rs, and σs), each as-built surface (S1–S5), and for each of the data-driven model and the multivariate regression benchmark. We highlight the ML algorithm that creates the data-driven model with the highest R2 and the lowest RMSE/MAE in gray for each deterministic surface topography parameter and as-built surface.

Table 2

Prediction error (R2, RMSE, and MAE) of data-driven models derived from ML algorithms and multivariate regression for each as-built surface (S1–S5)

Surface topography parameterPrediction error metrick-Nearest neighborsDecision treesRandom forestSupport vector machineArtificial neural networkMultivariate regression
Summit density, ηS1R20.5820.6210.6190.6350.5310.444
RMSE0.0110.0100.0110.0100.0120.013
MAE0.0090.0080.0080.0080.0090.010
S2R20.7840.8130.8030.8120.6540.623
RMSE0.0120.0110.0110.0110.0140.014
MAE0.0090.0080.0080.0080.0110.011
S3R20.7100.7410.6960.7180.6020.390
RMSE0.0110.0110.0110.0110.0130.016
MAE0.0090.0090.0090.0090.0110.013
S4R20.5960.6330.6190.6110.5420.532
RMSE0.0120.0110.0110.0110.0120.012
MAE0.0090.0080.0090.0090.0100.010
S5R20.7680.6970.6930.6620.6980.750
RMSE0.0190.0190.0190.0200.0190.019
MAE0.0150.0160.0160.0170.0160.016
Mean summit radius, RsS1R20.8220.7860.7790.7720.7180.635
RMSE0.0050.0050.0050.0050.0060.007
MAE0.0040.0040.0040.0040.0050.006
S2R20.9300.9520.9220.9450.7210.760
RMSE0.0050.0040.0050.0050.0110.011
MAE0.0040.0030.0040.0040.0080.008
S3R20.7520.7480.7190.7340.6440.340
RMSE0.0070.0070.0080.0080.0090.012
MAE0.0060.0060.0060.0060.0070.010
S4R20.6100.6710.6530.6110.4770.401
RMSE0.0080.0070.0070.0080.0090.009
MAE0.0060.0050.0060.0060.0070.007
S5R20.4980.4770.4920.4710.4720.430
RMSE0.0090.0100.0100.0100.0100.011
MAE0.0080.0080.0080.0080.0080.009
Standard deviation of summit heights, σsS1R20.8430.8200.8210.8080.8250.671
RMSE0.7040.7320.7200.7390.7151.124
MAE0.5650.5870.5710.6210.5670.943
S2R20.9110.9400.9170.6760.8960.825
RMSE0.7300.5830.7091.3910.7311.190
MAE0.5680.4650.5631.0380.5690.861
S3R20.7500.7530.7320.6680.7350.514
RMSE1.2681.2631.2981.3921.2971.714
MAE0.9830.9681.0231.1231.0111.411
S4R20.6670.6990.6820.5890.6950.568
RMSE1.1071.0241.0401.1941.0311.316
MAE0.8100.7420.7820.9300.7441.048
S5R20.7190.7250.7350.7000.7250.738
RMSE2.3522.3462.2962.3732.3462.111
MAE2.0122.0102.0002.0152.0101.799
Surface topography parameterPrediction error metrick-Nearest neighborsDecision treesRandom forestSupport vector machineArtificial neural networkMultivariate regression
Summit density, ηS1R20.5820.6210.6190.6350.5310.444
RMSE0.0110.0100.0110.0100.0120.013
MAE0.0090.0080.0080.0080.0090.010
S2R20.7840.8130.8030.8120.6540.623
RMSE0.0120.0110.0110.0110.0140.014
MAE0.0090.0080.0080.0080.0110.011
S3R20.7100.7410.6960.7180.6020.390
RMSE0.0110.0110.0110.0110.0130.016
MAE0.0090.0090.0090.0090.0110.013
S4R20.5960.6330.6190.6110.5420.532
RMSE0.0120.0110.0110.0110.0120.012
MAE0.0090.0080.0090.0090.0100.010
S5R20.7680.6970.6930.6620.6980.750
RMSE0.0190.0190.0190.0200.0190.019
MAE0.0150.0160.0160.0170.0160.016
Mean summit radius, RsS1R20.8220.7860.7790.7720.7180.635
RMSE0.0050.0050.0050.0050.0060.007
MAE0.0040.0040.0040.0040.0050.006
S2R20.9300.9520.9220.9450.7210.760
RMSE0.0050.0040.0050.0050.0110.011
MAE0.0040.0030.0040.0040.0080.008
S3R20.7520.7480.7190.7340.6440.340
RMSE0.0070.0070.0080.0080.0090.012
MAE0.0060.0060.0060.0060.0070.010
S4R20.6100.6710.6530.6110.4770.401
RMSE0.0080.0070.0070.0080.0090.009
MAE0.0060.0050.0060.0060.0070.007
S5R20.4980.4770.4920.4710.4720.430
RMSE0.0090.0100.0100.0100.0100.011
MAE0.0080.0080.0080.0080.0080.009
Standard deviation of summit heights, σsS1R20.8430.8200.8210.8080.8250.671
RMSE0.7040.7320.7200.7390.7151.124
MAE0.5650.5870.5710.6210.5670.943
S2R20.9110.9400.9170.6760.8960.825
RMSE0.7300.5830.7091.3910.7311.190
MAE0.5680.4650.5631.0380.5690.861
S3R20.7500.7530.7320.6680.7350.514
RMSE1.2681.2631.2981.3921.2971.714
MAE0.9830.9681.0231.1231.0111.411
S4R20.6670.6990.6820.5890.6950.568
RMSE1.1071.0241.0401.1941.0311.316
MAE0.8100.7420.7820.9300.7441.048
S5R20.7190.7250.7350.7000.7250.738
RMSE2.3522.3462.2962.3732.3462.111
MAE2.0122.0102.0002.0152.0101.799

