This paper deals with the systematic optimization method for multiple input variables (laser irradiation power and scanning speed) in a class of laser-aided powder deposition (LAPD) processes. These processes are normally described by a coupled system of nonlinear partial differential equations (PDEs). To begin with, a desired solid–liquid (S/L) interface geometry is first approximated from a few practical process target parameters that define the desired process properties. Then, the control problem is formulated as one of seeking the optimal combination of process inputs that achieves close tracking of the desired S/L interface in quasi-steady state. The paper details the derivation of the adjoint-based solution for this PDE-constrained multivariable control input optimization problem. The effectiveness of the proposed method is illustrated via a case study on a laser cladding process.

References

1.
Laeng
,
J.
,
Stewart
,
J.
, and
Liou
,
F. W.
,
2000
, “
Laser Metal Forming Processes for Rapid Prototyping—A Review
,”
Int. J. Prod. Res.
,
38
(
16
), pp.
3973
3996
.
2.
Gibson
,
I.
,
Rosen
,
D. W.
, and
Stucker
,
B.
,
2010
,
Additive Manufacturing Technologies
,
Springer
,
New York
.
3.
Majumdar
,
J. D.
, and
Manna
,
I.
,
2011
, “
Laser Material Processing
,”
Int. Mater. Rev.
,
56
(
5–6
), pp.
341
388
.
4.
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
.
5.
Fathi
,
A.
,
Toyserkani
,
E.
,
Khajepour
,
A.
, and
Durali
,
M.
,
2006
, “
Prediction of Melt Pool Depth and Dilution in Laser Powder Deposition
,”
J. Phys. D: Appl. Phys.
,
39
(
12
), pp.
2613
2623
.
6.
Song
,
L.
, and
Mazumder
,
J.
,
2011
, “
Feedback Control of Melt Pool Temperature During Laser Cladding Process
,”
IEEE Trans. Control Syst. Technol.
,
19
(
6
), pp.
1349
1356
.
7.
Hofman
,
J.
,
Pathiraj
,
B.
,
van Dijk
,
J.
,
de Lange
,
D.
, and
Meijer
,
J.
,
2012
, “
A Camera Based Feedback Control Strategy for the Laser Cladding Process
,”
J. Mater. Process. Technol.
,
212
(
11
), pp.
2455
2462
.
8.
Salehi
,
D.
, and
Brandt
,
M.
,
2006
, “
Melt Pool Temperature Control Using labview in Nd: YAG Laser Blown Powder Cladding Process
,”
Int. J. Adv. Manuf. Technol.
,
29
(
3–4
), pp.
273
278
.
9.
Fathi
,
A.
,
Khajepour
,
A.
,
Toyserkani
,
E.
, and
Durali
,
M.
,
2007
, “
Clad Height Control in Laser Solid Freeform Fabrication Using a Feedforward PID Controller
,”
Int. J. Adv. Manuf. Technol.
,
35
(
3–4
), pp.
280
292
.
10.
Morville
,
S.
,
Carin
,
M.
,
Peyre
,
P.
,
Gharbi
,
M.
,
Carron
,
D.
,
Le Masson
,
P.
, and
Fabbro
,
R.
,
2012
, “
2D Longitudinal Modeling of Heat Transfer and Fluid Flow During Multilayered Direct Laser Metal Deposition Process
,”
J. Laser Appl.
,
24
, p.
032008
.
11.
Wen
,
S.
, and
Shin
,
Y. C.
,
2010
, “
Modeling of Transport Phenomena During the Coaxial Laser Direct Deposition Process
,”
J. Appl. Phys.
,
108
(4), p.
044908
.
12.
Qi
,
H.
,
Mazumder
,
J.
, and
Ki
,
H.
,
2006
, “
Numerical Simulation of Heat Transfer and Fluid Flow in Coaxial Laser Cladding Process for Direct Metal Deposition
,”
J. Appl. Phys.
,
100
(2), p.
024903
.
13.
Han
,
L.
, and
Liou
,
F.
,
2004
, “
Numerical Investigation of the Influence of Laser Beam Mode on Melt Pool
,”
Int. J. Heat Mass Transfer
,
47
(
19
), pp.
4385
4402
.
14.
Saldi
,
Z. S.
,
2012
, “
Marangoni Driven Free Surface Flows in Liquid Weld Pools
,” Ph.D. thesis, Delft University of Technology, Delft, The Netherlands.
15.
Traidia
,
A.
,
2011
, “
Multiphysics Modeling and Numerical Simulation of GTA Weld Pools
,” Ph. D. thesis, Ecole Polytechnique, Palaiseau, France.
16.
Sethian
,
J.
, and
Smereka
,
P.
,
2003
, “
Level Set Methods for Fluid Interfaces
,”
Annu. Rev. Fluid Mech.
