18-04-23
Challenge the future
DelftUniversity ofTechnology
Stochastic FEM for analyzing static and dynamic pull-in of microsystems
Stephan Hannot, Clemens Verhoosel and Daniel Rixen
2Stochastic FEM for analyzing pull-in
Introduction
Microsystems or Micro-Electro-Mechanical Systems.
Typical dimensions 1~100 micrometers
Microsystems
3Stochastic FEM for analyzing pull-in
Introduction
At these small scales physical forces act different.
For instances electrostatic forces can deform and move things.
Electro-mechanical coupling
4Stochastic FEM for analyzing pull-in
IntroductionPull-in voltage
5Stochastic FEM for analyzing pull-in
IntroductionFinite element model
6Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
7Stochastic FEM for analyzing pull-in
Stochastic FEM
A material property is not fixed, definitely at the
microscale it can be an highly uncertain value.
•For instance in the 1D example• Assume k is random, but normally
distributed. • What happens to the pull-in voltage?
Problem definition
8Stochastic FEM for analyzing pull-in
Stochastic FEM
Generate N different values of k and compute N
pull-in voltages, subsequently determine the
distribution of the pull-in voltages.
•Advantages• Conceptually simple• Very robust
•Disadvantage• Computationally very expensive
Crude Monte Carlo Simulation
9Stochastic FEM for analyzing pull-in
Compute the sensitivities of V with respect to k, and use these to approximate the distribution.
•Advantages• Computationally very cheap
•Disadvantage• Design sensitivities required• Only information about mean and variance
Stochastic FEMPerturbation Stochastic FEM
10Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
11Stochastic FEM for analyzing pull-in
Static pull-inFEM model
12Stochastic FEM for analyzing pull-in
Static pull-in
Pull-in resembles limit pointbuckling, therefore the classic
limitpoint buckling sensitivity can
beused:
Sensitivities
13Stochastic FEM for analyzing pull-in
Static pull-in
The perturbation FEM will be compared with crude Monte
Carlo.
It is assumed that the Young’s modulus of material is distributed normally with the following characteristics:
Stochastic analysis
14Stochastic FEM for analyzing pull-in
Static pull-in
In that case MC gives
Stochastic analysis
15Stochastic FEM for analyzing pull-in
Static pull-in
And perturbation FEM gives:
Which is almost the same.
Stochastic analysis
16Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
17Stochastic FEM for analyzing pull-in
Dynamic pull-inFEM model
Step load of 40 Volt Step load of 41 Volt
18Stochastic FEM for analyzing pull-in
Dynamic pull-inFEM model
The transition is rather sharp
19Stochastic FEM for analyzing pull-in
Dynamic pull-inSensitivities
There is problem, mathematically it is difficult to define pull-in. However there is a work around.
20Stochastic FEM for analyzing pull-in
Dynamic pull-inUncertainty analysis
Monte Carlo Perturbation approach
21Stochastic FEM for analyzing pull-in
Dynamic pull-inReliability analysis
Monte Carlo Perturbation approach
What is the chance that the dynamic pull-in is below a critical value of V=37
Compute critical Ec, V(Ec)=37
What is the chance that E<Ec
22Stochastic FEM for analyzing pull-in
Contents
•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis
•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis
•Conclusions
23Stochastic FEM for analyzing pull-in
Conclusions
•Analytical sensitivities of Static and dynamic pull-in were derived.
•These sensitivities are sufficient for performing a Stochastic analysis.
•A more robust definition of dynamic pull-in would be nice for a more robust sensitivity computation.
24Stochastic FEM for analyzing pull-in
Thank you for your attention