Note: Models with the highest R2 and lowest RMSE and MAE are highlighted in gray.

We note that we maximize the prediction accuracy of the data-driven models by optimizing the parameters of the respective ML algorithms. Hence, we determine that the decision tree algorithm has unrestricted tree depth with a minimum of two deterministic surface topography measurements for every decision node and at least one deterministic surface topography measurement for every leaf node. The random forest algorithm averages six decision trees with the same parameters as the individual decision tree. The k-nearest neighbors algorithm estimates the deterministic surface topography parameters from the k = 3 nearest neighbors (using the Euclidean distance), with all neighbors weighted equally. The support vector machine uses a radial basis function kernel [42], and the ANN uses the limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) solver with twenty neurons [43]. Table 3 summarizes the deterministic surface topography parameters used to derive the data-driven models, and lists the average value of each parameter (over three replicate measurements) for each specimen and each as-built surface (S1–S5).

From Table 2, we observe that the prediction accuracy of the data-driven models derived from ML algorithms exceeds that of the multivariate regression benchmark for all data sets except the standard deviation of summit heights of S5. On average, the R2 metric of data-driven models exceeds that of the corresponding multivariate benchmark by 23.7%. This is likely because multivariate regression analysis requires the best-fit equation (e.g., linear, polynomial, or logarithmic) to apply to the entire parameter range and solution domain. In contrast, ML algorithms compartmentalize the parameter range and solution domain to determine the best fit in each compartment of the solution domain, thus affording more flexibility than multivariate regression analysis to relate the surface topography parameters to the L-PBF process parameters.

