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Research Papers

Thermal Modeling in Metal Additive Manufacturing Using Graph Theory

[+] Author and Article Information
M. Reza Yavari

Department of Mechanical and Materials Engineering,
University of Nebraska-Lincoln,
Lincoln, NE 68588-0526
e-mail: mrezayavari89@huskers.unl.edu

Kevin D. Cole

Department of Mechanical and Materials Engineering,
University of Nebraska-Lincoln,
Lincoln, NE 68588-0526
e-mail: kcole1@unl.edu

Prahalada Rao

Department of Mechanical and Materials Engineering,
University of Nebraska-Lincoln,
Lincoln, NE 68588-0526
e-mail: rao@unl.edu

1Corresponding author.

Manuscript received October 19, 2018; final manuscript received April 23, 2019; published online May 21, 2019. Assoc. Editor: Kevin Chou.

J. Manuf. Sci. Eng 141(7), 071007 (May 21, 2019) (20 pages) Paper No: MANU-18-1738; doi: 10.1115/1.4043648 History: Received October 19, 2018; Accepted April 25, 2019

The goal of this work is to predict the effect of part geometry and process parameters on the instantaneous spatiotemporal distribution of temperature, also called the thermal field or temperature history, in metal parts as they are being built layer-by-layer using additive manufacturing (AM) processes. In pursuit of this goal, the objective of this work is to develop and verify a graph theory-based approach for predicting the temperature distribution in metal AM parts. This objective is consequential to overcome the current poor process consistency and part quality in AM. One of the main reasons for poor part quality in metal AM processes is ascribed to the nature of temperature distribution in the part. For instance, steep thermal gradients created in the part during printing leads to defects, such as warping and thermal stress-induced cracking. Existing nonproprietary approaches to predict the temperature distribution in AM parts predominantly use mesh-based finite element analyses that are computationally tortuous—the simulation of a few layers typically requires several hours, if not days. Hence, to alleviate these challenges in metal AM processes, there is a need for efficient computational models to predict the temperature distribution, and thereby guide part design and selection of process parameters instead of expensive empirical testing. Compared with finite element analyses techniques, the proposed mesh-free graph theory-based approach facilitates prediction of the temperature distribution within a few minutes on a desktop computer. To explore these assertions, we conducted the following two studies: (1) comparing the heat diffusion trends predicted using the graph theory approach with finite element analysis, and analytical heat transfer calculations based on Green’s functions for an elementary cuboid geometry which is subjected to an impulse heat input in a certain part of its volume and (2) simulating the laser powder bed fusion metal AM of three-part geometries with (a) Goldak’s moving heat source finite element method, (b) the proposed graph theory approach, and (c) further comparing the thermal trends predicted from the last two approaches with a commercial solution. From the first study, we report that the thermal trends approximated by the graph theory approach are found to be accurate within 5% of the Green’s functions-based analytical solution (in terms of the symmetric mean absolute percentage error). Results from the second study show that the thermal trends predicted for the AM parts using graph theory approach agree with finite element analyses, and the computational time for predicting the temperature distribution was significantly reduced with graph theory. For instance, for one of the AM part geometries studied, the temperature trends were predicted in less than 18 min within 10% error using the graph theory approach compared with over 180 min with finite element analyses. Although this paper is restricted to theoretical development and verification of the graph theory approach, our forthcoming research will focus on experimental validation through in-process thermal measurements.

