Data-Driven Sensitivity Analysis for Static Mechanical Properties of Additively Manufactured Ti–6Al–4V

A different path from conventional subtractive manufacturing was first commercialized in the mid-1980 s known as rapid manufacturing or rapid prototyping [1,2]. More than 30 different procedures have been developed since then for both metals and plastics, which are now classified under a more generalized terminology of additive manufacturing (AM). Additive manufacturing uses a layer-on-layer fabrication technique wherein the material to be deposited is melted by a focused heat source such as laser power or electron beam, unlike its counterpart where the material is removed to develop the desired shape [3,4]. Computer-aided design (CAD) modeling allows an efficient fabrication of complex shapes and designs of any metal material as long as it can be deposited on the substrate by either fusion at the bed (powder bed fusion (PBF)) or depositing the fused material directly on the substrate (direct energy deposition (DED)) [5,6]. Table 1 illustrates several major categories of AM techniques for additively manufacturing metals and their alloys from existing open literature.

Selective laser melting (SLM) and direct metal laser sintering (DMLS) are the two PBF techniques that follow quite similar conceptual procedures, but they have a fundamental difference as the metal powder is sintered for DMLS while melted for SLM, which is a result of the working temperature of the respective lasers [15–17]. Sintering is generally carried at a temperature lower than the melting temperature (called sintering temperature) where the grain viscosity drops with temperature causing an interfacial kitting of the grains without fully melting them. Both DMLS and SLM processes begin with fixing up the base plate on which the rest of the build is being carried on. Inert gas is filled in the closed chamber. Then, a thin layer of spherical powder is optimally distributed and leveled to a predetermined thickness. The high-energy power source is used to scan the powder selectively melting or sintering the powder to fill in the design provided by the CAD data. After the first layer has attained the required shape, the next layer of powder is spread, and the process is repeated. Since the penetration power of the laser beam is deeper than one layer, each new layer gets welded to the previous layer. At the end of the process, the unused powder can be reused again or mixed with a new powder stack for another manufacturing unit [10,18].

Certain process parameters will affect the fabrication quality, such as the laser scanning speed, powder layer thickness, laser beam power, spot size, hatch spacing, and scanning strategy, which are schematically shown in Fig. 1 [19,20]. The parts fabricated by these laser-based processes tend to result in a high theoretical density, poor surface finish, and inherent residual stresses. Therefore, most of the fabrications undergo postfabrication processes where heat treatment and machining are performed. The heat treatment drastically changes the mechanical properties of the fabrication and thus becomes an important criterion for discussion.

From the past few decades, additively manufactured Ti–6Al–4V is one of the most investigated titanium alloys in diverse fields of engineering owing to its high strength, low density, low coefficient of thermal expansion, outstanding corrosion resistance, high cycle fatigue resistance, and biocompatibility. These characteristics of titanium alloys help in taking a decisive role in applications that warrant high reliability and end use of the products such as in surgery and medicine, aerospace, automotive, chemical plant, power generation, oil and gas extraction, sports, and other major industries. SLM, DMLS, electron beam melting (EBM), laser engineering net shaping, or any other technique, Ti–6Al–4V alloy captures more than half of the titanium alloy fabrication market [21,22].

Several experiments have been conducted over the years to efficiently fabricate a high-strength and high-ductility Ti–6Al–4V alloy part by varying the different process parameters in each of the SLM and DMLS techniques [23–27]. Other types of studies focused on understanding the cooling rate-dependent microstructural variation of AM fabrications. Alloy's tensile properties for both heat-treated and as-fabricated conditions have also been explored [28–33]. Experimental studies were done by Mierzejewska et al. [34] and Mierzejewska [35] aiming to find the effect of laser power, speed, and heat treatments on the mechanical behavior of DMLS fabricated Ti–6Al–4V alloy. Most existing studies focused on the data from an individual lab using a unique set of manufacturing parameters. Various studies have been conducted to develop a data-driven framework to optimize the AM manufacturing process. Wang et al. [36] developed a framework for EBM manufactured Ti–6Al–4V to develop equiaxial grains. Melt pool-based scan strategy predictive model has been attempted to attain an optimized melt pool for powder-based fusion AM techniques [37]. Process–structure–property relationships were studied for AM processes obtaining a data-driven multiscale multiphysics modeling approach [38]. This study explores a data-driven approach to estimate the tensile properties of Ti–6Al–4V alloy fabricated by either SLM or DMLS processes. For this purpose, data is collected from open literature for each of the SLM and DMLS fabricated Ti–6Al–4V alloys. These data are used to develop numerical models to identify any linear or nonlinear relationship between the process parameters and the tensile properties of Ti–6Al–4V alloy. The developed models are then subjected to both local and global sensitivity analysis (SA). Several conclusions and future work are identified based on the proposed study.

