The demand for lightweight and high-performance materials with intricate geometries has fueled the growth of additive manufacturing (AM) technologies [1][2][3][4]. Among the various AM processes, laser powder bed fusion (LPBF) has emerged as a leading method for the production of metallic components with exceptional mechanical properties and design flexibility [3,[5][6][7]. In LPBF, a thin layer of the feedstock powder is first placed on a substrate and is selectively melted by a laser beam at locations specified by a computer-aided design (CAD) model. The substrate is then lowered by one layer thickness and the processes of powder deposition and melting are repeated until the desired part is fabricated [3,4]. Due to the highly localized melting and strong temperature gradients and cooling rates, LPBF can provide non-namics of the melt pool including selective evaporation of alloying elements, turbulence, and convective motions (e.g., Marangoni flows), but also the cooling rates from the molten state and repeated thermal cycling as adjacent sections of the part are built [6,15,[17][18][19][20]. These phenomena are all controlled by dozens of process parameters, the most important of which include laser power, scan speed, shape and size of the laser spot, layer thickness, hatch spacing, and printing strategy [21][22][23].
Fully elucidating the complex material-specific relationships between process parameters, microstructural evolution and mechanical properties in LPBF, with a level of accuracy that enables optimization of process parameters, remains a formidable challenge [24]. At the same time, the number of process parameters is too large to optimize them via brute-force experimentation. Consequently, predictive modeling of process-property relations in LPBF has traditionally relied on domain knowledge and trial-and-error methods [25].
To date, the prevailing approach to process parameter optimization has been to distill a small number of physical quantities that embed the most critical parameters, and experimentally scan them to identify the optima. Volumetric Energy Density (VED) is one of the most popular of such features, which is defined as the laser power divided by the product of scan speed, hatch spacing (i.e. the distance between adjacent scan lines), and layer thickness [12]. Non-dimensional versions of VED have been introduced as an attempt to make this quantity materialindependent [26,27]. VED has been correlated with "print quality" for multiple materials: low values of VED generally result in lack-of-fusion (LOF) porosity, whereas high values can cause keyhole porosity. Hence, optimizing the process parameters for printed parts generally involves fabricating small samples over a range of VED values, characterizing their porosity to identify the optimal VED range for printing, and finally adopting combinations of parameters that result in this optimal VED. While this approach has been successfully demonstrated for multiple materials, two key challenges remain: (1) while high values of porosity are certainly deleterious to mechanical properties, other microstructural features mentioned above may play an equally significant role; (2) while VED has an appealing physical interpretation, there is no guarantee that it fully characterizes the effect of all process parameters on the material structure and properties.
To address some of these challenges, high-throughput (HT) techniques have been developed, which involve creating large arrays of samples with variations in composition or process parameters, followed by testing and screening to identify the conditions that yield optimal properties [25]. Compared to traditional approaches, HT techniques offer faster experimentation with reduced systematic errors and enhanced data reliability [25]. In the context of AM and especially LPBF, some efforts with HT approaches have been made to correlate process parameters, microstructure, and properties of fabricated materials [21][22][23]28].
Research in this area has involved automated tensile property characterization [28][29][30], alloy design by using feedstock materials with varying chemical composition [31], high-rate part fabrication [24,32], and sample design and characterization to link process parameters with material properties [21,23,28,33]. Even in the HT context, exploring a broad processing space is quite time-consuming, and hence most efforts have primarily focused on varying only laser power and scan speed over relatively small ranges and few different conditions [28,33], thus lacking insights into the effect of other process parameters or large parameter variations. Additionally, existing maps for parameter selection are either non-predictive [27] or validated only for specific materials [28,33].
As experimental data collection techniques advance, approaches based on machine learning (ML) are increasingly used to build datadriven process-property relations [34,35]. However, these ML models have some major limitations which are primarily due to the fact that printing and microstructurally/mechanically characterizing even a few dozen of samples is not only very expensive and time-consuming, but also produces stochastic data [21,[28][29][30]. Therefore, existing ML models focus on a limited subset of processing conditions, primarily laser power, laser speed, and occasionally layer thickness, and only consider a fixed and narrow range of values for each parameter. While this simplification might make it easier for ML models to learn patterns in the data, it likely results in less robust models that lack the ability to generalize beyond the narrowly defined conditions, potentially limiting their reliability and applicability in broader and more complex LPBF scenarios [36][37][38][39][40][41][42][43][44][45]. Naturally, these limitations could be addressed by using large datasets -often exceeding 1, 000 data points [42]. While larger datasets improve model performance, they are costly and time-intensive to generate, which does not resolve the core issue of ML models' reliance on extensive and expensive data. Simply increasing the dataset size allows any model to perform well, but it does not offer a methodological advantage [42].
In this work, we develop a novel ML approach coupled with HT printing and characterization investigations to optimize a wide range of LPBF
process parameters (laser power , laser scan speed , hatch spacing ℎ, powder layer thickness , and scan rotation between layers ) to achieve maximum combinations of material yield strength ( ) and ductility ( ). The framework is depicted in Fig. 1. As fabrication and testing of multiple dog bone specimens required for direct measurements of and is extremely costly and time-consuming, we propose a twostep experimental process to generate suitable training data for ML:
We print a large set of small cuboids spanning the entire processing parameter space and rapidly characterize their surface properties (here chosen as hardness and porosity). (2) We print a relatively small number of tensile dog bone specimens over a sub-set of the parameter space and test them via uniaxial loading to directly extract their and .
