The goal of this work is to understand the effect of process conditions on lack of fusion porosity in parts made using laser powder bed fusion (LPBF) additive manufacturing (AM) process, and subsequently, to detect the onset of process conditions that lead to lack of fusion-related porosity from in-process sensor data. In pursuit of this goal, the objectives of this work are twofold: (1) quantify the count (number), size and location of pores as a function of three LPBF process parameters, namely, the hatch spacing (H), laser velocity (V), and laser power (P); and (2) monitor and identify process conditions that are liable to cause porosity through analysis of in-process layer-by-layer optical images of the build invoking multifractal and spectral graph theoretic features. These objectives are important because porosity has a significant impact on the functional integrity of LPBF parts, such as fatigue life. Furthermore, linking process conditions to defects via sensor signatures is the first step toward in-process quality assurance in LPBF. To achieve the first objective, titanium alloy (Ti–6Al–4V) test cylinders of 10 mm diameter × 25 mm height were built under differing H, V, and P settings on a commercial LPBF machine (EOS M280). The effect of these process parameters on count, size, and location of pores was quantified based on X-ray computed tomography (XCT) images. To achieve the second objective, layerwise optical images of the powder bed were acquired as the parts were being built. Spectral graph theoretic and multifractal features were extracted from the layer-by-layer images for each test part. Subsequently, these features were linked to the process parameters using machine learning approaches. Through these image-based features, process conditions under which the parts were built were identified with the statistical fidelity over 80% (F-score).

## Introduction

### Background.

Powder bed fusion (PBF) refers to a family of additive manufacturing (AM) processes in which thermal energy selectively fuses regions of a powder bed [1]. Figure 1 shows the schematic of the PBF process. A layer of powder material is spread across a build plate. Certain areas of this layer of powder are then selectively melted (fused) with an energy source, such as a laser or electron beam. The bed is lowered and another layer of powder is spread over it and melted [2]. This cycle continues until the part is built. The PBF process embodied in Fig. 1 depicts a laser power source for melting the material; accordingly, the convention is to refer to the process as laser powder bed fusion (LPBF).

A galvanic mirror scans the laser across the powder bed. The laser is focused on the bed with a spot size on the order of 50–100 μm in diameter, the laser power is typically maintained in the range of 200–400 W, the linear scan velocity of the laser is varied in the 200–2000 mm/s range, and the distance between each stripe of the laser, called the hatch spacing, is maintained in the range of 100–200 μm. The distance through which the bed is lowered is termed the layer height and is typically in the range of 30–50 μm. [2]. Close to 50 other parameters are involved in the melting and solidification process in LPBF [3].

### Motivation.

The ability of LPBF to produce intricate geometry parts from hard-to-process materials, such as cobalt-chrome and nickel-based super alloys, has been conclusively demonstrated for a variety of demanding applications ranging from biomedical to aerospace [4,5]. Process repeatability and product quality, however, remain imposing barriers toward scaling LPBF to production environments [6]. Given the layer-by-layer nature of the process, a defect in a layer, if not averted, will be permanently sealed in by subsequent layers. These trapped defects adversely affect key functional properties of the part, such as its fatigue life and strength [7,8].

A major gap in the current research lies in the lack of quantitative models to correlate the effect of process conditions on specific defects, such as porosity via the data acquired from in situ sensors. Addressing this gap is the first-step toward in-process quality assurance in LPBF. Therefore, there is an urgent need to: (1) understand and quantify the effect of LPBF process conditions on defects, and (2) institute in-process sensing and monitoring to capture the onset of defects.

The following types of LPBF defects have attracted the most attention: porosity, surface finish, cracking, layer delamination, and geometric distortion. These defects are tracked to the following four root causes [9,10]:

• Poor part design, such as inadequately supported features [11].

• Machine and environmental factors, such as poor calibration of the bed and optics.

• Inconsistencies in the input powder material, such as contamination and deviations in particle distributions.

• Improper process parameter settings, for example, inordinately high laser power causes vaporization of the material leading to keyhole porosity, while insufficient laser power prevents powder particles from fusing together leading to large acicular pores [12,13]. This work specifically focuses on characterizing and detecting porosity in situ due to the improper selection of process parameters.

### Objectives.

The goal of this work is to quantify the effect of process conditions on part porosity in the LPBF process, and subsequently, detect the onset of porosity due to deviation in process conditions based on in-process sensor data. An example of such a possible deviation is the occlusion of the optics due to vaporization of the material during melting and its eventual condensation on the focusing lens. The gradual coating of residue on the laser will lead to loss of laser focus, and hence reduce the power delivered to the substrate without the knowledge of the operator. In extreme instances, because the residue deposited on the lens absorbs a significant portion of the incident energy, damage to the lens and optical train can occur [14].

In pursuit of this goal, the objectives of this work are twofold:

1. (1)

Quantify the effect of three LPBF process parameters, namely, laser power (P), hatch spacing (H), and velocity (V) on the size, count, and location of pores using X-ray computed tomography (XCT) scan data of the part.

