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rag_docs_final_review_tex_merged/beta_titanium_texture.tex

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+\documentclass[10pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage[version=4]{mhchem}
+\usepackage{stmaryrd}
+\usepackage{hyperref}
+\hypersetup{colorlinks=true, linkcolor=blue, filecolor=magenta, urlcolor=cyan,}
+\urlstyle{same}
+\usepackage{graphicx}
+\usepackage[export]{adjustbox}
+\graphicspath{ {./images/} }
+
+\title{Towards 3-D texture control in a $\beta$ titanium alloy via laser powder bed fusion and its implications on mechanical properties }
+
+
+\author{Sravya Tekumalla ${ }^{\mathrm{a},{ }^{*}}$, Jian Eng Chew ${ }^{\mathrm{b}}$, Sui Wei Tan ${ }^{\text {c }}$, Manickavasagam Krishnan ${ }^{\text {c }}$,\\
+Matteo Seita ${ }^{\mathrm{a}, \mathrm{b}, \mathrm{d}}$\\
+a School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore\\
+${ }^{\mathrm{b}}$ School of Materials Science and Engineering, Nanyang Technological University, Singapore 639798, Singapore\\
+${ }^{c}$ Advanced Remanufacturing Technology Centre (ARTC), Agency for Science, Technology and Research (A*STAR), Singapore 637143, Singapore\\
+${ }^{\mathrm{d}}$ Singapore Centre for 3D Printing, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore}
+\date{}
+
+
+%New command to display footnote whose markers will always be hidden
+\let\svthefootnote\thefootnote
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+    \else\svfootnotetext[#1]{#2}\fi%
+  \fi
+}
+
+\begin{document}
+\maketitle
+Research paper
+
+
+
+\section*{A R T I C L E I N F O}
+\section*{Keywords:}
+Crystallographic texture control
+
+Beta Ti alloy
+
+Elastic modulus
+
+Mechanical properties
+
+Deformation behavio
+
+\begin{abstract}
+A B S T R A C T Fusion-based additive manufacturing (AM) offers a new opportunity to design metallic materials with complex, direction-dependent mechanical properties by controlling the orientation distribution of the constituent grains-also known as texture. Texture control may be achieved site-specifically by varying the processing variables and tailoring the local melt pool geometry and solidification kinetics. However, the type and direction of textures currently achievable in AM alloys are limited by the melt pool geometry itself. In this work, we advance the capabilities of controlling crystallographic texture in laser powder bed fusion (LPBF) by producing specimens of $\beta$ titanium-niobium alloy with textures aligned along three different directions: the laser scan direction (SD), the build direction (BD), and-for the first time-the direction perpendicular to both BD and SD (i, e., PD). We achieve this three-dimensional (3-D) texture control by fine tuning the keyhole melt pool geometry and amount of overlap throughout the build. We test the tensile properties of the three different specimens along their respective texture axes and elucidate the relationships between crystallographic orientation, mesostructure, and mechanical behavior. We find that the novel PD texture exhibits the best combination of strength, strain hardenability, and ductility. We ascribe these results to the unique mesostructure of this specimen. This work opens new opportunities for designing novel materials with directional properties by achieving threedimensional (3-D) texture control during fusion-based AM.
+\end{abstract}
+
+\section*{1. Introduction}
+The combined freedom of design and ability to control the microstructure of materials site-specifically [1,2] make additive manufacturing (AM) a promising technology to produce metal parts with complex geometries [3,4] and tailored properties [5,6]. Several studies have demonstrated the possibility of tuning features such as porosity [7], grain structure and morphology [1,8,9], composition [10-13], and residual stresses [14] using different AM processes, and the beneficial effects that these features may have on part properties. Another promising strategy that has raised a lot of interest across the academic community relies on controlling the crystallographic texture of metal alloys produced via fusion-based AM processes [15-19]. Notable examples are previous works focused on face centered cubic (FCC) alloys-including stainless steel 316 L [17,18] and Inconel 718 [15]-and body-centered cubic (BCC) alloys-such as titanium-based alloys [19]—produced by laser powder bed fusion (LPBF). The interest behind controlling the crystallographic texture is motivated by the additional opportunities to impart directional mechanical properties to the build by aligning the resulting textures along specific directions [5, $16,19,20]$. This capability offers the opportunity to design and produce materials that combine microstructures with largely different mechanical behaviors, which may unlock novel deformation mechanisms $[5,21$, 22]. It may also be useful to site-specifically tailor materials properties that are indirectly related to texture, such as those governed by grain boundaries $[23,24]$.
+
+Strong crystallographic textures in powder bed fusion-based metal AM result from the directional solidification of metal alloys-in the form of cells or dendrites-which follow the steep thermal gradients promoted during the localized melting of powders [6,19]. To date, the main strategies for crystallographic texture control in powder bed fusion-based metal AM include tuning the processing parameters to vary
+\footnotetext{\begin{itemize}
+  \item Corresponding author.
+\end{itemize}
+
+\href{https://doi.org/10.1016/j.addma.2022.103111}{https://doi.org/10.1016/j.addma.2022.103111}
+
+Received 28 February 2022; Received in revised form 19 August 2022; Accepted 22 August 2022
+
+Available online 27 August 2022
+
+2214-8604/© 2022 Elsevier B.V. All rights reserved.
+}
+the ratio between thermal gradient and solidification velocity at the solid/liquid interface [15], or the melt pool shape and thus, the solidification direction [16-18,25].
+
+The current challenge in this texture control materials design paradigm, however, is that texture orientation in the build cannot be arbitrary. Strong, bulk $<100>$ crystallographic textures in cubic alloys are typically found parallel to the build direction (BD) [20,26] or along the laser scanning direction (SD) [16-18,27-32]. These $<100>$ textures result from epitaxial growth of crystals along these specific directions following local and global thermal gradients during production, which depend on the melt pool geometry. In cubic alloys, in fact, the easy growth direction of crystals is $\langle 100>$ [16,33-35]. Since thermal gradients are perpendicular to melt pool boundaries [16], it becomes difficult to align them along other arbitrary directions within the build and achieve three-dimensional (3-D) texture control. However, this capability would open a new degree of freedom in the design of structural alloys with controlled functionality of the materials such as mechanical [5,18,36], corrosion [18,37], and magnetic properties [38]. A notable application would be for cellular solids, where aligning crystallographic texture with strut orientation could lead to significant improvements in mechanical performance [39].
+
+This work aims at advancing texture control capabilities in LPBF by using $\beta$-type titanium-niobium (Ti-Nb) alloy. This class of Ti alloys exhibits low modulus and excellent bio-mechanical properties, making them suitable for biomedical implants owing to their high compatibility with human cortical bone $[20,26,40]$. In this context, 3-D texture control would be extremely valuable to produce structures with tuneable elastic properties, since the modulus of elasticity is a function of crystallographic texture $[20,26,40]$. We demonstrate the ability of promoting $<100>$ textures along all the three reference coordinate axes, namely the BD, the SD, and the direction perpendicular to both SD and BD (i.e., PD). We achieve this result by promoting keyhole melt pools in the alloy. We elucidate the different texture formation mechanisms at play in the three different cases and assess the corresponding mechanical properties of the material. While the elastic modulus along the $<100>$ axis is the same regardless of texture direction, we find that the samples with the $<100>$ texture along PD exhibit the highest strength, work hardening, and ductility. Our results open new opportunities for tuning texture-dependent properties of $\beta-\mathrm{Ti}$ alloys and may be extended to other cubic alloys such as stainless steel 316 L and Inconel 718.
+
+\section*{2. Materials and methods}
+\subsection*{2.1. Materials}
+Our experiments involve in-situ alloying [41] of gas atomised, pure, and spherical titanium powders (CP Ti, Grade 2 ASTM B348, supplied by Tecnisco Advance Material Pte Ltd, Singapore) and niobium powders ( $99 \%$ purity, supplied by Stanford Advanced Materials, California, USA) during LPBF process. The powder particle size distribution of Ti ranged between $20 \mu \mathrm{m}$ and $63 \mu \mathrm{m}$, with a median diameter, $\mathrm{d}_{50}$ of $42.9 \mu \mathrm{m}$. Since the melting temperature of $\mathrm{Nb}(\sim 2750 \mathrm{~K})$ is significantly higher than that of $\mathrm{Ti}(\sim 1941 \mathrm{~K})$, we chose Nb powders with reduced particle size distribution between $15 \mu \mathrm{m}$ and $45 \mu \mathrm{m}$ to ensure complete melting using our LPBF process parameters (see Section 2.2). Nb is a $\beta$-phase stabiliser in Ti alloys, with a molybdenum equivalency $\left(\mathrm{Mo}_{\mathrm{eq}}\right)$ contribution equivalent to $0.28(\mathrm{wt} \% \mathrm{Nb})$ [42]. Mo $\mathrm{eq}_{\text {eq }}$ is a metric that defines the stability of a $\beta$ phase in a Ti-base alloy at ambient temperatures [42] and is estimated as the sum of the weighted averages of the elements as given in Eq. (1) below:
+
+
+\begin{align*}
+M o_{e q .} & =1.0(\mathrm{wt} . \% \mathrm{Mo})+0.67(\mathrm{wt} . \% \mathrm{~V})+0.44 \quad(\mathrm{wt} . \% \mathrm{~W}) \\
+& +0.28(\mathrm{wt} . \% \mathrm{Nb})+0.22(\mathrm{wt} . \% \mathrm{Ta})+2.9 \quad(\mathrm{wt} . \% \mathrm{Fe}) \\
+& +1.6(\mathrm{wt} . \% \mathrm{Cr})+1.25(\mathrm{wt} . \% \mathrm{Ni})+1.70 \quad(\mathrm{wt} . \% \mathrm{Mn}) \\
+& +1.70(\mathrm{wt} . \% \mathrm{Co})-1.0(\mathrm{wt} . \% \mathrm{Al}) \tag{1}
+\end{align*}
+
+
+It may be noted that since Al is an $\alpha$ stabilizer, therefore, the minus sign in Eq. (1) is used to account for the negative weighted contribution from Al . Ti alloys with $10<\mathrm{Mo}_{\mathrm{eq}}<30$ are known to retain a metastable $\beta$ phase. Therefore, in this work, we chose an alloy composition of Ti$45 \mathrm{wt} \% \mathrm{Nb}$ ( $30 \mathrm{at} \%$ ) which translates to a Mo $\mathrm{Moq}_{\text {eq }}$ value of 12.6 (obtained by substituting $45 \mathrm{wt} \%$ in Eq. (1) as Nb 's contribution and 0 from the other elements since the alloy is binary).
+
+\subsection*{2.2. Material processing}
+We mixed the pure elemental Ti and Nb powders using a highcapacity tumbler mixer (Inversina 20 L , Bioengineering AG) in an argon atmosphere at a rotational speed of 30 rpm for 6 h . The resulting uniform powder blend is shown in Fig. 1a. We carried out our LPBF experiments using an EOS M290 system equipped with a 400 W Yb-fiber laser in Ar atmosphere. We maintained a constant layer thickness of 0.03 mm for the $\mathrm{Ti}-45 \mathrm{Nb}$ alloy powders and used a bidirectional scanning strategy with no rotation in-between layers across all experiments. We varied laser power, scan speed, and hatch spacing, and studied how these parameters affect the resulting microstructure and crystallographic texture. For the initial design of experiments, we printed cube samples with a side of 8 mm . We report a summary of all these experiments and the corresponding microstructural analysis in the Appendix and in Fig. A1. Here, we only show and discuss those which led to builds with low porosity and low fraction of un-melted Nb . From this database, we selected three sets of parameters giving rise to samples with strong $<100>$ textures along SD, BD, and PD. These textured samples were then printed again with dimensions of 23 (L) $\times 10$ (B) $\times 10(\mathrm{H}) \mathrm{mm}^{3}$, which we show in Fig. A1e, to carry out detailed characterization and testing. Hereafter, we refer to these textures as $<100>\| S D$, $<100>\| \mathrm{BD}$, and $<100>\| \mathrm{PD}$, respectively, and the corresponding samples as SD-sample, BD-sample, and PD-sample, respectively.
+
+\subsection*{2.3. Microstructural characterization and orientation mapping}
+We determined the relative densities and porosities of the printed cube Ti-45 Nb samples through Archimedes' principle using the XS204 Density Kit (Mettler Toledo, Columbus, Ohio, United States). To assess the alloys microstructure, we prepared the samples following standard metallographic techniques, which include grinding, polishing (using $\mathrm{OPS}+20 \% \mathrm{H}_{2} \mathrm{O}_{2}$ ), and etching with Kroll's Reagent ( 2 vol $\%$ Hydrofluoric acid, 6 vol $\%$ Nitric acid, 92 vol $\%$ water). We used an Axioscope 2 (Carl Zeiss AG, Germany) optical microscope to study the melt pool geometry and a JEOL 7800 F Prime field emission scanning electron microscope (FESEM) equipped with a Symmetry S2 detector by Oxford Instruments (UK) to visualize the microstructure of our specimens and carry out electron backscattered diffraction (EBSD) measurements on the YZ plane i.e., PD-BD plane. We run EBSD at an accelerating voltage of 20 kV and a probe current of 20 nA on the polished surfaces. We fixed the minimum number of pixels to be considered a grain to 5 , the grain tolerance angle to $5^{\circ}$, and the step size to $1 \mu \mathrm{m}$ for all the EBSD measurements. We determined the $\%$ of un-melted Nb from each SEM image using ImageJ software. We also identified and analyzed the constituent phases in our samples by means of X-ray diffraction (XRD, PANalytical
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-03(2)}
+\end{center}
+
+(a)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-03}
+\end{center}
+
+(b)
+
+Fig. 1. (a) Energy dispersive spectroscopy (EDS) map of a pre-mixed Ti-45 Nb alloy powder showing the distribution of Ti and Nb particles; (b) Schematics of the tensile specimens cut out of builds with different $<100>$ textures. The black rectangle on each tensile specimen in (b) shows the location of EBSD measurements.
+
+Empyrean, Netherlands), using a Cu-K $\alpha$ radiation and a step size of $0.01^{\circ}$.
+
+\subsection*{2.4. Mechanical testing}
+We performed uniaxial tensile tests on dog-bone shaped samples with gauge length of 6 mm , width of 2 mm and thickness of 1 mm using a Shimadzu AGX 10 kN equipped with a laser extensometer at a strain rate of $10^{-3} / \mathrm{s}$. We tested all samples along the direction parallel to their respective $<100\rangle$ crystallographic texture, as shown schematically in Fig. 1b. From the data, we extrapolated the alloys modulus of elasticity, yield strength, UTS, and elongation to failure. To understand the deformation behavior of the different samples, we polished the side surfaces of the fractured samples and analyzed the microstructure using both FESEM and EBSD.