Similarly, we use a two-step ML approach based on Gaussian processes (GPs) [46,47] to learn the complex correlations between process parameters and and : first, two GPs are built to relate process parameters to hardness and porosity; subsequently, two more GPs are trained that leverage the first two GPs as well as the additional tensile data. Finally, an optimization scheme is devised to identify the process parameters that maximize combinations of strength and ductility.
The rationale behind our approach is that the information embedded in easy-to-measure surface properties (hardness maps and porosity) is correlated with the properties of interest (strength and ductility). While such correlations are not surprising, their functional form is unknown to us and can only be quantified in a data-driven manner via ML models. In our case, the complexity of these correlations is very high because our two datasets based on surface and tensile measurements are (1) affected by LPBF process parameters in different ways, (2) based on samples whose shapes and sizes significantly differ (cuboid vs tensile coupons), and (3) unbalanced since we have far more data from cuboids which are easy to manufacture and test.
While the proposed approach is material agnostic, here we demonstrate it with 17-4 PH stainless steel (SS), a precipitation-hardened alloy with diverse industrial applications requiring high strength and corrosion resistance [6,9,20]. While in the conventional wrought form this steel is fully martensitic, multiple studies clearly indicate that the microstructure of LPBF-processed 17-4 PH SS is very complex, often consisting of combinations of martensite, ferrite, and occasionally residual austenite [6,9,16,20,[48][49][50][51]. The microstructure and mechanical properties of LPBF processed 17-4 PH SS are highly sensitive to the process parameters [6,9,15,20,49], making it a great candidate to demonstrate the power of the proposed approach.
Nitrogen atomized 17-4 PH SS powder with a particle size range of 15 -50 μm (Carpenter Additive, USA) was used as the feedstock material. Printing was carried out using an SLM Solutions 125HL printer, featuring a Yb-fiber laser with a maximum output of 400 W and a beam diameter of 80 μm. The build chamber operated in a 99.99% N 2 atmosphere where the build plate was preheated to 200 °C, and a "Stripe" scan strategy was used. Fill contour and border scans were not used to obtain a uniform microstructure across thin wall samples. These machine settings were used for the manufacturing of both cuboids and tensile coupons. A design of experiments (DOE) methodology based on the Sobol sequence was employed to generate 270 LPBF process parameter combinations, with laser power ( ) varying in the range 80 -400 W, laser scan speed ( ) in the range 150 -1500 mm/s, powder layer thickness ( ) in the range 20 -75 μm, hatch spacing (ℎ) in the range 70 -120 μm, and two possible scan rotation ( ) values of 67 or 90 degrees. The 270 combinations produced VED values ranging approximately from 10 to 1000 J/mm 3 and are presented in Table S1. These process settings were used to create 270 cube-shaped 2×2×2 mm samples which were numbered as 1, ⋯ , 270 based on their corresponding process parameter combination.
To enable HT manufacturing and surface characterization of cuboids, a unique island-based setup was designed. This setup allows for the manufacturing of samples with several varying layer thicknesses on one build plate, while facilitating sample removal via electrodischarge machining (EDM) and subsequent polishing and characterization of multiple samples concurrently. 15 islands of size 14 × 12 × 3 mm, each containing 9 cuboids with different processing conditions, were positioned in the chamber and numbered as shown in Fig. 2(a). This process was repeated a second time on a separate build plate to obtain islands 16 through 30. The layer thickness values in the 270 combinations were projected to 10 unique levels, i.e., ∈ {20, 26, 30, 38, 44, 50, 57, 60, 69, 75} μm, where the first 5 values were used for islands 1 through 15 and the rest for islands 16 through 30.
The island-based process is schematically illustrated in Fig. 2(a), where each color represents a specific layer thickness. First, the cuboids in gray were printed at a layer thickness of 20 μm where printing was concluded once an overall height of 2 mm was attained. Then, as shown via the yellow cuboids, printing was continued with a layer thickness of 26 μm to add another 2 mm of material to rows 2 through 5. This process was repeated for the three remaining layer thickness values. Hence, the cuboids shared the same layer thickness value if they were on islands that were positioned in the same row in Fig. 2(a), but the other four process parameters changed across all cuboids, see Table S1 for process parameters used for cuboids on each island. The cuboids on the second build plate were printed in a similar manner, except that the layer thickness values started from 50 μm and continued to 75 μm. It is noted that using this approach even higher numbers of layer thicknesses can be printed on the same build plate, based on the manufacturing needs and part size limits.
Tensile specimens were printed in the vertical direction, meaning the build direction was parallel to the tensile loading direction. A subset of 270 process parameter combinations, specifically the 54 combinations that correspond to layer thickness values of 30 and 60 μm, were used for manufacturing of the tensile specimens (see the highlighted rows in Table S1). These two layer thickness values were selected because they fall in the range of the layer thickness values for which cuboids were built. For each 54 process parameter combination, three replicas were printed to assess the variability of tensile properties under the same process parameters. The dimensions of the tensile specimens and images of the printed tensile coupons are provided in Fig. 2(b).
LPBF printed samples were removed from the build plate via wire electro-discharge-machining (EDM). All cuboids and tensile coupons were heat treated upon their removal from the build plate to increase their hardness and yield strength [52]. The samples were directly aged (with no prior solutionization step) at 482 °C for 1 h in a Nabertherm B400 furnace in an ambient atmosphere with a heating rate of 10 °C min -1 , and then air quenched.