2. (2)

Monitor and discriminate process deviations that are liable to cause porosity using in-process optical images of the powder bed invoking multifractal and spectral graph theoretic analysis.

The first objective is realized by simultaneously building nine titanium alloy cylinders on a commercial LPBF machine (EOS M280) at varying P, H, and V conditions, and quantifying their effect on the pore spatial distribution count, size and location using XCT images.

The second objective is achieved by acquiring layer-by-layer optical images of the parts while they are being built, and then extracting statistical, multifractal, and spectral graph theoretic features from these images. These features are subsequently used in various classification approaches such as neural networks to ascertain their ability to isolate process conditions that are liable to produce parts with severe pores.

The rest of this paper is structured as follows: A brief review of the literature focusing on porosity and in-process sensing in LPBF is presented in Sec. 2; Sec. 3 describes the experimental conditions and layer-by-layer acquisition of part images; Sec. 4 explains the spectral graph theory and multifractal analysis of in-process image data for feature extraction and process modeling; and conclusions and avenues for future work are presented in Sec. 5.

## Review of the Relevant Literature

The literature concerning the reasons and mechanisms of porosity formation and in-process sensing are summarized in Secs. 2.1 and 2.2, respectively.

### Effect of Laser Powder Bed Fusion Process Parameters on Porosity.

Of the various multiscale defects in LPBF, porosity and its attendant causes have garnered the most attention [10,1517]. According to Rao et al., voids or pores are empty spaces in a material and porosity is a measure of the volume occupied by these empty spaces over the total part volume [18]. Mechanical properties such as strength and fatigue performance of LPBF-processed parts are severely affected by porosity; pores cause high-stress concentration, which in turn results in crack formation [1922].

The formation of porosity is closely tied to and governed by the thermal phenomena at the meltpool-level [23]. Gong et al. have identified four distinctive regimes of melting contingent on the laser power (P) and velocity (V) process parameter settings. These regimes are demarcated as zone I (fully dense), zone II (over melting), zone III (incomplete melting), and overheating zone (OH) [20,24]. Visualizing a process map of laser power plotted on the ordinate axis, and the velocity on the abscissa, the region along the 45 degree slope falls under zone I, also termed as the conduction mode. In this region, parts with least porosity-related defects were obtained. Zone II is to the left of zone I, herein the laser power is higher for a given velocity compared to zone I. This region is home to the so-called keyhole mode melting, where, as experimentally and theoretically elucidated by King et al. material vaporization occurs due to excessive energy input [25]. Zone III is to the right of zone I, and is characterized by relatively higher velocity for a given power setting compared to zone I. In this zone (zone III), there is inadequate energy for the material to completely fuse.

While Gong et al. found that parts can be made in either of zones I, II, and III, however, parts could not be built in the OH zone, which is mapped to the left of zone II, because the layers tend to deform to such a high degree during the build that the deposition of subsequent layers is impeded. Gong et al. report that in their experiments, the recoater jams occurred in the OH zone due to contact with the part [20,24]. Similar process mapping results for other AM processes, such as powder and wire-fed directed energy deposition, and electron beam powder bed fusion are reported by Beuth et al. [2628]. Within the three melting zones, zone I–III, the mechanism, and nature of pores formed are distinctive.

Lack of fusion porosity occurs in zone III because the laser energy supplied is insufficient to fuse the adjacent tracks, and the current and previously deposited layers. Lack of fusion porosity results in the formation of large acicular pores of size in the range of 30–100 μm [4]. From an experimental perspective for Titanium alloy Ti–6Al–4V, Gong et al. correlate areal energy density ($EA=PH×V$ J/mm2) with porosity and observed the onset of lack of fusion porosity typically occurs for EA < 1.1 (approximately). Considering also the layer thickness T as a factor (maintained constant at 30 μm) the equivalent threshold for volumetric energy density $EV=PH×V×T$ is ≈ 36 J/mm3.

Keyhole-collapse porosity in zone II occurs due to vaporization of powder material [14]. King et al. elucidate through theoretical simulations and experimental studies that when the energy supplied by the laser is inordinately high, the laser melts through several layers of the powder vaporizing material in its path. The vapor cavity eventually collapses thus forming pores deep within the meltpool [25]. The pores resulting from operating in the keyhole melting mode are uniform and circular in shape and are typically on the scale of 10–20 μm [12]. Gong et al.'s studies indicate that as the energy density in the processing of Ti–6Al–4V increases beyond a threshold value (typically $EA$ > 2, $EV$ > 66), the process enters the keyhole melting mode [20,24].