+
+\section*{3. Experimental Results}
+\subsection*{3.1. Crystallographic texture control}
+Owing to a difference of 1082 K in melting temperatures between Ti and Nb , it is difficult to obtain Ti-Nb alloys that exhibit both high density and homogeneous composition (i.e., do not contain un-melted Nb particles) by LPBF in the as-built conditions. Several works discuss the porosity-unmelted particles trade off in $\mathrm{Ti}-\mathrm{Nb}$ alloys and propose strategies to mitigate it, including variations in volumetric energy density\\
+(VED) [26], cross hatching style with an angle of $74^{\circ}$ between the layers [40], and laser intensity profile [20]. Here, we leverage on the work of Fischer et al. [26] and perform a large matrix of experiments with varying VEDs to identify a processing window that yields highly dense $\mathrm{Ti}-45 \mathrm{Nb}$ builds with minimal un-melted Nb content. We provide a detailed description of the optimization process and process parameter windows in the Appendix. We confirm that the porosity and un-melted Nb generally scale inversely with VED and identify two working windows that are suitable for our alloy composition as given in Fig. A1. The first is characterized by VED between 125 and $200 \mathrm{~J} / \mathrm{mm}^{3}$ and relatively high laser power (from 300 W to 350 W ). In the second, VED ranges between 265 and $290 \mathrm{~J} / \mathrm{mm}^{3}$ and requires relatively low laser power ( $\leq$ 100 W ). Fig. 2a shows a SEM micrograph of an area of 2 mm by 1.5 mm from a Ti-45 Nb sample that is representative of the build quality we obtain when using the initially optimized process parameters as shown in the green zones in Fig. A1a and Fig. A1b. The material contains a negligible number of un-melted particles (indicated by the orange arrows) and pores (indicated by the white arrows). The XRD spectrum from the same representative sample (shown in Fig. 2b) confirms that the Ti-45 Nb alloy consists of a $100 \%$ BCC phase. This is because the $\mathrm{Mo}_{\text {eq }}$ of the alloy is 12.6 (see Eq. 1), which is greater than the critical value of $10 \%$ required for retaining the $\beta$ phase at room temperature. Moreover, the fast-cooling rates during the LPBF process further facilitate the $\beta$ phase retention at room temperature. Similar $\beta$ phase stability has been shown in other in-situ alloyed Ti-Nb alloys processed by LPBF techniques previously $[20,43]$.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-03(1)}
+\end{center}
+
+(b)
+
+Fig. 2. (a) Representative SEM image showing the build quality of the samples produced using optimized process parameters to minimize both porosity (indicated by white arrows) and fraction of un-melted Nb particles (indicated by orange arrows); (b) XRD pattern of the same sample as in (a) showing a 100\% $\beta$ phase in Ti45 Nb alloy.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-04}
+\end{center}
+
+Fig. 3. Inverse pole figure grain orientation maps and pole figures of (a-d) $<100>\|$ SD textured sample; (e-h) $<100>\|$ BD textured sample, (i-l) $<100>\|$ PD textured sample. Please note that in this work, we represent the build direction as Z and the scan direction as X. MUD in the texture intensity scale bar stands for Multiples of Uniform Density, and indicates the texture strength of a given material. The cross section for EBSD measurements is shown in Fig. 1b in the small rectangular insets on the tensile specimens.
+
+Table 1
+
+Process parameters used to obtain the strong $<100>$ crystallographic textures along scanning direction (SD), build direction (BD), perpendicular to scan direction (PD).
+
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|c|c|}
+\hline
+S. No. & Samples & Rotation Angle $\left({ }^{\circ}\right)$ & Laser power $(\mathrm{W})$ & Hatch spacing (mm) & Scan speed $(\mathrm{mm} / \mathrm{s})$ & Layer thickness (mm) & Volumetric Energy Density $\left(\mathrm{J} / \mathrm{mm}^{3}\right)$ \\
+\hline
+1 & $<100>\| S D$ & 0 & 100 & 0.12 & 100 & 0.03 & 277.78 \\
+\hline
+2 & $<100>\| \mathrm{BD}$ & 0 & 350 & 0.05 & 1200 & 0.03 & 194.44 \\
+\hline
+3 & $<100>\| \mathrm{PD}$ & 0 & 350 & 0.12 & 600 & 0.03 & 162.04 \\
+\hline
+\end{tabular}
+\end{center}
+
+Within the initially optimized process parameter windows, we ran an additional design of experiments (DOE) to investigate the crystallographic textures attainable and identify three different sets of process parameters that lead to a strong $<100>$ texture along each of the three principal axes, BD, SD, and PD (as shown in Fig. 3). We detail the specific combination of parameters that we used to produce these three samples in Table 1. To the best of our knowledge, a strong $<100>\|$ PD texture
+
+Table 2
+
+Grain size and aspect ratio of the $<100>\|\mathrm{SD},<100>\| \mathrm{BD}$, and $<100>\|$ PD textured samples.
+
+\begin{center}
+\begin{tabular}{llllll}
+\hline
+\begin{tabular}{l}
+S. \\
+No. \\
+\end{tabular} & Sample & \begin{tabular}{l}
+Mean \\
+Grain Size \\
+$(\mu \mathrm{m})$ \\
+\end{tabular} & \begin{tabular}{l}
+Min. Grain \\
+Diameter \\
+$(\mu \mathrm{m})$ \\
+\end{tabular} & \begin{tabular}{l}
+Max. Grain \\
+Diameter \\
+$(\mu \mathrm{m})$ \\
+\end{tabular} & \begin{tabular}{l}
+Grain \\
+Aspect \\
+ratio \\
+\end{tabular} \\
+\hline
+1 & $<100>\|$ & $32 \pm 29$ & 7 & 220 & 3.86 \\
+ & SD &  &  &  & $\pm 1.95$ \\
+2 & $<100>\|$ & $54 \pm 36$ & 15 & 216 & 5.82 \\
+ & \begin{tabular}{l}
+BD \\
+$<100>\|$ \\
+PD \\
+\end{tabular} & $29 \pm 24$ & 7 & 146 & 0.76 \\
+ &  &  &  & $\pm 0.41$ &  \\
+\hline
+\end{tabular}
+\end{center}
+
+has never been reported in the literature. By contrast, strong $<100>$ crystallographic textures along the SD and BD i.e., $<100>\|$ SD and $<100>\| \mathrm{BD}$ have been reported in the past in various BCC and FCC alloys [15,17-19,26].
+
+From visual inspection of the EBSD maps and the corresponding pole figures in Fig. 3, we observe a stark contrast in grain morphology among the three samples. While SD- and BD-samples exhibit long, columnar grains, the PD-samples consist of grains with lower aspect ratio and smaller average size. We ascertain the complex grain morphology of the three samples by estimating the grain size using four different metrics i. e., mean grain size, minimum grain diameter, maximum grain diameter, and grain aspect ratio, which we present in Table 2. To calculate the mean grain size, we approximate the grain by a circle and compute the area-weighted mean of equivalent diameters. From Table 2, we note the mean grain size and maximum grain diameter in SD- and BD-samples are significantly higher as compared to the PD-samples. However, it may be noted that these measurements are acquired along the build direction from micrographs in Fig. 3 and should not be taken as the "standard" (e. g., 3-D) grain size measurements. Since the PD samples exhibit the
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-05}
+\end{center}
+
+Fig. 4. 3-D representation of grain morphology in PD-samples.
+
+smallest grains with aspect ratio closest to 1 , we investigate the 3-D morphology of these grains by sectioning the PD-samples along the $\mathrm{XY}, \mathrm{XZ}$, and YZ planes and build a 3-D representation of the microstructure, which we report in Fig. 4. The representation confirms that the grains in PD-samples have small aspect ratio across all cross sections, which is atypical for LPBF alloys with highly columnar and textured microstructures. Typically, the grains in AM-produced samples with a strong $<100>$ texture are elongated, owing to the epitaxial growth of the cellular structures, resulting in columnar grained microstructures with large aspect ratio.
+
+\subsection*{3.2. Mechanical properties}
+To study the effects of the three different microstructures on the mechanical behavior of the alloy, we analyze the elastic modulus, strength, and ductility of our samples by means of tensile testing along the $<100>$ texture direction-i.e., $<100>\|$ PD, $<100>\|$ BD, $<100>\|$ SD-as shown in Fig. 1b. We measure the average elastic modulus along the $<100>$ texture in all the three cases to be in the range of 56-59 GPa (Table 3). This value is significantly lower compared to that of conventional Ti alloys such as Ti64 (110-120 GPa). Because of the low modulus of elasticity, $\beta$-Ti alloys are deemed particularly suitable for biomedical applications, where significant efforts are being made to produce biocompatible metallic implants with elastic moduli as close as possible to that of human cortical bone ( $\sim 30 \mathrm{GPa}$ ) to minimize stress shielding effects $[20,26,40]$. Our results confirm that the elastic modulus in $\beta$-TiNb alloys is insensitive to the mesostructure imparted by LPBF, including the variable and directional grain morphology, the cell structure, and the melt pool shape. Indeed, the elastic modulus is a material property that is a function on the interatomic bond force and the type of crystal structure formed, while it is independent of the microstructure, particularly in microcrystalline materials [44,45]. Previous studies, in fact, reported that the modulus of Ti-Nb alloys is mainly
+
+Table 3
+
+Results from tensile testing of the three Ti- 45 Nb alloys tested along their respective strongest $<100>$ crystallographic orientation directions.
+
+\begin{center}
+\begin{tabular}{|c|c|c|c|c|c|}
+\hline
+\begin{tabular}{l}
+S. \\
+No. \\
+\end{tabular} & \begin{tabular}{l}
+Loading \\
+direction \\
+\end{tabular} & \begin{tabular}{l}
+Elastic \\
+modulus \\
+$(\mathrm{GPa})$ \\
+\end{tabular} & \begin{tabular}{l}
+$0.2 \%$ offset \\
+yield strength \\
+(MPa) \\
+\end{tabular} & \begin{tabular}{l}
+Ultimate \\
+tensile \\
+strength \\
+(MPa) \\
+\end{tabular} & \begin{tabular}{l}
+Ductility \\
+(\%) \\
+\end{tabular} \\
+\hline
+1 & SD & $59.2 \pm 4.5$ & $687 \pm 5$ & $690 \pm 10$ & \begin{tabular}{l}
+21 \\
+$\pm 3.7$ \\
+\end{tabular} \\
+\hline
+2 & BD & $56.8 \pm 3.7$ & $596 \pm 19$ & $613 \pm 25$ & \begin{tabular}{l}
+17 \\
+$\pm 1.5$ \\
+\end{tabular} \\
+\hline
+3 & PD & $57.2 \pm 3.1$ & $668 \pm 1$ & $684 \pm 3$ & \begin{tabular}{l}
+23 \\
+$\pm 1.5$ \\
+\end{tabular} \\
+\hline
+\end{tabular}
+\end{center}
+
+influenced by the Nb content, which determines the stability of the BCC phase. The particularly low value of the elastic modulus we measure stems from the orientation-dependent elastic response of BCC Ti alloys. Indeed, the $<100>$ orientation is known to exhibit the lowest elastic modulus. Lee et al. [19], showed that the Young's modulus of a Ti-15Mo-5Zr-3Al alloy is the highest ( $\sim 120 \mathrm{GPa}$ ) along the $<111>$ orientation and the lowest ( $\sim 44.4 \mathrm{GPa}$ ) along the $<100\rangle$. Since the alloy composition of our samples is kept constant and the tensile axis is always parallel to the $<100>$ orientation, we find similar, low elastic moduli values across all three samples.
+
+We present the tensile stress-strain curves of the three samples in Fig. 5a and report the nominal mechanical properties in Table 3. We observe that the PD-samples exhibit the best combination of strength and ductility. By contrast, the BD-samples demonstrate the lowest strength and ductility. The SD-samples also exhibit high strength, a reasonable ductility of $21 \%$, and a dip in the strength post yielding, which is similar to the case of BD-samples. To interpret the differences in mechanical responses and investigate the deformation mechanisms at play in the three samples, we plot strain hardening curves (Fig. 5b) and analyze the microstructure evolution upon deformation (Figs. 6 and 7). From the strain hardening curves, it is evident that the three different samples display classical transformation induced plasticity (TRIP) behavior characterized by a conventional transition from elastic to the plastic region (stage I), followed by a large increase in the strain hardening rate from the elastic limit to a plastic strain, $\varepsilon$, of 0.06 (stage II), and finally a gradual decrease in strain hardening up to failure (stage III). In previous works on metastable $\beta$-Ti alloys [46,47], the "hump" seen in the strain hardening curves (stage II) was ascribed to the formation of stress-induced martensite ( $\alpha^{\prime \prime}$ ) (SIM) and mechanical twinning, which continuously interact with each other throughout the plastic deformation regime. This phenomenon is commonly referred to as transformation induced plasticity/twinning induced plasticity (TRIP/TWIP) [47], and is prominent in metastable $\beta$-Ti alloys. To assess whether $\beta$ to $\alpha^{\prime \prime}$ transformation occurs in our samples, we performed XRD post-deformation. The results, plotted in Fig. 6, provide unambiguous evidence of peaks corresponding to both $\beta$ and $\alpha$ " phases, confirming the SIM transformation/TRIP phenomenon in our Ti-Nb alloys, regardless of their microstructure.
+
+Based on the strain hardening and XRD curves, we speculate that the active deformation mechanism governing the plasticity of the alloy changes from dislocation glide to mechanical twinning alongside $\alpha^{\prime \prime}$ transformation in all the samples. To confirm the occurrence of mechanical twinning, we analyze the samples post-facto failure at the necked regions by means of EBSD. Fig. 7a-c show representative grain orientation maps and Fig. 7d-f show the corresponding kernel average misorientation (KAM) maps computed from the raw EBSD data. The KAM maps are computed by calculating the average misorientation between each pixel and a kernel $(3 \times 3)$ of surrounding pixels. From Fig. 7, we notice a dense network of deformation twins that spans across all the three samples in the necked region. It is reasonable to expect deformation twinning in $\beta$-Ti alloys, due to the low shear strain required to generate $\{332\}<113>$ twins upon deformation, which makes it an energetically favourable deformation mechanism [48,49]. Therefore, both $\alpha^{\prime \prime}$ SIM and deformation twinning dictate the shape of the strain-hardening curves in Fig. 5 b. The $\alpha^{\prime \prime}$ precipitation provides a significant contribution to the observed macroscopic strain-hardening, inducing a "plateau" shortly followed by a strong "hump" in the strain hardening curves (Fig. 5b). This "hump" can be interpreted as the elastic deformation contribution of the newly formed $\alpha^{\prime \prime}$ martensite to the overall stress-strain behavior during the plastic deformation of the BCC matrix, thus, playing the role of an "elastic inclusion". The elastic contribution ends when the martensite itself starts to deform plastically (at $\varepsilon \approx 0.06$ for SD-samples and PD-samples, and $\varepsilon \approx 0.04$ for BD-samples). This interpretation is also in agreement with previous works on other $\beta$-Ti alloys such as Ti-12Mo and Ti-8.5Cr-1.5Sn alloys [50].