The experimental data available in the open literature are collected and the influential processing parameters are determined based on the sufficiency of data and the researches carried out by various authors. The linear regression model and the nonlinear artificial neural network (ANN) model based on Bayesian regularization are developed based on the data collected. These models take four major processing parameters, namely, “scanning speed,” “hatch spacing,” “powder layer thickness,” and “beam power” along with some postfabrication processing treatments represented by “heating time,” “heating temperature,” and “hot isostatically pressed (HIPed) or not” as the input parameters and achieve the tensile properties, namely, “ultimate tensile strength (UTS),” “yield strength (YS),” and “elongation (El)” of the Ti–6Al–4V alloy as the output parameters. A total of 53 out of 89 and 100 out of 114 datasets of useful data for SLM and DMLS, respectively, were screened out such that the modeling requirements are met. This study deals with only the vertical build direction of the fabrication process. The data collection for each of the processes can be seen in Appendices  A and  B. The schematic illustration can be seen in Fig. 2. The data collected here have been focused on a common minimum data availability methodology where commonly reported data are used to develop the models and ignoring the various additional data reported by several authors. Additional data could be understood as fabrication equipment, powder size, fabrication environment, beam diameter, spot size, and scan strategy, etc. which could not be added to the model because of the unavailability of sufficient reported information or being non-numeric data. Some of this additional information as reported by various authors is referenced in Appendix  C.

where X1, X2, X3 … Xn are the independent variables (i.e., manufacturing parameters in this case) and β1, β2, β3 … βn are their corresponding coefficients for estimating the dependent variable Y (e.g., mechanical properties in this case) by minimizing the sum of the squared deviations. ε is the error term. Since the relationship is unknown and may have a strong nonlinearity, a flexible nonlinear model based on ANN is also explored. ANN is a widely used machine learning approach and is well documented in the literature. Thus, no details are given here. In this study, the ANN models developed are based on the multi-layer-perceptron class of feedforward ANN modeling [39]. The ANN-based model has the input layer consisting of seven neurons connected to two hidden layers of 16 and 64 neurons, respectively, which is then connected to the output layer. “Trainbr” Bayesian regularization provided by matlab utilizing the gradient descent with momentum weight and bias learning (learngdm) is used as an adaptation learning function while the performance or loss function is mean squared error (MSE) based. It should be noted that the ANN might have an overfitting issue with the small data size, although the used Bayesian regularization algorithm can mitigate the overfitting possibility. Detailed investigation using other alternative regression methods and validation are required to further improve the sensitivity analysis accuracy and robustness. Both linear regression and ANN models are widely documented in the open literature and no details are given here.

To understand the behavior of different parameters, SA is performed after the linear and nonlinear models are obtained. Both local and global SAs have been performed and discussion and assessment have been attempted for each.

A localized sensitivity analysis is based on understanding the behavior of output parameters developed against a small perturbation near the input parameters at a localized reference point [40]. A derivative-based localized sensitivity analysis is performed for each of the models where the median values of all input data are kept as the reference point. Since the minimum to maximum range for each input parameter is very different, z-score normalization is done to achieve a common comparable scale.

A global sensitivity analysis varies all input parameters in predefined regions to quantify their individual and interactive importance [41]. The Sobol method is one of the commonly used global SA methods and is used in this paper. It is a method based on variance decomposition. In the Sobol method, the total variance of the model output is composed of the variance from individual parameters, and the variances from interactive parameters. The proportion of variance resulting from individual and interactive parameters to the total variance corresponds to the first-order and interactive sensitivity indices, respectively [42]. The detailed calculation of Sobol sensitivity indices can be seen in Ref. [43].