To avoid oxidation of the powder, the LPBF process is carried out in a chamber filled with inert gas (usually argon or nitrogen) depending upon the material to be processed. The argon or nitrogen gas may get trapped in the powder and lead to the formation of gas pores [29]. Additionally, gas pores are also formed when bubbles are trapped in the meltpool during the solidification process [18]. Gong et al. also explain the formation of voids and pits due to the ejection of powder material as spatter on account of the thermal energy [20,24]. The ejected particles may settle within the boundary of the part, and on cooling may adhere to the surface of the powder bed. Further, as the next layer is being deposited, the adhered particles may subsequently be removed by the recoater leaving a pit or void in its place. Finally, lower melting impurities and constituents may vaporize given a sufficiently high energy density (and not due to keyhole collapse) leaving voids in the part [30]. Such types of pores are not restricted to one type melting zone and are stochastic in nature.

From the extensive experimental work of Gong et al., it is surmised that for Ti–6Al–4V material, the conduction melting mode typically occurs in the range of $1.1 J/mm2; or equivalently $36 J/mm3. Aboulkhair et al. [12,19] and Stucker and coworkers [3133] report extensive process optimization studies related to porosity in LPBF with conclusions in line with findings by Beuth et al. [2628]. While most of the existing process maps relate the effect of areal or volumetric energy density to porosity with the aid of XCT, a conspicuous gap remains in relating pore size, density, and location simultaneously with EA. This work addresses the foregoing gap through objective 1.

In closing this section, we note that the process zones and concomitant types of porosity reported in the literature are contingent on the presumptions of stable process operation and that the part geometry and its location on the build plate have negligible effect.

### Sensing and Monitoring in Laser Powder Bed Fusion.

Comprehensive review papers for in-process sensing are available in Refs. [9] and [3437]. Significant research in process sensing and control for metal AM processes is being done in academe and national laboratories [3843]. Nassar and coworkers experimented with imaging of the LPBF powder bed under various illumination conditions [40,43,44]. The resulting layer data were analyzed, and defects, such as voids caused by improper raking of the powder across the bed, were identified. Lane et al. at NIST integrated an LPBF machine (EOS M270) with thermal and high-speed cameras, and a photodetector [38]. NIST and Edison Welding Institute (EWI) are currently building a customized LPBF testbed instrumented with multiple sensors [42,45]. A large body of work in sensing and monitoring in LPBF is reported by the Kruth group [4648] and Witt group [4952] in Europe. Recent breakthroughs with in situ X-ray imaging of the LPBF process have been reported by scientists at Lawrence Livermore National Laboratories [53].

To detect evolving process anomalies, researchers have sought to incorporate sensing techniques such as vibration, CCD video imaging, infrared and ultraviolet imaging, pyrometers, photodiodes, ultrasonic wave generators in AM machines [49,5460]. An early example was presented by Melwin et al. [61], who used a video-micrography apparatus bearing band pass and polarizing filters for observing the meltpool in polymer LPBF.

In a series of related works, Craeghs et al. [4648] describe optical-based approaches for monitoring build quality in PBF by imaging the thermal behavior at the meltpool. Craeghs et al. were able to detect process defects, such as deformation and overheating using their optical system [47]. Bartkowiak [62] describes a PBF apparatus integrated with a spectrometer for in situ measurements of the layer melt characteristics, such as emissivity. Other researchers, e.g., Chivel et al. [63], and Jacobsmuhlen et al. [49], have also developed optical imaging systems for process monitoring in AM [63]. In a recent work, Rieder et al. [57] used an ultrasonic sensing system for tracking build status in PBF. A broadband ultrasonic sensor mounted on the underside of the build plate is used to detect voids, akin to acoustic microscopy.

Craeghs et al. [47,64,65] report that the amplitude of the photodiode signal is correlated with the melt-pool area and the melt-pool temperature. They subsequently use this information to identify process failures, such as detection of deformation due to thermal stresses and overheating at overhang structures, in each build layer. Further, they developed a feedback control sensor based on optical images. Chivel and Smurov [63] use two different wavelengths and selected temperature profiles to extract information of the bed temperature distribution, and the size of the meltpool for process monitoring.

Regarding the fidelity of the different sensing approaches for detecting defects specific to PBF AM processes, the viability of thermal imaging and optical spectroscopy-based techniques has been demonstrated in the literature. Recent work done by researchers at NIST aims to comprehensively capture the effect of meltpool shape and thermal gradients to defects. From the meltpool monitoring vista, a fast response thermal camera with a high framerate (> 1000 frames/second) and resolution in the micrometer range is typically used to circumvent blurring effects [66]. In recent work by EWI researchers, the meltpool-level thermal camera is coupled with another thermal camera that monitors the heat flux over the entire bed to detect large macro-scale defects, such as warping [45]. However, such high-fidelity thermal cameras are exceeding expensive, and moreover, they are appropriate for capturing thermal trends rather than the exact temperature of the target because the emissivity of the meltpool remains to be established. Dual color pyrometers can be used to circumvent the lack of emissivity information.