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-06}
+\end{center}
+
+Fig. 5. (a) True stress-true strain curves under tension and (b) work hardening curves of the three Ti-45 Nb alloys tested along their respective $<100>$ crystallographic orientation directions. TD refers to the tensile direction.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-06(1)}
+\end{center}
+
+Fig. 6. X-ray diffraction of the deformed Ti-45 Nb alloys.
+
+\section*{4. Discussion}
+\subsection*{4.1. Solidification microstructure and texture formation}
+We gain insights into the texture formation mechanisms that yield the microstructures shown in Fig. 3 by analyzing the melt pool shapes and the orientation of the cellular structure, which we reveal after chemical etching. Indeed, $<100>$ crystallographic textures in cubic alloys produced by fusion-based AM are typically parallel to the solidification direction, which is aligned with the maximum thermal gradient $[16,18]$. The melt pool shapes visible in the optical and electron micrographs in Fig. 8 indicate that all three samples entail a keyhole mode of melting, with deep melt pools spanning several layers in depth. This result is expected given that we produced all these textured samples using high VED (of $>160 \mathrm{~J} / \mathrm{mm}^{3}$ which facilitates the melting of Nb and homogenisation of the alloy). However, there are some obvious differences in melt pool geometry among the three samples, which stem from the different laser process parameters used (i.e., power, scan speed, and hatch spacing) and lead to the different textures shown in Fig. 3. It may be noted that the scan strategy i.e., bidirectional scanning without rotation between layers, is kept constant across all the samples.
+
+In SD-samples, produced by low laser power and higher VED, we observe a nail-head-like melt pool geometry in the YZ plane (see Fig. 8a). This mode of melt pool formation has been observed previously in alloys produced by LPBF when using the same scan strategy which we employ in this work-i.e., a bidirectional serpentine pattern without rotation between layers [51]. This specific melt pool geometry is attributed to the positive thermocapillary flow (i.e., the Marangoni effect) in the molten alloy. Due to the curvature of the melt pool walls, the direction of the thermal gradients varies across the melt pool [52]. Along the melt pool centerline, the heat flow is vertical and leads to segregation cells (and therefore, grains) that solidify with a $<100>$ orientation parallel to BD and grow epitaxially across multiple layers (see Fig. 3a). At either side of the centerline, however, the cells grow at $\pm 45^{\circ}$ with respect to the BD. Direct epitaxial growth of these cells across adjacent melt pools is ensured and is also assisted by side-branching on some parts of the melt pools [53], as seen in Fig. 8b. This phenomenon results in a $<110>$ texture component along the BD and consequently, a $<100>\|$ SD texture. This particular $\{100\}<110>$ bi-axial texture-also called rotated cube texture (rotated by $45^{\circ}$ )—has been reported in LPBF stainless steel 316 L [17,51] and other cubic alloys [54]. Because of this epitaxial growth of crystals across multiple layers, the resulting grains formed also have high aspect ratio, as seen in Fig. 3a-c and Table 2 .
+
+By contrast, in BD-samples we observe melt pools that overlap within the YZ (PD-BD) plane to a greater extent owing to the smaller hatch spacing used during processing (Fig. 5c). The increased overlap between subsequent melt tracks results in greater lateral remelting, which preserves growth of cells at the melt pool centerline but disrupts the symmetric $\pm 45^{\circ}$ crystal growth [55]. In other words, crystal growth along BD is preferred. This can be observed in Fig. 5d where some cells not growing along BD are disrupted by the cells growing along BD upon remelting (adjacent track). This results in the epitaxial growth of cells across adjacent melt pools [55] and leads to the establishment of a strong $<100>\|$ BD texture $[54,56]$ with a weak near-cube texture. Despite a near-cube texture, we refer to it as $<100>\| B D$ in this manuscript as the strongest $<100>$ texture component is along BD. Because of the copious remelting induced by a greater melt pool overlap,\\
+$<100>\|$ SD
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-07(2)}
+\end{center}
+
+(a)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-07}
+\end{center}
+
+(d) $<100>\|$ BD
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-07(4)}
+\end{center}
+
+(b)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-07(1)}
+\end{center}
+
+(e)
+
+$$
+<100>\| \text { PD }
+$$
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-07(3)}
+\end{center}
+
+(f)
+
+Fig. 7. EBSD after necking of (a) SD-sample; (b) BD-sample; (c) PD-sample; Corresponding kernel average misorientation (KAM) maps after necking in (d) SDsample; (e) BD-sample; (f) PD-sample.
+
+BD-samples exhibit large, columnar grains with high aspect ratio (Table 2), which we observe directly in Fig. 3e-g. Such grain structures have been commonly reported in additively manufactured parts and laser beam welded joints [57]. Moreover, the absence of side-branching is conducive to the highest texture strength amongst all samples investigated here (Fig. 3 h ). Since these solidification cell growth mechanisms for SD- and BD-samples are in line with what has been reported previously-for instance, in references [17,20], respectively-we do not discuss them further.
+
+Finally, PD-samples display the deepest keyhole melt pools amongst all our samples (as confirmed in Fig. A1d) owing to the highest laser power used to produce these specimens. While the direction of the thermal gradient changes within the melt pool generally, the long, vertical melt pool walls are conducive to thermal gradients that are within the plane of the powder layer, and thus to cells growing along the direction perpendicular to SD and BD, as indicated by the white solid arrows in Fig. 8f. The case of $<100>\|$ PD texture is an important result from this work as it shows a novel texture formation mechanism during LPBF. Fig. 8e and 8 f show that this $<100>\|$ PD growth extends across melt pool boundaries. This evidence suggests that epitaxial growth is maintained both within the layer (horizontally, across adjacent melt tracks) as well as across subsequent layers.
+
+To understand the evolution of this PD texture, we perform a detailed microstructural analysis by corelating the electron micrographs of the etched cell structure with the corresponding EBSD maps (Fig. 9). Based on this analysis, we propose that the PD texture forms as a result of keyhole melt pool "nesting", which leads to the concentric patterns seen in Fig. 9a. This phenomenon is a result of many remelting events across multiple layers. Any cross section of a melt pool reveals the deep "nesting" of multiple melt pools inside one another resembling stackable tall glass tumblers as shown in Fig. 9a. For this reason, we call this a "deep nesting" melting mode. During "deep nesting" mode of melting, a single melt pool penetrates deep inside the bulk, remelting most of the previously solidified-and cold-material directly below. Because of the relatively low temperatures of the surrounding metal, strong thermal gradients develop perpendicular to the vertical melt pool boundaries, promoting epitaxial growth of cells along PD (Fig. 9a). Because of the high density of melt pool boundaries-which stems from melt pool "deep nesting"-this PD growth is also remelted several times. As a result, grain selection occurs in a relatively small volume, leading to a strong $<100>$ texture along PD. Noteworthy is that PD samples also exhibit a weak $<100>$ texture along the BD (see pole figure in Fig. 3 1), which yields a near $<100>$ cube texture. Since the highest texture strength is along the PD direction, we refer to this texture as $<100>\|$ PD as indicated earlier. The weak texture along BD stems, once again, from the melt pool centerlines (Fig. 8f), which are characterized by cells growing along the BD as depicted by the dashed arrows in Fig. 8f. These centerlines interrupt the PD growth.
+
+We study the orientation relationship of centerlines with respect to the PD-textured grains using EBSD (Fig. 9b). Due to the presence of the centerline thermal gradient, a grain selection process in the region surrounding centerlines drives the growth of crystals with $<100>$ axes parallel to both BD and PD (see centerline 1 in Fig. 9b with the ideal $<100>$ orientation along BD and PD ). This phenomenon results in the establishment of a strong bi-axial texture, albeit only in these regions where the two perpendicular thermal gradients (PD and BD) co-exist. Not all centerlines, however, exhibit this biaxial texture (see centerline 2 and 3 in Fig. 9b). In some cases, epitaxial growth of centerline grains is interrupted by slight misalignments of melt pools in consecutive layers. This apparently stochastic misalignment of melt pools may be due to an irregular remelting during the multi-layer printing which results in the deviation from the ideal orientation [17]. In these cases, the centerline grains continue to grow along the dominant thermal gradient i.e., PD (following epitaxial growth), therefore continuing to maintain $\mathrm{a}<100>$ orientation along PD as seen from grains corresponding to centerlines 2 and 3 in Fig. 9b. From these observations, we conclude that the "deep nesting" mode of melting is conducive to the establishment of two prominent crystal growth directions: one along PD on either side of the melt pool and one along BD at the melt pool centerlines. We expect this crystal growth configuration to be specific to the PD-sample. Indeed, it is unlikely that the shallower keyhole melt pools seen in BD- and SD-samples may generate prominent thermal gradients and crystal growth directions parallel to PD.
+
+Another important difference in PD-samples is that grains are bound
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-08(1)}
+\end{center}
+
+(a)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-08(4)}
+\end{center}
+
+(c)\\
+\includegraphics[max width=\textwidth, center]{2024_07_13_1419f509943821a0a42ag-08(2)}
+
+(SD)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-08}
+\end{center}
+
+(b)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-08(3)}
+\end{center}
+
+(d)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-08(5)}
+\end{center}
+
+(f)
+
+Fig. 8. Optical micrographs, melt pool evolution simulations, and crystal growth orientation of (a, b) SD-samples; (c, d) BD-samples; (e, f) PD-samples
+
+by centerlines (as seen in Fig. 4), which interrupt their growth and thus limit their size (Table 2). Indeed, these centerlines are also present in the SD-samples, where they interrupt grain growth along the Y axis (Fig. 3ac). The centerline formation mechanism is discussed in detail in other works $[17,31,57]$. However, the key difference between the SD- and PDsamples, despite both exhibiting centerline grains, is the growth direction of the solidification cells. In the case of SD-samples, since the growth direction is $\pm 45^{\circ}$, they grow epitaxially between layers, therefore resulting in columnar grain growth along the Z direction. However, in the case of PD-samples, since the growth direction is $90^{\circ}$, grains remain short because restricted between two melt pools centerlines and within a few layers.
+
+Beside the effects of the "deep nesting" mode of melting on epitaxial growth, texture direction, and grain morphology, we also observe marked differences in the density of geometrically necessary dislocations (GNDs), It is commonly understood that LPBF processes result in metal parts with a GND density $\sim 2$ orders of magnitude higher than their annealed counterparts $[36,58,59]$. We assess the density of GNDs, $\rho$, in our samples from KAM maps computed from the EBSD data (Fig. 10) by using [60]:
+
+$\rho=\frac{\alpha \theta}{X b}$
+
+Here, $\alpha=2$ for high angle grain boundaries, $\theta$ corresponds to the local misorientation angle, X is the step-size used during the EBSD scan, and $b$ corresponds to the Burgers vector's magnitude. We notice that the GND density (Fig. 10d) varies as: $\rho$ (PD) $<\rho$ (SD) $<\rho$ (BD). Since GNDs originate from the thermal stresses caused by the repeated thermal cycles $[59,61]$ and rapid cooling rates, to understand the origin of the
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-09(1)}
+\end{center}
+
+(a)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-09}
+\end{center}
+
+Fig. 9. (a) Schematics of the novel growth mechanism in "deep nesting" melt pools similar to the stackable glass tumblers that results in a $<100>$ texture perpendicular to SD and BD, (b) Electron micrograph and EBSD maps of the melt pools of PD-samples depicting the orientation relationship of the centerline grains with the PD textured grains.
+
+differences in GNDs among the three samples, we estimate the cooling rates these samples underwent from the analysis of the solidification microstructure (cell spacing) shown in Fig. 8. Indeed, the higher the cooling rate, $\dot{T}$, the smaller the size of the solidification microstructure-whether it consists of dendrites or cells [62,63]. This relationship is qualitatively described by the empirical power-law [64]:
+
+$\lambda_{c}=K|\dot{T}|^{n}$
+
+Here, $\lambda_{c}$ is the primary spacing of solidification cells and K and n are material constants. To the best of our knowledge, no previous work derived the K and n values for the Ti- 45 Nb alloy system used here. However, in Ti alloys, there is an inverse correlation between the cell spacing and cooling rate, with n values ranging between -0.31 to -0.39 and K assumes a value of 49 [62,63]. From our SEM images (Fig. $8 \mathrm{~b}, \mathrm{~d}, \mathrm{f}$ ), we measure $\lambda_{\mathrm{c}}$ in the three samples from grains having the same orientation and using the line intercept method. Table 4 lists these values, which follow the order $\lambda_{c}$ (PD) $>\lambda_{c}$ (SD) $>\lambda_{c}$ (BD); with PD-samples having nearly twice as large $\lambda_{c}$ value than that measured in BD-samples. Therefore, we conclude that the three samples undergo significantly different cooling rates, following the order $\dot{T}$ (PD) $<\dot{T}$ (SD) $<\dot{T}$ (BD). Although oversimplified, this analysis is in line with the observed differences in the initial GND density across the three samples. Indeed, higher cooling rates result in higher initial dislocation densities in the additively manufactured materials.
+
+\subsection*{4.2. The effect of microstructure on yielding}
+Loading the three samples along their fixed $<100>$ orientation allows us to probe the individual contributions of different microstructural features to the mechanical properties of our complex LPBF Ti-45 Nb alloys. Such features include melt pool geometry, grain structure, solidification structure, and crystallographic textures.
+
+As seen from the Fig. 8 and discussed in Section 3.2, despite the samples nominally having the same crystallographic texture along the tensile direction, they exhibit markedly different yield strength. Specifically, PD-samples have the highest yield strength, followed by SDand BD-samples. While the $<001>$ axis is constant along the tensile loading direction for all the three samples, both the in-plane grain orientation distribution as well as the grain structure differ substantially in each sample and may explain the differences in yield strength.
+
+We first consider the effect of grain orientation. Since the initial deformation mechanisms at play in $\beta-\mathrm{Ti}$ alloys is dislocation slip, we compute the Schmid factors for $\{110\}<111>$ slip systems for the SD, BD , PD samples from the EBSD measurements to identify the samples showing most favourability to slip. The results are reported in Fig. 11. We note that the average Schmid factors are nearly the same for the three samples, with PD samples showing a marginally lower value. Since this difference is not significant, we conclude that the trends in yield strength cannot be ascribed to the type of $<100>$ texture and the change in the in-plane orientation.