The total order sensitivity index measures the overall effect of input on the output including both its main effect and its interactions with the remaining inputs. The difference between total order sensitivity index S_Ti and first-order sensitivity index S_i calculates how much the parameter x_i interacts with other parameters, i.e., the joint effect of the parameters [44]. To calculate the sensitivity indices, the Monte Carlo algorithm provides a feasible method using values of model outputs only [43,46]. Saltelli sampler method, based on Monte Carlo, is adopted in this work for generating efficient estimations of model outputs [44]. Fourteen thousand samples are generated with Saltelli's sampler.

All the data are collected from the open literature. Data for tensile testing of Ti–6Al–4V alloy are available for three different build directions, namely, flat, edge, and vertical. The data collected and the model built in this study consider only the build direction that is “vertical” or “parallel to the worktable movement.”

SLM tensile data available from the literature included variations with scan speed, laser power, powder layer thickness, heating temperature, heating time, and if the process was HIPed or not. Detailed raw data and associated references are listed in Appendix  A. Similarly, detailed raw data and associated references for DMLS are listed in Appendix  B.

One typical challenge for the data collection from literature is due to the “missing data.” Some papers did not include everything either because the authors did not perform or did not report. Sometimes, only descriptive language is used in the context and no quantitative data is reported. Thus, data augmentation and preprocessing are necessary in order to make the data analysis feasible. For example, some experimental data are for an as-fabricated Ti–6Al–4V alloy that undergoes no heat-treatment process after fabrication. In order to use these data for regression, we augmented these data with a “dummy” heat treatment of room temperature (30 °C) for 3 h. Another case is for HIPed AM specimen. HIPed is a postfabrication heat treatment process where the sample is externally pressurized while heating, aiming to achieve a 100% theoretical density. The data collection included several HIPed samples heated at different heat temperatures higher than 920 °C but nearly the same external pressure of about 100 MPa. Many other data do not include HIPed and there are not enough data for a reliable continuous representation of HIPed. Thus, we treated HIPed as a binary variable. HIPed is either 1 to be set as a “yes” or 0 as “no.”

Another challenge is due to the diversity of testing design. Since all collected data are not from a single lab and are not designed coordinately, the distribution of input manufacturing parameters and output properties may be skewed, which is due to the different design of experiment (DOE) rules used by different researchers for various objectives. Readers need to pay more attention to the collected data range and its distribution for proper interpretation of the results. Thus, detailed empirical cumulative distribution function (CDF) together with the best-fitted distribution functions can be seen in Figs. 3 and 4 for the output parameters, UTS, YS, and El. The corrected Akaike Information Criterion (AICc), a version of AIC that considers small sample correction to determine the fitting criteria, is used to identify the data distributions for output parameters in this study [47].

Tables 2 and 3 show distribution details for the output parameters data for SLM and DMLS, respectively. The input parameters show clustered behavior due to DOE from collected papers, which makes it difficult to be fit into distribution functions. Thus, only basic statistical interpretation of input parameter data is represented in Table 4 while the attempted probability distribution functions for SLM and DMLS-based collected input parameters can be referenced to Appendices D and E, respectively.

It should be noted that Figs. 3 and 4 show the cumulative probability distribution (CDF) of the output parameters for SLM and DMLS collected data, respectively, while Tables 2 and 3 represent the details for the best fit distribution for the CDFs. These details also serve as the information on the sources of uncertainties associated with the study. The main source of uncertainties in the current study includes multiple types: variability across different labs, intrinsic AM process uncertainties, and data collection uncertainty. The first type of uncertainty comes from the machine setup, procedure, and raw materials used at different labs. Since the data are collected from open literature, there is no control of the labs' setting. The variability among labs will cause uncertainties in the reported data. The second category of uncertainty includes natural variation in powder absorptivity, fluctuation in temperature boundary, uncertainty in powder particle properties, and many others [48]. This represents the intrinsic randomness of AM materials even there are made in a single lab with the same nominal settings. The last type of uncertainty comes from the data collection itself. Since we collect data from open literature, not all data are reported in tabular format and some data are missing. Thus, the data collection procedure itself (e.g., digitization from a graph and data imputation for missing data) will cause additional uncertainties.

