The increased use of micro-electro-mechanical systems (MEMS) as key components for actuation and sensing purposes in novel devices and systems emphasizes the need for optimal design methods. Stochastic variations in manufacturing and operational conditions must be considered in order to meet performance goals. This study proposes a reliability based design optimization methodology for the design of geometrically complex electrostatically actuated MEMS. The first order reliability method is used for reliability analysis of fully-coupled electrostatic-mechanical problems. A general methodology for predicting the instability phenomenon of pull-in and incorporating it into an automatic optimization process is proposed and verified with analytical and experimental results. The potential of this methodology is illustrated with the design of an analog micromirror.

1.
Mi
,
B.
,
Smith
,
D. A.
,
Kahn
,
H.
,
Merat
,
F. L.
,
Heuer
,
A. H.
, and
Phillips
,
S. M.
, 2005, “
Static and Electrically Actuated Shaped MEMS Mirrors
,”
J. Microelectromech. Syst.
1057-7157,
14
(
1
), pp.
29
36
.
2.
Kong
,
J. S.
,
Franpgopol
,
D.
,
Raulli
,
M.
,
Maute
,
K.
,
Saravanan
,
R.
,
Liew
,
L. -A.
, and
Raj
,
R.
, 2004, “
A Methodology for Analyzing the Variability in the Performance of a MEMS Actuator Made From a Novel Ceramic
,”
Sens. Actuators, A
0924-4247,
116
(
2
), pp.
336
344
.
3.
Osterberg
,
P.
, and
Senturia
,
S.
, 1997, “
M-Test: A Test Chip for MEMS Material Property Measurement Using Electrostatically Actuated Test Structures
,”
J. Microelectromech. Syst.
1057-7157,
6
, pp.
107
118
.
4.
Enevoldsen
,
I.
, and
Sorensen
,
J.
, 1994, “
Reliability-Based Optimization in Structural Engineering
,”
Struct. Safety
0167-4730,
15
, pp.
169
196
.
5.
Chandu
,
S.
, and
Grandhi
,
R.
, 1995, “
General Purpose Procedure for Reliability Based Structural Optimization Under Parametric Uncertainties
,”
Adv. Eng. Software
0965-9978,
23
, pp.
7
14
.
6.
Yu
,
X.
,
Choi
,
K.
, and
Chang
,
K.
, 1997, “
A Mixed Design Approach for Probabilistic Structural Durability
,”
Struct. Multidiscip. Optim.
1615-147X,
14
(
2–3
), pp.
81
90
.
7.
Frangopol
,
D.
, and
Maute
,
K.
, 2003, “
Life-Cycle Reliability-Based Optimization of Civil and Aerospace Structures
,”
Comput. Struct.
0045-7949,
81
, pp.
397
410
.
8.
Allen
,
M.
,
Raulli
,
M.
, and
Maute
,
K.
, 2004, “
Reliability Based Analysis and Design Optimization of Electrostatically Actuated MEMS
,”
Comput. Struct.
0045-7949,
82
(
13–14
), pp.
1007
1020
.
9.
Meral
,
F.
, and
Basdogan
,
I.
, 2007, “
Design Methodology for Microelectromechanical Systems. Case Study: Torsional Scanner Mirror
,”
ASME J. Mech. Des.
0161-8458,
129
(
10
), pp.
1023
1030
.
10.
Agarwal
,
N.
, and
Aluru
,
N. R.
, 2008, “
Stochastic Modeling of Coupled Electromechanical Interaction for Uncertainty Quantification in Electrostatically Actuated Mems
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
197
(
43–44
), pp.
3456
3471
.
11.
Abdalla
,
M.
,
Reddy
,
C.
,
Faris
,
W.
, and
Gürdal
,
Z.
, 2005, “
Optimal Design of an Electrostatically Actuated Microbeam for Maximum Pull-In Voltage
,”
Comput. Struct.
0045-7949,
83
, pp.