+
+Next, we analyze grain size and morphology, since dislocation glide is governed by the interactions between slip bands and grain boundaries [65]. The PD-samples exhibit grains with lower aspect ratio and finer mean grain size compared to the SD- and BD-samples (Fig. 3, Table 2). In fact, the maximum grain size of the PD-samples is about $70-80 \mu \mathrm{m}$ lower than that of the SD- and BD- samples. Therefore, it is reasonable to expect higher yield strength according to the Hall Petch relationship [66]. Interestingly, we find that SD-samples also have a high yield strength, which is comparable to that of PD-samples (Table 3) despite
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-10(4)}
+\end{center}
+
+(a)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-10}
+\end{center}
+
+(c)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-10(1)}
+\end{center}
+
+(b)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-10(2)}
+\end{center}
+
+(d)
+
+Fig. 10. Kernel average misorientation (KAM) maps in as-printed (a) SD-sample; (b) BD-sample; (c) PD-sample and (d) estimated average GND density of the three samples computed from the KAM maps.
+
+Table 4
+
+Measurement of cell spacing by the line intercept method from the scanning electron microscopic (SEM) images in Fig. 8.
+
+\begin{center}
+\begin{tabular}{lll}
+\hline
+S. No. & Material & Cell spacing $\left(\lambda_{c}\right)(\mu \mathrm{m})$ \\
+\hline
+1 & $<100>\| \mathrm{SD}$ & $0.45 \pm 0.06$ \\
+2 & $<100>\| \mathrm{BD}$ & $0.37 \pm 0.06$ \\
+3 & $<100>\| \mathrm{PD}$ & $0.78 \pm 0.10$ \\
+\hline
+\end{tabular}
+\end{center}
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-10(3)}
+\end{center}
+
+Fig. 11. A histogram of Schmid factor (SF) distribution with the respective average SFs and corresponding ranges for $\{110\}<111>$ slip system for the SD, BD-, and PD- samples. consisting of coarser grains. By contrast, BD-samples exhibit a relatively inferior yield strength ( $\sim 80-100$ MPa lower than the SD-, PD-samples). To understand the significant differences in strength among the BD-, SDand PD-samples, we probe into the solidification structure, since several previous works have demonstrated how solidification cells in LPBF alloys contribute to yield strength, albeit to a lower extent in comparison to grain boundaries $[67,68]$. Indeed, solidification cell walls are decorated with solute elements-such as Nb in our case-and copious dislocations, which provide additional resistance to dislocation motion and, thus, increased strength [6]. We note that the SD-samples and BD-samples, appear to have finer solidification structures compared to PD-samples, as shown in Table 4. Since both SD- and BD-samples have relatively large grain size and fine cell size, the fine solidification cell size does not explain the difference in the yield strength. We, therefore, ascribe the additional strengthening in the SD- and PD-samples to the presence of numerous fine centerline grains (with a size range of $10-30 \mu \mathrm{m}$ ). These centerline grains, formed as a result of the vertical heat flow at the center of the melt pools (Section 4.1), act as additional barriers for dislocation motion as illustrated in Fig. 12, therefore bringing about additional strength in the SD- and PD-samples. Since the BD samples consist of only columnar grains without any additional centerline grains, the resistance to dislocation motion is lower, therefore exhibiting a relatively inferior yield strength.
+
+From the tensile stress-strain curves in Fig. 5, we also notice that SDand BD-samples undergo a slight dip in the stress levels during yielding by $\sim 10-25 \mathrm{MPa}$, which is not the case in PD-samples. This phenomenon-referred to as yield point phenomenon-was previously observed in other $\beta$-Ti alloy systems, both conventionally produced $[69,70]$ as well as fabricated by LPBF [65]. In BCC alloys, this phenomenon is typically associated with dislocations overcoming interstitials [67]. However, we expect no appreciable difference in interstitials-e.g., oxygen atoms-in our samples, since only two out of three exhibit this phenomenon and all of them were fabricated with the same LPBF machine in Ar environment.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-11(1)}
+\end{center}
+
+(a)
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_07_13_1419f509943821a0a42ag-11}
+\end{center}
+
+(b)\\
+S: Source
+
+$\perp$ Dislocation
+
+\section*{/ Grain \\
+ Boundary}
+\includegraphics{smile-lyjj8pm1sntflvxr4s.png}
+
+
+(SD)
+
+Fig. 12. Schematic illustration of the grain structure and dislocation motion in the (a) SD- and PD- samples, where the presence of fine centerline grains further restrict dislocation motion; and (b) BD-sample where such centerline grains are not present. All the boundaries in the illustration represent high angle grain boundaries.
+
+It is plausible that other barriers to dislocation motion may exist to a different extent in the three samples. As discussed in the foregoing, the SD- and BD-samples-which exhibit the yield point phenomenon-consist of a solidification microstructure that is finer compared to that found in PD-samples (Table 4). The higher density of cell boundaries in these samples may require higher stress levels to reach the upper yield point, after which dislocations could cut through the cell boundaries, causing a sudden drop in the lower yield point. By contrast, because of the coarser solidification cell size in PD-samples, we speculate the yield point phenomenon not to be as prominent as in the other two microstructures. This difference justifies the plateau we find in the stress-strain curves of BD- and SD-samples (Fig. 5a). Furthermore, both these samples are characterized by a relatively higher GND density compared to PD-samples (as discussed in Section 4.1). This difference may also contribute-although less significantly-to the upper yield point by obstructing the movement of newly generated dislocations upon yielding. These speculations are in line with the observations reported recently in a different $\beta$-Ti alloy produced by LPBF [65]. Further investigations on the yield point phenomenon are beyond the scope of this work and will be explored in a follow-up study.
+
+\subsection*{4.3. The effect of microstructure on work hardening}
+Despite the similarities in texture orientation along the tensile axis and the deformation mechanisms at play, PD-samples exhibit a remarkably higher work hardening rate ( $\sim 900 \mathrm{MPa})$ compared to the other two samples ( $\sim 400-500 \mathrm{MPa}$ ), and a peculiar "hump" in the curve (Fig. 5). To elucidate the origin of these differences, we analyze the fraction of $\alpha^{\prime \prime}$ and the local GND density from the XRD measurements (Fig. 6) and the KAM maps (Fig. 7d-f) computed from the high magnification EBSD scans in Fig. 7a-c. The XRD patterns suggest the presence of a much higher volume fraction of $\alpha^{\prime \prime}$ SIM and deformation twinning in PD-samples compared to the others, as observed from the higher XRD peak in Fig. 6. We believe the higher SIM fraction in this sample to be one of the factors leading to the "hump" in the corresponding strain hardening curve seen in the PD-samples. Furthermore, the KAM maps indicate higher local misorientation within twin variants across the matrix, suggesting that high GND densities are stored within the twins and where twins intersect grain boundaries. This heterogeneous strain localisation has been observed before and is known to be due to the increased plastic activity caused as a result of the interaction between twins and grain boundaries [71].
+
+We also observe unindexed regions in the EBSD images in Fig. 7 at the intersections between the same twin variants. Based on our XRD results (Fig. 6), we confirm that these correspond to the $\alpha^{\prime \prime}$ needles and we attribute the lack of signal to the high local strains and to the presence of a network of martensitic $\alpha^{\prime \prime}$ needles that form within the twin lamella at every twin intersection, and which makes indexing at the junctions very challenging by means of EBSD [71]. The reason for their occurrence at grain boundaries is that martensitic $\alpha^{\prime \prime}$ needles typically form to accommodate the local stress/strain incompatibilities between the highly hardened twinned grains and the soft $\beta$ matrix neighbours. Therefore, TRIP and TWIP modes operate simultaneously in the three different textures, as observed previously [48], albeit to a different extent in each case.
+
+Despite all the samples exhibiting concurrent TRIP and TWIP phenomenon, we still capture the differences between the PD-, BD-, and SDsamples in terms of the twin density. We note that the twin multiplication (twin density) and volume fraction of $\alpha^{\prime \prime}$ SIM are significantly higher in PD-samples, as evidenced from Fig. 7. We believe that this is caused by the smaller grain size, particularly by the low aspect ratio of grains in PD-samples. Indeed, it is known that decreasing the grain size of the parent phase lowers the martensitic start temperature (Ms), as a result of lowering the elastic and friction energy [72,73]. Therefore, we see higher volume fraction of $\alpha^{\prime \prime}$ SIM and twin density in PD-samples which, in turn, result in higher strain hardening compared to what we measure in SD- and BD-samples. As twins multiply in the microstructure across the grains, they lead to the formation of additional boundaries that act as barriers to dislocation motion and they decrease the dislocation mean free path, further improving the strain hardening of the alloy following the dynamic Hall-Petch mechanism [71,74]. While these twin boundaries are also sites of dislocation pile-up and stress concentration (which could potentially lead to strain localization), the presence of $\alpha^{\prime \prime}$ SIM results in a synergistic cooperation between the hardening mechanism (by twins) and strain accommodation mechanism (by $\alpha^{\prime \prime}$ SIM). The result is a stable plastic flow and a strong strain hardening. Similar observations have been made previously in TRIP/TWIP Ti alloys [75]. Therefore, we attribute the high strain hardening and ductility in PD-samples to the presence of higher $\alpha^{\prime \prime}$ SIM and twin fraction. While there is appreciable strain hardening in the SD- and BD-samples, their larger grain size is not conducive to the same extent of $\alpha^{\prime \prime}$ SIM, twin multiplication, and dynamic Hall-Petch effect seen in PD-samples, resulting in lower strain hardening and early failure [76]. We attribute the lowest ductility of BD-samples to the higher density of interfaces across the sample's mesostructure-such as melt pool boundaries-which are aligned along the $<100>\|$ BD. Since these boundaries are perpendicular to the loading axis (which is parallel to the BD in these samples), they act as strain localization sites impeding\\
+uniform elongation and leading to premature failure [77]. Thus, the excellent combination of plasticity and strain-hardening in PD-samples stem from the reduced dislocation mean free path via the synchronous activation of TRIP and TWIP, as well as from the backstresses generated by dislocation pile-up at obstructive interfaces [48]. However, it may be noted that due to the relatively higher $\operatorname{MoE}(\sim 12.6)$ and $\beta$ phase stability of the $\mathrm{Ti}-45 \mathrm{Nb}$ alloy, the extent of stress-induced transformation is limited [78], and the deformation is mediated through both slip as well as TRIP/TWIP. Therefore, we see an overall limited ductility than what is typical of TRIP/TWIP Ti alloys [76].
+
+We believe that the novel $<100>\|$ PD texture with low aspect ratio (near-equiaxed) grains is a promising microstructure to enable a combination of low modulus, superior strength, strain hardenability, and ductility in the current $\mathrm{Ti}-45 \mathrm{Nb}$ alloy system. As such we envision this microstructure to be of interest for biomedical applications. The novel PD-texture associated with this microstructure may also be extended to other alloy systems to decouple the high aspect ratio grains and strong textures to result in favourable mechanical properties in AM alloys.
+
+\section*{5. Conclusions}
+In this work, we demonstrate the feasibility of producing a strong $<100>$ crystallographic texture along three perpendicular directions in an in-situ alloyed cubic Ti alloy by means of laser powder bed fusion (LPBF). We achieve this capability by varying the melt pool shapes in the keyhole melting regime to promote directional, epitaxial growth along the laser scan direction (SD), the build direction (BD), and the direction perpendicular to both SD and BD, which we call PD. The key findings of our work are summarized below:
+
+\begin{enumerate}
+  \item We find a novel $<100>$ texture along PD in our $\beta \mathrm{Ti}$ alloy, which we explain on the basis of a "deep nesting" mode of melting.
+
+  \item By contrast to BD- and SD-samples-which exhibit long columnar grains with high aspect ratio (> 3.8)—we find low aspect ratio grains ( 0.76) in PD-samples. This difference stems from the unique melt pool shapes and texture formation mechanism found in these samples.
+
+  \item Owing to this unique microstructure, the $<100>\|$ PD textured samples show the best combination of strength, strain hardening, and ductility amongst all samples.
+
+\end{enumerate}
+
+Our work extends the scope of AM by providing opportunities to design novel components with engineered crystallographic textures that can be spatially and directionally varied in all the three dimensions (3D) to adapt to the local stress fields and enable significantly improved functionality of metallic materials.
+
+\section*{CRediT authorship contribution statement}
+Sravya Tekumalla: Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization, Writing - original draft, Writing - review \& editing. Jian Eng Chew: Methodology, Investigation. Sui Wei Tan: Methodology. Manickavasagam Krishnan: Methodology. Matteo Seita: Conceptualization, Writing - review \& editing.
+
+\section*{Declaration of Competing Interest}
+The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
+
+\section*{Data Availability}
+Data will be made available on request.
+
+\section*{Acknowledgments}
+This work was supported by the NTU Presidential Postdoctoral Fellowship (Grant number: 04INS000761C160). Access to shared experimental facilities used in this work was provided by the School of Mechanical and Aerospace Engineering and the Facility for Analysis Characterization Testing and Simulation (FACTS) at NTU.
+
+\section*{Appendix A. Supporting information}
+Supplementary data associated with this article can be found in the online version at doi:10.1016/j.addma.2022.103111.
+
+\section*{References}
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+
+\begin{itemize}
+  \item 
+\end{itemize}
+
+
+\end{document}

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+\title{Investigation of Melt Pool Geometry Control in Additive Manufacturing Using Hybrid Modeling }
+
+
+\author{Sudeepta Mondal 1,2, Daniel Gwynn 1, Asok Ray 1,2(1) and Amrita Basak 1,*\\
+1 Department of Mechanical Engineering, Pennsylvania State University, University Park, PA 16802, USA;\\
+sbm5423@psu.edu (S.M.); dug221@psu.edu (D.G.); axr2@psu.edu (A.R.)\\
+2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA\\
+* Correspondence: aub1526@psu.edu; Tel.: +1-814-863-1323}
+\date{}
+
+
+\begin{document}
+\maketitle
+Article
+
+Received: 4 April 2020; Accepted: 18 May 2020; Published: 22 May 2020
+
+\begin{abstract}
+Metal additive manufacturing (AM) works on the principle of consolidating feedstock material in layers towards the fabrication of complex objects through localized melting and resolidification using high-power energy sources. Powder bed fusion and directed energy deposition are two widespread metal AM processes that are currently in use. During layer-by-layer fabrication, as the components continue to gain thermal energy, the melt pool geometry undergoes substantial changes if the process parameters are not appropriately adjusted on-the-fly. Although control of melt pool geometry via feedback or feedforward methods is a possibility, the time needed for changes in process parameters to translate into adjustments in melt pool geometry is of critical concern. A second option is to implement multi-physics simulation models that can provide estimates of temporal process parameter evolution. However, such models are computationally near intractable when they are coupled with an optimization framework for finding process parameters that can retain the desired melt pool geometry as a function of time. To address these challenges, a hybrid framework involving machine learning-assisted process modeling and optimization for controlling the melt pool geometry during the build process is developed and validated using experimental observations. A widely used 3D analytical model capable of predicting the thermal distribution in a moving melt pool is implemented and, thereafter, a nonparametric Bayesian, namely, Gaussian Process (GP), model is used for the prediction of time-dependent melt pool geometry (e.g., dimensions) at different values of the process parameters with excellent accuracy along with uncertainty quantification at the prediction points. Finally, a surrogate-assisted statistical learning and optimization architecture involving GP-based modeling and Bayesian Optimization (BO) is employed for predicting the optimal set of process parameters as the scan progresses to keep the melt pool dimensions at desired values. The results demonstrate that a model-based optimization can be significantly accelerated using tools of machine learning in a data-driven setting and reliable a priori estimates of process parameter evolution can be generated to obtain desired melt pool dimensions for the entire build process.