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References

Schmidt, M., Merklein, M., Bourell, D., Dimitrov, D., Hausotte, T., Wegener, K., Overmeyer, L., Vollertsen, F., and Levy, G. N., 2017, “Laser Based Additive Manufacturing in Industry and Academia,” CIRP Ann., 66(2), pp. 561–583. [CrossRef]
Gibson, I., Rosen, D. W., and Stucker, B., 2010, Additive Manufacturing Technologies: Rapid Prototyping to Direct Digital Manufacturing, Springer, Berlin.
Tofail, S. A. M., Koumoulos, E. P., Bandyopadhyay, A., Bose, S., O’Donoghue, L., and Charitidis, C., 2018, “Additive Manufacturing: Scientific and Technological Challenges, Market Uptake and Opportunities,” Mater. Today, 21(1), pp. 22–37. [CrossRef]
Khoda, B., Benny, T., Rao, P. K., Sealy, M. P., and Zhou, C., 2017, “Applications of Laser-Based Additive Manufacturing,” Laser-Based Additive Manufacturing of Metal Parts, CRC Press, Boca Raton, FL, pp. 253–298.
Badiru, A. B., Valencia, V. V., and Liu, D., 2017, Additive Manufacturing Handbook: Product Development for the Defense Industry, CRC Press, Boca Raton, FL.
Bauereiß, A., Scharowsky, T., and Körner, C., 2014, “Defect Generation and Propagation Mechanism During Additive Manufacturing by Selective Beam Melting,” J. Mater. Process. Technol., 214(11), pp. 2522–2528. [CrossRef]
Liu, Q. C., Elambasseril, J., Sun, S. J., Leary, M., Brandt, M., and Sharp, P. K., 2014, “The Effect of Manufacturing Defects on the Fatigue Behaviour of Ti-6Al-4V Specimens Fabricated Using Selective Laser Melting,” Proceedings of the Advanced Materials Research, Melbourne, Australia, Mar. 2–7, pp. 1519–1524. Trans Tech Publications.
Gorelik, M., 2017, “Additive Manufacturing in the Context of Structural Integrity,” Int. J. Fatigue, 94(Part 2), pp. 168–177. [CrossRef]
Seifi, M., Gorelik, M., Waller, J., Hrabe, N., Shamsaei, N., Daniewicz, S., and Lewandowski, J. J., 2017, “Progress Towards Metal Additive Manufacturing Standardization to Support Qualification and Certification,” JOM, 69(3), pp. 439–455. [CrossRef]
Lewandowski, J. J., and Seifi, M., 2016, “Metal Additive Manufacturing: A Review of Mechanical Properties,” Annu. Rev. Mater. Res., 46(1), pp. 151–186. [CrossRef]
DebRoy, T., Wei, H. L., Zuback, J. S., Mukherjee, T., Elmer, J. W., Milewski, J. O., Beese, A. M., Wilson-Heid, A., De, A., and Zhang, W., 2018, “Additive Manufacturing of Metallic Components—Process, Structure and Properties,” Prog. Mater. Sci., 92(3), pp. 112–224. [CrossRef]
Foteinopoulos, P., Papacharalampopoulos, A., and Stavropoulos, P., 2018, “On Thermal Modeling of Additive Manufacturing Processes,” CIRP J. Manuf. Sci. Technol., 20(1), pp. 66–83. [CrossRef]
Sames, W. J., List, F., Pannala, S., Dehoff, R. R., and Babu, S. S., 2016, “The Metallurgy and Processing Science of Metal Additive Manufacturing,” Int. Mater. Rev., 61(5), pp. 315–360. [CrossRef]