1320
1329
.
12.
Bochobza-Degani
,
O.
, and
Nemirovsky
,
Y.
, 2004, “
Experimental Verification of a Design Methodology for Torsion Actuators Based on a Rapid Pull-In Solver
,”
J. Microelectromech. Syst.
1057-7157,
13
(
1
), pp.
121
130
.
13.
Younis
,
M.
,
Abdel-Rahman
,
E.
, and
Nayfeh
,
A.
, 2003, “
A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS
,”
J. Microelectromech. Syst.
1057-7157,
12
(
5
), pp.
672
680
.
14.
Chen
,
K. -S.
,
Ou
,
K. -S.
, and
Li
,
L. -M.
, 2004, “
Development and Accuracy Assessment of Simplified Electromechanical Coupling Solvers for MEMS Applications
,”
J. Micromech. Microeng.
0960-1317,
14
, pp.
159
169
.
15.
Senturia
,
S.
,
Harris
,
R.
,
Johnson
,
B.
,
Kim
,
S.
,
Nabors
,
K.
,
Shulman
,
M.
, and
White
,
J.
, 1992, “
A Computer-Aided Design System for Microelectromechanical Systems (MEMCAD)
,”
J. Microelectromech. Syst.
1057-7157,
1
(
1
), pp.
3
13
.
16.
Mukherjee
,
S.
,
Bao
,
Z.
,
Roman
,
M.
, and
Aubry
,
N.
, 2005, “
Nonlinear Mechanics of MEMS Plates With a Total Lagrangian Approach
,”
Comput. Struct.
0045-7949,
83
, pp.
758
768
.
17.
Zhulin
,
V.
,
Owen
,
S.
, and
Ostergard
,
D.
, 2000, “
Finite Element Based Electrostatic-Structural Coupled Analysis With Automated Mesh Morphing
,”
Technical Proceedings of the 2000 International Conference on Modeling and Simulation of Microsystems
,
Computational
,
Boston
, pp.
501
504
.
18.
Rochus
,
V.
,
Rixen
,
D.
, and
Golinval
,
J. -C.
, 2006, “
Monolithical Modeling of Electro-Mechanical Coupling in Micro-Structures
,”
Int. J. Numer. Methods Eng.
0029-5981,
65
(
4
), pp.
461
493
.
19.
Shaul
,
D.
, and
Sumner
,
T.
, 2004, “
Estimating Accuracy of Electrostatic Finite Element Models
,”
Commun. Numer. Methods Eng.
1069-8299,
20
, pp.
313
321
.
20.
Farhat
,
C.
,
Degand
,
C.
,
Koobus
,
B.
, and
Lesoinne
,
M.
, 1998, “
Torsional Springs for Two-Dimensional Dynamic Unstructured Fluid Meshes
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
163
, pp.
231
245
.
21.
Johnson
,
A.
, and
Tezduyar
,
T.
, 1994, “
Mesh Update Strategies in Parallel Finite Element Computations of Flow Problems With Moving Boundaries and Interfaces
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
119
, pp.
73
94
.
22.
Farhat
,
C.
,
Lesoinne
,
M.
, and
LeTallec
,
P.
, 1998, “
Load and Motion Transfer Algorithms for Fluid/Structure Interaction Problems With Non-Matching Discrete Interfaces: Momentum and Energy Conservation, Optimal Discretization and Application to Aeroelasticity
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
157
, pp.
95
114
.
23.
Maute
,
K.
,
Lesoinne
,
M.
, and
Farhat
,
C.
, 2000, “
Optimization of Aeroelastic Systems Using Coupled Analytical Sensitivities
,”
38th Aerospace Science Meeting and Exhibit
, Reno, NV, Jan. 10–13, Paper No. AIAA 2000-0560.
24.
Enevoldsen
,
I.
, and
Sorensen
,
J.
, 1994, “
Reliability-Based Optimization as an Information Tool
,”
Mech. Struct. Mach.