+\end{abstract}
+
+Keywords: additive manufacturing; melt pool dimension control; machine learning; Gaussian process modeling; Bayesian Optimization; surrogate-assisted modeling
+
+\section*{1. Introduction}
+Metal additive manufacturing (AM) facilitates direct fabrication of near-net-shape metallic components, prototypes, or both under rapid solidification conditions [1]. AM is currently used in manufacturing a wide variety of components with increasing complexity, for example, fuel nozzles, rocket injectors, and lattice structures [2]. The concept of AM is built on the principle of incremental layer-by-layer material consolidation through localized melting and resolidification of feedstock materials by using high-power energy sources [3]. The localized heating causes the formation of\\
+a melt pool that controls the microstructure and, therefore, the properties of the manufactured component [3]. Due to the cyclic nature of the deposition process as AM components continue to gain thermal energy, the thermal gradient $(G)$ and the solidification velocity $(R)$ inside the melt pool continuously change, resulting in significant alterations in the melt pool properties (e.g., geometry, thermal profile, and flow field among others) between the initial and final layers [4,5] as illustrated in Figure 1a. Figure $1 \mathrm{~b}$ shows that in spite of fixing the $G / R$ ratio at a value meant for yielding a columnar microstructure, at the leading edge (e.g., beginning of the scan), the grains are columnar; however, at the trailing edge (e.g., towards the end of the scan), the grains become equiaxed [6] due to progressive heating. The primary reason for controlling melt pool geometry in metal AM is to allow a part to be built with consistent melt pool dimensions, even as thermal conditions change continuously [7]. Understanding the fundamental physics of the melt pool evolution is, therefore, a key requirement for AM process development, optimization, and control. Although high-fidelity computational modeling of AM processes can provide reliable estimates of the melt pool evolution during the build process, such elaborate models are nearly-intractable when coupled with an optimization code for melt pool geometry control due to the computational costs [8]. Therefore, while these models are well suited for understanding the physical phenomena, challenges exist in using them for performing process design, optimization, and control [9].
+
+In this respect, machine learning (ML) shows potential in assisting and automating the process of manufacturing [10] through reliable prediction of melt pool geometries [11]. For example, Neural Networks (NN) [12] are widely used as a popular choice in prediction problems by modeling nonparametric input-output relationships. Despite the simplicity of usage, most of these techniques based on complex NN architectures suffer from issues of interpretability [13]. Moreover, the applications of such methods in limited datasets are rare, and lack of data can often result in poor predictive models that lack generalizability due to overfitting [14]. Moreover, a vast majority of the ML techniques used in AM relies on point estimates of the quantities of interest, without catering for uncertainty in the predictions. In critical applications involving high stakes associated with mispredictions, the estimates of uncertainty are particularly important. An attractive alternative, therefore, is to use probabilistic ML techniques like Gaussian Processes (GPs) [15] that offer the advantages of interpretability and applicability in limited data regimes. However, their applications in surrogate-based modeling and optimization are relatively less explored in multi-physics problems in AM. An implementation of surrogate modeling through the construction of computationally efficient approximations that can be used in lieu of the original simulation model [16], therefore, holds significant promise in metal AM.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-02}
+\end{center}
+
+Figure 1. (a) Comparison between experimentally observed and computationally predicted build profile in 316 stainless steel, with constant beam power of $210 \mathrm{~W}$ and the scanning speed of $12.7 \mathrm{~mm} / \mathrm{s}$ (Reproduced from [4,5], with permissions of Elsevier, 2009 and Springer Nature, 2016). (b) Simulation results showing transition from columnar grains at the beginning of the scan to equiaxed grains towards the end of the scan in an Inconel 718 specimen with a $G / R$ ratio meant to yield columnar microstructure (Reproduced from [6], with permission of Elsevier, 2019).
+
+Surrogate-assisted modeling and optimization techniques are popularly used in various applications that involve expensive computational models [17-23] for function evaluations. Among the different types of surrogate models, GPs [15] are used profusely for modeling black-box functions whereby a fully Bayesian approach allows for probabilistic estimates of the target functions [24-27]. With a relatively small number of measurements, a GP surrogate can be learnt to serve as a proxy to an expensive objective function [28] (e.g., prediction of melt pool dimensions for a range of process parameters in metal AM [29]). Under the settings of a GP surrogate, a Bayesian Optimization (BO) set-up can be invoked for gradient-free global optimization of an objective function under budget limitations [30] (e.g., prediction of optimal process parameters for controlling temporal melt pool dimensions in metal AM within $N$ number of iterations; $N$ being an user input). Moreover, with nonlinearities in the objective function, a search for optimum would require significant amount of sampling in the search space, particularly in high dimensions. In such settings, BO is found to be quite successful [31].
+
+In the field of AM, there are very few applications of GPs as surrogate models for expensive experiments and simulations. For example, Tapia et al. [32] used experimental data to learn spatial GPs for predicting porosity in metal AM produced during the laser powder bed fusion process. In another work, Tapia at al. [16] demonstrated the usage of GPs in predicting melt pool depth as a function of different process parameters, which were used to describe regimes of operation where the process was expected to be robust. Seede et al. [33] used a GP framework to develop a calibrated surrogate model for predicting optimal process parameters for building porosity free parts with low-alloy martensitic steel AF9628. However, to the best of the authors' knowledge, no work exists in the field of AM that leverages the surrogate-based predictions and uses them as a basis for active learning strategies for optimizing a desired objective, e.g., controlling melt pool dimensions over time under computational budget constraints in laser powder bed fusion process. The design and deployment of such predictive methodologies will immensely augment the response time of feedback or feedforward control strategies as the current response times are rather long compared to the process time scales [34].
+
+To address this gap, this paper proposes a novel framework of controlling the melt pool geometry in laser powder bed fusion process by formulating it as an optimization problem through the integration of the tools of physics-based analytical modeling and data-driven analysis. The cardinal contribution of this work in AM is to bridge the gap in model-assisted prediction and control of melt pool geometry by using ML techniques that can potentially accelerate the process of melt pool geometry control. The results demonstrate that by using a low-cost surrogate-assisted modeling using GP and BO, it is possible to obtain an excellent estimation of process parameter evolution as a function of time in order to maintain the desired meltpool geometry (e.g., dimensions) throughout the build process. The framework consists of three critical steps: (i) evaluation of the thermal field using an experimentally validated 3D analytical melt pool evolution model which serves as a source of data for formulating a GP surrogate, (ii) development of GPs with flexible kernel structures capable of handling anisotropies in the dataset, and (iii) establishment of a BO framework involving the GP surrogate that aims to solve a global optimization problem in a gradient-free setting under a constrained budget. The computational budget is predefined in most practical problems. With an appropriate selection of initial design points and acquisition function for active learning of optimal design points, this work achieves estimates of process parameters with limited model iterations. The GP surrogates, being based on Bayesian inference, offer principled estimates of uncertainty in predictions. While the present work uses a 3D analytical model for demonstrating the efficacy of the proposed approach, the framework described can be applied to any other prediction models of users' choice.
+
+The paper is organized in five sections including the current one. Section 2 describes the development of a hybrid modeling framework consisting of the 3D analytical model and its validation against experimental data. Section 3 presents the results and discusses the significant findings. The paper is summarized and concluded in Section 4 along with recommendations for future research.
+
+Supplemental information in Appendix A provides the mathematical background for ML aspects of modeling and optimization.
+
+\section*{2. Development of a Hybrid Modeling Framework}
+\subsection*{2.1. Simulation Model}
+A well-known analytical model developed by Eagar and Tsai [35] that solves for the 3-dimensional temperature field produced by a traveling distributed heat source on a semi-infinite plate is used in the present work. This model has been previously used by several different researchers for evaluating melt pool evolution in laser powder bed fusion (L-PBF) [33] and directed energy deposition (DED) [36] AM processes when extensive evaluation of process parameter space is required. It is a distributed heat source modification of the Rosenthal's $[37,38]$ solution for the temperature distribution produced by a traveling point heat source. As compared to Rosenthal's solution, Eagar-Tsai's model provides a significant improvement in prediction of temperature in the near heat source regions. Figure 2 explains the coordinate system used in the model. The heat source is traveling with a uniform speed of $v$ in the $X$-direction, and is assumed to be a 2D surface Gaussian:
+
+
+\begin{equation*}
+Q(x, y)=\frac{P}{2 \pi \sigma^{2}} e^{-\frac{\left(x^{2}+y^{2}\right)}{2 \sigma^{2}}} \tag{1}
+\end{equation*}
+
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-04}
+\end{center}
+
+Figure 2. Schematic illustrating the coordinate system of the analytical model.
+
+Here, $Q(x, y)$ is the power distribution per unit area on the $X-Y$ plane of the specimen produced by the Gaussian beam of peak power $P$ with a distribution parameter $\sigma$. According to Eagar-Tsai's model, the temperature $T(x, y, z, t)$, at a particular location $(x, y, z)$ and time $t$, with the initial temperature of the substrate being $T_{0}$, is denoted as
+
+
+\begin{equation*}
+T(x, y, z, t)-T_{0}=\frac{\alpha_{L} P}{\pi \rho c(4 \pi a)^{1 / 2}} \int_{0}^{t} \frac{d t^{\prime}\left(t-t^{\prime}\right)^{-1 / 2}}{2 a\left(t-t^{\prime}\right)+\sigma^{2}} e^{-\frac{\left(x-v t^{\prime}\right)^{2}+y^{2}}{4 a\left(t-t^{\prime}\right)+2 \sigma^{2}}-\frac{z^{2}}{4 a\left(t-t^{\prime}\right)}} \tag{2}
+\end{equation*}
+
+
+Here, $\alpha_{L}$ is the absorptivity of the laser beam, $a \triangleq \frac{k}{\rho c}$ is the thermal diffusivity, $\rho$ is the density, and $c$ is the specific heat capacity of the material of the specimen. The primary assumptions of the model include the following.
+
+\begin{enumerate}
+  \item Convective and radiative heat transfer from the substrate to the environment are ignored. As the process deals with metals which are good conductors, heat transfer through radiation is negligible [39] as compared to that due to conduction.
+
+  \item The temperature dependence of the thermo-physical properties is not taken into account.
+
+  \item The substrate is semi-infinite, therefore the increase in the surface temperature $T_{0}$ with time is negligible.
+
+  \item Phase change of the material is not taken into consideration.
+
+\end{enumerate}
+
+Eagar-Tsai's model, despite its limitations (e.g., its inability to model keyholing effect [33]), is widely used in literature for parameter space exploration through design of experiments ( $\mathrm{DoE}$ )-based approaches. Given the wide range of process parameters in metal AM machines (e.g., L-PBF has over 130 parameters that can affect the final part quality [40]), such a DoE approach would require an exponential number of samples in the dimension of the parameter space to fully explore the design space. This makes it prohibitively expensive to explore the design space for building complex parts through experimentation or high-fidelity simulations. Computationally efficient surrogates can reduce the computational burden by a significant amount. Eagar-Tsai's model, when appropriately calibrated, provides one such alternative as a low-cost simulation model that are extensively used by several researchers in recent times [33,41,42]. This model especially works well for single-track and single-layer AM depositions. Owing to these characteristics, the validation and control experiments outlined in this paper are all single-track and single-layer.
+
+\subsection*{2.2. Experimental Validation of the Simulation Model}
+Eagar-Tsai presented experimental validation of their model with a range of process parameters in welding carbon steel plates [35]. Other researchers validated this model for a range of different materials such as nickel- [43], iron- [33], and titanium-based [44] alloys among others to obtain satisfactory accuracy with the experimental data across a range of AM processes, e.g., L-PBF and DED [45], in recent times. However, most of the experimental validation of this model, so far, has been performed under steady state conditions. Being a transient model, Eagar-Tsai's formulation can potentially be used to simulate the melt pool dimensions as a function of time as well. In this section, both steady and unsteady state validation of the Eagar-Tsai's formulation are presented.
+
+\subsection*{2.2.1. Steady-State Experimental Validation}
+For the steady-state model validation, the melt pool dimensions calculated by Eagar-Tsai's model are compared with those obtained using finite-element simulations and experiments for a popular nickel-based superalloy CMSX-4 ${ }^{\circledR}$ reported by Wang \href{http://et.al}{et.al} [36]. This alloy was processed by several different researchers using L-PBF [46,47], electron beam powder bed fusion (E-PBF) [48], and DED [49], and therefore a considerable amount of experimental data is available in the open literature. The thermo-physical properties of CMSX-4 ${ }^{\circledR}$ chosen in this work are those reported by Gäumann et al. [50]: $k=22 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}), \rho=8700 \mathrm{~kg} / \mathrm{m}^{3}, c=690 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K})$, and the liquidus temperature $T_{L}=1660 \mathrm{~K}$. Three surface melting experiments were reported by Wang et al. under different processing parameters, as enlisted in Table $1 . \mathrm{A} \mathrm{CO}_{2}$ laser beam with radius $\mathrm{W}$ was used in their work.
+
+Table 1. Processing parameters as reported in Wang et al. [36].
+
+\begin{center}
+\begin{tabular}{cccc}
+\hline
+Specimen & A & B & C \\
+\hline
+$T_{0}(K)$ & 300 & 300 & 300 \\
+$P(\mathrm{~W})$ & 900 & 900 & 450 \\
+$v(\mathrm{~mm} / \mathrm{s})$ & 2 & 6 & 6 \\
+$\mathrm{~W}(\mathrm{~mm})$ & 0.39 & 0.39 & 0.20 \\
+$\alpha_{L}$ & 0.114 & 0.114 & 0.114 \\
+\end{tabular}
+\end{center}
+
+Figure 3 shows the comparison between the current work with those obtained experimentally and using finite element simulations by Wang et al. [36] showing excellent agreement. Figure 4 shows a transverse cross section $(Y-Z$ plane, with $x=0)$ of the analytically calculated melt pool geometry juxtaposed on the fusion zone produced by the process parameters in the CMSX-4 ${ }^{\circledR}$ Specimen A, as reported by Wang et al. [36]. Similar agreements are also observed for the other two specimens (i.e., B and C), as evident from the results reported in Figure 3. Cross-sectional images for specimens B and $\mathrm{C}$ are omitted for brevity.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-06}
+\end{center}
+
+Figure 3. Comparison of experimentally measured melt pool dimensions with simulations results provided by Wang et al. (blue markers) [36] and the present work (red markers). Squares correspond to Specimen A, circles to Specimen B, and Diamonds to Specimen C. The filled markers indicate the melt pool depths and the unfilled ones indicate the melt pool width.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-06(2)}
+\end{center}
+
+Figure 4. Comparison of experimental and calculated melt pools for the CMSX-4 ${ }^{\circledR}$ Specimen A of Wang et al [36] (Reproduced from [36], with permission of Elsevier, 2017.). L corresponds to the liquid zone of the melt pool and s corresponds to the solid substrate.