The prediction accuracy of models derived from interpretable ML algorithms exceeds those derived from non-interpretable algorithms in all cases except one (η of S1), which could be the result of the simplicity of the dataset (i.e., a linear fit resulted in the highest R2 with multivariate regression) [41]. The data-driven models derived from the decision tree ML algorithm show the highest prediction accuracy of all interpretable ML algorithms, with an R2 metric that on average exceeds the multivariate regression benchmark by 28.3%. Decision tree ML algorithms are more tolerant to missing and irrelevant values than other algorithms, including KNN. In addition, random forest algorithms sometimes over-segregate a small data set [44]. Interpretable ML algorithms are also beneficial because they allow a user to comprehend how the data-driven model arrives at its prediction. The model derived from the ANN algorithm shows the highest prediction accuracy of all non-interpretable ML algorithms, with an R2 metric that on average exceeds the multivariate regression benchmark by 15.0%. ANN algorithms typically outperform SVM algorithms when handling multiple input parameters [44].

We show typical results obtained from the data-driven models to illustrate the results of Table 2. We define the nondimensional power Pnd as the ratio of contour and bulk laser power and nondimensional scan speed vnd as the ratio of the contour and bulk laser scan speed. Figure 3 depicts the summit density η as a function of the nondimensional laser scan speed vnd and the nondimensional laser power Pnd for S1–S4 based on data-driven models derived using (a) the ANN algorithm (R2 = 0.534) and (b) the decision tree algorithm (R2 = 0.587). Figure 3 illustrates the results of the data-driven models. Figure 3(a) shows continuous results, whereas Fig. 3(b) depicts discrete results that correspond to the different leaves of the decision tree. However, we qualitatively observe similar trends in the results from both models.

Fig. 3
Summit density η as a function of the nondimensional laser scan speed vnd and the nondimensional laser power Pnd derived from (a) the ANN algorithm and (b) the decision tree algorithm for S1–S4
Fig. 3
Summit density η as a function of the nondimensional laser scan speed vnd and the nondimensional laser power Pnd derived from (a) the ANN algorithm and (b) the decision tree algorithm for S1–S4
Close modal

Figure 4 illustrates a decision tree for summit density η of S1–S4. The algorithm is trained on 80% of the data (252 data points) and predicts the summit density η based on Pnd and vnd. We prune the decision tree to only show the first nine decision nodes (white) to reduce figure size and complexity. Each decision node shows the number of data points n at that node, and each end node or leaf (gray) shows the predicted summit density η for that tree branch and the number of data points at that leaf. The decision tree illustrates the hierarchy of decision-making to predict the summit density η, based on the L-PBF process parameters. In this example, we observe that the decision nodes with the highest hierarchy involve vnd rather than Pnd, indicating that vnd has a more substantial effect on η than Pnd.

Fig. 4
Typical example of a decision tree, pruned to the first nine nodes for clarity and showing the decision nodes that lead to predicting the summit density η of S1–S4. Each decision node (white) shows the decision and the number of data points n at that node. Each end node or leaf (gray) shows the summit density η and the number of data points n used to train that prediction.
Fig. 4
Typical example of a decision tree, pruned to the first nine nodes for clarity and showing the decision nodes that lead to predicting the summit density η of S1–S4. Each decision node (white) shows the decision and the number of data points n at that node. Each end node or leaf (gray) shows the summit density η and the number of data points n used to train that prediction.
Close modal

3.2 Physical Phenomena Determining the As-Built Surface Topography.

The surface topography measurements and the results of the data-driven models show that the as-built surface topography depends on the nondimensional laser scan speed vnd and the nondimensional laser power Pnd, which determine the melt pool geometry (width and depth) and the temperature gradient between the melt pool and the previously melted tracks [45]. Specifically, we explain the relationship between the as-built surface topography and the melt pool geometry and temperature gradient through thermocapillary convection and melt track overlap [45] based on results from the data-driven models (see, e.g., Fig. 3), and we provide experimental evidence in support of this discussion in Fig. 6.