,
35
(
1
), pp.
341
372
.
17.
Pinkerton
,
A. J.
, and
Lin
,
L.
,
2004
, “
Modeling Powder Concentration Distribution From a Coaxial Deposition Nozzle for Laser-Based Rapid Tooling
,”
ASME J. Manuf. Sci. Eng.
,
126
(
1
), pp.
33
41
.
18.
Pal
,
D.
,
Patil
,
N.
,
Zeng
,
K.
, and
Stucker
,
B.
,
2014
, “
An Integrated Approach to Additive Manufacturing Simulations Using Physics Based, Coupled Multiscale Process Modeling
,”
ASME J. Manuf. Sci. Eng.
,
136
(
6
), p.
061022
.
19.
Cheng
,
B.
,
Price
,
S.
,
Lydon
,
J.
,
Cooper
,
K.
, and
Chou
,
K.
,
2014
, “
On Process Temperature in Powder-Bed Electron Beam Additive Manufacturing: Model Development and Validation
,”
ASME J. Manuf. Sci. Eng.
,
136
(
6
), p.
061018
.
20.
Safdar
,
S.
,
Pinkerton
,
A. J.
,
Li
,
L.
,
Sheikh
,
M. A.
, and
Withers
,
P. J.
,
2013
, “
An Anisotropic Enhanced Thermal Conductivity Approach for Modeling Laser Melt Pools for Ni-Base Super Alloys
,”
Appl. Math. Modell.
,
37
(
3
), pp.
1187
1195
.
21.
Kamara
,
A.
,
Wang
,
W.
,
Marimuthu
,
S.
, and
Li
,
L.
,
2011
, “
Modeling of the Melt Pool Geometry in the Laser Deposition of Nickel Alloys Using the Anisotropic Enhanced Thermal Conductivity Approach
,”
Proc. Inst. Mech. Eng., Part B: J. Eng. Manuf.
,
225
(
1
), pp.
87
99
.
22.
Le Guen
,
E.
,
Carin
,
M.
,
Fabbro
,
R.
,
Coste
,
F.
, and
Le Masson
,
P.
,
2011
, “
3D Heat Transfer Model of Hybrid Laser Nd: Yag-MAG Welding of S355 Steel and Experimental Validation
,”
Int. J. Heat Mass Transfer
,
54
(
7
), pp.
1313
1322
.
23.
Peyre
,
P.
,
Aubry
,
P.
,
Fabbro
,
R.
,
Neveu
,
R.
, and
Longuet
,
A.
,
2008
, “
Analytical and Numerical Modelling of the Direct Metal Deposition Laser Process
,”
J. Phys. D: Appl. Phys.
,
41
(
2
), p.
025403
.
24.
Zhang
,
W.
,
Kim
,
C.-H.
, and
DebRoy
,
T.
,
2004
, “
Heat and Fluid Flow in Complex Joints During Gas Metal Arc Welding—Part II: Application to Fillet Welding of Mild Steel
,”
J. Appl. Phys.
,
95
(
9
), pp.
5220
5229
.
25.
Tang
,
L.
, and
Landers
,
R. G.
,
2011
, “
Layer-to-Layer Height Control for Laser Metal Deposition Process
,”
ASME J. Manuf. Sci. Eng.
,
133
(
2
), p.
021009
.
26.
Song
,
L.
,
Bagavath-Singh
,
V.
,
Dutta
,
B.
, and
Mazumder
,
J.
,
2012
, “
Control of Melt Pool Temperature and Deposition Height During Direct Metal Deposition Process
,”
Int. J. Adv. Manuf. Technol.
, 58(1–4), pp.
247
256
.
27.
Jameson
,
A.
,
1988
, “
Aerodynamic Design Via Control Theory
,”
J. Sci. Comput.
,
3
(
3
), pp.
233
260
.
28.
Repke
,
S.
,
Marheineke
,
N.
, and
Pinnau
,
R.
,
2010
,
On Adjoint-Based Optimization of a Free Surface Stokes Flow
,
Fraunhofer-Institut für Techno-und Wirtschaftsmathematik, Fraunhofer (ITWM)
.
29.
Vossen
,
G.
,
Hermanns
,
T.
, and
Schüttler
,
J.
,
2013
, “
Analysis and Optimal Control for Free Melt Flow Boundaries in Laser Cutting With Distributed Radiation
,”
ZAMM—J. Appl. Math. Mech./Z. Angew. Math. Mech.
,
95
(
3
), pp.
297
316
.
30.
Bernauer
,
M. K.
, and
Herzog
,
R.
,
2011
, “
Optimal Control of the Classical Two-Phase Stefan Problem in Level Set Formulation
,”
SIAM J. Sci. Comput.
