Abstract

In this study, we carry out robust optimal design for the machining operations, one key process in wafer polishing in chip manufacturing, aiming to avoid the peculiar regenerative chatter and maximize the material removal rate (MRR) considering the inherent material and process uncertainty. More specifically, we characterize the cutting tool dynamics using a delay differential equation (DDE) and enlist the temporal finite element method (TFEM) to derive its approximate solution and stability index given process settings or design variables. To further quantify the inherent uncertainty, replications of TFEM under different realizations of random uncontrollable variables are performed, which however incurs extra computational burden. To eschew the deployment of such a crude Monte Carlo (MC) approach at each design setting, we integrate the stochastic TFEM with a stochastic surrogate model, stochastic kriging, in an active learning framework to sequentially approximate the stability boundary. The numerical result suggests that the nominal stability boundary attained from this method is on par with that from the crude MC, but only demands a fraction of the computational overhead. To further ensure the robustness of process stability, we adopt another surrogate, the Gaussian process, to predict the variance of the stability index at unexplored design points and identify the robust stability boundary per the conditional value at risk (CVaR) criterion. Therefrom, an optimal design in the robust stable region that maximizes the MRR can be identified.

References

1.
McLain
,
S.
,
2021
, “
Auto Makers Retreat From 50 Years of ‘Just in Time’ Manufacturing
,”
Wall Str. J.
Accessed 21 December, 2021. [Online]. Available: https://www.wsj.com/articles/auto-makers-retreat-from-50-years-of-just-in-time-manufacturing-11620051251
2.
Cheng
,
C.
,
Wang
,
Z.
,
Hung
,
W.
,
Bukkapatnam
,
S. T. S.
, and
Komanduri
,
R.
,
2015
, “
Ultra-Precision Machining Process Dynamics and Surface Quality Monitoring
,”
Procedia Manuf.
,
1
, pp.
607
618
.
3.
Bukkapatnam
,
S. T. S.
,
Rao
,
P. K.
,
Lih
,
W.-C.
,
Chandrasekaran
,
N.
, and
Komanduri
,
R.
,
2007
, “
Process Characterization and Statistical Analysis of Oxide CMP on a Silicon Wafer With Sparse Data
,”
Appl. Phys. A
,
88
(
4
), pp.
785
792
.
4.
Shamsan
,
A.
, and
Cheng
,
C.
,
2019
, “
Intrinsic Multiplex Graph Model Detects Incipient Process Drift in Ultraprecision Manufacturing
,”
J. Manuf. Syst.
,
50
, pp.
81
86
.
5.
Ringgaard
,
K.
,
Mohammadi
,
Y.
,
Merrild
,
C.
,
Balling
,
O.
, and
Ahmadi
,
K.
,
2019
, “
Optimization of Material Removal Rate in Milling of Thin-Walled Structures Using Penalty Cost Function
,”
Int. J. Mach. Tools Manuf.
,
145
, p.
103430
.
6.
Che
,
Y.
, and
Cheng
,
C.
,
2018
, “
Uncertainty Quantification in Stability Analysis of Chaotic Systems With Discrete Delays
,”
Chaos Solitons Fractals
,
116
, pp.
208
214
.
7.
Che
,
Y.
,
Liu
,
J.
, and
Cheng
,
C.
,
2019
, “
Multi-Fidelity Modeling in Sequential Design for Stability Identification in Dynamic Time-Delay Systems
,”
Chaos Interdiscip. J. Nonlinear Sci.
,
29
(
9
), p.
093105
.
8.
Schmitz
,
T. L.
,
Karandikar
,
J.
,
Ho Kim
,
N.
, and
Abbas
,
A.
,
2011
, “
Uncertainty in Machining: Workshop Summary and Contributions
,”
ASME J. Manuf. Sci. Eng.
,
133
(
5
), p.
051009
.
9.
Li
,
X.
,
Yang
,
Y.
,
Li
,
L.
,
Zhao
,
G.
, and
He
,
N.
,
2020
, “
Uncertainty Quantification in Machining Deformation Based on Bayesian Network
,”
Reliab. Eng. Syst. Saf.
,
203
, p.
107113
.
10.