Thus, Table 4 shows the range and statistics of all the above-mentioned uncertainties. Table 4 also can be viewed as the design space of all collected data from various labs, which represents the tendency in the open literature for AM process design. It will be best to separate the uncertainties from different labs versus from different AM processing parameters. For example, the intrinsic uncertainty quantification of operator-specified laser power (e.g., mean value of 200 W with a standard deviation of 5 W) will be very valuable to investigate the material property variation with respect to the uncertainties of AM process parameters. Unfortunately, most collected data do not contain this information and we are not able to separate different sources of uncertainties. This is a great future research direction for uncertainty reduction. The current investigation did not separate the uncertainties from different labs with the intrinsic uncertainties within the AM process. This is partially due to the missing description for individual lab settings, equipment parameters, and raw material properties. Future study is suggested to have that information reported/collected to reduce the uncertainties in the model predictions.

The analysis has subsequently been carried out through two regression models described above with 53 and 100 datasets for SLM and DMLS, respectively. R-square value is presented in Table 5 for the two models. It is observed that the ANN has a much higher R-squared value compared to that of linear regression, which indicates the nonlinear relationship of the mechanical properties with respect to manufacturing parameters. This is also confirmed by the parametric studies shown later.

It is also observed that the correlation result for elongation (especially for DMLS) is not as good as strength results. Thus, further study for the ductility of AM material is required. It should be noted that the high R-square value does not indicate a good prediction capability of the ANN model and it only indicates a good correlation of properties within the investigated parameter range. The current ANN is deterministic. We used the ANN as a surrogate model to predict the outputs (material properties in this case) giving a random input variable vector (AM process in this case). Detailed model uncertainty investigation is suggested to use a stochastic model, especially when the model uncertainties are not stationary with respect to different AM process parameters. In the current investigation, we ignore the model uncertainty. However, since ANN shows a much better correlation, the following parametric study is based on the ANN model.

The training and test performance of the Bayesian regularized ANN for SLM and DMLS can be seen in Figs. 5(a) and 5(b), respectively. Out of a total of 53 input datasets for SLM, 42 are used for training the model while 11 are used to test it. In the case of DMLS, out of a total of 100 data sets, 80 were used as training datasets while 20 were used for testing the model.

where $n is the number of datasets, Y(normalizing factor) is theypredicted_max or (ypredicted_max−ypredicted_min)$⁠, $ytarget_i is theith value of the target output parameter, and$$ypredicted_i is the ith value of the predicted output parameter$

This study aims to understand how each fabrication parameter influences the final AM build and its tensile properties. The results for the parametric study are conducted in two folds, one is the local sensitivity analysis where derivative-based analysis is performed by keeping all but one fabrication parameters constant and analyzing the behavior of that one parameter against the tensile properties and global sensitivity analysis following Sobol's approach where the overall impact of the parameters individually or along with their corresponding parametric interactions is discussed over the whole parametric range.

The data collected for each of the processes are from various sources which do not account for similar fabrication density and hence there could be variable porosities in the fabricated sample which are unknown at this stage because of the absence of proper porosity profiles and micro-CT images. Any inherent defects in the fabricated sample could distort the final results and the model does not account for such variability. Another aspect to look at is that the microstructure of Ti–6Al–4V AM fabrications is different than the traditional subtractive manufacturing fabrications because different grain profiles can be generated upon varying the solidification rate and the thermal gradient. This study does not focus on the microstructural changes which would be occurring during the fabrication or after subsequent heat treatment but an estimate of what might happen to the microstructure can surely be made after analyzing the results from the model alongside proper sample imaging.

It should also be noted that HIPed (yes or no) is a discrete input value and we used values 1 and 0 to represent these two conditions in the regression analysis, respectively. Since they are discrete variables, we did not perform local sensitivity analysis as there are no derivatives defined. Also, most data sets do not have HIPed procedures and we focus on these datasets and the global sensitivity analysis does not include HIPed either.

