Optimal Design of Electrostatically Actuated Microsystems

209 Pages·2005·4.82 MB·English

Optimal Design of Electrostatically Actuated
Microsystems
by
Michael Raulli
Bachelor of Science, Villanova University, 2000
Master of Science, University of Colorado, 2002
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Department of Aerospace Engineering Sciences
2005
This thesis entitled:
Optimal Design of Electrostatically Actuated Microsystems
written by Michael Raulli
has been approved for the Department of Aerospace Engineering Sciences
Kurt Maute
Martin Dunn
Date
The final copy of this thesis has been examined by the signatories, and we find that
both the content and the form meet acceptable presentation standards of scholarly
work in the above mentioned discipline.
Raulli, Michael (Ph.D., Aerospace Engineering Sciences)
Optimal Design of Electrostatically Actuated Microsystems
Thesis directed by Dr. Kurt Maute
This thesis addresses the application of numerical optimization techniques to
devices classified as Micro-Electro-Mechanical-Systems (MEMS). The finite element
(FEM) based analysis and sensitivity analysis capabilities for the fully-coupled electro-
mechanicalproblem,whichresultsfromelectrostaticallyactuatedMEMS,aredeveloped
and implemented. Specifically, shape and topology optimization, using gradient based
algorithms, are applied to the design of MEMS. Topology optimization generates opti-
mal topology without the requirement of close-to-optimal initial designs. It is proposed
toextendthecurrentstateoftheartintopologyoptimizationtoafully-coupledelectro-
mechanical problem, in which the electrostatic and structural domains occupy distinct
regionsofspace.Thedifficultyinallowingtheconductinginterfacebetweenthedomains
to freely evolve during the topology optimization process is addressed. Additionally, the
prediction of the instability phenomena known as ‘pull-in’, by an eigenvalue analysis, is
incorporatedintotheproposeddesignmethodology.Thedesignoptimizationofrealistic
MEMSdevicesisthedrivingforcebehindtheapplicationofalldevelopedcomputational
tools.
Acknowledgements
I would like to acknowledge, first and foremost, my advisor Kurt Maute for the
extensive amount of time, effort and expertise he contributed to this thesis. Also, my
committee members: Martin Dunn, Carlos Felippa, Jim Allen and Victor Bright, for
their insight. I would also like to acknowledge the support of the Center for Aerospace
Structures.SpecialthanksaregiventoRajeshPoolaSubranmanyaswamyforgenerating
the computational meshes used in the example problem of section 5.2.2, and studying
the behavior of the device. Special thanks are given to Wissam Houchaime for his work
on the formulation of the stability eigenvalue problem. Most importantly though, the
support of my wife, Corrie, made this thesis possible.
Contents
Chapter
1 Introduction 1
1.1 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Original Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Manuscript Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Computational Framework for Coupled Optimization Problem 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Three Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Electro-mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Electro-mechanical State Equations . . . . . . . . . . . . . . . . 15
2.3.2 Discretization of the Electrostatic Field . . . . . . . . . . . . . . 18
2.3.3 Solution of coupled Electro-mechanical System . . . . . . . . . . 20
2.4 Electro-mechanical Sensitivity Analysis . . . . . . . . . . . . . . . . . . 24
2.4.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Adjoint Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Evaluation of Off-diaganol Jacobian Terms . . . . . . . . . . . . 36
2.4.4 Evaluation of Residual Partial Derivatives . . . . . . . . . . . . . 41
2.4.5 Evaluation of Criteria Partial Derivatives . . . . . . . . . . . . . 46
2.5 Stability Analysis of Electro-mechanical systems . . . . . . . . . . . . . 47
vi
3 Topology Optimization, Theory and Applications 53
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Topology Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Classical Problem Definition . . . . . . . . . . . . . . . . . . . . 54
3.2.2 Ill-posedness and Regularization . . . . . . . . . . . . . . . . . . 56
3.3 Active Research in Topology Optimization . . . . . . . . . . . . . . . . . 60
3.3.1 Multiphysics in Topology Optimization . . . . . . . . . . . . . . 61
3.3.2 Design Dependent Loads in Topology Optimization . . . . . . . . 63
3.4 Methodology for Electro-mechanical Topology Optimization . . . . . . . 65
3.4.1 Changing Electrostatic Domain . . . . . . . . . . . . . . . . . . . 69
3.4.2 Changing Voltage Boundary Conditions . . . . . . . . . . . . . . 73
3.4.3 Changing Electrostatic Force . . . . . . . . . . . . . . . . . . . . 76
4 Verification of Methodology and Implementation 80
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Verification of Electrostatic Element Implementation . . . . . . . . . . . 81
4.3 Verification of Coupled Analysis Algorithm . . . . . . . . . . . . . . . . 85
4.3.1 Analytical Verification . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.2 Numerical Verification . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.3 Experimental Verification . . . . . . . . . . . . . . . . . . . . . . 89
4.4 Verification of Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . 94
4.5 Verification of Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . 97
4.6 Verification of Soft-Voltage Boundary Conditions . . . . . . . . . . . . . 102
4.7 Verification of Topology Optimization Parameterization . . . . . . . . . 103
4.7.1 Two-dimensional Verification . . . . . . . . . . . . . . . . . . . . 104