A far less expensive alternative to thermal imaging for detection of micrometer-level defects is through the use of photodetectors and spectrometers. Nassar et al. in a series of papers demonstrate the use of such optical emission spectroscopy-based sensing [41,67,68]. The key idea is to measure the intensity (amplitude) of the line-to-continuum ratio emission spectra of the material being processed and relate the readings to part defects. For this purpose, two photodetectors are coupled through a 50:50 beam splitter, and focused upon the entire bed area. Each of the photodetectors is fitted with an optical bandpass filter that captures light corresponding to the emission spectra of a particular element in the alloy being processed. For instance, for detecting anomalies in LPBF of Inconel 718, Nassar et al. used a 520 ± 5 nm and 530±5 nm optical bandpass filters corresponding to the continuum and line spectra, respectively, of Cr I emissions [67].

Instead of using two photodetectors to capture formation of porosity, Montazeri et al. in two papers published in this journal have used a single photodetector to capture the onset of material contamination, and also to distinguish the process signatures emanating for different feature geometries, such as overhang-related features [30,69]. While photodetectors and spectrometers present a cost advantage over thermal imaging, and are capable of sampling rates nearing 1 MHz, their main drawback is that the output is in terms of a time series or frequency spectrum, which have far limited information compared to thermal imaging.

In this context, the use of optical imaging for detection of conditions liable to produce porosity is a novel contribution of this work. Optical imaging cameras are significantly less expensive than their thermal and high-speed counterparts. However, the challenge of capturing pores directly from the layerwise optical images, as opposed, to the anomalous process conditions has not yet been attempted. In closure, we note that Abdelrahman et al. [44] have used optical imaging data to capture the large-scale (>100 μm) defects, which were deliberately introduced during the build.

The main drawback in most of these studies is that they do not connect practical process conditions to defects, but rather focus on artificially inducing flaws by way of catastrophic process anomalies. Furthermore, the analytical techniques rely on classical time-series signal processing techniques, which may not be effective in capturing subtle defects. Recent progress to overcome this limitation is reported by the Clare group at Nottingham University who have used spatially resolved acoustic spectroscopy to detect porosity ex situ in LPBF, wherein the amplitude of a surface acoustic wave generated by laser is correlated with the location and severity of porosity at different laser power settings [70,71]. The current work addresses this extant gap through objective 2.

## Experimental Setup and Data Acquisition

Experiments were conducted on an EOS M280 LPBF machine. The input material was a Titanium alloy, ASTM B348 Grade 23 Ti–6Al–4V powder material whose particle size ranges from 14 $μm$ to 45 $μm$. The parts analyzed in this study are cylinders, which were printed by varying the hatch spacing (H), scan velocity (V) and laser power (P). The cylinders are 25 mm in length and 10 mm in diameter, showing the seven process parameter settings, which were used to print these cylinders. The nominal settings are labeled as H0 = 0.12 mm, V0 = 1250 mm/s, and P0 = 340 W. The layer height maintained is constant at T = 60 μm. Hatch spacing and laser print velocity are increased by 25% and 50%, and laser powder has been decreased by 25% and 50% from their nominal settings. The three process settings are aggregated in terms of the areal energy density applied for melting called the Andrew number: EA = $(P/H×V)$ J/mm2 or the volumetric energy density $EV=(P/H×V×T)$. Comparing the EV values reported in Table 1 with the experimental results of Gong et al. [20,24], we note that barring the nominal settings, which is set in the conduction regime (zone I), all other experimental treatment combinations fall within the lack of fusion (zone III) regime where acicular pores are expected (EV < 36).

A digital single-lens reflex camera (DSLR, Nikon D800E) along with multiple flash-lamps placed inside the build chamber is used to capture the layer-by-layer powder bed images. Images are captured at two instances in every layer, namely, post laser scan and post re-coat. The camera shutter is controlled by a proximity sensor that registers the location of the recoater blade. Five images of the powder bed images are captured under bright-field and dark-field flash settings. The layout of the camera and flash-lamp location is shown in Fig. 2, and the representative images under the five light schemes are shown in Fig. 3. In this work, images from the bright-field light scheme in Fig. 3(a) are analyzed. Details of the experimental setup are available in Ref. [44].

## Methodology and Results

As shown in Fig. 4, the LPBF process data are analyzed in two phases, namely, (1) offline analysis of XCT data in Sec. 4.1; and (2) analysis of in situ images of the powder bed in Sec. 4.2.

### Phase 1: Offline Analysis of Porosity.

This section aims to analyze the effect of hatch spacing (H), laser velocity (V), and laser power (P) on the count, size, and location of pores. Representative XCT images of parts under different P, H and V conditions are shown in Fig. 5. A visual inspection of the XCT scans shows that the size and number (count) of the pore is inversely proportional to the areal energy density (EA).

As the areal energy density (Andrew's number, EA) is reduced, we observed that the size and number of the pores become larger. However, we caution that, although the critical process parameters, such as laser power (P, Watt), hatch spacing (H, mm), scan velocity (V, mm/s), and layer height (mm), can be optimized for certain part geometries, and aggregated in terms of the global volumetric energy density (EA) pores can still occur. This is because, EA does not account for the thermal aspects in the part (heat flux), which is contingent on the part geometry, orientation, and its location on the build plate. For instance, parts in the far edge of the build platen (near the end of the recoater action) may suffer from insufficient powder feed (powder shorting); likewise, the laser spot size is liable to change as the laser tends to defocus on the outer edge of the build platen leading to lack-of-fusion-related porosity.