The median values of all collected data are used for the SLM Model are considered as the local reference point for performing the local sensitivity analysis and parametric study, see Table 7. A small perturbation in the input parameters (1%) is used to estimate the sensitivity [40]. Each of the parameters is varied once keeping the others constant to understand the impact of that parameter at the median values (Fig. 22).

The sensitivity result for SLM can be seen in Fig. 6, which shows that the scan speed and laser power have opposite sensitivity to each other. An increase in the scan speed leads to a decrease in the strength compensated by an increase in the elongation and the opposite is the case for laser power. At the median values as the local point, the powder layer thickness behaves opposite to the hatch spacing however is more impactful than the latter. The observed trend for heat temperature and heat time agrees with the reported behavior in the literature [31]. It is also observed that scan speed and laser power show a similar trend with that of heat temperature and heat time, respectively. The results shown in Fig. 6 correspond to a single reference point (median of the data) and a parametric study is used to investigate the sensitivity change with respect to the change of manufacturing parameters in the entire range of the collected data. Details are shown below:

• Material properties versus scan speed

  The parametric studies for scan speed are shown in Fig. 7. It is observed that the small to medium scan speed has a small effect on the strength. An increase in scan speed beyond a certain range leads to a rapid decrease in strength. This might be because, at a very high scanning speed, subsequent melt pools do not overlap enough thus generating lack of fusion porosity, in turn, resulting in a decreased strength. The loss of strength is compensated by an increase in elongation. The nonmonotonicity of the elongation trend is reflective of the inflection point in the strength profile.

• Material properties versus laser power

  Laser power has a direct relation with the strength, which is due to the fact that an increase in laser power ensures an efficient melt pool leading to minimal porosity. This behavior can also be observed from the model output where at lower power values, the strength of Ti64 is lower than the strength at higher power values as can be seen in Fig. 8. Once the power is sufficiently high, the strength does not change very much.

  Following the general trend, elongation behaved inversely to the behavior of strength. However, it should be noted that a very high-power laser alone or a medium laser power at a low scanning speed could develop keyhole pores defect, which leads to reduced strength. The data analysis for SLM did not show such behavior which might be due to the combined effect of median scanning speed and other parameters for this localized parametric study.

• Material properties versus hatch spacing

  Figure 9 shows that the strength starts to decrease as the hatch spacing is increased beyond a certain value while the elongation was observed to have a nonmonotonic behavior. The hatch spacing determines how far the new round of scan begins from the previous round in the same layer. This spacing is advisable to have around half the size of the beam spot diameter because an efficient overlap between consecutive scans leads to a stable and efficient melt pool. Such a melt pool leads to lower cooling rates developing longer grains at solidification which in turn imparts higher strength to the build. However, an increase in the hatch spacing beyond a certain value leads to an improper fusion of the powder in consecutive scans, i.e., the developed melt pools are spread further apart and generates the lack of fusion defects within the fabrication, therefore, leading to increased porosity causing a reduction in the strength. Elongation profile behaves in a compensating manner to the strength as the initial increase of strength caused by increasing hatch spacing leads to a decline in the elongation, however, at high hatch spacing values where strength declined rapidly, elongation increased nearly at the same rate.

• Material properties versus powder layer thickness

  Figure 10 shows that the strength decreases as the powder layer thickness increases. Powder layer thickness plays an inverse relation to the overall energy being imparted to the melt pool. An increased powder layer thickness demands more power input to perform the proper melting of the layer in each scan. For the same power, increasing the layer thickness would cause improper melting of the powder leading to a porous fabrication after solidification, which would have a lower strength. Elongation showed a decline for declining strengths at lower powder layer thickness values; however, at higher values of powder layer thickness, elongation showed compensating trends to the strength. Possible reasoning for such a behavior is the absence of appropriate continuous data for powder layer thickness.

• Material properties versus heat temperature

  Figure 11 shows that the strength generally decreases with the increase in the heat treatment temperature. A rapid transition happens between 650 and 900 deg. The strength does not change much below 600 deg and above 900 deg. The elongation variation also supports this transition zone and is observed to increase at high heat treatment temperatures. This is likely due to the microstructure change as it is well known that heat treatments are used to develop variations in the microstructures where grain sizes elongate, and the alloy loses its resistivity to the plastic flow.

