A line input (LI) model has been developed, which makes the accurate modeling of powder bed processes more computationally efficient. Goldak's ellipsoidal model has been used extensively to model heat sources in additive manufacturing (AM), including lasers and electron beams. To accurately model the motion of the heat source, the simulation time increments must be small enough such that the source moves a distance smaller than its radius over the course of each increment. When the source radius is small and its velocity is large, a strict condition is imposed on the size of time increments regardless of any stability criteria. In powder bed systems, where radii of 0.1 mm and velocities of 500 mm/s are typical, a significant computational burden can result. The line heat input model relieves this burden by averaging the heat source over its path. This model allows the simulation of an entire heat source scan in just one time increment. However, such large time increments can lead to inaccurate results. Instead, the scan is broken up into several linear segments, each of which is applied in one increment. In this work, time increments are found that yield accurate results (less than 10% displacement error) and require less than 1/10 of the central processing unit (CPU) time required by Goldak's moving source model. A dimensionless correlation is given that can be used to determine the necessary time increment size that will greatly decrease the computational time required for any powder bed simulation while maintaining accuracy.

References

1.
Khaing
,
M.
,
Fuh
,
J.
, and
Lu
,
L.
,
2001
, “
Direct Metal Laser Sintering for Rapid Tooling: Processing and Characterisation of EOS Parts
,”
J. Mater. Process. Technol.
,
113
(
1
), pp.
269
272
.
2.
Zhang
,
P.
,
Toman
,
J.
,
Yu
,
Y.
,
Biyikli
,
E.
,
Kirca
,
M.
,
Chmielus
,
M.
, and
To
,
A.
,
2015
, “
Efficient Design-Optimization of Variable-Density Hexagonal Cellular Structure by Additive Manufacturing: Theory and Validation
,”
ASME J. Manuf. Sci. Eng.
,
137
(
2
), p.
021004
.
3.
Kolossov
,
S.
,
Boillat
,
E.
,
Glardon
,
R.
,
Fischer
,
P.
, and
Locher
,
M.
,
2004
, “
3D FE Simulation for Temperature Evolution in the Selective Laser Sintering Process
,”
Int. J. Mach. Tools Manuf.
,
44
(
2
), pp.
117
123
.
4.
Yin
,
J.
,
Zhu
,
H.
,
Ke
,
L.
,
Lei
,
W.
,
Dai
,
C.
, and
Zuo
,
D.
,
2012
, “
Simulation of Temperature Distribution in Single Metallic Powder Layer for Laser Micro-Sintering
,”
Comput. Mater. Sci.
,
53
(
1
), pp.
333
339
.
5.
Paul
,
S.
,
Gupta
,
I.
, and
Singh
,
R.
,
2015
, “
Characterization and Modeling of Microscale Preplaced Powder Cladding Via Fiber Laser
,”
ASME J. Manuf. Sci. Eng.
,
137
(
3
), p.
031019
.
6.
Bugeda
,
G.
,
Cervera
,
M.
, and
Lombera
,
G.
,
1999
, “
Numerical Prediction of Temperature and Density Distributions in Selective Laser Sintering Processes
,”
Rapid Prototyping J.
,
5
(
1
), pp.
21
26
.
7.
Contuzzi
,
N.
,
Campanelli
,
S.
, and
Ludovico
,
A.
,
2011
, “
3D Finite Element Analysis in the Selective Laser Melting Process
,”
Int. J. Simul. Modell. (IJSIMM)
,
10
(
3
), pp.
113
121
.
8.
Roberts
,
I.
,
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
), pp.
916
923
.
9.
Morgan
,
R.
,
Sutcliffe
,
C.
, and
O'neill
,
W.
,
2004
, “
Density Analysis of Direct Metal Laser Re-Melted 316l Stainless Steel Cubic Primitives
,”
J. Mater. Sci.
,
39
(
4
), pp.
1195
1205
.
10.
O'neill
,
W.
,
Sutcliffe
,
C.
,
Morgan
,
R.
,
Landsborough
,
A.
, and
Hon
,
K.
,
1999
, “
Investigation on Multi-Layer Direct Metal Laser Sintering of 316l Stainless Steel Powder Beds
,”
CIRP Ann. Manuf. Technol.
,
48
(
1
), pp.
151
154
.
11.
Zhang
,
L.
,
Reutzel
,
E.
, and
Michaleris
,
P.
,
2004
, “
Finite Element Modeling Discretization Requirements for the Laser Forming Process
,”
Int. J. Mech. Sci.
,
46
(
4
), pp.
623
637
.
12.
Fell
,
A.
, and
Willeke
,
G.