Furthermore, there is the possibility of a complex, nonlinear interaction between P, V, and H, which remains as yet undiscovered and therefore not captured in the relationship representing the areal energy density (EA). For instance, in the equation for EA, all terms are assumed to be equal in weight, i.e., the exponent P, V, and H is unity (=1) and therefore the relationship between EA and the process parameters is implicitly assumed to be a simple linear relationship. The following inference is made based on Fig. 5. For instance, while the severity of pores is influenced by all three process parameters, laser power (P) seems to have an inordinately high effect. This observation is further quantified by extracting count, size, and location attributes by analyzing the XCT scan images through the steps shown in Fig. 6.

• Figure 6(a)—XCT scans for 30 randomly chosen cross-sectional areas are analyzed.

• Figures 6(b) and 6(c)—The XCT scan images are binarized based on a heuristically determined threshold. Some information is inevitably compromised during the binarization process. A complement of the binary image is taken to return a black background, which makes computation easier as the image matrix becomes sparse.

• Figure 6(d)—To reduce noise induced due to binarization the nearest neighborhood approach is used [72]. We note that while it is customary to refer to voxels in the context of XCT, because the images are converted to binary images (binarized), we revert to using the term pixel. In this procedure, a binarized XCT pixel is labeled as a defect only if it is connected to the eight-nearest pixels. In other words, if the eight nearest neighboring pixels of a particular pixel are also bright (i.e., 1), then the pixel is deemed to represent part of a defect.

Next, the pore count, size, and location are extracted as follows:

• Pore count—The number of eight-connected binarized XCT pixel over a layer translates to the pore count.

• Size of pores—The size of a pore is grouped into one of five classes contingent on its radius. Each pore is considered as an annular structure on the noise reduced image, and then, the number of pixels within each annulus is calculated. Depending on the number of pixels in the annulus, the pores are classified into various radii, namely 1–5 pixel radii. A radius of one-pixel unit equates to a pore radius of 16 μm on the part.

• Pore location—The pore location is determined by segmenting the XCT scan image into 5 concentric areas as shown in Fig. 7. The number of pores in each 1 mm thick segment of the XCT scan image is then counted. This establishes the distance of the pores from the center of the cylinder.

#### Effect of process parameters on count and size of pores.

Analysis of the XCT scan images shows that decrease in the areal energy density (EA) leads to an increase in the count (number of pores) and size of pores. This effect of laser power (P), hatch spacing (H), and laser print velocity (V) on pore count and size are exemplified in Fig. 8 from which the following inferences are drawn. In Fig. 8, the x-axis is the pore size, and the y-axis is the mean count (or number) of the pore observed on 30 randomly selected slices of the XCT scan. These results are also detailed in Table 2, which reports the mean number of pores, rounded to the nearest integer, along with the standard deviation for 30 randomly chosen layers.

• Referring to Fig. 8(a), the pore distribution in terms of count versus pore size is plotted for different levels of laser power (P). The decrease in laser power by 50% (170 W) leads to almost a 100-fold increase in the number of pores. Further, parts produced under P-50% (170 W) have pores ranging from 1 pixel to 4 pixels in size, i.e., 28 μm to 112 μm, whereas parts produced under nominal power (P0 = 340 W) and P -25% (270 W) have pores of radius 2 pixels (∼ 32 μm) at most.

• Referring to Fig. 8(b), increasing the hatch spacing (H) leads to an increase in both the count and size of pores. The magnitude of the effect of laser hatch spacing is significantly smaller than that of laser power. In case of varying hatch spacing (Fig. 8(b)), the highest number of pores is seen in the cylinder, which is printed with H +50%, i.e., 0.18 mm hatch spacing. From Fig. 8(b), for all the three levels of hatch spacing, the largest pore radius observed is 2 pixels.

• Referring to Fig. 8(c), akin to hatch spacing, increase in laser print velocity (V) leads to increase in count and size of pores. The largest pore size of radius 3 pixels (∼48 μm) was recorded in the cylinder printed with V +50% (1875 mm/s). The effect of velocity on porosity is least consequential of the three factors studied in this work.

#### Effect of Process Parameters on the Location of Pores.

The location of pores in the test cylinders is determined by segmenting the XCT scan image of a cylinder into five concentric parts as described previously in the context of Fig. 7. This establishes the distance of the pores from the center of the cylinder. The mean and standard deviation of pores in each segment of the part for 30 randomly chosen layers are reported in Table 3 and depicted in Fig. 9, from which the following inferences are drawn:

• Referring to Fig. 9(a), it is evident that as the laser power decreases, more number of pores are recorded in the L2 (4–8 mm) to L4 (8–12 mm) segment, of the cylinder. Figure 9(a) further reveals that the cylinder printed with nominal laser power (340 W) has most number of pores in the first two annular segments of length L1 (0–4 mm) and L2 (4–8 mm), which indicates that the pores are located close to the center. This trend is also observed in the cylinder printed with P -25% laser power (270 W). In contrast, the cylinder printed with -50% laser power has most number of pores in the third segment (8–12 mm).