• Material properties versus heat time

The model developed plots for a positive variation of heat time leading to a decline in both the strength and the elongation of the alloy can be seen in Fig. 12. Keeping in mind the median values for this analysis takes 650 °C as the median temperature where usually the stress-relieving process for Ti–6Al–4V is operated. The relationship presented here cannot be explained at this stage as it could either be a result of a microstructural variation or a model-generated anomaly. The dummy data added to the original data earlier to fit the as-fabricated samples in the same bracket as the heat-treated samples could have been one of the promising reasons for such an anomaly.

Similar to the analysis presented for the SLM model, localized sensitivity analysis as a parametric study is performed where 1% perturbation in individual parameters are analyzed against the remaining kept constant at the median values for the DMLS model. The median values for the DMLS data used to build the model can be seen in Table 8 and the sensitivity results are shown in Fig. 13.

At the median value of the collected data, scan speed and laser power again have completely opposite behavior to each other. An increase in the scan speed leads to a decrease in the strength and elongation, whereas an inverse behavior is the case for laser power. Hatch spacing and heat temperature have a large and similar impact behavior on the output properties. An increase in either leads to a decrease in strength compensating the elongation considering the localized median point. Powder layer thickness also reflects a similar behavior but with a much lower influence. Heat time seems to have an almost negligible effect on mechanical properties (Fig. 23).

1. Material properties versus scan speed

   The strength attains a maximum value for medium-range scan speeds from 500 to 600 mm/s. At very low scan speeds, the strength is somewhat smaller which could be explained by the formation of keyhole defects due to excess melting of powder leading to rounded porosities. For higher scan speeds, the strength drops below 900 MPa as a suspected result of the unmelted powder contributing to the lack of fusion defects leading to irregular porosities as can be seen from Fig. 14.

   The elongation plot presented behaves differently than the usual elongation trend of generating a positive variation in elongation for a decline in strength. The reasons for this behavior are not clear. One hypothesis is that the increased speed will introduce a lack of fusion defect and both strength and ductility will reduce due to increased defect density while the other hypothesis is that since DMLS does not offer proper melting and works around sintering of the alloy powder; this might be the actual case how strength and elongation profile behave for the process. However, these hypotheses cannot be validated using the collected data and need detailed microstructure imaging observations.

2. Material properties versus laser power

   Figure 15 shows that the maximum strength was obtained in the power range of 210–240 W when the remaining median parameters were kept constant. This behavior is supported by the general physics for melting using the laser. At very low power values, the powder material gets insufficient heat input and is unable to completely melt the material which leads to a high porosity fabrication, in turn, resulting in a decline in the strength. However, increasing the power too much also leads to a decline in the strength of the alloy which is caused by the development of extra internal stress concentration sites in the form of keyhole pores due to excessive energy being imparted to melt the powder.

   The behavior of elongation with power variation presented in Fig. 14 also suggests that the defect mechanism is dominating in the strength and ductility trend. It is well known that the strength increase will usually accompany with ductility loss. This is in general agreement with the grain structure characteristics. For example, increased grain size during the heat treatment will cause the strength to decrease with an increased ductility. If both strength and ductility show the same trend, it indicates the major mechanism is the defect. If defect density increases/decreases, both strength, and ductility will decrease/increase.

3. Material properties versus hatch spacing

   Strength reduces with an increase in the hatch space and the elongation was compensated for the decrease in the strength which agrees with the general physics of hatch spacing, see Fig. 16. However, this plot behavior cannot be considered as the ideal behavior of the strength and elongation for the DMLS process with the hatch spacing because the model is built on 90% hatch spacing values of 100 μm and a few 120 μm values. It can essentially be said that the increment in hatch spacing results in an increment in elongation and a decrement of the strength.