Figure 5(a) schematically shows a cross-sectional view of a single laser track, illustrating the temperature gradient between the hot melt pool and the cold solidified melt track. The surface tension of the molten material decreases with increasing temperature [46], and thus, the temperature gradient within the melt pool creates a surface tension gradient. As a result, the liquid melt pool flows from low to high surface tension, i.e., toward previously melted and solidified material, and it forms a smooth surface deviation or the so-called globule. This phenomenon is often referred to as thermocapillary convection [47], and its importance increases with increasing melt pool temperature or with increasing melt pool size, which occurs with increasing Pnd or decreasing vnd. Globules decrease the summit density η and increase the standard deviation of summit heights σs, which we schematically illustrate in Fig. 5(a).

Fig. 5
(a) Cross-sectional view of a single laser track illustrating the effect of thermocapillary convection on the surface topography of an L-PBF as-built surface, indicating the temperature difference between the hot melt pool and the cold solidified melt track, (b) top view and cross-sectional view illustrating insufficient melt pool overlap, showing unmelted metal powder particles between adjacent melt tracks, and (c) top view and cross-sectional view of remelting of previously melted tracks
Fig. 5
(a) Cross-sectional view of a single laser track illustrating the effect of thermocapillary convection on the surface topography of an L-PBF as-built surface, indicating the temperature difference between the hot melt pool and the cold solidified melt track, (b) top view and cross-sectional view illustrating insufficient melt pool overlap, showing unmelted metal powder particles between adjacent melt tracks, and (c) top view and cross-sectional view of remelting of previously melted tracks
Close modal

Figure 5(b) schematically shows a top view and cross-sectional view of a laser melt track, illustrating the effect of insufficient melt pool overlap on the as-built surface topography. When the melt pool size does not fully cover the previously melted track, insufficient overlap exists between adjacent melt tracks, which may break up the melt track (so-called balling driven by minimizing the surface energy of the melt pool), create spatter, and leave trenches with unmelted metal powder particles between adjacent melt tracks. Decreasing melt track overlap increases the standard deviation of summit heights σs as a result of trenches with unmelted metal powder particles between adjacent melt tracks. In addition, it decreases the summit density η because the unmelted metal powder particle density increases with decreasing melt track overlap and each unmelted metal powder particles covers multiple summits on the as-built surface, as the radius of an unmelted metal powder particle (mean radius, 39.98 µm) is much larger than that of the mean summit radius (on the order of 0.1–1.0 µm, see, e.g., Fig. 6). Furthermore, the mean summit radius Rs decreases because the melt pool temperature decreases with decreasing melt track overlap, thus increasing surface tension and increasing roughness [46]. Even though the number of unmelted metal powder particles increases with decreasing melt track overlap, its effect on the mean summit radius Rs is negligible as the number of unmelted metal powder particles is several orders of magnitude smaller than the number of summits.

Fig. 6
(a) Optical microscopy images and (b) confocal microscopy images of the as-built S1 surface of specimen 14 (Pnd = 0.367 and vnd = 0.369), specimen 24 (Pnd = 0.767 and vnd = 0.415), specimen 14 (Pnd = 0.367 and vnd = 0.369), and specimen 1 (Pnd = 0.366 and vnd = 0.269), from top left to bottom right, with globules highlighted in red. (c) and (e) Optical microscopy and confocal microscopy images of the as-built S1 surface of specimen 23 (Pnd = 0.367 and vnd = 0.369) on the left and specimen 12 (Pnd = 0.435 and vnd = 0.654) on the right with unmelted metal powder particles highlighted in red and (d) a trace of the surface topography from specimen 23 (top) and specimen 12 (bottom) between unmelted metal powder particles. (f) and (h) Optical microscopy images and confocal microscopy images of the as-built S1 surface of specimen 18 (Pnd = 0.143 and vnd = 0.143) on the left and specimen 11 (Pnd = 0.581 and vnd = 0.269) on the right with unmelted metal powder particles highlighted in red and (d) a trace of the surface topography from specimen 18 (top) and specimen 11 (bottom).
Fig. 6
(a) Optical microscopy images and (b) confocal microscopy images of the as-built S1 surface of specimen 14 (Pnd = 0.367 and vnd = 0.369), specimen 24 (Pnd = 0.767 and vnd = 0.415), specimen 14 (Pnd = 0.367 and vnd = 0.369), and specimen 1 (Pnd = 0.366 and vnd = 0.269), from top left to bottom right, with globules highlighted in red. (c) and (e) Optical microscopy and confocal microscopy images of the as-built S1 surface of specimen 23 (Pnd = 0.367 and vnd = 0.369) on the left and specimen 12 (Pnd = 0.435 and vnd = 0.654) on the right with unmelted metal powder particles highlighted in red and (d) a trace of the surface topography from specimen 23 (top) and specimen 12 (bottom) between unmelted metal powder particles. (f) and (h) Optical microscopy images and confocal microscopy images of the as-built S1 surface of specimen 18 (Pnd = 0.143 and vnd = 0.143) on the left and specimen 11 (Pnd = 0.581 and vnd = 0.269) on the right with unmelted metal powder particles highlighted in red and (d) a trace of the surface topography from specimen 18 (top) and specimen 11 (bottom).
Close modal