,
2010
, “
Fast Simulation Code for Heating, Phase Changes and Dopant Diffusion in Silicon Laser Processing Using the Alternating Direction Explicit (ADE) Method
,”
Appl. Phys. A
,
98
(
2
), pp.
435
440
.
13.
Michaleris
,
P.
,
2014
, “
Modeling Metal Deposition in Heat Transfer Analyses of Additive Manufacturing Processes
,”
Finite Elem. Anal. Des.
,
86
, pp.
51
60
.
14.
Zienkiewicz
,
O. C.
, and
Zhu
,
J. Z.
,
1987
, “
A Simple Error Estimator and Adaptive Procedure for Practical Engineering Analysis
,”
Int. J. Numer. Methods Eng.
,
24
(
2
), pp.
337
357
.
15.
Picasso
,
M.
,
2003
, “
An Anisotropic Error Indicator Based on Zienkiewicz–Zhu Error Estimator: Application to Elliptic and Parabolic Problems
,”
SIAM J. Sci. Comput.
,
24
(
4
), pp.
1328
1355
.
16.
Berger
,
M. J.
, and
Oliger
,
J.
,
1984
, “
Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations
,”
J. Comput. Phys.
,
53
(
3
), pp.
484
512
.
17.
Berger
,
M. J.
, and
Colella
,
P.
,
1989
, “
Local Adaptive Mesh Refinement for Shock Hydrodynamics
,”
J. Comput. Phys.
,
82
(
1
), pp.
64
84
.
18.
Bell
,
J.
,
Berger
,
M.
,
Saltzman
,
J.
, and
Welcome
,
M.
,
1994
, “
Three-Dimensional Adaptive Mesh Refinement for Hyperbolic Conservation Laws
,”
SIAM J. Sci. Comput.
,
15
(
1
), pp.
127
138
.
19.
Bank
,
R. E.
,
Sherman
,
A. H.
, and
Weiser
,
A.
,
1983
, “
Some Refinement Algorithms and Data Structures for Regular Local Mesh Refinement
,”
Sci. Comput. Appl. Math. Comput. Phys. Sci.
,
1
, pp.
3
17
.
20.
Shepherd
,
J. F.
,
Dewey
,
M. W.
,
Woodbury
,
A. C.
,
Benzley
,
S. E.
,
Staten
,
M. L.
, and
Owen
,
S. J.
,
2010
, “
Adaptive Mesh Coarsening for Quadrilateral and Hexahedral Meshes
,”
Finite Elem. Anal. Des.
,
46
(
1
), pp.
17
32
.
21.
Prasad
,
N. S.
, and
Narayanan
,
S.
,
1996
, “
Finite Element Analysis of Temperature Distribution During Arc Welding Using Adaptive Grid Technique
,”
Weld. J.
,
75
(
4
), pp.
123
128
.
22.
Runnemalm
,
H.
, and
Hyun
,
S.
,
2000
, “
Three-Dimensional Welding Analysis Using an Adaptive Mesh Scheme
,”
Comput. Methods Appl. Mech. Eng.
,
189
(
2
), pp.
515
523
.
23.
Liu
,
X.
,
Lan
,
S.
, and
Ni
,
J.
,
2015
, “
Thermal Mechanical Modeling of the Plunge Stage During Friction-Stir Welding of Dissimilar Al 6061 to Trip 780 Steel
,”
ASME J. Manuf. Sci. Eng.
,
137
(
5
), p.
051017
.
24.
Choi
,
W.
, and
Chung
,
H.
,
2015
, “
Variation Simulation of Compliant Metal Plate Assemblies Considering Welding Distortion
,”
ASME J. Manuf. Sci. Eng.
,
137
(
3
), p.
031008
.
25.
Patil
,
N.
,
Pal
,
D.
,
Rafi
,
K.
,
Zeng
,
K.
,
Moreland
,
A.
,
Hicks
,
A.
, and
Beeler
,
D.
,
2015
, “
A Generalized Feed Forward Dynamic Adaptive Mesh Refinement and Derefinement Finite Element Framework for Metal Laser Sintering—Part I: Formulation and Algorithm Development
,”
ASME J. Manuf. Sci. Eng.
,
137
(
4
), p.
041001
.
26.
Garrido
,
I.
,
Lee
,
B.
,
Fladmark
,
G.
, and
Espedal
,
M.
,
2006
, “
Convergent Iterative Schemes for Time Parallelization
,”
Math. Comput.
,
75
(
255
), pp.
1403
1428
.
27.
Farhat
,
C.
, and
Chandesris
,
M.