4. Material properties versus powder layer thickness

   Similar to hatch spacing, the strength and elongation profiles for the powder layer thickness showed a decline in strength for an increased powder layer thickness getting compensated by the increase in elongation as shown in Fig. 17. Again, similar to the case for the hatch spacing, the model is mostly built on a powder layer thickness of 30 μm and a few cases of 60 μm; therefore, valuable information for the trend of powder layer thickness with the strength and elongation is still missing but an increase in the layer thickness reduces the strength and increases the elongation of the fabrication.

5. Material properties versus heat temperature

   Heat treatment alters the already generated microstructure during the fabrication process, which is determined by the melt solidification rate. The new microstructure is determined by the temperature at which the alloy is heated and the cooling rate at which it comes back to room temperature. The model generated heat temperature behavior can be seen in Fig. 18 where an increased heating temperature resulted in a reduction of the alloy strength and an increase in the elongation. The data for DMLS consisted of an annealing process where slow cooling is performed after heat treatment and after the heat treatment, the alloy is observed to show higher ductility but lower strength which supports the results from the model.

6. Material properties versus heat time

   Following a similar behavior as in the heat temperature, increasing heating time reduced the alloy strength and increased the elongation, see Fig. 19. However, this behavior cannot be confirmed just by data analysis because of the introduction of the dummy data discussed previously. More data and experimentation would be necessary to reach a definite conclusion regarding the impact of heat time on the tensile properties of the alloy.

Sobol's indices are used for global sensitivity analysis, which includes first-order indices (S1), second-order indices (S2), and total order indices (ST). S1 represents the individual fractional contribution of the input parameters while S2 represents the fractional contribution of parameter interactions in determining the output parameters globally over the range considered for building the model. ST sums up the overall impact of parameters considering the first, second, and other higher order indices. To perform Sobol's analysis, 14,000 Saltelli samples are generated using the python library “SALib.” These Saltelli inputs are put through the developed matlab model and the outputs are recorded. These outputs along with the inputs are used in another python code to obtain the sensitivity indices. Despite being less influential, S2 indices are also calculated for each of the SLM and DMLS models and are presented in Appendices F and G, respectively.

Figures 20(a) and 20(b) show the respective S1 and ST values for the SLM based ANN model. First, discussing the S1 indices for the study, individually, laser power is the most dominant parameter on the UTS, YS, and elongation for SLM fabricated Ti–6Al–4V alloy indicating that a variation in laser power alone would drastically impact the tensile properties of the alloy. Heating temperature and hatch spacing influence the ultimate tensile followed by the laser power the most while heat time, layer thickness, and scan speed have minimal impact. Similarly, hatch spacing follows the laser power for influence on the YS while scan speed, heat time, heat temperature, and layer thickness have minimal impact. In the case of elongation, heat temperature and heat time are the stronger influencers after the laser power while scan speed, layer thickness, and hatch spacing are the weaker influencers.

The differences in ST and S1 values suggest that there are interactions between the considered parameters and the influence of these interactions can be seen in the Appendix  F. Impact of the interactions can be noticed in the ST values of the Sobol's analysis where scan speed and layer thickness registering negligible impact individually contribute considerably when all the interactions are considered. Now we discuss the ST indices obtained from the study. The influence order for UTS and YS stays the same as was observed in the S1 indices however, in the case of elongation, layer thickness, and scan speed become the most influential parameters while the remainder contributes nearly equally to any variation in the output.

In a nutshell, alloy's UTS is most dependent on laser power, hatch spacing, and heating temperature. Similarly, laser power, hatch spacing, and scan speed should be carefully considered for YS variations and in the case of elongation, scan speed and layer thickness would be the major deciders; however, the remaining parameters would also be considerably influential here.

S1 and ST results obtained from Sobol's analysis on DMLS data are presented in Figs. 21(a) and 21(b), respectively. A major variation compared to the SLM results can be noticed here. Firstly, discussing the S1 indices obtained. Clearly, laser power is the only major influencer for the UTS while layer thickness followed by scan speed impacts the YS majorly. Laser power followed by scan speed and heat temperature equally impacts the elongation characteristics of the alloy. All the other parameters are almost negligibly impactful when compared to the up stated parameters.

Again, a clear contract is visible in S1 and ST values suggesting interactions between the parameters and these interaction influences can be seen in Appendix  G providing the S2 indices.