Figure 5(c) schematically shows a top view and cross-sectional view of multiple laser tracks, illustrating the effect of melt track overlap on the as-built surface topography. Increasing overlap increases remelting of adjacent melt tracks, thus decreasing the depth of trenches between them, melting unmelted metal powder particles and, thus, “smoothening” the as-built surface. As a result, the mean summit radius Rs increases and the standard deviation of summit heights σs decreases with increasing melt track overlap. Furthermore, the summit density η increases because the unmelted metal powder particle density decreases with increasing melt track overlap and each unmelted metal powder particle covers multiple summits on the as-built surface.

Figure 6 shows the two parameters (thermocapillary effect and overlap between adjacent melt tracks) and the three deterministic surface topography parameters (η, Rs, and σs) in matrix format. We use confocal and optical microscopy images and surface topography measurements to substantiate the effects we schematically illustrate in Fig. 5. In Fig. 6, the confocal and optical microscopy images, as well as the surface traces, represent indicative subsets of the entire surface, whereas the surface topography parameters reflect the entire surface topography measurement. Figure 6(a) shows optical microscopy and Fig. 6(b) shows confocal microscopy images of as-built surface S1 of specimen 14 (Pnd = 0.367 and vnd = 0.369) compared to specimen 24 (Pnd = 0.767 and vnd = 0.415); i.e., we increase the laser power but maintain a constant laser scan speed, and specimen 14 (Pnd = 0.367 and vnd = 0.369) compared to specimen 1 (Pnd = 0.366 and vnd = 0.269); i.e., we maintain constant laser power but decrease laser scan speed. In both comparisons, we increase the temperature gradient between the melt pool and previously solidified material, thus increasing the formation of globules (highlighted in red) due to thermocapillary convection. We observe that the summit density η decreases as a result of the presence of globules. Similarly, we observe that the standard deviation of summit heights σs increases with an increasing number of globules.