For microstructural characterization and hardness testing, islands of cuboids were embedded in epoxy/resin mounts, ground, and polished via standard procedures for stainless steels down to 1 μm with diamond polishing suspensions (MetaDi, Buehler). Finally, samples were chemically-mechanically polished with 0.05 μm Alumina suspension (MasterPrep Alumina, Buehler). Etching was done using Waterless Kalling's, also known as Kalling's No.2 Reagent (ES Laboratory, LLC) to distinguish between phases in 17-4 PH SS. Samples were submerged in the etchant for approximately 24 s, immediately rinsed, sonicated in water for 1 min, and air dried. Microstructural analyses of 270 cuboids were performed using an Olympus DSX10-UZH Digital Optical Microscope. Polished and etched surfaces of cuboids were imaged for analysis of the defects (pores and cracks) and microstructure phases.
A robust porosity measurement approach was developed to extract the porosity content of each sample based on the OM images obtained from the as-polished surfaces. To refine the images, mitigate noise ef- fects, and enhance clarity and quality in this process, preprocessing techniques such as blurring and cropping were first used. Then, based on the distribution of the pixel values shown in Fig. A.2, a threshold
of 75 was selected to distinguish pores, i.e., pixels whose brightness is below 75 were classified as pores. Subsequently, porosity was computed as the ratio of pore area to the total image area. While multiple techniques exist to properly and accurately measure volumetric porosity (e.g., Archimedes technique or CT scanning), they are time intensive and do not align with the HT approach proposed in this manuscript.
Here, we only measure and report area fraction of pores, and use it as a proxy for volumetric porosity. While area and volumetric porosity are likely different in absolute values, their trends with processing parameters are expected to be the same. More details on image processing and porosity calculations are provided in Appendix A.
Vickers microindentation hardness mapping was selected as the rapid HT mechanical property characterization. Instrumented indentation is well-suited for HT testing because it can quickly measure location-specific mechanical responses, has automation capability, and only requires a flat polished surface for testing [23]. The Vickers microindentation hardness test utilizes a calibrated machine to apply a square-based pyramidal-shaped diamond indenter with face angles of 136° to the material's surface [53]. Test forces range from 1 to 1000 gf (9.8 × 10 -3 to 9.8 N) and the resulting indentation impressions (diagonals) are measured using a light microscope upon the load removal. Then, the Vickers hardness (kgf/mm 2 ) is determined as = 1.8544 ∕ 2 [53] where is force (kgf) and is the mean diagonal length of the indentations (mm). Vickers hardness measurements were obtained via a Buehler Wilson VH3300 automated indenter for the 270 cuboids. Each cuboid underwent 36 measurements spaced 280 μm apart, using a 0.5 kgf load and a 10 s hold time for each indent. The median of these 36 measurements was recorded as the hardness value for each sample. Median was used rather than the mean to reduce the effect of outliers.
Tensile tests were performed on an Instron 5985 load frame equipped with a 250 kN load cell. Tensile specimens were tested with their surfaces in the as-printed condition following the heat treatment, without any surface machining or polishing prior to the test. Each specimen was marked with two white circular fiducial marks setting the gauge length limits for strain tracking. An AVE2663 -901 video extensometer with a Fujinon HF16HA-1S lens was used to track the strain of the gauge section. Tests were conducted according to ASTM E8 standards at a quasi-static strain rate of 0.001 s -1 . The obtained stress-strain curves were assessed to extract the 0.2% offset yield strength ( ), strain to failure ( ), and ultimate tensile strength ( ). These three parameters were extracted using a built-in software in the Instron 5985.
To emphasize the impact of this HT approach to fabrication and testing, we estimate that the entire set of prints employed approximately 6 kg of material and took approximately 14 hr. For reference, if we had printed exclusively tensile dog-bones with the same number of different process parameters (270), with three repetitions per condition, we would have needed to print 810 samples, on at least 10 platform. We estimate that this would have required approximately 22 kg of material, and a print time of about 48 hrs. Hence, our approach resulted in a 3.5 reduction in both material cost and print time (a very conservative estimate, as we are not factoring in the EDM time as well as LPBF preparation time, which scales linearly with number of platforms), as well as a 5 reduction in tensile testing time.
We propose an ML framework to leverage auxiliary features obtained from hardness maps and OM images of cuboids towards the overarching represent the median experimental and predicted values for property ∈ {H, EP, , }, respectively. The categorical feature is added for data fusion. 1 and 2
indicate the number of auxiliary and tensile data points, respectively. We also note that and are equivalent to and .
goal of predicting the mechanical properties of tensile coupons. As illustrated in Fig. 3, our framework has a hierarchical nature, where we first learn to predict the auxiliary features as functions of process parameters and then use their predicted values as additional inputs in the GPs that estimate tensile properties. Our ML framework is designed based on two critical assumptions: (i) the auxiliary features and tensile properties are naturally related (since they are material characteristics), and depend on the same set of process parameters; (ii) hardness and porosity are measured within an HT scheme which leads to far more samples on the auxiliary features than on and which are obtained via tensile tests. Hence, we do not learn the dependence of these two sets of properties on process parameters with a single ML model because such a model would have to be trained on the combined dataset, which is highly imbalanced. Such a dataset would cause the ML model to primarily focus on hardness and porosity, while our ultimate goal is to predict tensile properties as functions of process parameters. To mitigate this issue while leveraging the relation between the two sets of properties, we build predictive models for the auxiliary features first, and subsequently use them in ML models that predict and .