Kruth, J. P., Froyen, L., Van Vaerenbergh, J., Mercelis, P., Rombouts, M., and Lauwers, B., 2004, “Selective Laser Melting of Iron-Based Powder,” J. Mater. Process. Technol., 149(1), pp. 616–622. [CrossRef]
Raghavan, N., Dehoff, R., Pannala, S., Simunovic, S., Kirka, M., Turner, J., Carlson, N., and Babu, S. S., 2016, “Numerical Modeling of Heat-Transfer and the Influence of Process Parameters on Tailoring the Grain Morphology of IN718 in Electron Beam Additive Manufacturing,” Acta Mater., 112(11), pp. 303–314. [CrossRef]
Everton, S. K., Hirsch, M., Stravroulakis, P., Leach, R. K., and Clare, A. T., 2016, “Review of In-Situ Process Monitoring and In-Situ Metrology for Metal Additive Manufacturing,” Mater. Des., 95(7), pp. 431–445. [CrossRef]
Maskery, I., Aboulkhair, N. T., Corfield, M. R., Tuck, C., Clare, A. T., Leach, R. K., Wildman, R. D., Ashcroft, I. A., and Hague, R. J. M., 2016, “Quantification and Characterisation of Porosity in Selectively Laser Melted Al–Si10–Mg Using X-Ray Computed Tomography,” Mater. Charact., 111(1), pp. 193–204. [CrossRef]
Hadadzadeh, A., Amirkhiz, B. S., Li, J., and Mohammadi, M., 2018, “Columnar to Equiaxed Transition During Direct Metal Laser Sintering of AlSi10Mg Alloy: Effect of Building Direction,” Addit. Manuf., 23(5), pp. 121–131. [CrossRef]
Roberts, I. A., Wang, C., Esterlein, R., Stanford, M., and Mynors, D., 2009, “A Three-Dimensional Finite Element Analysis of the Temperature Field During Laser Melting of Metal Powders in Additive Layer Manufacturing,” Int. J. Mach. Tools Manuf., 49(12–13), pp. 916–923. [CrossRef]
Markl, M., and Körner, C., 2016, “Multiscale Modeling of Powder Bed-Based Additive Manufacturing,” Annu. Rev. Mater. Res., 46(1), pp. 93–123. [CrossRef]
King, W. E., Anderson, A. T., Ferencz, R., Hodge, N., Kamath, C., Khairallah, S. A., and Rubenchik, A. M., 2015, “Laser Powder Bed Fusion Additive Manufacturing of Metals; Physics, Computational, and Materials Challenges,” Appl. Phys. Rev., 2(4), p. 041304. [CrossRef]
Khairallah, S. A., Anderson, A. T., Rubenchik, A., and King, W. E., 2016, “Laser Powder-Bed Fusion Additive Manufacturing: Physics of Complex Melt Flow and Formation Mechanisms of Pores, Spatter, and Denudation Zones,” Acta Mater., 108(7), pp. 36–45. [CrossRef]
Bourell, D., Kruth, J. P., Leu, M., Levy, G., Rosen, D., Beese, A. M., and Clare, A., 2017, “Materials for Additive Manufacturing,” CIRP Ann., 66(2), pp. 659–681. [CrossRef]
Seifi, M., Salem, A., Beuth, J., Harrysson, O., and Lewandowski, J. J., 2016, “Overview of Materials Qualification Needs for Metal Additive Manufacturing,” JOM, 68(3), pp. 747–764. [CrossRef]
O’Regan, P., Prickett, P., Setchi, R., Hankins, G., and Jones, N., 2016, “Metal Based Additive Layer Manufacturing: Variations, Correlations and Process Control,” Proc. Comput. Sci., 96(19), pp. 216–224. [CrossRef]
Huang, Y., Leu, M. C., Mazumder, J., and Donmez, A., 2015, “Additive Manufacturing: Current State, Future Potential, Gaps and Needs, and Recommendations,” ASME J. Manuf. Sci. Eng., 137(1), p. 014001. [CrossRef]