+
+\subsection*{2.2.2. Transient Experimental Validation}
+After obtaining excellent agreement with the steady-state results, the analytical model is further validated using transient melt pool results available in the open literature [51]. Figure 5 shows the longitudinal cross section of an L-PBF-processed CMSX-4 ${ }^{\circledR}$ specimen. This specimen is fabricated by consolidating a powder layer thickness of $1.4 \mathrm{~mm}$ on a CMSX-4 ${ }^{\circledR}$ substrate having dimensions of $35.56 \mathrm{~mm}$ (length) $\times 6.86 \mathrm{~mm}$ (width) $\times 2.54 \mathrm{~mm}$ (thickness). The process parameters reported are $750 \mathrm{~W}$ laser power, $12.7 \mu \mathrm{m}$ raster scan spacing, and $600 \mathrm{~mm} / \mathrm{s}$ raster scan speed. The raster scanning speed $\left(v_{R}\right)$ is related to the linear velocity ( $\left.\mathrm{v}\right)$ of the laser in the $X$ direction by $v=\frac{\text { scan spacing } \times v_{R}}{2 \times \text { width }}$.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-06(1)}
+\end{center}
+
+Figure 5. Longitudinal cross section of a CMSX-4 ${ }^{\circledR}$ specimen fabricated with $750 \mathrm{~W}$ laser power and scan speed of $600 \mathrm{~mm} / \mathrm{s}$ showing increase in melt pool depth along the length of the specimen (Reproduced with permission from the authors of [51].).
+
+Figure 6 shows the comparison between the analytically calculated and experimentally observed melt pool depth for the CMSX-4 ${ }^{\circledR}$ specimen as illustrated in Figure 5. The maximum absolute relative error between the experimental and analytical melt pool depths is $\sim 8 \%$ at the start of the scan period, with the mean absolute error during the $30 \mathrm{~s}$ period being $2.71 \%$. The results also show the transient\\
+nature of the melt pool, with the depth continuously increasing as a function of time with a fixed set of process parameters.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-07}
+\end{center}
+
+Figure 6. Comparison of analytically calculated transient melt pool evolution with experimental data.
+
+\subsection*{2.3. Hybrid Model}
+The validation results show that Eagar-Tsai's model can provide excellent low-cost estimates of melt pool dimensions for single-track and single-layer AM depositions. The goal of this work is to obtain model-based estimates of the required temporal variations in process parameters for achieving target melt pool dimensions during the build process. In order to achieve this, an optimization problem that finds process parameters as a function of time is solved with an objective of minimizing the deviation from the desired melt pool dimensions. As the true functional form that relates the process parameters to the melt pool dimensions is unknown, the information regarding gradients is not readily accessible, therefore the optimization problem is in a black-box setting. Several such black-box optimization techniques are well-studied by different researchers [52], e.g., stochastic process-based approaches (kriging methods) [53], evolutionary algorithms [54], trust-region based algorithms [55], and random search [56]. These approaches typically involve iterative sampling of the objective function in the search space. Such sampling often proves to be prohibitively expensive, particularly in high dimensions, when the involved process model is computationally very expensive.
+
+To address this challenge, in the present work, an application of gradient-free optimization under budget limitations is implemented by solving the melt pool geometry control problem using BO that employs computationally inexpensive GP surrogate models learnt via sparse sampling of the space of process parameters (e.g., $P$ and $v$ ). The details of the mathematical formulations of GP and BO can be found in Appendix A. Although all optimization demonstrations in this paper involve the Eagar-Tsai's model as the process model, it can very well be replaced by any other process models of users' choice, without any fundamental change in the optimization algorithm.
+
+\section*{3. Results and Discussion}
+This section presents the results and discusses the significant findings therefrom.
+
+\subsection*{3.1. Prediction of Melt Pool Dimensions Using GP}
+The melt pool dimensions in the L-PBF AM process continuously increase due to progressive heating of the specimen caused by the layer-wise fabrication as evinced by simulation results [57] as well as experimental observation [47]. Figure 7 shows the variation in melt pool dimensions as a function of time for a CMSX-4 ${ }^{\circledR}$ specimen fabricated using a laser power of $900 \mathrm{~W}$ and linear scan velocity of $0.25 \mathrm{~mm} / \mathrm{s}$. The melt pool depth increases from $1.814 \mathrm{~mm}$ at $t=1 \mathrm{~s}$ to $3 \mathrm{~mm}$ at $t=20 \mathrm{~s}$, while the melt pool width increases from $3.82 \mathrm{~mm}$ to $6.11 \mathrm{~mm}$. The percentage increase in melt pool\\
+depth is $\sim 65 \%$, and the increase in width is $\sim 59 \%$. The melt pool reaches a steady state at $\sim 20 \mathrm{~s}$. If the melt pool dimensions need to be maintained at desired values, the process parameters should be adjusted during this transience period. However, performing simulations for a range of process parameters over the entire deposition process is computationally expensive when coupled with an optimization framework.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-08}
+
+Figure 7. Temporal variation of melt pool dimensions for a CMSX-4 ${ }^{\circledR}$ specimen fabricated using $P=900 \mathrm{~W}$ and $v=0.25 \mathrm{~mm} / \mathrm{s}$ : (a) longitudinal cross section with the liquidus isotherm showing the variation in melt pool dimensions with time and $(\mathbf{b})$ variation of melt pool depth and width with time.
+
+In applications where data is scarce, particularly when each simulation can be potentially computationally expensive, it is often infeasible to have a large number of training points. Moreover, inherent uncertainties in the physical process and the simulation models (e.g., uncertainties in the thermophysical properties of the material) mandate the need of uncertainty quantification with the predictions $[33,58]$. Surrogates, like GPs, can be very useful in such cases in order to have probabilistic estimates of the quantity of interest with limited datasets, whereby the posterior mean and variance can not only be informative for prediction at unknown input locations, but the information can also be used for budget-constrained optimization, as described in the following subsection. A surrogate modeling and optimization technique as discussed in Appendix A is adopted in this work to find the model-based estimates of the process parameters required for controlling the melt pool dimensions.
+
+In order to build the surrogate using GPs, two-hundred (200) Latin hypercube sampling (LHS) simulations are performed to obtain the values of melt pool depth and width at each time instant for combinations of $P$ and $v$ in the range of 300 to $1200 \mathrm{~W}$ and 0.5 to $2.5 \mathrm{~mm} / \mathrm{s}$, respectively. This parameter range is chosen to avoid any keyhole mode of melt pool formation in CMSX-4 ${ }^{\circledR}$ where the Eagar-Tsai's model shows limited accuracy $[59,60]$. The cross section of melt pools created in conduction mode is generally semicircular, as predicted by Eagar and Tsai's conduction mode model [35]. LHS is selected as this is one of the most commonly used statistical methods for DoE. It allows for a good spread of the initial DoE over the design region with limited iterations due to its high sampling efficiency [61]. For every time instant under consideration, each DoE point comprises a combination of process parameters $(P, v)$ and its corresponding melt pool depth and width values.
+
+The effect of training data size on the regression performance of the GP surrogate in predicting melt pool depth and width is of critical interest. Out of the 200 initial LHS points, a test set of randomly selected 100 points is set aside for testing the regression performance of the GP models. Different GPs are trained at different time instants with randomly selected samples from the training set. The training data size is varied from 10 to 100 , in steps of 10 samples. Ten random selections of the training data is chosen for each training size, in order to find the average behavior of the regression performance for each training size. The prediction performances are tested on the set aside 100 samples for each trained GP model.
+
+The metric for gauging the prediction performance is chosen as the Relative Squared Error (RSE) $\triangleq \frac{\left\|\hat{y}-y^{*}\right\|^{2}}{\left\|y^{*}\right\|^{2}}$, where $\hat{y}$ is the prediction and $y^{*}$ is the true value (i.e., ground truth as obtained by the Eagar-Tsai's model). A lower RSE suggests that the most likely prediction (mean of the predicted posterior Gaussian distribution) of the depth/width in the test set matches closely with the true value. Figure 8 shows that the RSE is quite low for all the time steps, while there is a slight increase in RSE from $2 \mathrm{~s}$ to $20 \mathrm{~s}$ for both depth and width. RSEs for the melt pool depth and width prediction show a sharp drop from training size of 10 to 20 samples, after which the RSE saturates for almost all the higher training sizes. This indicates that a training size of at least 20 LHS DoE points is in general sufficient for a satisfactory prediction of the melt pool depth and width in the selected process parameter space. This provides an estimate of the number initial DoE points required for learning the surrogate model for the subsequent $\mathrm{BO}$ steps at each time instant.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-09}
+
+Figure 8. Relative Squared Error in prediction of melt pool depth and width at (a) $t=2 \mathrm{~s}$, (b) $t=5 \mathrm{~s}$, (c) $t=10 \mathrm{~s}$, and (d) $t=20 \mathrm{~s}$.
+
+Figure 9 shows the regression performance (on the test dataset of 100 samples) in predicting the melt pool depth and width at $2 \mathrm{~s}$ with a training set size of 20. Sample numbers in Figure 9a,c refer to the test samples, which are different $(P, v)$ combinations. The test samples are arranged in an ascending order of their true depth values. The $2 \sigma$-band coverage percentage is the proportion of test points for which the true amplitude lies within $\pm 2 \sigma$ of the predicted mean, which corresponds to the $95 \%$ confidence interval, as the output distribution is a Gaussian. The parity plots in Figure 9b,d show the comparison of the predicted means with the true values. The high $R^{2}$ (coefficient of determination) scores for the depth (0.96) and width (0.99) predictions at 2 s indicate high reliability of the GP model in predicting the melt pool dimensions. Similar results are shown in Figure 10 for the melt pool depth and width predictions at $20 \mathrm{~s}$, with a training set size of 20 samples. The $R^{2}$ score decreases to 0.82 for depth prediction and 0.88 for width prediction, which is also explained by the higher RSE in prediction at $20 \mathrm{~s}$ (Figure 8). The variability in the predicted values from the true values is explained by the\\
+wider $\pm 2 \sigma$ band at the test locations. The lower $R^{2}$ score at $20 \mathrm{~s}$ as compared to $2 \mathrm{~s}$ can be attributed to the higher variability in the steady state values of depth and width as compared to the initial stages as a function of the process parameters. From the perspective of applicability of the surrogate predictions, GPs provide the end user with not only a computationally inexpensive way of predicting the melt pool dimensions at different operating conditions for which experiments and simulations are not performed, but also with an estimate of uncertainty quantification (UQ) for predictions from the variance associated at the query points.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-10(1)}
+
+Figure 9. Melt pool depth and width prediction performance at $2 \mathrm{~s}$ with a training set size of 20 samples. $(\mathbf{a}, \mathbf{b})$ Probabilistic prediction of depth and corresponding parity plot. (c,d) Probabilistic prediction of width and corresponding parity plot. Samples arranged in ascending order of the true values in panels $(\mathbf{a}, \mathbf{c})$. Predicted values at the points of query are represented as follows; GP mean by blue dots and $\pm 2 \sigma$ bands by vertical bars, where $\sigma$ is the standard deviation of the GP's posterior prediction at a query point. Red stars indicate the true value of the melt pool depth at those points.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-10}
+
+Figure 10. Cont.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-11}
+
+Figure 10. Melt pool depth and width prediction performance at $20 \mathrm{~s}$ with a training set size of 20 samples. (a,b) Probabilistic prediction of depth and corresponding parity plot. (c,d) Probabilistic prediction of width and corresponding parity plot. Samples arranged in ascending order of the true values in panels $(\mathbf{a}, \mathbf{c})$. Predicted values at the points of query are represented as follows; GP mean by blue dots and $\pm 2 \sigma$ bands by vertical bars, where $\sigma$ is the standard deviation of the GP's posterior prediction at a query point. Red stars indicate the true value of the melt pool depth at those points.
+
+\subsection*{3.2. Optimization of Process Parameters for Melt Pool Dimension Control: Objective Function}
+As discussed in Sections 1 and 2, appropriate control of the process parameters, e.g., $P$ and $v$ are required for maintaining desired melt pool dimensions during the deposition process. The goal of controlling the melt pool dimensions is formulated as an objective function that needs to be maximized in the setting of BO, as described in Appendix A.3. In order to pose the melt pool control problem in an optimization setting, an appropriately defined objective function $(J)$ is to be maximized at every discrete time instant of control,
+
+
+\begin{equation*}
+J \triangleq-\left(c_{1} \frac{\left|d-d^{*}\right|}{\left|d^{*}\right|}+c_{2} \frac{\left|w-w^{*}\right|}{\left|w^{*}\right|}+c_{3} \frac{\left|P-P_{\min }\right|}{\left|P_{\max }-P_{\min }\right|}\right) \tag{3}
+\end{equation*}
+
+
+where $d$ and $\mathrm{W}$ denote the depth and width, respectively, at a particular time instant, whereas $d^{*}$ and $w^{*}$ represent the desired depth and width, respectively, during the deposition process. $P$ indicates the power at time instants of control, whereas $P_{\max }$ and $P_{\min }$ denote the maximum and minimum values of the range of laser power in the space of process parameters. Therefore, $\frac{\left|P-P_{\min }\right|}{\left|P_{\max }-P_{\min }\right|}$ denotes the normalized power input at a particular time instant, and incorporating it into the objective functional serves as a penalty term for high power values. This is introduced since it is desired to achieve the controlled process along with avoiding processing conditions involving very high levels of power.
+
+$c_{1}, c_{2}$, and $c_{3}$ in Equation (3) denote the relative weights of the components of the objective function that controls the depth, width, and power, respectively. By varying the relative values of $c_{1}, c_{2}$, and $c_{3}$, it is possible to preferentially weigh the objective function to meet the requirements. Formulating the objective functional $J$ is a key step in the optimization process, and the optimization routine should be designed in such a way that undesirable process parameters result in low values of objectives. This formulation conforms to the mathematical characteristics of the Matern covariance function [15] used in this work, which assumes local smoothness of the inputs, so that input process parameters that are close together in the $(P, v)$ space are expected to have similar objective values. As undesirable parameter combinations have low objectives, points in the close vicinity of them will have low acquisition potential (see Appendix A) during the optimization steps, and therefore will have lower priority for the incumbent selection.