The relation between the auxiliary features and tensile properties is hidden, i.e., we rely on data to model this relation as there is no analytic physics-based formula that can relate hardness and porosity of cuboids to tensile properties of dog-bone specimens in LPBF. The complexity of this hidden relation is especially high in our case due to the sizeeffect: while the auxiliary features are obtained from surface of small cuboid samples, and are based on tensile tests performed on much larger coupons. To unveil these complex hidden relations, we use GPs in our framework because they can distill highly complex input-output relations even from small datasets.
Below, we first review some technical details on GPs in Section 2.3.1 and then elaborate on how they are used in our hierarchical learning framework in Section 2.3.2. Our framework is implemented via the open-source Python package GP+ [46].
GPs are probabilistic models that assume the training data follows a multivariate normal distribution. They are defined by parametric mean and covariance functions whose parameters can be systematically optimized via maximum likelihood estimation (MLE) or maximum a posteriori (MAP). Once the parameters are estimated, closed-form conditional distribution formulas are used for probabilistic prediction [54][55][56][57][58]. GPs are particularly suited for our application as they (1) can naturally handle noise, (2) do not rely on big data, and (3) can efficiently learn complex input-output relations [59][60][61].
In this work, we design the mean and covariance functions of the GPs to seamlessly (1) handle the categorical variable in the process parameter space, and (2) enable data fusion or multi-fidelity (MF) modeling, which refers to the process of jointly learning from multiple datasets that share some mutual information. MF modeling is essential in this work as we aim to combine hardness/porosity and tensile data together to leverage their connection and, in turn, reduce the reliance on expensive tensile data.
As detailed in Appendix B, traditional GPs cannot handle categorical variables directly, as they are not naturally endowed with a distance metric. To address this limitation, GP+ first uses the user-defined fixed function ( ) to transform the categorical variable into the quantitative representation . To reduce the dimensionality of while capturing its effects on the response, is then passed through the parametric embedding function ℎ ( ; ℎ ) with parameters ℎ . Since the outputs of ℎ ( ; ℎ ) are low-dimensional and quantitative, they can be easily integrated with the mean and covariance functions of the GP. Upon this integration, all model parameters are estimated via MAP.
GP+ enables MF modeling by simply augmenting the input space with an additional categorical feature which merely indicates the source of a data point, see Fig. 4 where
, } is the added two- level categorical variable, are the numerical inputs, and denotes the output. This approach provides the option of having a mean function ( , ; ) with parameters that is either dependent or independent of the data source (Fig. 4 shows the former case as affects the mean function through ). As detailed in Section 2.3.2, MF modeling is used to fuse the hardness/porosity and tensile datasets. To this end, the two-
space and is converted to and then via grouped one-hot encoding and matrix multiplication, respectively. Additionally, the mean function is modeled via a feed-forward fully-connected neural network (FFNN) and all model parameters are estimated via MAP.
As explained earlier, the GPs for the auxiliary features are built first, and then used by the GPs that predict tensile properties. To devise the hierarchy among the four variables, we calculate the Pearson correlation coefficient [62] between any pair of variables. Pearson coefficient provides values between -1 and 1, where the sign and magnitude indicate the direction (direct or inverse) and strength of the correlation, respectively. While these correlations are linear, they provide a good starting point for our nonlinear analyses. To account for any nonlinear correlation between the variables, we also calculate the Spearman cor-
To fuse the cuboid data with tensile data using GP+, we add a two-level categorical variable = { , } (source indicator) to the input space. We use grouped one-hot encoding and matrix multiplication to convert to its low-dimensional quanti- tative representation . Then, these mapped values ( ) are concatenated with the quantitative input features and fed into the mean and covariance functions. To capture more complex relations in the data, we use a FFNN as a mean function and all the model parameters are estimated via MAP. The final step involves learning ductility, which is more challenging due to its lower correlation with other properties and its significant inherent variability. To address these challenges, we augment the process parameters with all previously modeled properties (i.e., predicted hardness, predicted engineered porosity, and predicted tensile strength), and also fuse the two datasets obtained from tensile and cuboid samples. We denote the resulting model with which, similar to , benefits from data fusion and predicts a dummy hardness variable.
We note that' ' in the input space of and is a categorical variable that differentiates data types and enables data fusion as detailed in Appendix B. This allows and to predict two distinct values based on the value assigned to' '. For example, ′ ′ ∈ { ′ ′ , ′ ′ } in the case of which predicts ̂ for samples with' ′ = ′ ' and a dummy hardness for' ′ = ′ '. It is also noted that we design the mean function of each GP in Fig. 3 based on the dataset size and complexity.
These details are included in Section 3 where we also evaluate the ad-vantages of augmenting tensile models with auxiliary features and using data fusion.
To evaluate the accuracy of the GP models, 5-fold cross-validation (CV) is used by partitioning the data into 5 folds and then iterating over them. In each iteration, one fold serves as the validation set while the remaining 4 folds are used for training [64][65][66]. We measure the emulation accuracy in each iteration via mean squared error (MSE): For evaluation, we also report the coefficient of determination (R 2 ) which, unlike MSE, is calculated based on the entire data via:
where ̄ indicates the average of (the median of) property over the samples. An R 2 close to 1 indicates that the trained model can adequately explain the variability of the output with respect to the inputs. The obtained R 2 values by the GPs are R 2 = 0.93, R 2 = 0.98, R 2 = 0.94, and R 2 = 0.68.