,
33
(
1
), pp.
342
363
.
31.
Volkov
,
O.
, and
Protas
,
B.
,
2009
, “
An Inverse Model for a Free-Boundary Problem With a Contact Line: Steady Case
,”
J. Comput. Phys.
,
228
(
13
), pp.
4893
4910
.
32.
Hinze
,
M.
, and
Ziegenbalg
,
S.
,
2007
, “
Optimal Control of the Free Boundary in a Two-Phase Stefan Problem
,”
J. Comput. Phys.
,
223
(
2
), pp.
657
684
.
33.
Vossen
,
G.
, and
Hermanns
,
T.
,
2014
, “
On an Optimal Control Problem in Laser Cutting With Mixed Finite-/Infinite-Dimensional Constraints
,”
J. Ind. Manage. Optim.
,
10
(
2
), pp.
503
519
.
34.
Volkov
,
O.
,
Protas
,
B.
,
Liao
,
W.
, and
Glander
,
D. W.
,
2009
, “
Adjoint-Based Optimization of Thermo-Fluid Phenomena in Welding Processes
,”
J. Eng. Math.
,
65
(
3
), pp.
201
220
.
35.
Wu
,
C.
,
Chen
,
J.
, and
Zhang
,
Y.
,
2007
, “
Numerical Analysis of Both Front-and Back-Side Deformation of Fully-Penetrated GTAW Weld Pool Surfaces
,”
Comput. Mater. Sci.
,
39
(
3
), pp.
635
642
.
36.
Hofman
,
J. T.
,
2009
,
Development of an Observation and Control System for Industrial Laser Cladding
,
University of Twente
,
Enschede, The Netherlands
.
37.
Medina
,
I.
, and
Ramses
,
J.
,
2013
, “
Development and Application of a CFD Model of Laser Metal Deposition
,” Ph.D thesis, University of Manchester, Manchester.
38.
Al-Khalidy
,
N.
,
1997
, “
Application of Optimization Methods for Solving Inverse Phase-Change Problems
,”
Numer. Heat Transfer
,
31
(
4
), pp.
477
497
.
39.
Pham
,
X.-T.
,
2013
, “
Two-Dimensional Rosenthal Moving Heat Source Analysis Using the Meshless Element Free Galerkin Method
,”
Numer. Heat Transfer, Part A: Appl.
,
63
(
11
), pp.
807
823
.
40.
Cao
,
X.
, and
Ayalew
,
B.
,
2015
, “
Control-Oriented MIMO Modeling of Laser-Aided Powder Deposition Processes
,”
IEEE American Control Conference
(
ACC
), Chicago, IL, July 1–3.
41.
Pinkerton
,
A. J.
,
2007
, “
An Analytical Model of Beam Attenuation and Powder Heating During Coaxial Laser Direct Metal Deposition
,”
J. Phys. D: Appl. Phys.
,
40
(
23
), pp.
7323
7334
.
42.
Toyserkani
,
E.
,
Khajepour
,
A.
, and
Corbin
,
S.
,
2004
, “
3-D Finite Element Modeling of Laser Cladding by Powder Injection: Effects of Laser Pulse Shaping on the Process
,”
Opt. Lasers Eng.
,
41
(
6
), pp.
849
867
.
43.
Kumar
,
A.
, and
Roy
,
S.
,
2009
, “
Effect of Three-Dimensional Melt Pool Convection on Process Characteristics During Laser Cladding
,”
Comput. Mater. Sci.
,
46
(
2
), pp.
495
506
.
44.
De Oliveira
,
U.
,
Ocelik
,
V.
, and
De Hosson
,
J. T. M.
,
2005
, “
Analysis of Coaxial Laser Cladding Processing Conditions
,”
Surf. Coat. Technol.
,
197
(
2
), pp.
127
136
.
45.
Li
,
S.
, and
Petzold
,
L.
,
2004
, “
Adjoint Sensitivity Analysis for Time-Dependent Partial Differential Equations With Adaptive Mesh Refinement
,”
J. Comput. Phys.
,
198
(
1
), pp.
310
325
.
46.
Tröltzsch
,
F.
,
2010
,
Optimal Control of Partial Differential Equations: Theory, Methods, and Applications
,
American Mathematical Soc.
,
Providence, RI
.
47.
Neittaanmaki
,
P.
, and
Tiba
,
D.
,
1994
,
Optimal Control of Nonlinear Parabolic Systems: Theory: Algorithms and Applications
,
CRC Press
,
New York
.
48.
Han
,
L.
,
Phatak
,
K.
, and
Liou
,
F.
,
2004
, “
Modeling of Laser Cladding With Powder Injection
,”
Metall. Mater. Trans. B
,
35
(
6
), pp.
1139
1150
.
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