Wang
,
Z. C.
, and
Cleghorn
,
W. L.
,
2002
, “
Stability Analysis of Spinning Stepped-Shaft Workpieces in a Turning Process
,”
J. Sound Vib.
,
250
(
2
), pp.
356
367
.
11.
Baker
,
J. R.
, and
Rouch
,
K. E.
,
2002
, “
Use of Finite Element Structural Models in Analyzing Machine Tool Chatter
,”
Finite Elem. Anal. Des.
,
38
(
11
), pp.
1029
1046
.
12.
Mahdavinejad
,
R.
,
2005
, “
Finite Element Analysis of Machine and Workpiece Instability in Turning
,”
Int. J. Mach. Tools Manuf.
,
45
(
7
), pp.
753
760
.
13.
Quintana
,
G.
, and
Ciurana
,
J.
,
May 2011
, “
Chatter in Machining Processes: A Review
,”
Int. J. Mach. Tools Manuf.
,
51
(
5
), pp.
363
376
.
14.
Mahnama
,
M.
, and
Movahhedy
,
M. R.
,
2010
, “
Prediction of Machining Chatter Based on FEM Simulation of Chip Formation Under Dynamic Conditions
,”
Int. J. Mach. Tools Manuf.
,
50
(
7
), pp.
611
620
.
15.
Turkes
,
E.
,
Orak
,
S.
,
Neseli
,
S.
, and
Yaldiz
,
S.
,
2011
, “
Linear Analysis of Chatter Vibration and Stability for Orthogonal Cutting in Turning
,”
Int. J. Refract. Met. Hard Mater.
,
29
(
2
), pp.
163
169
.
16.
Landers
,
R. G.
, and
Ulsoy
,
A. G.
,
2007
, “
Nonlinear Feed Effect in Machining Chatter Analysis
,”
ASME 2007 International Manufacturing Science and Engineering Conference
,
Atlanta, GA
,
Oct. 15–18
, pp.
17
26
.
17.
Greis
,
N. P.
,
Nogueira
,
M. L.
,
Bhattacharya
,
S.
, and
Schmitz
,
T.
,
2020
, “
Physics-Guided Machine Learning for Self-Aware Machining
,”
2020 AAAI Spring Symposium on AI and Manufacturing
,
Palo Alto, CA
,
Mar. 23–25
.
18.
Altintaş
,
Y.
, and
Budak
,
E.
,
1995
, “
Analytical Prediction of Stability Lobes in Milling
,”
CIRP Ann.
,
44
(
1
), pp.
357
362
.
19.
Khasawneh
,
F. A.
, and
Munch
,
E.
,
2016
, “
Chatter Detection in Turning Using Persistent Homology
,”
Mech. Syst. Signal Process.
,
70–71
, pp.
527
541
.
20.
Schmitz
,
T. L.
,
Medicus
,
K.
, and
Dutterer
,
B.
,
2002
, “
Exploring Once-Per-Revolution Audio Signal Variance as a Chatter Indicator
,”
Mach. Sci. Technol.
,
6
(
2
), pp.
215
233
.
21.
Axinte
,
D. A.
,
Gindy
,
N.
,
Fox
,
K.
, and
Unanue
,
I.
,
2004
, “
Process Monitoring to Assist the Workpiece Surface Quality in Machining
,”
Int. J. Mach. Tools Manuf.
,
44
(
10
), pp.
1091
1108
.
22.
Denkena
,
B.
,
Bergmann
,
B.
, and
Reimer
,
S.
,
2020
, “
Analysis of Different Machine Learning Algorithms to Learn Stability Lobe Diagrams
,”
Procedia CIRP
,
88
, pp.
282
287
.
23.
Friedrich
,
J.
,
Hinze
,
C.
,
Renner
,
A.
,
Verl
,
A.
, and
Lechler
,
A.
,
2015
, “
Estimation of Stability Lobe Diagrams in Milling With Continuous Learning Algorithms
,”
Robot. Comput.-Integr. Manuf.
,
43
, pp.
124
134
.
24.
Friedrich
,
J.
,
Torzewski
,
J.
, and
Verl
,
A.
,
2018
, “
Online Learning of Stability Lobe Diagrams in Milling
,”
Procedia CIRP
,
67
, pp.
