19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static...

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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

Introduction

At these small scales physical forces act different.

For instances electrostatic forces can deform and move things.

Electro-mechanical coupling

IntroductionPull-in voltage

IntroductionFinite element model

Contents

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

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

•Conclusions

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

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

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

Contents

•Conclusions

Static pull-inFEM model

Static pull-in

Pull-in resembles limit pointbuckling, therefore the classic

limitpoint buckling sensitivity can

beused:

Sensitivities

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

Static pull-in

In that case MC gives

Stochastic analysis

Static pull-in

And perturbation FEM gives:

Which is almost the same.

Stochastic analysis

Contents

•Conclusions

Dynamic pull-inFEM model

Step load of 40 Volt Step load of 41 Volt

Dynamic pull-inFEM model

The transition is rather sharp

Dynamic pull-inSensitivities

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

Dynamic pull-inUncertainty analysis

Monte Carlo Perturbation approach

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

Contents

•Conclusions

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.

Thank you for your attention

19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static...

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