• Referring to Figs. 9(b) and 9(c), in cylinders printed with varying hatch spacing (H) and laser print velocity (V), respectively, it is observed that parts produced at +50% hatch spacing (0.18 mm) and laser print velocity (1875 mm/s) have the highest number of pores at the radial distance with L3 (8–12 mm). Pores in the cylinders printed with +25% and nominal hatch spacing and laser print velocity are mainly located in the first two segments 0–1 mm and 1–2 mm.

The sharp drop in porosity in L5 is likely due to the reason that the external boundary of the part is scanned with increased EA after the rest of the part (postcontour melting). The added heat at the periphery mitigates porosity in L5. Further, the concentration of heat in the core of the part may explain the reduced porosity toward the center (L1). Finally, the effect of thresholding to convert may lead to a loss of information; this last reason can be largely discounted in the light of Figs. 6(a) and 6(d), wherein pores in the boundaries are captured appreciably.

### Phase 2: Analysis of Online Data of Laser Powder Bed Fusion Process.

This section aims to link the process conditions to the layer-by-layer images of the parts as they are melted. This will allow detection of process drifts in their early stages. For this purpose, two methods are proposed, the first based on spectral graph theory, and the second using multifractal and lacunarity analysis.

#### Application of Spectral Graph Theory for Part Image Analysis.

Spectral graph theoretic Laplacian eigenvalues extracted from online images are used to identify the process conditions under which a part is produced. The approach has the following two steps.

Step 1: Representing the image of each part as a graph.

A layerwise image obtained from the DSLR camera for a laser sintered cylinder layer with M × N pixels can be represented by a matrix XM×N. As shown in Fig. 10, each row of the matrix X is considered as a row vector and it represents a node or vertex (V) of an undirected graph, which is denoted as $G≡V,E$, where E is the edges in the graph [73]. The M row vectors of the matrix $X$ are represented as $aK,K=1,2,…M}$.

Further, a pairwise comparison is performed between each of the row vectors through a kernel function $Ω$ [74]. A pairwise comparison along the columns has been shown to lead to similar results as long as the image is homogeneous [75]
$wpq=Ω(ap,aq)∀p,q∈K$
(1)
The kernel function $Ω$ used in this study to compute the pairwise comparison is the radial basis kernel function (Eqs. (2) and (3))
$wpq=e−EσX2$
(2)

$E=ap−aq2$
(3)
where $σX$ is the overall standard deviation of $E$. Next, a binary similarity matrix $S=wpq$ is created with help of a threshold function. This threshold function $θ$ when applied to $wpq$ converts it into binary form [76]
$Θwpq=wpq=0,1$
(4)
This threshold function facilitates in determining whether there is a connection between two nodes [76]. $wpq$= 1 if there is a connection and otherwise it is zero.
$Θwpq=wpq=1,wpq≤r0,wpq>r$
(5)
Here, $r$ is given by
$r=∑p=1p=M∑q=1q=MwpqM2$
(6)

Step 2: Extracting features from the graph.

Once a graph is formulated from the image, topological features are extracted from the graph. These features are useful in classification of parts which are made with different process parameters. The first step toward feature extraction is computing the degree $dp$ of a node $p$, i.e., the number of edges that pass through the node $p$. The degree of node $p$ is computed by summing each row in the similarity matrix $S$. From the degree of node $dp$, a diagonal degree matrix $D$ is formed as follows:
$D=defdiagd1,…,dM$
(7)
Now, with the help of the degree $D$ matrix and the similarity matrix $S$, the normalized Laplacian $L$ of the graph is defined as follows:
$L=defD−12×D−S×D−12$
(8)

where $D−12=diag1/d1,…,1/dM.$

Finally, the Eigen spectra of the Laplacian are computed as follows [77]:
$Lv=λ*v$
(9)
The eigenvalues (λ) of the Laplacian are used in the classification of LPBF parts per their processing conditions. In this work, the first five smallest nonzero eigenvalues are used. Also, the Kirchhoff index for each graph is computed as follows, where $λi$ are the nonzero eigen values of the Laplacian:
$Kf=2×ε×∑i=2Mλi−1$
(10)

where $ε=∑i=1i=M∑j=1j=MSij/2$

The nonirradiated part of the part image, i.e., the unsintered powder, is fairly homogenous, so when the image undergoes a row-wise comparison, the distance kernel function becomes zero. The nodes which are far apart from each other are connected on the graph.