278
283
.
25.
Huang
,
X.
,
Hu
,
M.
,
Zhang
,
Y.
, and
Lv
,
C.
,
2016
, “
Probabilistic Analysis of Chatter Stability in Turning
,”
Int. J. Adv. Manuf. Technol.
,
87
(
9–12
), pp.
3225
3232
.
26.
Park
,
S. S.
, and
Qin
,
Y. M.
,
2007
, “
Robust Regenerative Chatter Stability in Machine Tools
,”
Int. J. Adv. Manuf. Technol.
,
33
(
3
), pp.
389
402
.
27.
Totis
,
G.
,
2009
, “
RCPM—A New Method for Robust Chatter Prediction in Milling
,”
Int. J. Mach. Tools Manuf.
,
49
(
3–4
), pp.
273
284
.
28.
Löser
,
M.
,
Otto
,
A.
,
Ihlenfeldt
,
S.
, and
Radons
,
G.
,
2018
, “
Chatter Prediction for Uncertain Parameters
,”
Adv. Manuf.
,
6
(
3
), pp.
319
333
.
29.
Kong
,
F.
, and
Yu
,
J.
,
May 2007
, “
Study of Fuzzy Stochastic Limited Cutting Width on Chatter
,”
Int. J. Adv. Manuf. Technol.
,
33
(
7–8
), pp.
677
683
.
30.
Emmerich
,
M. T. M.
,
Giannakoglou
,
K. C.
, and
Naujoks
,
B.
,
2006
, “
Single - and Multiobjective Evolutionary Optimization Assisted by Gaussian Random Field Metamodels
,”
IEEE Trans. EComput.
,
10
(
4
), pp.
421
439
.
31.
Zhou
,
Z.
,
Ong
,
Y. S.
,
Nair
,
P. B.
,
Keane
,
A. J.
, and
Lum
,
K. Y.
,
2007
, “
Combining Global and Local Surrogate Models to Accelerate Evolutionary Optimization
,”
IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.
,
37
(
1
), pp.
66
76
.
32.
Ponweiser
,
W.
,
Wagner
,
T.
, and
Vincze
,
M.
,
2008
, “
Clustered Multiple Generalized Expected Improvement: A Novel Infill Sampling Criterion for Surrogate Models
,”
2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence)
,
Hong Kong, China
,
June 1–6
, pp.
3515
3522
.
33.
Lookman
,
T.
,
Balachandran
,
P. V.
,
Xue
,
D.
, and
Yuan
,
R.
,
2019
, “
Active Learning in Materials Science With Emphasis on Adaptive Sampling Using Uncertainties for Targeted Design
,”
npj Comput. Mater.
,
5
(
1
).
34.
Liu
,
B.
,
Zhang
,
Q.
, and
Gielen
,
G. G. E.
,
2014
, “
A Gaussian Process Surrogate Model Assisted Evolutionary Algorithm for Medium Scale Expensive Optimization Problems
,”
IEEE Trans. E Comput.
,
18
(
2
), pp.
180
192
.
35.
Scott
,
W.
,
Frazier
,
P.
, and
Powell
,
W.
,
2011
, “
The Correlated Knowledge Gradient for Simulation Optimization of Continuous Parameters Using Gaussian Process Regression
,”
SIAM J. Optim.
,
21
(
3
), pp.
996
1026
.
36.
van Houtum
,
G. J. J.
, and
Vlasea
,
M. L.
,
2021
, “
Active Learning Via Adaptive Weighted Uncertainty Sampling Applied to Additive Manufacturing
,”
Addit. Manuf.
,
48
(
PB
), p.
102411
.
37.
Che
,
Y.
, and
Cheng
,
C.
,
2021
, “
Active Learning and Relevance Vector Machine in Efficient Estimate of Basin Stability for Large-Scale Dynamic Networks
,”
Chaos Interdiscip. J. Nonlinear Sci.
,
31
(
5
), p.
053129
.
38.
Botcha
,
B.
,
Iquebal
,
A. S.
, and
Bukkapatnam
,
S. T. S.
,
2021
, “
Efficient Manufacturing Processes and Performance Qualification Via Active Learning: Application to a Cylindrical Plunge Grinding Platform
,”
Procedia Manuf.