+
+\subsection*{3.3. Optimization Routine}
+A sequential global-local optimization thread [62] is employed in this work for the selection of process parameters for controlling the melt pool geometry. The terms "global" and "local" correspond to the search spaces with respect to which BO is performed. In the global optimization thread, a potential optimal process parameter combination is selected, around which a refinement is made during the local optimization thread for the final selection of the optimal process parameter combination. The outline of the process is described in the flowchart as depicted in Figure 11. Details of the optimization algorithm can be found in Appendix B. Entities capped with "tilde" $(\sim)$ correspond to parameters of the local optimization thread.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-12}
+\end{center}
+
+Figure 11. Flowchart depicting the global-local optimization process.
+
+The control problem is designed to maintain the melt pool depth $\left(d^{*}\right)$ at $2.5 \mathrm{~mm}$ and width $\left(w^{*}\right)$ at $5.0 \mathrm{~mm}$. The initial global GPs are learnt with $20 N_{\text {init }}$ LHS points in the $(P, v)$ space, with $P \in[300,1200] \mathrm{W}$ and $v \in[0.5,2.5] \mathrm{mm} / \mathrm{s}$. The ranges of the process parameters and the controlled geometrical parameters chosen in this problem are often dictated by the experimental requirements and constraints, and can be modified as needed. The optimization problem over the entire duration of the L-PBF process can be considered as a sequence of optimization problems performed at some discrete intervals of time, which motivates the choice of learning separate surrogates for each discrete time instant for which the melt pool needs to be controlled.
+
+With the initial trained GP from 20 LHS samples for each time instant, low-cost surrogate predictions are made over the large global search space $X_{\text {star }}$, consisting of 2000 uniformly chosen points. This methodology forms the key to the surrogate-based optimization processes: predictions are made over an extensive search space by employing a low-cost surrogate by avoiding functional evaluations (physics-based simulations in this case) at the search points. Based on the mean-variance trade-off of the GP's posterior predictive distribution, the acquisition function EI (See Appendix A) guides the search iteratively to subsequent optimization points until the computational budget is depleted. $N_{\text {GlobalIter }}=10$ optimization steps are chosen for the global thread, after which the optimal process parameter combination with the highest objective value is selected.
+
+For all the time instants of control, it was possible to obtain optimal process parameters within the global thread that can yield the melt pool depth and width close to the desired values.
+
+However, for finer adjustments in $P$ and $v$ values in order to achieve the micron-level control in the melt pool geometry, the local optimization thread is executed. Here, the local search space $\tilde{X}_{\text {star }}$ is built around $\tilde{\mathbf{x}}^{*}$, the optimum with the highest objective value from the global thread, in the following way; $P$ is varied within $\pm 20 \mathrm{~W}$ and $v$ within $\pm 0.15 \mathrm{~mm} / \mathrm{s}$ of the corresponding values in $\tilde{\mathbf{x}}^{*} .20 \tilde{\mathrm{N}}_{\text {init }}$ LHS points are chosen within the limits of $\tilde{X}_{\text {star }}$, the optimization routine is performed within this local search space for $\tilde{N}_{\text {Locallter }}=10$ steps. This yields the refinement of the process parameter values that result in melt pool dimensions very close to the ones desired. The number of optimization iterations in both the local and global threads (i.e., $N_{\text {Globaliter }}$ and $\tilde{N}_{\text {Localtter }}$ ) are kept as design parameters in this paper, which are expected to be driven by computational budget for practical applications. The scaling factors for the objective function components are chosen as $c_{1}=1, c_{2}=0.1$, and $c_{3}=0.1$, which are based on the greater relative importance of maintaining the depth as close to $d^{*}$ as possible during the build process, so that a uniform deposit is maintained.
+
+Figure 12 shows the performance of the surrogate based optimization method. Figure 12a,b shows the variation of the controlled melt pool depth and width from the desired $d^{*}$ and $w^{*}$ values as a function of time. It is seen that the maximum variation in the meltpool depth is $1.1 \mu \mathrm{m}$ which is $0.04 \%$ of the desired depth, while that for width is $173 \mu \mathrm{m}$, which is $3.46 \%$ of the desired width. The process parameters change from $P=948.07 \mathrm{~W}, v=0.40 \mathrm{~mm} / \mathrm{s}$ at $t=2 \mathrm{~s}$ to $P=737.81 \mathrm{~W}, v=0.37 \mathrm{~mm} / \mathrm{s}$ at $t=20 \mathrm{~s}$ (Figure 12c,d). It is to be noted that penalization of $P$ in the objective function allows us to have a smoothly variation of $P$ over $20 \mathrm{~s}$ with lower $P$ values, which changes by $\sim 22 \%$ of the starting value at $t=2 \mathrm{~s}$. The maximum change in $v$ is $\sim 27 \%$. It is possible to have process parameters with higher $P$ values (and higher $v$ values) that control the melt pool, but those conditions are avoided by $P$ penalization in the objective function. Similar formulation of the objective function can be pursued by $v$ penalization, if smother $v$ transition is desired.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-13}
+
+Figure 12. Time-varying control of process parameters at intervals of $2 \mathrm{~s}$ for maintaining $d^{*}=2.5 \mathrm{~mm}$ and $w^{*}=5.0 \mathrm{~mm}$ : (a) variation in melt pool depth $\left(d-d^{*}\right)$ at the control steps, (b) variation in melt pool width $\left(w-w^{*}\right)$ at the control steps, (c) optimal laser power, and (d) optimal laser velocity.
+
+\subsection*{3.4. Validation with Experimental Results of Melt Pool Depth Control}
+The model-based control strategy is validated with experimental results reported in the open literature [63]. Figure 13a shows the longitudinal cross section of an L-PBF processed René 80 specimen fabricated using a powder layer thickness of $1.4 \mathrm{~mm}$ on a substrate of dimensions $35.56 \mathrm{~mm} \times 6.86 \mathrm{~mm} \times 2.54 \mathrm{~mm}$. The experiment was carried out with the raster scanning speed of $450 \mathrm{~mm} / \mathrm{s}$ and scan spacing of $25.4 \mu \mathrm{m}$. In reality, experiments are extremely challenging to perform with melt pool depth as a controlled variable as there exists no easy way of measuring the melt pool depth in situ. During the conduction mode, the surface temperature control of melt pools was found to correlate well with the melt pool depth experimentally by Bansal et al. [63]. The observation was also made by Raghavan et al. in E-PBF of Incone ${ }^{\circledR} 718$ [64]. The mean value of the melt pool depth during the control period is around $1200 \mu \mathrm{m}$ from experiments. Accordingly, $d^{*}$ is set at $1200 \mu \mathrm{m}$ for the surrogate-based optimization routine involving only $P$ as outlined in the experiments.
+
+The melt pool simulation is performed by employing the Eagar-Tsai's model, with the thermo-physical properties of solid René $80: k=24.56 \mathrm{~W} /(\mathrm{m} \cdot \mathrm{K}), \rho=7604 \mathrm{~kg} / \mathrm{m}^{3}, c=600 \mathrm{~J} /(\mathrm{kg} \cdot \mathrm{K})$, and the liquidus temperature $T_{L}=1607 \mathrm{~K}$ obtained using the software JMatPro ${ }^{\circledR}$ [65], with the alloy composition of René 80 provided by the vacuum alloy product catalog of Cannon Muskegon [66]. The objective function involves two components in this case:
+
+
+\begin{equation*}
+J \triangleq-\left(c_{1} \frac{\left|d-d^{*}\right|}{\left|d^{*}\right|}+c_{2} \frac{\left|P-P_{\min }\right|}{\left|P_{\max }-P_{\min }\right|}\right) \tag{4}
+\end{equation*}
+
+
+with $c_{1}=1$ and $c_{2}=0.1$.
+
+Figure 13 shows that the predicted melt pool depth with the surrogate assisted control scheme remains very close to the desired $d^{*}$. A variation of $\sim 200 \mu \mathrm{m}$ is observed in the experimental results, whereby the melt pool depth shows a slightly increasing trend towards the end of the control process, reflected by the slight increase in power input from $10 \mathrm{~s}$ till $14 \mathrm{~s}$. The trend of the surrogate predicted controlled power input matches closely with the experimental results till $8 \mathrm{~s}$, which is explained by the fact that for a constant laser speed, the power input should decrease as a function of time in order to maintain a constant melt pool depth as the specimen gains thermal energy continuously. Nonetheless, the validation results indicate the efficacy of the surrogate-based optimization routine in predicting process parameters as a function of time for achieving melt pool dimension control during the deposition process.
+
+(a)\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-14}
+
+Figure 13. (a) Longitudinal cross section showing a René 80 specimen fabricated using an adaptive control scheme (Reproduced from [63], with permission of the author). (b) Melt pool depth as a function of time. (c) Controlled laser power as a function of time, obtained by the adaptive control scheme of Bansal [63], compared with the results from surrogate based optimization method. The black markers in $(\mathbf{b}, \mathbf{c})$ show the variation in the melt pool depth and laser power as a function of time during the time of control. The red markers in $(\mathbf{b}, \mathbf{c})$ show the variation in the melt pool depth and laser power obtained by the BO routine.
+
+\section*{4. Summary and Conclusions}
+This paper develops a novel hybrid methodology for control of melt pool geometry in AM in the framework of ML-assisted modeling and optimization. The continuous changes in the melt pool geometry are predicted by a low-cost GP surrogate developed using an experimentally validated analytical 3D model. The uncertainties in the predictions for the melt pool geometry are quantified using GPs. Reliable estimates of the optimal process parameter evolution are obtained through active learning via BO by devising appropriate objective functions that quantify the control requirements. The methodology provides an estimation of process parameter variation during the AM deposition process in order to maintain a desired melt pool geometry under continuously varying thermal conditions.
+
+Being data-driven, the reliability of the optimized process parameters obtained from the algorithm is based on the accuracy of the underlying physical model in predicting the melt pool geometry in the range of process parameters considered. However, with a high-fidelity model, the computational cost involved in the optimization process can be significantly high. For example, the cost of running a Netfabb ${ }^{\circledR}$ [67] DED model for a single-track and single-layer process on a CMSX-4 ${ }^{\circledR}$ specimen as described in this paper is $\sim 10$ times as expensive as Eagar-Tsai's model at $t=2 \mathrm{~s}$, and $\sim 150$ times at $t=10 \mathrm{~s}$. Although such high-cost simulation is prohibitive in practical applications, in the future, the methodology will be enhanced by using information from different fidelities [68] of AM computational models to develop a multifidelity modeling and optimization framework for prediction and control of melt pool geometries. For example, a Netfabb ${ }^{\circledR}$ model can be combined with inexpensive observations from Eagar-Tsai's model to develop a two-fidelity GP that can be used for melt pool geometry prediction and optimization with an expected reduction in computational cost. Additional investigations are also planned as summarized below.
+
+\begin{enumerate}
+  \item Usage of process parameters (other than $P$ and $v$ ) such as scan spacing (i.e., hatch spacing), scan pattern (i.e., hatch pattern), build plate temperature, and powder layer thickness (for powder bed AM) and powder feed rate (for directed energy AM) as control inputs.
+
+  \item Incorporation of design constraints in the surrogate assisted modeling framework, e.g., tackling harder problems whereby a melt pool geometry needs to be controlled, along with the final microstructure such as columnar grains in CMSX-4 ${ }^{\circledR}$.
+
+  \item Development of heterogeneous design spaces in formulating multifidelity modeling framework, which can be useful in catering to optimization problems where the varied levels of fidelities have different input spaces. As an example, a HF L-PBF process model can have several process parameters, such as powder distribution properties, hatch spacing, scan strategy, etc., apart from $P$ and $v$ of the laser (which are the only process parameters for a LF model like the Eagr-Tsai's model), which can potentially affect the build characteristics. Optimization in such a framework can be possible via heterogeneous transfer learning [69] to learn from a common subspace of the inputs.
+
+  \item Integration of the surrogate modeling framework developed in this work with microstructure prediction framework using an open source SPPARKS code [70] to optimize the process parameters for obtaining a desired microstructure.
+
+  \item Implementation of the surrogate modeling framework for checking its efficacy towards mitigating unintended melt pool behavior such as keyholing effect. However, in such a case, the computational model needs to be a high fidelity one capable of predicting such a behavior.
+
+\end{enumerate}
+
+It is envisioned that the newly developed framework can be implemented in developing feedback strategies for melt pool control during AM processes with shorter response time due to improved prediction capability [71].
+
+Author Contributions: Conceptualization, S.M. and A.B.; methodology, S.M.; software, S.M. and D.G.; validation, S.M., A.B. and D.G.; formal analysis, S.M.; investigation, S.M. and A.B.; resources, A.B. and A.R.; writing-original draft preparation, S.M. and A.B.; writing-review and editing, A.B., S.M. and A.R.; visualization, S.M.; supervision, A.B. and A.R.; funding acquisition, A.R. and A.B. All authors have read and agreed to the published version of the manuscript.
+
+Funding: This research was funded in part by U.S. Air Force Office of Scientific Research (AFOSR) under Grant \# FA9550-15-1-0400 in the area of dynamic data-driven application systems (DDDAS). Any opinions, findings, and conclusions in this paper are those of the authors and do not necessarily reflect the views of the sponsoring agencies.
+
+Conflicts of Interest: The authors declare no conflict of interest.
+
+\section*{Appendix A. Mathematical Background}
+This section provides the mathematical background for machine learning (ML)-assisted modeling and optimization of additive manufacturing (AM) processes. While the details are available in standard literature (see, e.g., in [72,73]), the following three subsections succinctly present the core concepts of ML for completeness of the paper.
+
+\section*{Appendix A.1. Surrogate Modeling: Gaussian Processes}
+This subsection provides a brief overview of the surrogate modeling technique using Gaussian Processes (GPs), which is the foundation of the Bayesian Optimization (BO) framework used in this work for control of melt pool geometry. Salient properties of GPs modeling are delineated below.
+
+\begin{enumerate}
+  \item GPs belong to a class of stochastic processes based on the assumption of a multivariate jointly Gaussian distribution for any finite collection of random variables.
+
+  \item GPs are nonparametric models that do not assume a predefined functional relationship between the inputs and outputs (unlike polynomial regression models [73], for example).
+
+  \item GPs are flexible for modeling nonlinear functions. As the underlying concept is fully Bayesian, the predictions are made via a posterior probability distribution, which is a normal distribution in case of GPs, completely specified by its mean and covariance.