Huang, Y., and Leu, M., 2013, “Frontiers of Additive Manufacturing Research and Education—Report of NSF Additive Manufacturing Workshop,” National Science Foundation, Arlington, VA.
NIST, 2013, Measurement Science Roadmap for Metal-Based Additive Manufacturing - Report Prepared by Energetics Corporation,” National Institute of Standards and Technology, Gaithersburg, MD.
Mazumder, J., 2015, “Design for Metallic Additive Manufacturing Machine With Capability for ‘Certify as You Build’,” Proc. CIRP, 36(10), pp. 187–192. [CrossRef]
Edgar, T., Davis, J., and Burka, M., 2015, “NSF Workshop on Research Needs in Advanced Sensors, Controls, Platforms, and Modeling (ASCPM) for Smart Manufacturing,” National Science Foundation, Atlanta, GA.
Simpson, T. W., Williams, C. B., and Hripko, M., 2017, “Preparing Industry for Additive Manufacturing and Its Applications: Summary & Recommendations From a National Science Foundation Workshop,” Addit. Manuf., 13(1), pp. 166–178. [CrossRef]
Gu, H., Gong, H., Pal, D., Rafi, K., Starr, T., and Stucker, B., 2013, “Influences of Energy Density on Porosity and Microstructure of Selective Laser Melted 17-4PH Stainless Steel,” Proceedings of the 2013 Solid Freeform Fabrication Symposium., University of Texas, Austin, Aug. 12–14, pp. 474–489.
Gong, H., Rafi, K., Gu, H., Starr, T., and Stucker, B., 2014, “Analysis of Defect Generation in Ti–6Al–4V Parts Made Using Powder Bed Fusion Additive Manufacturing Processes,” Addit. Manuf., 1–4(1), pp. 87–98. [CrossRef]
Gong, H., Rafi, K., Starr, T., and Stucker, B., 2012, “Effect of Defects on Fatigue Tests of As-Built Ti-6Al-4V Parts Fabricated by Selective Laser Melting,” Proceedings of the Solid Freeform Fabrication Symposium, University of Texas, Austin, Aug. 6–8, pp. 499–506.
Montazeri, M., Yavari, R., Rao, P., and Boulware, P., 2018, “In-Process Monitoring of Material Cross-Contamination Defects in Laser Powder Bed Fusion,” ASME J. Manuf. Sci. Eng., 140(11), p. 111001. [CrossRef]
Montazeri, M., and Rao, P., 2018, “Sensor-Based Build Condition Monitoring in Laser Powder Bed Fusion Additive Manufacturing Process Using a Spectral Graph Theoretic Approach,” ASME J. Manuf. Sci. Eng., 140(9), p. 091002. [CrossRef]
Fox, J. C., Moylan, S. P., and Lane, B. M., 2016, “Effect of Process Parameters on the Surface Roughness of Overhanging Structures in Laser Powder Bed Fusion Additive Manufacturing,” Proc. CIRP, 45(6), pp. 131–134. [CrossRef]
Strano, G., Hao, L., Everson, R., and Evans, K., 2013, “A New Approach to the Design and Optimisation of Support Structures in Additive Manufacturing,” Int. J. Adv. Manuf. Technol., 66(9–12), pp. 1247–1254. [CrossRef]
Thomas, D., 2009, “The Development of Design Rules for Selective Laser Melting,” Ph.D. Dissertation, University of Wales. http://hdl.handle.net/10369/913.
Jamshidinia, M., and Kovacevic, R., 2015, “The Influence of Heat Accumulation on the Surface Roughness in Powder-Bed Additive Manufacturing,” Surf. Topogr.: Metrol. Prop., 3(1), p. 014003. [CrossRef]