Figure 6(c) shows optical microscopy and Fig. 6(e) shows confocal microscopy images of as-built surface S1 of specimen 23 (Pnd = 0.367 and vnd = 0.369) and specimen 12 (Pnd = 0.435 and vnd = 0.654); i.e., we increase the laser scan speed, which reduces the size of the melt pool and, correspondingly, reduces the overlap between adjacent melt tracks. We identify and highlight unmelted metal powder particles in red (specimen 23: 69.1 unmelted metal powder particles/mm2 and specimen 12: 251.9 unmelted metal powder particles/mm2), which primarily reside in the trenches between adjacent melt tracks. We observe that summit density η decreases with decreasing melt track overlap because the increasing number of unmelted metal powder particles cover multiple summits. Furthermore, the presence of unmelted metal powder particles and trenches between adjacent tracks increases the “spikiness” of the surface (i.e., the mean summit radius Rs decreases and the standard deviation of summit heights σs increases), as illustrated in the traces shown in Fig. 6(d) and the standard deviation of summit heights σs shown in Fig. 6(e). Figure 6(f) shows optical microscopy and Fig. 6(h) shows confocal microscopy images of as-built surface S1 of specimen 18 (Pnd = 0.143 and vnd = 0.143) and specimen 11 (Pnd = 0.581 and vnd = 0.269); i.e., we increase the laser power, which increases the size of the melt pool and increases overlap between adjacent melt tracks, thus increasing remelting and, correspondingly, reducing the number of unmelted metal powder particles. We highlight unmelted metal powder particles in red (specimen 18: 177.8 unmelted metal powder particles/mm2 and specimen 11: 90.1 unmelted metal powder particles/mm2). We observe that the summit density η increases and the mean summit radius Rs increases with increasing melt track overlap. Figure 6(h) shows confocal microscopy images of both surfaces and illustrates that the standard deviation of summit heights σs decreases with increasing melt track overlap because the size of the trenches between adjacent melt tracks, and correspondingly, the number of unmelted metal powder particles decreases.

4 Conclusions

We have derived data-driven models that relate deterministic surface topography parameters to L-PBF process parameters using five supervised ML algorithms, and we have compared the prediction accuracy of the data-driven models to a multivariate regression benchmark model. We conclude that:

  1. All supervised ML algorithms considered in this work derive data-driven models with prediction accuracy that exceeds that of the multivariate regression model, using the R2, RMSE, and MAE metrics, because a multivariate regression model does not always capture the complex relationship between the deterministic surface topography parameters and the L-PBF process parameters in a single best-fit equation.

  2. The data-driven models derived from decision tree (interpretable) and ANN (non-interpretable) algorithms provide the highest prediction accuracy.

  3. We show experimental evidence that thermocapillary convection and overlap between adjacent melt tracks are the main drivers of the deterministic surface topography parameters, i.e., the summit density η, the mean summit radius Rs, and the standard deviation of summit heights σs.

Acknowledgment

We gratefully acknowledge support from the Department of Defense, Office of Local Defense Community Cooperation, under award no. ST1605-21-04.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Appendix

Table 3 shows the deterministic surface topography parameters (η, Rs, and σs) for each specimen and as-built surface S1–S5. The value of each deterministic surface topography parameter is the average of the three replicates of each as-built surface (surface topography measurement locations are shown in Fig. 2).

Table 3

Average of the deterministic surface topography parameters for each specimen and as-built surface (S1–S5)

