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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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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