Denlinger, E. R., Irwin, J., and Michaleris, P., 2014, “Thermomechanical Modeling of Additive Manufacturing Large Parts,” ASME J. Manuf. Sci. Eng., 136(6), p. 061007. [CrossRef]
Bandyopadhyay, A., and Traxel, K. D., 2018, “Invited Review Article: Metal-Additive Manufacturing—Modeling Strategies for Application-Optimized Designs,” Addit. Manuf., 22(4), pp. 758–774. [CrossRef] [PubMed]
Denlinger, E. R., Gouge, M., and Michaleris, P., 2018, Thermo-Mechanical Modeling of Additive Manufacturing, Butterworth-Heinemann, London.
Francois, M. M., Sun, A., King, W. E., Henson, N. J., Tourret, D., Bronkhorst, C. A., Carlson, N. N., Newman, C. K., Haut, T., Bakosi, J., Gibbs, J. W., Livescu, V., Vander Wiel, S. A., Clarke, A. J., Schraad, M. W., Blacker, T., Lim, H., Rodgers, T., Owen, S., Abdeljawad, F., Madison, J., Anderson, A. T., Fattebert, J. L., Ferencz, R. M., Hodge, N. E., Khairallah, S. A., and Walton, O., 2017, “Modeling of Additive Manufacturing Processes for Metals: Challenges and Opportunities,” Curr. Opin. Solid State Mater. Sci., 21(4), pp. 198–206. [CrossRef]
Cheng, B., Shrestha, S., and Chou, Y. K., 2016, “Stress and Deformation Evaluations of Scanning Strategy Effect in Selective Laser Melting,” Proceedings of the ASME 2016 11th International Manufacturing Science and Engineering Conference, Blacksburg, VA, June 27–30, p. V003T008A009.
Williams, R. J., Davies, C. M., and Hooper, P. A., 2018, “A Pragmatic Part Scale Model for Residual Stress and Distortion Prediction in Powder Bed Fusion,” Addit. Manuf., 22(4), pp. 416–425. [CrossRef]
Zeng, K., Pal, D., Gong, H. J., Patil, N., and Stucker, B., 2015, “Comparison of 3DSIM Thermal Modelling of Selective Laser Melting Using New Dynamic Meshing Method to ANSYS,” Mater. Sci. Technol., 31(8), pp. 945–956. [CrossRef]
Luo, Z., and Zhao, Y., 2018, “A Survey of Finite Element Analysis of Temperature and Thermal Stress Fields in Powder Bed Fusion Additive Manufacturing,” Addit. Manuf., 21(3), pp. 318–332. [CrossRef]
Michaleris, P., 2014, “Modeling Metal Deposition in Heat Transfer Analyses of Additive Manufacturing Processes,” Finite Elem. Anal. Des., 86(9), pp. 51–60. [CrossRef]
Peng, H., Ghasri-Khouzani, M., Gong, S., Attardo, R., Ostiguy, P., Gatrell, B. A., Budzinski, J., Tomonto, C., Neidig, J., Shankar, M. R., Billo, R., Go, D. B., and Hoelzle, D., 2018, “Fast Prediction of Thermal Distortion in Metal Powder Bed Fusion Additive Manufacturing: Part 1, A Thermal Circuit Network Model,” Addit. Manuf., 22, pp. 852–868. [CrossRef]
Ganeriwala, R., and Zohdi, T. I., 2014, “Multiphysics Modeling and Simulation of Selective Laser Sintering Manufacturing Processes,” Proc. CIRP, 14(2), pp. 299–304. [CrossRef]
Ganeriwala, R., and Zohdi, T. I., 2016, “A Coupled Discrete Element-Finite Difference Model of Selective Laser Sintering,” Granul. Matter, 18(2), p. 21. [CrossRef]