We provide a detailed analysis of the properties of the cuboids and tensile coupons in Sections 3.1 and 3.2, respectively. As detailed in Section 2.3, we link these properties to process parameters via four GP emulators that are trained hierarchically. We illustrate the prediction accuracy of these GPs using the metrics defined in Section 2.4. These emulators are used in Section 3.3 to optimize the combination of yield strength and ductility of a tensile sample which is then built and tested to assess the effectiveness of our framework.
Throughout, we use the median of the properties to mitigate the effects of outliers. Specifically, hardness refers to the median value observed in the hardness map of a cuboid. For the tensile coupons, ,
, and (strain to failure/ ductility) refer to the respective median values observed in the three tensile test repetitions that are carried out for each process parameter combination. For brevity, we drop the word median when referring to these properties. We highlight that and represent the same properties as and , respectively.
Representative VED is widely used in the literature to correlate mechanical properties and defect content to the LPBF process parameters [6,28,67,68]. To assess the predictive power of VED in our case, we plot the porosity and hardness of the cuboids against their corresponding VEDs, see Fig. 6(c) and Fig. 6(d). As shown in Fig. 6(c), porosity initially decreases at a high rate as energy density increases and then it plateaus once the energy density exceeds roughly 100 J/mm 3 due to the removal of LOF porosity. Expectedly, we observe an opposite trend for hardness in Fig. 6(d),
where it increases and then nearly plateaus as VED increases. The initial rapid change in hardness is due to the increase in the structure's density and reduction in the LOF porosity. For equally dense structures in Fig. 6(d), we observe that hardness shows a roughly increasing trend as VED exceeds ∼ 70 J/mm 3 and then plateaus at ∼ 400 J/mm 3 VED.
We attribute this trend to the increased martensitic nature of the microstructure at higher VED [6]. Importantly, however, the increase in hardness with VED does not apply to all the processing conditions: the inset in Fig. 6(d) shows a broad set of data where a specific relationship between hardness and VED cannot be inferred.
As evidenced by Fig. 6(c) and Fig. 6(d), it is notable that samples with identical energy densities can display significant variations in porosity and hardness. This result clearly implies that VED cannot fully capture the impact of process parameters on hardness for 17-4PH steel. To further elaborate this point, we leverage curve fitting to regress the data via a wide range of analytic functions 3 and report the one with the highest R 2 in Fig. 6(c) and Fig. 6(d). Since the VED of the majority of the sam- ples is below 300 J/mm 3 , the fitted curves prioritize these regions and hence fail to provide accurate predictions at high VEDs, where much smaller property variations are observed. Hence, even the best fitted curves provide low R 2 values, supporting the conclusion that VED alone is insufficient to accurately predict porosity or hardness.
The wide variations of porosity and hardness, coupled with the insufficiency of VED in accurately linking them to the process parameters, motivate the use of ML models. As explained in Section 2.3.2, we first relate hardness to the process parameters via a GP model and then use this model while building another GP that predicts the porosity of cuboids as a function of process parameters and the predicted hardness. To assess features' importance and potentially reduce the input dimensionality, we calculate Sobol's sensitivity indices (SI) based on the GP model that predicts hardness. As detailed in Appendix D, these indices are based on variance analysis and quantify the importance of a variable on the output either solely on its own (main SI) or including its interactions via other variables (total SI). These SI indices are enumerated in Table 2, and indicate that the most important process parameters are laser power and speed, followed by layer thickness and hatch spacing. Scan rotation has a negligible effect on hardness, and hence can be excluded from the ensuing studies. As the difference between main and total SIs indicates the effect of variable interactions on the output, we deduce from the numbers in Table 2 that the four process parameters affect hardness in a non-trivial manner. For instance, we observe that the effect of hatch spacing (or layer thickness) on hardness increases by about five times as other process parameters vary. Additionally, since the main SIs are 3
We use polynomials, (⋅), (⋅), and combinations thereof. insets show the magnified image of a smaller region of energy density and property. not zero (especially in the case of laser power and speed), we can once again conclude that a single variable such as VED (which only consists of variable interactions) cannot explain the variability of the output as the inputs change. Upon excluding scan rotation from the inputs, we conduct 5-fold CV via GPs and report the corresponding MSEs in the top row of Table 3 (the CV plots are included in Fig. S4). By comparing the MSEs to ̂ 2 = 5 × 10 -3 , we observe that these values are quite close. This observation, in conjunction with the large R 2 = 0.93, indicates that GPs can effectively predict hardness as a function of process parameters.
Additionally, the consistent MSE values across different folds show the robustness of the trained model. We note that in two cases the reported MSEs are smaller than noise variance which is due to the fact that the latter is also "estimated". We obtain our final model for hardness prediction by fitting a GP to the entire 270 samples and henceforth denote it by .
As shown in Fig. 3, we use the estimated hardness values from to augment the input and output (i.e., the engineered porosity) spaces of the porosity data. The sensitivity indices in the bottom row of Table 2 illustrate similar trends as in hardness; laser power and speed are the We conduct 5-fold CV via GPs trained on the first four process parameters concatenated with the predicted hardness values from .