,
2003
, “
Time-Decomposed Parallel Time-Integrators: Theory and Feasibility Studies for Fluid, Structure, and Fluid–Structure Applications
,”
Int. J. Numer. Methods Eng.
,
58
(
9
), pp.
1397
1434
.
28.
Cortial
,
J.
, and
Farhat
,
C.
,
2009
, “
A Time-Parallel Implicit Method for Accelerating the Solution of Non-Linear Structural Dynamics Problems
,”
Int. J. Numer. Methods Eng.
,
77
(
4
), pp.
451
470
.
29.
Christlieb
,
A.
, and
Ong
,
B.
,
2011
, “
Implicit Parallel Time Integrators
,”
J. Sci. Comput.
,
49
(
2
), pp.
167
179
.
30.
Wu
,
S.
,
Shi
,
B.
, and
Huang
,
C.
,
2009
, “
Parareal–Richardson Algorithm for Solving Nonlinear ODEs and PDEs
,”
Commun. Comput. Phys.
,
6
(
4
), pp.
883
902
.
31.
Wanxie
,
Z.
,
Zhuang
,
X.
, and
Zhu
,
J.
,
1998
, “
A Self-Adaptive Time Integration Algorithm for Solving Partial Differential Equations
,”
Appl. Math. Comput.
,
89
(
1
), pp.
295
312
.
32.
Horton
,
G.
, and
Vandewalle
,
S.
,
1995
, “
A Space-Time Multigrid Method for Parabolic Partial Differential Equations
,”
SIAM J. Sci. Comput.
,
16
(
4
), pp.
848
864
.
33.
Fischer
,
P.
,
Romano
,
V.
,
Weber
,
H.
,
Karapatis
,
N.
,
Boillat
,
E.
, and
Glardon
,
R.
,
2003
, “
Sintering of Commercially Pure Titanium Powder With a Nd:Yag Laser Source
,”
Acta Mater.
,
51
(
6
), pp.
1651
1662
.
34.
Shiomi
,
M.
,
Yoshidome
,
A.
,
Abe
,
F.
, and
Osakada
,
K.
,
1999
, “
Finite Element Analysis of Melting and Solidifying Processes in Laser Rapid Prototyping of Metallic Powders
,”
Int. J. Mach. Tools Manuf.
,
39
(
2
), pp.
237
252
.
35.
Wang
,
X.
,
Laoui
,
T.
,
Bonse
,
J.
,
Kruth
,
J.-P.
,
Lauwers
,
B.
, and
Froyen
,
L.
,
2002
, “
Direct Selective Laser Sintering of Hard Metal Powders: Experimental Study and Simulation
,”
Int. J. Adv. Manuf. Technol.
,
19
(
5
), pp.
351
357
.
36.
Tolochko
,
N. K.
,
Arshinov
,
M. K.
,
Gusarov
,
A. V.
,
Titov
,
V. I.
,
Laoui
,
T.
, and
Froyen
,
L.
,
2003
, “
Mechanisms of Selective Laser Sintering and Heat Transfer in TI Powder
,”
Rapid Prototyping J.
,
9
(
5
), pp.
314
326
.
37.
Zeng
,
K.
,
Pal
,
D.
, and
Stucker
,
B.
,
2012
, “
A Review of Thermal Analysis Methods in Laser Sintering and Selective Laser Melting
,”
Solid Freeform Fabrication Symposium
, Vol.
23
, p.
796
.
38.
Ibraheem
,
A. K.
,
Derby
,
B.
, and
Withers
,
P. J.
,
2003
, “
Thermal and Residual Stress Modelling of the Selective Laser Sintering Process
,” DTIC Document,
Contract No. N00014-02-1-0820
.
39.
Liu
,
H.
,
2014
, “
Numerical Analysis of Thermal Stress and Deformation in Multi-Layer Laser Metal Deposition Process
,”
Master's thesis
, Missouri University of Science and Technology, Rolla, MO.
40.
Song
,
J.
,
Shanghvi
,
J.
, and
Michaleris
,
P.
,
2004
, “
Sensitivity Analysis and Optimization of Thermo-Elasto-Plastic Processes With Applications to Welding Side Heater Design
,”
Comput. Methods Appl. Mech. Eng.
,
193
(
42
), pp.
4541
4566
.
41.
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
.
42.
Denlinger
,
E. R.
,
Heigel
,
J. C.
, and
Michaleris
,
P.
,
2014
, “
Residual Stress and Distortion Modeling of Electron Beam Direct Manufacturing Ti–6AL–4V
,”
Proc. Inst. Mech. Eng., Part B
,
229
(
10
), pp.
1803
1813
.
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