Solomon, J., 2015, “PDE Approaches to Graph Analysis,” preprint arXiv:1505.00185.
Belkin, M., Sun, J., and Wang, Y., 2008, “Discrete Laplace Operator on Meshed Surfaces,” Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, College Park, MD, June 9–11, pp. 278–287.
Zhang, F., and Hancock, E. R., 2008, “Graph Spectral Image Smoothing Using the Heat Kernel,” Pattern Recognit., 41(11), pp. 3328–3342. [CrossRef]
Silling, S. A., and Askari, E., 2005, “A Meshfree Method Based on the Peridynamic Model of Solid Mechanics,” Comput. Struct., 83(17), pp. 1526–1535. [CrossRef]
Chen, Z., Niazi, S., Zhang, G., and Bobaru, F., 2017, “Peridynamic Functionally Graded and Porous Materials: Modeling Fracture and Damage,” Handbook of Nonlocal Continuum Mechanics for Materials and Structures, G. Z. Voyiadjis, ed., Springer International Publishing, Cham, pp. 1–35.
Sun, Y.-S., and Li, B.-W., 2010, “Spectral Collocation Method for Transient Conduction-Radiation Heat Transfer,” J. Thermophys. Heat Transf., 24(4), pp. 823–832. [CrossRef]
Rahmati, A. R., and Niazi, S., 2012, “Simulation of Microflows Using the Lattice Boltzmann Method on Nonuniform Meshes,” Nanosci. Technol., 3(1), pp. 77–97.
Kondor, R. I., and Lafferty, J. D., 2002, “Diffusion Kernels on Graphs and Other Discrete Input Spaces,” Proceedings of the 19th International Conference on Machine Learning., San Francisco, CA, July 8–12, pp. 315–322.
Saito, N., 2013, “Tutorial: Laplacian Eigenfunctions—Foundations and Applications,” University of California, Davis, Graduate University for Advanced Studies, National Institute of Fusion Science, Japan.
Chung, F. R. K., 1997, Spectral Graph Theory, American Mathematical Society, Providence, RI.
Bai, X., and Hancock, E. R., Heat Kernels, Manifolds and Graph Embedding, Springer, Berlin Heidelberg, pp. 198–206.
Goldak, J. A., and Akhlaghi, M., 2005, “Computer Simulation of Welding Processes,” Computational Welding Mechanics, pp. 16–69.
Goldak, J., Chakravarti, A., and Bibby, M., 1984, “A New Finite Element Model for Welding Heat Sources,” Metall. Trans. B, 15(2), pp. 299–305. [CrossRef]
Cole, K. D., Beck, J. V., Haji-Sheikh, A., and Litkouhi, B., 2010, Heat Conduction Using Green’s Functions, CRC Press, Boca Raton, FL.
Cole, K. D.,2018, “Parallelepiped with Insulated Boundaries and Piecewise Initial Condition” EXACT Analytical Conduction Toolbox, Oct. 18 www.exact.unl.edu.
Nunes, A., 1983, “An Extended Rosenthal Weld Model,” Weld. J., 62(6), pp. 165s–170s.
Karayagiz, K., Elwany, A., Tapia, G., Franco, B., Johnson, L., Ma, J., Karaman, I., and Arróyave, R., 2018, “Numerical and Experimental Analysis of Heat Distribution in the Laser Powder Bed Fusion of Ti-6Al-4V,” IISE Trans., 51(2), pp. 136–152. [CrossRef]
Rubenchik, A., Wu, S., Mitchell, S., Golosker, I., LeBlanc, M., and Peterson, N., 2015, “Direct Measurements of Temperature-Dependent Laser Absorptivity of Metal Powders,” Appl. Opt., 54(24), pp. 7230–7233. [CrossRef] [PubMed]

