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

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Introduction

Microsystems or Micro-Electro-Mechanical Systems.

Typical dimensions 1~100 micrometers

Microsystems

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Introduction

At these small scales physical forces act different.

For instances electrostatic forces can deform and move things.

Electro-mechanical coupling

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IntroductionPull-in voltage

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IntroductionFinite element model

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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

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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

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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

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Contents

•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis

•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis

•Conclusions

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Static pull-inFEM model

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Static pull-in

Pull-in resembles limit pointbuckling, therefore the classic

limitpoint buckling sensitivity can

beused:

Sensitivities

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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

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Static pull-in

In that case MC gives

Stochastic analysis

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Static pull-in

And perturbation FEM gives:

Which is almost the same.

Stochastic analysis

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Contents

•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis

•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis

•Conclusions

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Dynamic pull-inFEM model

Step load of 40 Volt Step load of 41 Volt

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Dynamic pull-inFEM model

The transition is rather sharp

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Dynamic pull-inSensitivities

There is problem, mathematically it is difficult to define pull-in. However there is a work around.

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Dynamic pull-inUncertainty analysis

Monte Carlo Perturbation approach

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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

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Contents

•Stochastic Finite Element Method•Static pull-in• FEM computation• Sensitivities• Stochastic analysis

•Dynamic pull-in• FEM computation • Sensitivities • Stochastic analysis

•Conclusions

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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.

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Thank you for your attention