After considering the interactions between the input parameters, laser power, and scan speed turn out to be the major influencers for each of the output parameters while the remainder accounts for negligible influence except for heat temperature which has somewhat impact on the elongation of the alloy. Therefore, based on this study, it can be said that the laser power and scan speed alone would be the deciding factors for the tensile properties of the alloy in a DMLS process.

The local sensitivity calculates partial derivatives of the output functions with respect to the input variables. The input parameters vary within a small interval of variation around a reference point. However, the main drawbacks of local sensitivity analysis are that derivatives provide information only at the base point, where they are computed; therefore, they do not provide an exploration of the whole space of the input [50]. In contrast to the local sensitivity, global sensitivity analysis provides an understanding of how the model outputs respond to changes in the inputs by taking a sampling approach from probability density functions (PDFs). Furthermore, all the model parameters are varied simultaneously, and the sensitivities are calculated over the entire range of each of the input parameters [51]. However, more number of simulations is required to get a valuable sensitivity index estimation compared with the local method [52]. The knowledge accumulated in this study could be used to tailor the overall tensile properties of the alloy in general, i.e., without specific value for a component design. Results obtained from Sobol's analysis indicate which parameters should be worked around carefully to get the optimal tensile properties. If the specific design values are known for a component/structure, the trends suggested by the local sensitivity analysis could be used to vary one or two input variables such that a certain required variation about the mean design values in the output variables could be obtained.

Data-driven analysis of two major laser-based PBF AM techniques, namely, SLM and DMLS are performed for the fabrication of Ti–6Al–4V alloy. Several data relating the fabrication parameters to the tensile properties of the alloy are collected for each of the processes and later used as input parameters to develop both linear regression and nonlinear ANN models.

Major conclusions drawn from the above study are summarized as follows:

1. The ANN model is found to deliver higher R-square values than the linear regression model and therefore, further data analytics are performed using the ANN model.

2. Localized sensitivity analysis executed at the median data of each process suggests that the SLM fabricated alloy is most sensitive to laser power, scanning speed, and heat treatment temperature while the heat treatment temperature, hatch spacing, and laser power are the prominent parameters for influencing the DMLS fabricated alloy.

3. Sobol's global sensitivity analysis suggests that SLM-fabricated alloy is most sensitive to laser power followed by heating temperature and hatch spacing as far as strength is concerned and powder layer thickness followed by scanning speed prominently impacts the elongation properties of the build. On the other hand, laser power, and scan speed are found to be the most impactful input parameters for tensile properties of the alloy while heating time turned out to be the least affecting parameter for DMLS fabricated Ti–6Al–4V alloy.

4. Sobol's first-order indices and total indices are different for the investigated dataset and manufacturing process, which indicate the interaction between the process parameters.

Some limitations and future work are listed below. First, despite the accumulation of nearly mostly published data for SLM and DMLS AM processes for Ti–6Al–4V alloy discussing the fabrication parameters and tensile behavior, one of the drawbacks of the current model developments is the small sample data, especially due to the missing data. Sample data comprise seven input parameters and three output parameters. If one of those parameters is not published, the whole dataset becomes invalid and not utilized. Therefore, additional statistical analysis, such as missing data imputation, could help to improve the data collection efficiency. The heat treatment data needed to be adjusted in the model because of the limited data availability and there could be some model generated errors induced because of the dummy data added externally. A situation-specific alternative regression method would be needed to rectify this limitation where text-based information could be incorporated along with the other input parameters. In addition, the current study only focuses on the static mechanical properties; time-dependent fatigue properties are also critical for AM design and should be investigated. Some sensitivity trends are being interpreted based on principles and rigorous model verification and validation will need extensive imaging analysis to support the hypotheses. Unfortunately, the most existing study did not perform/report the complete imaging results, especially 3D microstructure imaging. Thus, synchronized mechanical testing and 3D imaging (e.g., using Micro-CT) will be very valuable.

• Naval Air Systems Command, NAVAIR through Technical Data Analysis, Inc. (Contract No. N68335-20-C-0477, Funder ID: 10.13039/100010464).