,
53
, pp.
716
725
.
39.
Shim
,
J.
,
Kang
,
S.
, and
Cho
,
S.
,
2020
, “
Active Learning of Convolutional Neural Network for Cost-Effective Wafer Map Pattern Classification
,”
IEEE Trans. Semicond. Manuf.
,
33
(
2
), pp.
258
266
.
40.
Gal
,
Y.
,
Islam
,
R.
, and
Ghahramani
,
Z.
,
2017
, “
Deep Bayesian Active Learning With Image Data
,”
Proceedings of the 34th International Conference on Machine Learning
,
Sydney, NSW, Australia
,
Aug. 6–11
, pp.
1183
1192
.
41.
Bayly
,
P. V.
,
Halley
,
J. E.
,
Mann
,
B. P.
, and
Davies
,
M. A.
,
2003
, “
Stability of Interrupted Cutting by Temporal Finite Element Analysis
,”
ASME J. Manuf. Sci. Eng.
,
125
(
2
), pp.
220
225
.
42.
Carlberg
,
K.
,
Barone
,
M.
, and
Antil
,
H.
,
2017
, “
Galerkin v. Least-Squares Petrov–Galerkin Projection in Nonlinear Model Reduction
,”
J. Comput. Phys.
,
330
, pp.
693
734
.
43.
Shang
,
B.
, and
Apley
,
D. W.
,
Mar. 2021
, “
Fully-Sequential Space-Filling Design Algorithms for Computer Experiments
,”
J. Qual. Technol.
,
53
(
2
), pp.
173
196
.
44.
Qian
,
P. Z. G.
,
2012
, “
Sliced Latin Hypercube Designs
,”
J. Am. Stat. Assoc.
,
107
(
497
), pp.
393
399
.
45.
Rennen
,
G.
,
Husslage
,
B.
,
Van Dam
,
E. R.
, and
Den Hertog
,
D.
,
2010
, “
Nested Maximin Latin Hypercube Designs
,”
Struct. Multidiscip. Optim.
,
41
(
3
), pp.
371
395
.
46.
Ankenman
,
B.
,
Nelson
,
B. L.
, and
Staum
,
J.
,
2010
, “
Stochastic Kriging for Simulation Metamodeling
,”
Oper. Res.
,
58
(
2
), pp.
371
382
.
47.
Che
,
Y.
,
Guo
,
Z.
, and
Cheng
,
C.
,
2021
, “
Generalized Polynomial Chaos-Informed Efficient Stochastic Kriging
,”
J. Comput. Phys.
,
445
, p.
110598
.
48.
Totis
,
G.
, and
Sortino
,
M.
,
2020
, “
Polynomial Chaos-Kriging Approaches for an Efficient Probabilistic Chatter Prediction in Milling
,”
Int. J. Mach. Tools Manuf.
,
157
, p.
103610
.
49.
Bukkapatnam
,
S. T. S.
, and
Cheng
,
C.
,
2010
, “
Forecasting the Evolution of Nonlinear and Nonstationary Systems Using Recurrence-Based Local Gaussian Process Models
,”
Phys. Rev. E
,
82
(
5
), p.
056206
.
50.
Zhan
,
D.
, and
Xing
,
H.
,
2020
, “
Expected Improvement for Expensive Optimization: A Review
,”
J. Glob. Optim.
,
78
(
3
), pp.
507
544
.
51.
Jones
,
D. R.
,
Schonlau
,
M.
, and
Welch
,
W. J.
,
1998
, “
Efficient Global Optimization of Expensive Black-Box Functions
,”
J. Glob. Optim.
,
13
(
4
), pp.
455
492
.
52.
Yin
,
J.
,
Ng
,
S. H.
, and
Ng
,
K. M.
,
2011
, “
Kriging Metamodel With Modified Nugget-Effect: The Heteroscedastic Variance Case
,”
Comput. Ind. Eng.
,
61
(
3
), pp.
760
777
.
53.