#### Multifractal and Lacunarity Analysis of Part Images.

The fractal dimension has been extensively used to characterize the texture and patterns of manufactured surfaces [17,7881]. This work goes beyond the traditional methods that extract a single fractal dimension from the surface image, but rather assume the irregularity and nonhomogeneity of image data are due to the presence of several fractal dimensions [81]. As such, we extract a spectrum of multifractal features to characterize the layer-by-layer images obtained in LPBF. A fractal is defined as a shape that embodies geometric similarity across multiple scales [8284]. Assuming that a fractal object occupies a limited area in the Euclidean space, then the object can be covered by N measure elements with size $ε$ as follows:
$Nε=ε−D$
(11)
where D is the fractal dimension. The box-counting method is widely used to estimate the fractal dimension of an irregular object. This method covers a fractal set with measure elements (e.g., box) at different sizes and observes how the number of boxes varies with its size [85]. This procedure is repeated using different boxes of size l. Once the l becomes sufficiently small, N(l) being the number of boxes that are needed to cover a fractal object with the size l, then the box-counting dimension $D0$ is defined as
$D0=liml→0lnNlln1l$
(12)

For example, Fig. 11 shows three types of fractal objects called multifractal trees that are constructed with the iterated function systems (IFS) method. These fractal trees are labeled T1, T2, and T3. The estimates of fractal dimension ($D0$) using the box-counting method in Fig. 11 are $D0=2.0449$ for all three fractal trees. However, three trees show high levels of self-similarity, irregularity and heterogeneity due to the presence of a spectrum of fractal dimensions. This demonstrates that the traditional box-counting fractal dimension is limited in the ability to fully characterize the patterns of multifractal objects [86]. Multifractal analysis provides a means to overcome this limitation of traditional fractal dimensions. The procedure to estimate the multifractal spectrum from image data is as follows:

Step 1: Estimating the local densities function ($PiL$).
$PiL=NilNT$
(13)

where $Nil$ is the number of mass or pixels in the ith box of size l, $NT$ is the total mass of a set and $Pil$ is the probability in the ith box.

Step 2: Calculating singularity strength exponent ($lαi$).
$Pil∼lαi$
(14)
where $αi$ reflects the local behavior of Pi (l) in the ith box with size l and it can be derived as
$αi=liml→0lnPillnl$
(15)

Step 3: Estimating multifractal spectrum ($fα$).

The multifractal spectrum $fα$ is the fractal dimension of the set of locations that have same values for singularity strength exponents $αi$. Given the number of boxes $Nα$ where the probability $Pil$ has exponent values between $α$ and $α+dα$ the multifractal spectrum $fα$ can be calculated as follows:
$fα=liml→0lnNαln(1l)$
(16)

Step 4: Characterizing multifractal measures (Dq).

Multifractal measures are characterized by the scaling of the qth moments of $Pil$ distributions as
$∑i=1NlPiql=lτq$
(17)
where $τq$ is called the mass exponent of qth-order moment. Then, the generalized fractal dimensions $Dq$ can be written as
$Dq=τqq−1$
(18)
Then, the Legendre transformation is used to derive the multifractal spectrum as
$fαq=qαq−τq$
(19)

$αq=dτqdq$
(20)
However, Legendre transformations are computationally demanding in the calculation of $fα$. Also, this approach requires smoothing the $Dq$ curve e, which causes errors in the estimated $fα$ [87]. To overcome this limitation and bypass intermediate smoothing steps in estimating $fα$, a family of normalized measures $μiq,l$ as qth moments of mass probability Pi(l) are introduced in Eq. (21). A constant l range is also used to avoid multifractal properties over a small interval of scales.
$μiq,l=Piql∑i=1NlPiql$
(21)
As such, the multifractal spectrum $fα$ and the average singularity strength exponent $αq$ can be written as
$fαq=liml→0∑i=1Nlμiq,llnμiq,llnl$
(22)

$αq=liml→0∑i=1Nlμiq,llnPiqllnl$
(23)

Figure 12 shows the multifractal spectra for three IFS trees in Fig. 11. It is evident that multifractal features effectively distinguish the differences in the three IFS trees that were not captured using the traditional fractal dimension. Note that the tail of the third IFS tree T3 is longer than other two IFS trees. Because T3 has more pixels with lower values (value toward 0 or black pixels) in comparison to the other two trees and the $fαq$ spectrum intensifies the effect of pixels with lower values.

Furthermore, lacunarity complements multifractal analysis by characterizing the manner or distribution in which the fractal objects fill the space [88,89]. Lacunarity and multifractal analysis jointly describe the irregularity and non-homogeneity in fractal objects as well as how they fill the space that cannot be otherwise achieved by traditional box-counting dimension or statistical features. To obtain the lacunarity measure, a unit box of size l is placed over the object and the number of set points s (black pixels) in the image is counted—this is called the box mass. Next, the box is translated one space along the set, and the box mass is again determined. This process is repeated over the entire set, creating a frequency distribution of the box masses represented as N(s,l). This frequency distribution is converted into a probability distribution Q(s,l) by dividing by the total number of boxes N(l) of a given size l [90,91]
$Qs,l=Ns,lNl$
(24)
The first and second moments of this distribution can be written respectively as
$Z1=∑sQs,l$
(25)

$Z2=∑s2Qs,l$
(26)
The lacunarity method with box size l can be computed as
$Λl=Z2Z12$
(27)

In Eq. (27), Λ(l) represents the lacunarity for the box size l. This procedure is repeated for different box sizes, and a log-log plot of the lacunarity versus the size of the box is traced. Figure 13 shows T3 has higher lacunarity values in comparison to the two other trees. The distribution of gap sizes is termed as lacunarity.