Deterministic surface topography parameters (η/Rs/σs)
SpecimenS1S2S3S4S5
10.142/0.100/6.6120.125/0.078/9.2880.150/0.096/6.2290.151/0.099/6.207
20.135/0.109/6.1980.114/0.086/8.8640.122/0.098/7.3290.122/0.097/7.227
30.110/0.084/10.370.113/0.090/9.5800.128/0.083/13.05
40.109/0.080/11.160.120/0.097/8.2550.118/0.087/9.7820.133/0.101/6.987
50.104/0.075/11.700.120/0.100/7.5150.131/0.097/7.0780.123/0.086/8.726
60.112/0.078/11.430.116/0.088/10.050.142/0.079/14.716
70.113/0.068/12.8350.129/0.079/10.630.160/0.082/13.29
80.113/0.077/12.0790.126/0.089/9.9610.150/0.098/8.482
90.144/0.106/6.7500.134/0.062/11.740.148/0.076/9.0480.145/0.071/10.17
100.157/0.091/7.6750.127/0.069/10.750.146/0.074/9.2340.125/0.076/9.938
110.158/0.113/5.5850.116/0.076/10.600.150/0.104/5.9080.135/0.090/7.927
120.128/0.077/10.030.135/0.094/7.7150.122/0.081/10.040.149/0.104/6.262
130.139/0.097/7.5610.119/0.075/10.880.127/0.072/11.460.138/0.084/9.354
140.177/0.109/5.0170.121/0.087/8.6130.142/0.103/6.8380.141/0.101/6.686
150.145/0.106/6.9080.134/0.095/8.3950.198/0.087/8.813
160.167/0.116/5.0970.141/0.098/6.7600.184/0.108/5.3900.164/0.106/6.177
170.138/0.098/7.1190.166/0.119/4.9870.165/0.106/6.2660.158/0.105/6.119
180.137/0.093/7.8700.148/0.095/7.0030.159/0.105/6.2310.151/0.102/6.283
190.144/0.103/6.3260.168/0.118/5.0930.146/0.109/6.1440.149/0.105/6.278
200.139/0.095/7.2130.164/0.116/5.1330.164/0.110/5.4880.157/0.109/5.604
210.159/0.113/5.6760.160/0.116/5.3490.183/0.096/7.097
220.157/0.103/5.9930.171/0.118/4.9080.164/0.107/5.7660.171/0.117/4.855
230.143/0.100/6.5490.161/0.115/5.2980.170/0.099/6.3990.157/0.097/6.953
240.155/0.107/5.6770.168/0.118/4.8790.156/0.092/7.4800.179/0.098/6.361
Deterministic surface topography parameters (η/Rs/σs)
SpecimenS1S2S3S4S5
10.142/0.100/6.6120.125/0.078/9.2880.150/0.096/6.2290.151/0.099/6.207
20.135/0.109/6.1980.114/0.086/8.8640.122/0.098/7.3290.122/0.097/7.227
30.110/0.084/10.370.113/0.090/9.5800.128/0.083/13.05
40.109/0.080/11.160.120/0.097/8.2550.118/0.087/9.7820.133/0.101/6.987
50.104/0.075/11.700.120/0.100/7.5150.131/0.097/7.0780.123/0.086/8.726
60.112/0.078/11.430.116/0.088/10.050.142/0.079/14.716
70.113/0.068/12.8350.129/0.079/10.630.160/0.082/13.29
80.113/0.077/12.0790.126/0.089/9.9610.150/0.098/8.482
90.144/0.106/6.7500.134/0.062/11.740.148/0.076/9.0480.145/0.071/10.17
100.157/0.091/7.6750.127/0.069/10.750.146/0.074/9.2340.125/0.076/9.938
110.158/0.113/5.5850.116/0.076/10.600.150/0.104/5.9080.135/0.090/7.927
120.128/0.077/10.030.135/0.094/7.7150.122/0.081/10.040.149/0.104/6.262
130.139/0.097/7.5610.119/0.075/10.880.127/0.072/11.460.138/0.084/9.354
140.177/0.109/5.0170.121/0.087/8.6130.142/0.103/6.8380.141/0.101/6.686
150.145/0.106/6.9080.134/0.095/8.3950.198/0.087/8.813
160.167/0.116/5.0970.141/0.098/6.7600.184/0.108/5.3900.164/0.106/6.177
170.138/0.098/7.1190.166/0.119/4.9870.165/0.106/6.2660.158/0.105/6.119
180.137/0.093/7.8700.148/0.095/7.0030.159/0.105/6.2310.151/0.102/6.283
190.144/0.103/6.3260.168/0.118/5.0930.146/0.109/6.1440.149/0.105/6.278
200.139/0.095/7.2130.164/0.116/5.1330.164/0.110/5.4880.157/0.109/5.604
210.159/0.113/5.6760.160/0.116/5.3490.183/0.096/7.097
220.157/0.103/5.9930.171/0.118/4.9080.164/0.107/5.7660.171/0.117/4.855
230.143/0.100/6.5490.161/0.115/5.2980.170/0.099/6.3990.157/0.097/6.953
240.155/0.107/5.6770.168/0.118/4.8790.156/0.092/7.4800.179/0.098/6.361

Note: “–“ indicates that an as-built surface did not exist for a specific specimen.

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