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Stochastic FEM for analyzing static and dynamic pull-in of microsystems

Stochastic FEM for analyzing static and dynamic pull-in of microsystems. Stephan Hannot, Clemens Verhoosel and Daniel Rixen. Introduction. Microsystems or Micro-Electro-Mechanical Systems. Typical dimensions 1~100 micrometers. Microsystems. Introduction.

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Stochastic FEM for analyzing static and dynamic pull-in of microsystems

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  1. Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens Verhoosel and Daniel Rixen

  2. Introduction Microsystems or Micro-Electro-Mechanical Systems. Typical dimensions 1~100 micrometers Microsystems

  3. Introduction At these small scales physical forces act different. For instances electrostatic forces can deform and move things. Electro-mechanical coupling

  4. Introduction Pull-in voltage

  5. Introduction Finite element model

  6. Contents • Stochastic Finite Element Method • Static pull-in • FEM computation • Sensitivities • Stochastic analysis • Dynamic pull-in • FEM computation • Sensitivities • Stochastic analysis • Conclusions

  7. Stochastic FEM Problem definition A material property is not fixed, definitely at the microscale it can be an highly uncertain value. • For instance in the 1D example • Assume kis random, but normally distributed. • What happens to the pull-in voltage?

  8. Stochastic FEM Generate Ndifferent 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

  9. 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 FEM Perturbation Stochastic FEM

  10. Contents • Stochastic Finite Element Method • Static pull-in • FEM computation • Sensitivities • Stochastic analysis • Dynamic pull-in • FEM computation • Sensitivities • Stochastic analysis • Conclusions

  11. Static pull-in FEM model

  12. Static pull-in Pull-in resembles limit point buckling, therefore the classic limit point buckling sensitivity can be used: Sensitivities

  13. 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

  14. Static pull-in In that case MC gives Stochastic analysis

  15. Static pull-in And perturbation FEM gives: Which is almost the same. Stochastic analysis

  16. Contents • Stochastic Finite Element Method • Static pull-in • FEM computation • Sensitivities • Stochastic analysis • Dynamic pull-in • FEM computation • Sensitivities • Stochastic analysis • Conclusions

  17. Dynamic pull-in FEM model Step load of 40 Volt Step load of 41 Volt

  18. Dynamic pull-in FEM model The transition is rather sharp

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

  20. Dynamic pull-in Uncertainty analysis Monte Carlo Perturbation approach

  21. Dynamic pull-in Reliability analysis What is the chance that the dynamic pull-in is below a critical value of V=37 Monte Carlo Perturbation approach Compute critical Ec, V(Ec)=37 What is the chance that E<Ec

  22. Contents • Stochastic Finite Element Method • Static pull-in • FEM computation • Sensitivities • Stochastic analysis • Dynamic pull-in • FEM computation • Sensitivities • Stochastic analysis • Conclusions

  23. 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.

  24. Thank you for your attention

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