The calculated MSEs are presented in Table 3 and plots depicting its performance on different folds are available in Fig. S4. Similar to hardness, the close similarity between MSEs and ̂ 2 = 2 × 10 -4 , high R 2 = 0.98, and the consistency of MSEs among different folds illustrate the effectiveness and robustness of GPs to predict (and hence ductility). We denote the final model for predicting as and obtain it by fitting a GP to the entire data which is augmented with the predicted hardness values.
We build tensile specimens based on 54 process parameter combinations which are also used to manufacture cuboids 55 -81 and 190 -216 as enumerated in Table S1. We provide representative stressstrain curves in Fig. 7 and refer the reader to Fig. S5 for the complete set. In Fig. 7 each plot also contains the hardness map and OM image of the corresponding cuboid. We extract , , and from the 54 stress-strain curves, and present the values in Table S3. Given the strong correlation between and observed in Table 1, we only analyze . The histograms and plots of the variation of and against each other and VED is shown in Fig. 8, which indicates that these two properties (1) vary quite substantially across the different samples, and (2) are nonlinearly related, whereby first increases but then decreases as increases. As indicated by the whiskers in Fig. 8(c) and Fig. 8(d),
shows higher stochasticity than , as commonly reported for AM metals [29,30].
The plots in Fig. 7 clearly show the diversity of samples in terms of tensile responses and microstructures. For instance, cuboid and tensile specimens 192 are built with a very small VED of 22.1 J/mm 3 , which does not provide sufficient energy for proper material consolidation.
Such a VED results in a cuboid with many large pores, a small hardness value of 368.7 HV0.5, and a scattered hardness map depending on the indentation location, see also Fig. 6. This processing condition also produces weak tensile specimens whose yield stress and ductility are roughly below 100 MPa and 1%, respectively. Further increase in VED to the level that LOF porosity is lowered to its plateau value increases the For example, processing condition 195 in Fig. 7 with a very high VED of 234.2 J/mm 3 has a of 1, 310 MPa and of 2.3%, respectively.
Overall, we observe that samples printed with VEDs of roughly 100 to 200 J/mm 3 exhibit enhanced ductility and tensile strength (see specimens 70 and 60 in Fig. 7) compared to samples printed with VED values outside this range. However, there is no clear relationship between VED variations and and , either within or outside this range. To better demonstrate this point, we consider process parameters 66 and 206 in Fig. 7, which result in similar VEDs but the corresponding cuboids and tensile coupons have dissimilar microstructures, surface properties, and stress-strain curves. Specifically, we observe that tensile specimen 66 produces stress-strain curves with higher ductility and larger stochasticity but lower strength. Such property variations in 17-4 PH SS happen mainly due to the changes in martensite/austenite/ferrite ratios (see Fig. S1), porosity percentage, precipitation, and elemental evaporation and chemical composition with changes in the processing conditions [6,16,20,48,50,51].
Considering the complex relationship between process parameters and tensile properties and variabilities in the measured properties, we arrive at a similar conclusion to Section 3.1 in that VED fails to fully explain the dependency of and on the process parameters and can only achieve modest R 2 values, as observed from the fitted regression models shown in Fig. 8(a) and Fig. 8(b). Hence, we rely on our GP-based hierarchical framework to distill these relations from the datasets. As demonstrated in Fig. 8(a), is less stochastic than , and more correlated with hardness and porosity. Therefore, we begin by linking to process parameters and predicted hardness and porosity (see Fig. 3).
Then, we use the trained GP, along with the GPs built-in Section 3.1, to predict as a function of process parameters and previously predicted properties.
As explained in Section 2.3.2, we leverage the relation between of tensile coupons and cuboid properties by (1) augmenting the process parameters with the predicted hardness and the engineered porosity, and (2) fusing the two datasets from tensile and cuboid samples. The resulting model is denoted by . To assess the impact of these choices, we compare the performance of to two baseline GPs denoted by ′ and ′′ . ′ leverages neither of the above steps (i.e., it only uses the 54 tensile samples to link to process parameters), while ′′ only excludes the data fusion step.
The SIs for are shown in the top row of Table 4 and indicate that except scan rotation, all other process parameters affect . This trend is similar to Table 3 but the interplay between the remaining four parameters is higher in this case (compare the total SIs across the two tables). We exclude scan rotation from the input space and conduct a 5-fold CV study via ′ . The MSEs are presented in the top row of S3 for processing conditions).
Table 5 Accuracy assessment based on MSEs: The errors are reported for 5-fold CV for both yield strength and ductility. Model MSE Fold 1 Fold 2 Fold 3 Fold 4 Fold 5 ′ 0.294 0.059 0.475 0.681 0.076 ′′ 0.085 0.046 0.342 0.463 0.182 0.096 0.016 0.296 0.0701 0.166 ′ 0.194 0.238 0.451 0.416 0.417 0.388 0.146 0.258 0.401 0.334 To improve the performance of the model, we augment the process parameters with the predicted hardness and porosity to train trade-off between and , with the axes showing their distributions among the 54 tensile specimens, highlighting the significant dependency of these properties on the laser process parameters.
We highlight that while we use an FFNN as the mean function within the GP models for predicting and , the FFNNs in our approach differ from conventional FFNN models that are trained directly on the data. In our case, the FFNN is integrated into the GP framework, and its parameters are learned jointly with the hyper-parameters of the GP's kernel.