Figures

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Fig. 1

The schematic of the (a) laser powder bed fusion (LPBF) and, (b) blown powder directed energy deposition (DED) metal AM processes

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Fig. 2

The salient heat transfer modes in LPBF and DED encompassing complex interactions among the part, material, energy source, and environment (surrounding inert gas)

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Fig. 3

LPBF knee implant with an overhang feature shows poor surface finish and coarse microstructure

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Fig. 4

The four steps in the spectral graph-theoretic approach used to estimate the temperature distribution in the part layer-by-layer. Here, we show an embodiment of the LPBF process. The powder particles surrounding the part are not shown in this figure.

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Fig. 5

The flowchart of four steps in the graph-theoretic approach in context of LPBF process

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Fig. 6

The cube with an initial heated region and insulated boundaries (Neumann boundary condition)

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Fig. 7

Analytical diffusion at observation point 1: {0.25, 0.25, 0.25} and observation point 2: {0.75, 0.75, 0.75} from the origin

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Fig. 8

The three steps toward the error calculation and verification with the analytical method

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Fig. 9

(a) Comparison of the heat diffusion trend between graph theory and analytical method (result of Case 2 with 800 selected nodes) and (b) absolute error comparison for different amount of nodes at observation point 1

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Fig. 10

The three steps of FEA toward the error calculation and verification with the analytical method

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Fig. 11

(a) C-shaped, (b) C-shaped with support, and (c) pyramid dimensions in millimeters

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Fig. 12

(a) Side view of C-shaped part, (b) side view of C-shaped part with supports, (c) scanning strategy of the two C-shaped parts from top view, (d) side view of pyramid part, and (e) scanning strategy of pyramid part from its top view

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Fig. 13

(a) The C-shaped part and the normalized temperature trends observed at three locations on the part: (1) in the bottom left, (2) bottom right, and (3) center, respectively, corresponding to three node densities per hatch: (a) 20, (b) 80, and (c) 120 nodes. The neighborhood size is ε = 2 mm and the gain factor is set at g = 2.2 × 106.

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Fig. 14

Explanation for the temperature trends and spikes observed at the center location. (a) The hatch demarcation of the part into three sections, and hatch pattern simulated, (b) the temperature trends observed at the bottom center, and (c) zoomed in view of section T1, noting that the temperature spikes correspond to the spatiotemporal location of the laser in relation to the observation point (sensor location) demarcated in (a). The neighborhood size is ε = 2 mm and the number of nodes per hatch is held constant at 80, and the gain factor is set at g = 2.2 × 106.

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Fig. 15

Effect of gain factor on the RMSE and SMAPE in simulating three layers of section T1 from the C-shaped part. The neighborhood size is ε = 2 mm and the number of nodes per hatch is held constant at 40.

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Fig. 16

The effect of gain factor (g) on the thermal trends observed in the center of the first three layers of the C-shaped part with 40 nodes per hatch and neighborhood distance ε = 2 mm. At lower gain factor (a) g = 0.2 × 106 and (b) g = 0.7 × 106, the rate of heat diffusion (heat flux) is restrained, as the gain factor is increased to (c) g = 2.2 × 106 and (d) g = 4 × 106 the rate of diffusion is comparatively faster.

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Fig. 17

The effect of neighborhood distance (ε) on the RMSE and SMAPE in simulating three layers of section T1 from the C-shaped part. A neighborhood distance between 2 mm and 3 mm is suggested given the characteristic length of the part.

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Fig. 18

The effect of the neighborhood distance (ε) on the thermal trends observed in the first three layers of the C-shaped part when compared with the FE solution. The number of nodes per hatch is set at 80, and gain factor g = 2.2 × 106. At lower values of neighborhood distance (a) ε = 0.6 mm and (b) g = 1.2 mm, the number of nodes is too sparse to capture the heat transfer phenomena. At larger values of the neighborhood distance of (c) ε = 2 mm and (d) ε = 3 mm, the individual spokes in a layer are matched temporally with the FE solution; however, the computation time increases exponentially, as the number of nodes connected are larger.

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Fig. 19

The C-shaped part from Fig. 13 is modified with two supports to provide a path for the heat in the overhang section to dissipate. The temperature trends observed at three locations on the part, in (a) left, (b) right, and (c) center. A zoomed in plot corresponding to the section T1 from (c) is shown in (d). The neighborhood size is ε = 2 mm and the number of nodes per hatch is held constant at 80, and the gain factor is set at g = 2.2 × 106.

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Fig. 20

The temperature profile for an observation point located at the center of layer one, and 40 nodes per hatch, and neighborhood distance ε = 0.25 mm, and gain factor g set at 2.2 × 106

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Fig. 21

Temperature distribution of three parts which is compared by three different methods; graph theory (50 nodes per hatch), FE analysis (abaqus), and commercial software (netfabb)

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