Figure 14 shows the singularity strength exponent α(q) and multifractal spectrum f(α(q)) estimated from 3132 layerwise images in the LPBF process. There are 1044 images in EA = 2.27; 696 in EA = 1.81; 348 in EA = 1.70; 696 in EA = 1.51; and 348 in EA = 1.13. Note that multifractal spectra of these images show significant variations with respect to the different Andrew's numbers.

### Application of Multifractal and Spectral Graph Theory to Online Images.

Further, the parts built under the different EA conditions described in Table 1 were classified using different machine learning approaches with various types of input features. A 70%-15%-15% split for training, testing, and validation data were imposed. The classification fidelity is reported in terms of the F-score, which is an aggregate of the types I and II statistical errors. The results are summarized in Table 4.

Three types of input features are used: (1) statistical image features, namely, intensity (mean) of an image, and local standard deviation of an image in 3 × 3 neighborhood, (2) spectral graph theoretic features, namely, the first five nonzero Eigenvalues and the Kirchhoff index, and (3) the multifractal and lacunarity features. It is observed that irrespective of the classification approaches used, the spectral graph and multifractal and lacunarity features outperform the conventional statistical features. Furthermore, combining the spectral graph and multifractal features results in F-score around 80%. The results reported in Table 4 show that the spectral graph theoretic and multifractal features discriminate the part quality with higher fidelity than traditional statistical analysis. This is valuable from the in-process quality monitoring viewpoint. In a practical scenario, images of the parts can be used to conclude the process within an optimal window.

## Conclusions and Future Work

This paper presents the modeling and analysis of in-process layerwise images in LPBF to investigate the effect of LPBF process conditions on the severity, size, and location of porosity, and further connects the process conditions to sensor signatures. This is an indirect way to monitor the LPBF process. The specific outcomes of the work are as follows:

1. (1)

Three process parameters, namely, laser power (P), hatch spacing (H), and scan velocity (V), were varied during the LPBF of Ti–6Al–4V powder material. The effect of varying these parameters on porosity was characterized offline using XCT. Based on the analysis of the XCT images, the following inference is tendered. Decreasing the laser power by 50% from 340 W to 170 W leads to almost a threefold increase in the average number of pores, compared to an equivalent percentage increase in hatch spacing, and tenfold increase compared to scan velocity. Hence, the control of laser power is most consequential for avoiding porosity.

2. (2)

Online visible spectrum images of the part were acquired as they are built using a still camera. These images were analyzed using multifractal and graph theoretic approaches. The features extracted by applying these approaches were subsequently used within various machine learning techniques. The aim was to distinguish the process conditions under which the parts were built given an image of the part. It is observed that combining multifractal and graph theoretic analysis leads to as much as 30% increase in the accuracy of discriminating process conditions compared to using traditional statistical measurements. Using this approach, the process conditions can be isolated with F-score approaching 80%. From a practical perspective, although the P, H, and V settings are predetermined for each material in terms of the Andrew number (EA), the laser power, particularly, is liable to drift due to occlusion of the focusing optics; the vaporized material tends to condense on the lens especially during long builds.

One limitation of this work is that it does not relate the sensor signatures directly to the defects, but rather isolates the process condition that leads to porosity. This is mainly due to the fact that the resolution of the camera is not sufficient to identify pores, which are in the 1665 μm range, from the images directly. To overcome this drawback, data from multiple sensors will be combined (e.g., thermography and meltpool monitoring) to not only capture multiple types of defects simultaneously but also improve upon the detection fidelity. Furthermore, in the future work, the authors will endeavor to understand the effect of process parameters on other type of defects, such as distortion and geometric inaccuracy.

## Acknowledgment

The authors acknowledge the help provided by Mr. Cheng-Bang Chen (Penn State), Mr. Ben Bevans (UNL), and Ms. Emily Curtis (UNL) in the analysis of X-ray computed tomography images. The author (HY) also thanks Harold and Inge Marcus Career Professorship for additional financial support. We gratefully acknowledge the valuable contributions of the faculty, staff, and students at Penn State's Center for Innovative Materials Processing through Direct Digital Deposition (CIMP-3D) for providing the data utilized in this research.

The authors finally thank the two anonymous reviewers, and associate editor – Dr. Sam Anand; their diligence, time, and insightful suggestions have gone a long way in making this work more rigorous.

## Funding Data

• Directorate for Engineering National Science Foundation Grant Nos. CMMI-1646660, CMMI-16171488, CMMI-1719388, CMMI-1739696, and CMMI-1752069.

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