As a result, the FFNN does not have a separate loss function; that is, its loss function is part of the loss function of the GP as explained in as de- The MSEs of are presented in the third row of Table 5 and the calculated noise variance and R 2 are 99 × 10 -4 and 0.94, respectively (see Fig. S4 for a graphical error representation). The high R 2 value and similarity between MSEs and the estimated noise variance indicate that is effectively leveraging data fusion and feature augmentation.
Henceforth, we denote this model as and note that, expectedly, it provides higher errors compared to and since has more variability, is a more complex property to predict, and there are fewer samples on it (54 vs 270).
Following Fig. 3, the final step involves learning , where process parameters are augmented with all of the previously predicted properties ( ̂ , ̂ , ̂ ) and tensile data are fused with cuboid data. To evaluate the effectiveness of these two choices, we compare the performance of to a baseline GP denoted as ′ which does not incorporate these enhancements.
The Sobol SIs for in the bottom row of Table 4 highlight layer thickness and laser speed as the most important process parameters, followed by laser power, hatch spacing, and scan rotation. This ordering is different than the ones in Table 2 which highlights the complexity of data fusion, i.e., our framework is expected to leverage hardness, porosity, and tensile strength features even though they are not affected by the process parameters in the same way as ductility.
The higher importance of layer thickness over laser power can be attributed to the fact that the GP model has only seen layer thickness values {30, 60} and incurs some errors when predicting other values which, in turn, affects the SIs. Similar to previous cases, we exclude scan rotation from the input space (as its main and total SIs are small) and perform a 5-fold CV using ′ . The MSEs are reported in the forth row of Table 5, with estimated R 2 and noise variance values being 0.76 and 0.16, respectively. Considering the inherent challenges of learning ductility, these metrics indicate that the GPs perform relatively well but cannot generalize accurately. This issue is illustrated in Fig. 9, where we generate 10, 000 random process parameters and predict their cor- we deem these results satisfactory and proceed to build the final model for ductility on all tensile and cuboid samples. We denote this model by and test its generalizability in Section 3.3.
We now use the trained GP models to identify the process parameters that optimize the combination of yield strength and ductility of a tensile coupon. In the literature, Bayesian optimization (BO) or metamodelbased searches are commonly employed for optimization but these methods are not suitable to our problem. BO finds the optimum of blackbox functions by iteratively sampling the most informative points in the parameter space [61,[69][70][71][72] while we aim to build and test only one sample. Metamodel-based methods leverage surrogates (GPs in our case) in an optimization package that are either gradient-based (e.g., Adam or L-BFGS) or heuristic (e.g., Genetic algorithm). We found these methods to provide poor performance in our case as their reported optima strongly depended on the initialization. We attribute this poor performance to the fact that we use four hierarchically linked GPs to predict tensile properties given the process parameters, and such a linkage produces a cascade of uncertainties that are not easy to quantify (see Appendix E for more clarification).
We address the above challenges via the following simple and intuitive approach, see Fig. 11. We first generate 10, 000 process parameter combinations by uniformly sampling from the ranges used to experimentally build the samples. Then we use the trained GPs to predict the ′ : (a) displays the 10, 000 generated samples color-coded based on their VED. In (b), we narrow these samples to those within the optimal domain and color-code them based on the prediction uncertainty, highlighting the significant uncertainty of the model in this region. : Each sample undergoes three repeated tensile tests. The squares represent the experimental ductility values obtained from each test case versus their predicted median. The pink "X" symbols denote the model's predictions on test data, while the gray circles depict the model's performance on the training data. Each horizontal line in the plots highlights the variability of the test cases for each sample. corresponding yield strength and ductility for these 10, 000 process conditions. The result is shown in Fig. 11(a) where the points are color coded based on the VED. We then narrow the search space to include the points with (1) VEDs between 100 and 200 J/mm 3 (based on the findings of Sections 3.1 and 3.2), and (2) high yet feasible ( ̂ > 1000 MPa) and ( ̂ > 12%), see Fig. 11(b). To consider the prediction uncertainties of the candidate points in Fig. 11(b), we color code them based on the predicted uncertainties for the individual objectives (i.e., ductility and yield strength), see Fig. 11(c) and (d). The error bars showing these uncertainties are also depicted in Fig. S6. We finally identify process settings with small uncertainties, see the small black box in Fig. 11(c) and (d), and randomly select one of them for the manufacturing process.
The chosen setting is marked by "OP" in Fig. 11(e), where the points are color coded based on the uncertainty of the objective functions that considers both ductility and yield strength. More specifically, this total uncertainty is calculated by first normalizing 4 the uncertainty of each feature to take out the effects of the scale differences, and then summing the normalized values. We also select two random points outside of the optimal window, marked as "T1" and "T2" in Fig. 11(e), to assess the overall predictive power of our framework. For "T2" we consider two scan rotations to evaluate our prediction based on the Sobol SIs that this process parameter insignificantly affects the mechanical properties.
4 min-max normalization [73].
The three selected process parameter sets are used to build and test tensile coupons, see Table 6. The stress-strain curves of these samples are shown in Fig. 12 where we also provide the -scatter plot of the 54 training samples to put the obtained results into perspective. It is observed that the experimental measurements ( and ) match with model predictions ( ̂ and ̂ ) quite well and the 70% prediction intervals in all cases contain the experimental measurements. We also observe that changing the scan rotation from 90 or 67 degrees has an insignificant impact on the mechanical properties. This finding is consistent with the literature where 67 is reported to show only slightly lower , and for 316L SS [74].