4.7.2 Three-dimensional Verification . . . . . . . . . . . . . . . . . . . 108
vii
5 Numerical Results and Applications 111
5.1 Electro-mechanical Shape Optimization . . . . . . . . . . . . . . . . . . 111
5.1.1 Experimental Actuator Stress-Displacement Optimization . . . . 112
5.1.2 Experimental Actuator Mass-Stiffness Optimization . . . . . . . 115
5.1.3 Comb Drive Actuator Shape Optimization . . . . . . . . . . . . . 115
5.2 Electro-mechanical Reliability Based Design Optimization . . . . . . . . 120
5.2.1 Experimental Actuator RBDO Problem . . . . . . . . . . . . . . 124
5.2.2 Torsional Mirror RBDO Problem . . . . . . . . . . . . . . . . . . 128
5.3 Electrostatic-Fluid-Structure Shape Optimization . . . . . . . . . . . . . 134
5.3.1 Electrostatic-Fluid-Structure Methodology. . . . . . . . . . . . . 135
5.3.2 Electrostatically Actuated Channel Optimization . . . . . . . . . 141
5.3.3 Electrostatically Actuated Aeroelastic Wing Optimization . . . . 145
5.4 Electro-mechanical Topology Optimization. . . . . . . . . . . . . . . . . 149
5.4.1 Two-dimensional Membrane Mass-Stiffness Topology Optimization151
5.4.2 Three-dimensional Plate Mass-Stiffness Topology Optimization . 160
5.4.3 Two-dimensional Membrane Force Inverter Topology Optimization168
5.4.4 Three-dimensional Plate Force Inverter Topology Optimization . 173
6 Summary 180
6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Bibliography 184
Appendix
A Notation Summary 191
Tables
Table
2.1 Optimization criteria used in thesis . . . . . . . . . . . . . . . . . . . . . 12
2.2 Electro-mechanical solution algorithm . . . . . . . . . . . . . . . . . . . 23
2.3 Direct method algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Adjoint method algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Power method algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1 Parameters for test problem of figure 4.1 . . . . . . . . . . . . . . . . . . 81
4.2 Results for 2-d test problem of figure 4.1 . . . . . . . . . . . . . . . . . . 82
4.3 Results for 3-d test problem of figure 4.1 . . . . . . . . . . . . . . . . . . 84
4.4 Mesh details for cantilever beam numerical comparison . . . . . . . . . . 88
4.5 Electrostatic and structural properties for cantilever beam . . . . . . . . 88
4.6 Electrostatic and structural properties for two-dimensional plate example 92
4.7 Mesh details for experimental comparison . . . . . . . . . . . . . . . . . 93
4.8 Parameters for soft voltage test problem . . . . . . . . . . . . . . . . . . 102
4.9 Agreement between strict and soft voltage for different weighting factors 103
4.10 Electrostatic and structural properties for verification of 2-d topology
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.11 Numerical results from 2-d exact and topology models . . . . . . . . . . 107
4.12 Electrostatic and structural properties for verification of 3-d topology
approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
ix
4.13 Numerical results from 3-d exact and topology models . . . . . . . . . . 110
5.1 Experimental actuator stress-displacement optimization summary . . . . 113
5.2 Experimental actuator mass-stiffness optimization summary . . . . . . . 115
5.3 Electrostatic and structural properties for comb drive example . . . . . 117
5.4 Summary of computational problem for comb drive example . . . . . . . 118
5.5 Summary of optimization problem and results for comb drive example . 120
5.6 Random variables for RBDO . . . . . . . . . . . . . . . . . . . . . . . . 128
5.7 Electrostatic and structural properties for mirror example . . . . . . . . 131
5.8 Summary of computational problem for mirror example . . . . . . . . . 131
5.9 Summary of optimization problem and results for comb drive example . 132
5.10 Random variables for RBDO . . . . . . . . . . . . . . . . . . . . . . . . 133
5.11 Summary of mirror RBDO . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.12 Algorithm for electro-fluid-elastic analysis . . . . . . . . . . . . . . . . . 138
5.13 Free stream conditions, electrostatic parameters, and structural proper-
ties of channel problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.14 Discretization of fluid, electrostatic, and structural domains of channel
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.15 Results of optimization for channel problem . . . . . . . . . . . . . . . . 145
5.16 Free stream conditions, electrostatic parameters, and structural proper-
ties; wing example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.17 Discretization of fluid, electrostatic, and structural domains; wing example148
5.18 Optimization results for micro-wing . . . . . . . . . . . . . . . . . . . . 149
5.19 Electrostaticandstructuralpropertiesfortwo-dimensionalmembraneex-
ample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.20 Summaryofcomputationalproblemfortwo-dimensionalmembrane,cases
1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
x
5.21 Summary of computational problem for two-dimensional membrane ex-
ample, case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.22 Summary of optimization problem and results for two-dimensional mem-
brane example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.23 Electrostatic and structural properties for three-dimensional plate example162
5.24 Summaryofcomputationalproblemforthree-dimensionalplateexample,
cases 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.25 Summaryofoptimizationproblemandresultsforthree-dimensionalplate
example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.26 Summary of optimization problem and results for two-dimensional inver-
sion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.27 Summary of optimization problem and results for three-dimensional in-
version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177