0890-5452,
22
(
1
), pp.
117
135
.
25.
Yu
,
X.
,
Choi
,
K.
, and
Chang
,
K.
, 1997, “
Reliability and Durability Based Design Sensitivity Analysis and Optimization
,” Center for Computer Aided Design, University of Iowa, Technical Report No. R97-01.
26.
Yu
,
X.
,
Chang
,
K.
, and
Choi
,
K.
, 1998, “
Probabilistic Structural Durability Prediction
,”
AIAA J.
0001-1452,
36
(
4
), pp.
628
637
.
27.
Grandhi
,
R.
, and
Wang
,
L.
, 1998, “
Reliability-Based Structural Optimization Using Improved Two-Point Adaptive Nonlinear Approximations
,”
Finite Elem. Anal. Des.
,
29
, pp.
35
48
. 0168-874X
28.
Luo
,
X.
, and
Grandhi
,
R. V.
, 1997, “
Astros for Reliability-Based Multidisciplinary Structural Analysis and Optimization
,”
Comput. Struct.
0045-7949,
62
, pp.
737
745
.
29.
Reh
,
S.
,
Lethbridge
,
P.
, and
Ostergaard
,
D.
, 2000, “
Quality Based Design and Design for Reliability of Micro Electro Mechanical Systems (MEMS) Using Probabilistic Methods
,”
2000 International Conference on Modeling and Simulation of Microsystems
, San Diego, CA.
30.
Ren
,
H.
,
Jog
,
A.
, and
Fair
,
R.
, 2001, “
Statistical Optimal Design of Microelectromechanical Systems (Mems)
,”
2001 International Conference on Modeling and Simulation of Microsystems, MSM
, San Diego, CA.
31.
Mawardi
,
A.
, and
Pitchumanid
,
R.
, 2005, “
Design of Microresonators Under Uncertainty
,”
J. Microelectromech. Syst.
1057-7157,
14
(
1
), pp.
63
69
.
32.
Wittwer
,
J.
,
Baker
,
M.
, and
Howell
,
L.
, 2006, “
Robust Design and Model Validation of Nonlinear Compliant Micromechanisms
,”
J. Microelectromech. Syst.
1057-7157,
15
(
1
), pp.
33
41
.
33.
Lepage
,
S.
,
Weickum
,
G.
, and
Maute
,
K.
, 2008, “
Stochastic Finite Element Modeling of Thermoelastic Quality Factor of Micro-Beams
,”
EuroSimE 2008, International Conference on Thermal, Mechanical and Multi-Physics Simulation and Experiments in Microelectronics and Micro-Systems
.
34.
Kahn
,
H.
, 1956, “
Use of Different Monte Carlo Sampling Techniques
,”
Proceedings of the Symposium on Monte Carlo Methods
,
H. A.
Meyer
, ed.,
Wiley
,
New York
.
35.
Shinozuka
,
M.
, 1983, “
Basic Analysis of Structural Safety
,”
J. Struct. Eng.
0733-9445,
109
(
3
), pp.
721
740
.
36.
Wu
,
Y. -T.
,
Millwater
,
H.
, and
Cruse
,
T.
, 1990, “
Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions
,”
AIAA J.
0001-1452,
12
, pp.
255
276
.
37.
Melchers
,
R.
, 1990, “
Radial Importance Sampling for Structural Reliability
,”
J. Eng. Mech.
0733-9399,
116
(
1
), pp.
189
203
.
38.
Madsen
,
H.
,
Krenk
,
S.
, and
Lind
,
N.
, 1986,
Methods of Structural Safety
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
39.
Lemaire
,
E.
,
Duysinx
,
P.
,
Rochus
,
V.
, and
Golinval
,
J. -C.
, 2006, “
Improvement of Pull-In Voltage of Electromechanical Microbeams Using Topology Optimization
,”
Third European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering
, Lisbon, Portugal, Jun.