Gramacy
,
R. B.
, and
Lee
,
H. K. H.
,
2008
, “
Bayesian Treed Gaussian Process Models With an Application to Computer Modeling
,”
J. Am. Stat. Assoc.
,
103
(
483
), pp.
1119
1130
.
54.
Chaudhuri
,
A.
,
Kramer
,
B.
,
Norton
,
M.
,
Royset
,
J. O.
, and
Willcox
,
K.
,
2021
, “
Certifiable Risk-Based Engineering Design Optimization
,”
AIAA J.
,
60
(
2
), pp.
1
15
.
55.
Zhu
,
S.
, and
Fukushima
,
M.
,
2009
, “
Worst-Case Conditional Value-at-Risk With Application to Robust Portfolio Management
,”
Oper. Res.
,
57
(
5
), pp.
1155
1168
.
56.
Royset
,
J. O.
,
Bonfiglio
,
L.
,
Vernengo
,
G.
, and
Brizzolara
,
S.
,
2017
, “
Risk-Adaptive Set-Based Design and Applications to Shaping a Hydrofoil
,”
ASME J. Mech. Des.
,
139
(
10
), p.
101403
.
57.
Chaudhuri
,
A.
,
Marques
,
A. N.
, and
Willcox
,
K.
,
2021
, “
mfEGRA: Multifidelity Efficient Global Reliability Analysis Through Active Learning for Failure Boundary Location
,”
Struct. Multidiscip. Optim.
,
64
(
2
), pp.
797
811
.
58.
Bichon
,
B. J.
,
Eldred
,
M. S.
,
Mahadevan
,
S.
, and
McFarland
,
J. M.
,
2013
, “
Efficient Global Surrogate Modeling for Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
135
(
1
), p.
011009
.
59.
Li
,
H.-S.
,
Zhao
,
A.-L.
, and
Tee
,
K. F.
,
2016
, “
Structural Reliability Analysis of Multiple Limit State Functions Using Multi-Input Multi-Output Support Vector Machine
,”
Adv. Mech. Eng.
,
8
(
10
), p.
1687814016671447
.
60.
Alibrandi
,
U.
,
Alani
,
A. M.
, and
Ricciardi
,
G.
,
2015
, “
A New Sampling Strategy for SVM-Based Response Surface for Structural Reliability Analysis
,”
Probabilistic Eng. Mech.
,
41
, pp.
1
12
.
61.
Basudhar
,
A.
, and
Missoum
,
S.
,
2008
, “
Adaptive Explicit Decision Functions for Probabilistic Design and Optimization Using Support Vector Machines
,”
Comput. Struct.
,
86
(
19
), pp.
1904
1917
.
62.
Basudhar
,
A.
,
Missoum
,
S.
, and
Harrison Sanchez
,
A.
,
2008
, “
Limit State Function Identification Using Support Vector Machines for Discontinuous Responses and Disjoint Failure Domains
,”
Probabilistic Eng. Mech.
,
23
(
1
), pp.
1
11
.
63.
Zhang
,
G.
,
Li
,
J.
,
Chen
,
Y.
,
Huang
,
Y.
,
Shao
,
X.
, and
Li
,
M.
,
2014
, “
Prediction of Surface Roughness in End Face Milling Based on Gaussian Process Regression and Cause Analysis Considering Tool Vibration
,”
Int. J. Adv. Manuf. Technol.
,
75
(
9–12
), pp.
1357
1370
.
64.
Kawai
,
K.
,
Irino
,
N.
,
Iiyama
,
K.
,
Matsubara
,
A.
,
Mizuyama
,
H.
,
Mori
,
K.
, and
Fujishima
,
M.
,
2021
, “
A Prediction Model for High Efficiency Machining Conditions Based on Machine Learning
,”
Proc. CIRP
,
101
, pp.
54
57
.
65.
Feng
,
J.
,
Hou
,
N.
,
Jian
,
Z.
,
Sun
,
Z.
, and
Zhang
,
J.
,
2020
, “
An Efficient Method to Predict the Chatter Stability of Titanium Alloy Thin-Walled Workpieces During High-Speed Milling by Considering Varying Dynamic Parameters
,”
Int. J. Adv. Manuf. Technol.
,
106
(
11–12
), pp.
5407
5420
.
You do not currently have access to this content.