+
+\end{enumerate}
+
+Remark A1. The main advantage in having a probabilistic prediction method is that it naturally provides a measure of uncertainty quantification (UQ) associated with these predictions through the variance of the distribution. Moreover, having a probabilistic estimate instead of a fixed estimate provides the end user with a confidence level of the predictions.
+
+For a finite subcollection $\left\{\mathbf{x}_{1}, \mathbf{x}_{2}, \cdots, \mathbf{x}_{N}\right\}$ of the random input $\mathbf{x}$, the corresponding objective values
+
+$$
+\left\{f\left(\mathbf{x}_{1}\right), f\left(\mathbf{x}_{2}\right), \ldots, f\left(\mathbf{x}_{N}\right)\right\}
+$$
+
+are assumed to have a multivariate jointly Gaussian distribution:
+
+\[
+\left[\begin{array}{c}
+f\left(\mathbf{x}_{1}\right)  \tag{A1}\\
+\vdots \\
+f\left(\mathbf{x}_{N}\right)
+\end{array}\right] \sim \mathcal{N}\left(\left[\begin{array}{c}
+m\left(\mathbf{x}_{1}\right) \\
+\vdots \\
+m\left(\mathbf{x}_{N}\right)
+\end{array}\right],\left[\begin{array}{c}
+k\left(\mathbf{x}_{1}, \mathbf{x}_{1}\right) \cdots k\left(\mathbf{x}_{1}, \mathbf{x}_{N}\right) \\
+\vdots \\
+k\left(\mathbf{x}_{N}, \mathbf{x}_{1}\right) \cdots k\left(\mathbf{x}_{N}, \mathbf{x}_{N}\right)
+\end{array}\right]\right)
+\]
+
+where the mean $m(\mathbf{x}) \triangleq E[f(\mathbf{x})]$ and the covariance $k\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \triangleq E\left[(f(\mathbf{x})-m(\mathbf{x}))\left(f\left(\mathbf{x}^{\prime}\right)-m\left(\mathbf{x}^{\prime}\right)\right)\right]$, and $E(\cdot)$ indicates the expectation of a random variable.
+
+Let $D^{t r n}=\left\{\left(\mathbf{x}_{i}^{t r n}, y_{i}^{t r n}\right)\right\}, i=1, \cdots, N$, be the training set that are available for model formulation. For a noisy regression model, it is assumed that
+
+
+\begin{equation*}
+y_{i}^{t r n}=f\left(\mathbf{x}_{i}^{t r n}\right)+\varepsilon_{i} \tag{A2}
+\end{equation*}
+
+
+where the additive noise $\varepsilon_{i}$ are independent and identically distributed (iid) zero-mean Gaussian random variables, i.e., $\varepsilon_{i} \sim \mathcal{N}\left(0, \sigma^{2}\right)$. By incorporating the noise term, the joint distribution of the observed values and the functional values at the test locations are
+
+\[
+\left[\begin{array}{l}
+\mathbf{y}^{t r n}  \tag{A3}\\
+\mathbf{y}^{t s t}
+\end{array}\right] \sim \mathcal{N}\left(\mathbf{0},\left[\begin{array}{cc}
+k\left(\mathbf{x}^{t r n}, \mathbf{x}^{t r n}\right)+\sigma^{2} I & k\left(\mathbf{x}^{t r n}, \mathbf{x}^{t s t}\right) \\
+k\left(\mathbf{x}^{t s t}, \mathbf{x}^{t r n}\right) & k\left(\mathbf{x}^{t s t}, \mathbf{x}^{t s t}\right)
+\end{array}\right]\right)
+\]
+
+where $D^{t s t}=\left\{\left(\mathbf{x}^{t s t}, y^{t s t}\right)\right\}$ is the test set in which $y^{t s t}$ is online observed data and $\mathbf{x}^{t s t}$ is the unknown variables to be estimated. Therefore, by the property of multivariate normal distributions, the conditional distribution of the function values at the test location is Gaussian [72]. In particular,
+
+
+\begin{equation*}
+\mathbf{y}^{t s t} \mid \mathbf{y}^{t r n}, \mathbf{x}^{t r n}, \mathbf{x}^{t s t} \sim \mathcal{N}\left(\mu^{t s t}, \Sigma^{t s t}\right) \tag{A4}
+\end{equation*}
+
+
+where
+
+\[
+\begin{array}{r}
+\mu^{t s t}=K\left(\mathbf{x}^{t s t}, \mathbf{x}^{t r n}\right)\left[K\left(\mathbf{x}^{t r n}, \mathbf{x}^{t r n}\right)+\sigma^{2} I\right]^{-1} \mathbf{y}^{t r n} \\
+\Sigma^{t s t}=K\left(\mathbf{x}^{t s t}, \mathbf{x}^{t s t}\right)-K\left(\mathbf{x}^{t s t}, \mathbf{x}^{t r n}\right)\left[K\left(\mathbf{x}^{t r n}, \mathbf{x}^{t r n}\right)\right. \\
+\left.+\sigma^{2} I\right]^{-1} K\left(\mathbf{x}^{t r n}, \mathbf{x}^{t s t}\right) \tag{A6}
+\end{array}
+\]
+
+Thus, the algorithm predicts the mean and covariance for the posterior distribution that models the output at every test data point.
+
+\section*{Appendix A.2. Kernel Function}
+The choice of the covariance function is one of the most critical aspects of the model selection process in GP-based surrogates. It is assumed that the objective function is locally smooth and that the GP priors belong to the class of automatic relevance determination (ARD) Matérn covariance functions [15]. The ARD formulation facilitates learning a length-scale for each input dimension to deal with directional anisotropies in the data set. The Matern class of covariance functions has a shape parameter that can be tuned in order to control the smoothness of the correlation function in the input space. In this work, a Matérn shape parameter of $5 / 2$ is used, which results in the form $k\left(\mathbf{x}, \mathbf{x}^{\prime} ; \boldsymbol{\theta}\right)=\sigma_{f}^{2}\left(1+\sqrt{5} r+\frac{5}{3} r^{2}\right) \exp (-\sqrt{5} r)$, where $r=\sqrt{\sum_{m=1}^{D} \frac{\left(x_{k}-x_{k}^{\prime}\right)^{2}}{\sigma_{m}^{2}}}$. The hyperparameters in the kernel are $\boldsymbol{\theta}=\left[\sigma_{f}^{2},\left(\sigma_{m}\right)_{k=1}^{D}\right]$, where $D$ is the dimensionality of the data set, and $\sigma_{f}$ and $\sigma_{m}$ are the scaling coefficient and the characteristic length-scales, respectively.
+
+Instead of squared exponential kernels that are generally best suited for interpolating smooth functional relationships, Matern kernels are chosen here because length-scales associated with the latter are less prone to be affected by the presence of non-smooth regions in the data set, which might yield poor extrapolation results in the smoother regions [74].
+
+\section*{Appendix A.3. Bayesian Optimization}
+In this work, the term optimization is used to denote maximization of a target function, without any loss of generality. A minimization problem can be posed similarly by taking the negative of the target function. Therefore, in order to optimize an objective function $f$, the following solution is sought.
+
+
+\begin{equation*}
+\mathbf{x}^{*}=\underset{\mathbf{x} \in \mathcal{X}}{\operatorname{argmax}} f(\mathbf{x}) \tag{A7}
+\end{equation*}
+
+
+If the functional form of $f$ is unknown, it may not be possible to have gradient-based optimization for solving the problem; in that case, gradient-free or black-box optimization might be suitable. $\mathrm{BO}$ is one such black-box optimization technique [31] that leverages the predictions through a surrogate model for sequential active learning (AL) to find the global optima of the objective function. The AL strategies\\
+are aimed at finding a trade-off between exploration and exploitation in possibly noisy settings [52,75,76], which facilitates a balance between global search and local optimization through acquisition functions. Usage of GPs in the surrogate modeling framework for BO often leads to closed-form analytical expressions for acquisition functions, which are inexpensive to compute.
+
+In the above formulation, the acquisition functions are designed in the following way. The potential of performance improvement is driven by the predicted mean function (which is in the category of exploitation), whereas the uncertainty prediction is manifested by regions of high variance (which is in the category of exploration). A trade-off between these two requirements is achieved by the acquisition functions iteratively, as a sequential optimization process.\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-18(1)}\\
+\includegraphics[max width=\textwidth, center]{2024_04_13_f3867aee40a0cd4a18c2g-18}
+
+Figure A1. Sequence of Bayesian Optimization iterations for maximizing $f(x)$. (a) Iteration \#1 : Prediction based on initial DOE of 4 points. (b) Iteration \#2. (c) Iteration \#5. (d) Iteration \#7. Filled blue circles indicate initial DOE. Filled red circles denote points selected in optimization iterations.
+
+One commonly used acquisition function in Bayesian optimization is Expected Improvement (EI), which is employed in this work. According to the formulation of Mockus et al. [77] and Jones et al. [52], the EI acquisition function can be written as
+
+\[
+E I(\mathbf{x})=\left\{\begin{array}{rr}
+\left(\mu(\mathbf{x})-f\left(\mathbf{x}^{+}\right)-\xi\right) \Phi(Z)+\sigma(\mathbf{x}) \phi(Z)  \tag{A8}\\
+0 & \text { if } \sigma(\mathbf{x})>0 \\
+\text { if } \sigma(\mathbf{x})=0
+\end{array}\right.
+\]
+
+where
+
+\[
+Z= \begin{cases}\frac{\left(\mu(\mathbf{x})-f\left(\mathbf{x}^{+}\right)-\xi\right)}{\sigma(\mathbf{x})} & \text { if } \sigma(\mathbf{x})>0  \tag{A9}\\ 0 & \text { if } \sigma(\mathbf{x})=0\end{cases}
+\]
+
+and $\mathbf{x}^{+}=\operatorname{argmax}_{\mathbf{x}_{i} \in \mathbf{x}_{1: k}} f\left(\mathbf{x}_{i}\right)$ is the input corresponding to the maximum functional value sampled until iteration $k$; the parameter $\xi>0$ controls the trade-off between exploration and exploitation [31]; $\mu(\mathbf{x})$ and $\sigma(\mathbf{x})$ are the mean and variance, respectively, predicted by the GPs for the an input point $\mathbf{x}$;\\
+and $\phi(\cdot)$ and $\Phi(\cdot)$ are the probability distribution function (PDF) and cumulative distribution function (CDF) of the standard normal distribution, respectively.
+
+If the objective function is noise corrupted, instead of using the best observation, the point with the highest expected value is defined as the incumbent, i.e., $f\left(\mathbf{x}^{+}\right)$replaced by $\mu^{+}$, which is defined as
+
+
+\begin{equation*}
+\mu^{+} \triangleq \underset{\mathbf{x}_{i} \in \mathbf{x}_{1: k}}{\operatorname{argmax}} \mu\left(\mathbf{x}_{i}\right) \tag{A10}
+\end{equation*}
+
+
+The optimization algorithm proceeds sequentially by sampling $\hat{\mathbf{x}}=\operatorname{argmax}_{\mathbf{x}} E I(\mathbf{x})$ at every step of the iteration process to add on to the dataset, after which the GP model is retrained with the new data set to predict the acquisition potential for the next iterative step. This process continues until an optimum is reached, or the computational budget is extinguished. As the acquisition potential is predicted over the entire search space by the surrogate, BO can achieve fast predictions without a lot of function calls in the search space (i.e., without having to run the simulations to obtain the objectives at all the search locations), which, otherwise, might be computationally infeasible when the search space is high-dimensional and the simulations are expensive.
+
+Figure A1 demonstrates the optimization process through a simple problem of maximizing a black-box function $f(x)$ in the input space of $[-2,2]$ (denoted by solid blue line). The process is started with an initial design of experiments (DoE) of 4 points (filled blue circles) chosen in the search domain by Latin hypercube sampling (LHS) [61], which allows the initial selection of points to be well-spread in the search space. Figure Ala shows the GP prediction in the initial stage, which is characterized by the mean function (red dotted line), and the variance associated with the prediction (denoted by the orange band). The variance is high globally at the initial stage, because only four points are sampled on the true function. The maximizer of the acquisition function $E I(x)$ shows the location to sample for the next point at every iteration (filled red circles). Figure A1b shows that the predicted mean of the function in Figure A1a drives the selection of the first optimization point. It is interesting to note that in Figure A1c, although the true maximizer of $f(x)$ is already sampled, $E I(x)$ points towards a region where the variance is high. This is because of the exploration-exploitation trade-off which allows the algorithm to explore the search space instead of greedily searching for the optimum, a property that helps the algorithm avoid being stuck at a local optima. Eventually after seven optimization iterations, the routine converges at the true optima, as shown in Figure A1d. Moreover, the mean function predicted at the last stage is very close to the true function, and the uncertainty band also reduces globally, which indicates that the algorithm not only finds the maximizer of the function, but in doing so it also learns a pretty accurate surrogate model of the function, which can now be used as a low-cost approximation of the true function.
+
+\section*{Appendix B. Optimization Algorithm}
+Algorithm A1 shows the surrogate-based optimization routine that is employed in this paper for solving the melt pool geometry control problem, which is elucidated in Section 3.3. The global thread picks up an optimal point $\tilde{\mathbf{x}}^{*}$, which is refined in the local thread to give $\mathbf{x}^{*}$, as the final optimal point chosen by the algorithm.
+
+\begin{center}
+\includegraphics[max width=\textwidth]{2024_04_13_f3867aee40a0cd4a18c2g-20}
+\end{center}
+
+\section*{Local Thread}
+Require: Start with surrogate local GP for the objective functional $J$ with $\tilde{N}_{\text {init }}$ LHS initializations for $k=1$ to $\tilde{N}_{\text {Localiter }}$ optimization steps do
+
+\begin{itemize}
+  \item Use the trained local GP to predict the posterior distribution of the objective function in the local search space $\tilde{X}_{\text {star }}$
+  \item Sample the optimized process parameter from $\tilde{X}_{\text {star }}$ as $\tilde{\mathbf{x}}_{\mathbf{k}}^{*}=\operatorname{argmax}_{\tilde{\mathbf{x}}} E I(\tilde{\mathbf{x}})$
+  \item Compute the objective function $J_{k}$ at the chosen $\tilde{\mathbf{x}}_{\mathbf{k}}^{*}$
+  \item Add the pair $\left(\tilde{\mathbf{x}}_{\mathbf{k}^{\prime}}^{*} J_{k}\right)$ to the local surrogate model's input and output sets respectively
+  \item Retrain the local surrogate model
+\end{itemize}
+
+$$
+\text { end for }
+$$
+
+\begin{itemize}
+  \item Select $\mathbf{x}^{*}=\operatorname{argmax}_{\tilde{\mathbf{x}}_{k}} J$
+\end{itemize}
+
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+(C) 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (\href{http://creativecommons.org/licenses/by/4.0/}{http://creativecommons.org/licenses/by/4.0/}).
+
+
+\end{document}