40.
Chen
,
W.
, and
Du
,
X.
, 2002, “
Sequential Optimization and Reliability Assessment Method for Efficient Probabilistic Design
,”
Proceedings of the ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2002
, Montreal, Canada.
41.
Wang
,
L.
, and
Kodiyalam
,
S.
, 2002, “
An Efficient Method for Probabilistic and Robust Design With Non-Normal Distribution
,”
Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference
, Denver, CO, Apr. 22–25.
42.
Kuschel
,
N.
, and
Rackwitz
,
R.
, 2000, “
A New Approach for Structural Optimization of Series Systems
,”
Proc. of the ICASP8 Conf.
, Sidney, 12–15 Dec., 1999, R. E. Melchers and M. G. Stewart, eds., Balkema, Rotterdam, 2000, Vol.
2
, pp.
987
994
.
43.
Kharmanda
,
G.
,
Mohamed
,
A.
, and
Lemaire
,
M.
, 2002, “
Efficient Reliability-Based Design Optimization Using a Hybrid Space With Application to Finite Element Analysis
,”
Struct. Multidiscip. Optim.
1615-147X,
24
, pp.
233
245
.
44.
Rong
,
H.
,
Huang
,
Q.
,
Nie
,
M.
, and
Li
,
W.
, 2004, “
An Analytical Model for Pull-In Voltage of Clamped-Clamped Multilayer Beams
,”
Sens. Actuators, A
0924-4247,
116
, pp.
15
21
.
45.
Chowdhury
,
S.
,
Ahmadi
,
M.
, and
Miller
,
W. C.
, 2005, “
A Closed-Form Model for the Pull-In Voltage of Electrostatically Actuated Cantilever Beams
,”
J. Micromech. Microeng.
0960-1317,
15
(
4
), pp.
756
763
.
46.
Long
,
D.
,
Shannon
,
M.
, and
Aluru
,
N.
, 2000, “
A Novel Approach for Determining Pull-In Voltages in Micro-Electro-Mechanical Systems (MEMS)
,”
International Conference on Modeling and Simulation of Microsystems
, pp.
481
484
.
47.
Raulli
,
M.
, 2004. “
Optimal Design of Electrostatically Actuated Microsystems
,” Ph.D. thesis, University of Colorado at Boulder, Boulder, CO.
48.
Senturia
,
S.
, 2001,
Microsystem Design
,
Kluwer
,
The Netherlands
.
49.
Gupta
,
R.
, 1997, “
Electrostatic Pull-In Test Structure Design for In-Situ Mechanical Property Measurements of Microelectromechanical Systems (MEMS)
,” Ph.D. thesis, Massachusetts Institute of Technology, MA.
50.
Militello
,
C.
, and
Felippa
,
C.
, 1991, “
The First Andes Elements: 9-Dof Plate Bending Triangles
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
93
, pp.
217
246
.
51.
Young
,
J.
, 2001, “
Design, Modeling, and Experiment on Two Degree-of-Freedom MEMS Micromirrors
,” MS thesis, University of California, Irvine.
52.
Samareh
,
J.
, 1999, “
A Survey of Shape Parameterization Techniques
,”
Proceedings of the CEAS/AIAA/ICASE/NASA Langley International Forum on Aeroelasticity and Structural Dynamics
, Williamsburg, VA, Jun. 22–25.
53.
Raulli
,
M.
, and
Maute
,
K.
, 2002, “
Symbolic Geometric Modeling and Parameterization for Multiphysics Shape Optimization
,”
Ninth AIAA/ISSMO Conference on Multidisciplinary Analysis and Optimization
.
54.
Schittkowski
,
K.
, 1985, “
NLPQL: A FORTRAN Subroutine for Solving Constrained Nonlinear Programming Problems
,”
Ann. Operat. Res.
0254-5330,
5
, pp.
485
500
.
You do not currently have access to this content.