As shown in Fig. 12(a), it is observed that yield strength and ductility are not necessarily inversely correlated in our data, as is regularly the case for most conventionally processed structural materials. The strength-ductility trade-off is usually related to intrinsic strengthening mechanisms such as solid solution, precipitation, grain boundary, or second-phase strengthening which tend to reduce ductility. However, the large process parameter space that we explore can change this trend as defects, complex microstructures, and variations arise. Nonetheless, our results show that tuning the process parameters can increase both yield strength and ductility.
Compared to previous studies [30,[75][76][77][78][79], we have designed an LPBF processed 17-4 PH SS with improved tensile strength and ductility even though our tensile specimens were created vertically and the loading direction during the tensile test was the same as the build direction. Table 6 Optimized processing conditions with corresponding predicted and experimentally measured tensile properties: and represent the experimental and , respectively, while ̂ and ̂ denote the corresponding predictions. The uncertainties for and are also included. Sample name (W) (mm/s) (μm) ℎ (μm) ( • ) VED (J/mm 3 ) ̂ (MPa) (MPa) ̂ (%) (%) Std Std OP 81 325 20 77 90 153.4 1199 1180.1 16.8 16.2 4.7 155.2 T1 233 1471 20 71 90 111.4 1073 1168.5 14.2 11.1 3.9 145.8 T2 90 227 1080 20 72 90 155.4 1130 1213.3 15.6 13.7 4.1 141.9 T2 67 227 1080 20 72 67 155.4 1130 1232.1 15.6 11.7 4.1 141.9
Vertically-built specimens exhibit reduced ductility and tensile strength compared to those built horizontally [75][76][77] since defects, which primarily form between layers, are perpendicular to the tensile loading direction in vertical specimens which, in turn, facilitates void growth [75]. Additionally, the shorter time interval between melted layers and decreased cooling rates in vertical samples coarsen the grain sizes and lower retained austenite content [75], which reduces the strength. Also, our samples are tested with their surfaces in the as-printed condition without any machining or polishing as opposed to other studies where surface roughness is removed by grinding or polishing [48,79]. Notwithstanding these strength/ductility reducing factors, we still achieved improved properties compared to the literature [30,[75][76][77][78][79].
Finally, we use the trained GPs to plot design maps for the properties. LPBF parameter window that yields ̂ greater than 12%. To optimize the combination of and , the optimization surface (i.e., the logarithm of the products of and ) is depicted in Fig. 13(c), where the region in yellow marks the optimized process window. We note that the anomalous behavior in the lower right corner of the plots corresponds to a combination of very low power and very high velocity, where
material consolidation is poor and hence the collected data (if any) are noisy, and (2) the model has larger errors because of the limited number of samples available for training in these regions. We also note that our GPs leverage an FFNN as their mean functions, so they can provide rough prediction profiles even though they use smooth kernels (i.e., the Gaussian covariance function). The combination of these factors results in the rough design maps observed in Fig. 13.
We develop a process optimization framework that integrates HT experiments and hierarchical ML to identify the process parameters that optimize the mechanical properties of LPBF-built parts. Our approach reduces the reliance on expensive tensile tests and streamlines process 90 , and (e) T2 67 . Note: T2 90 and T2 67 are the same laser processing conditions with one printed with a scan rotation of 90 and the other with 67 degrees. to show the optimization surface.
optimization by systematically integrating computational techniques (statistical sensitivity analysis, feature augmentation, data fusion, and emulation) with experiments that include large data from cuboid samples and small data from tensile coupons. While the proposed approach is material-agnostic and fully generic, in this paper we apply it to 17-4 PH SS which is a technologically important material.
We demonstrate that conventional process optimization approaches that solely rely on hand-crafted features, such as VED, fail to fully capture the effects of process parameters on the material properties and limit the search to small parameter spaces. Our framework ad-dresses these gaps because it directly and automatically predicts mechanical properties as functions of all process parameters including laser power, laser scan speed, hatch spacing, powder layer thickness, and scan rotation. Additionally, by founding these predictions on larger "easy to collect" and smaller "labor-intensive" data sets, our approach is much less dependent on expensive fabrication and characterization procedures, enabling the exploration of a large set of process parameters.
We demonstrate that our framework is very efficient at learning the complex relationships between multiple process parameters (vary-= 1, ⋯ , . With this augmentation, the datasets are concatenated as follows:
where the subscripts 1, 2, ..., correspond to the data sources, is the number of samples obtained from source , and are, respectively, the × ( + ) feature matrix and the × 1 vector of responses obtained from ( ), and' ' is a categorical vector of size × 1 whose elements are set to' '. Then, the GP model can be trained on { , } using the kernel-based method described above to effectively integrate and fuse data. Fig. C.3. between auxiliary features and tensile properties.
Y.T. Pei, Laser powder bed fusion of 17-4 PH stainless steel: a comparative study on the effect of heat treatment on the microstructure evolution and mechanical properties, Addit. Manuf. 46 (Oct. 2021) 102176,
Materials & Design 252 (2025) 113705
Since the predicted hardness of this model is never used, we treat it as a dummy variable which merely enables the data fusion. Materials & Design 252 (2025) 113705