1. 19-5-2015 Challenge the future Delft University of Technology Stochastic FEM for analyzing static and dynamic pull-in of microsystems Stephan Hannot, Clemens Verhoosel and Daniel Rixen

2. 2 Stochastic FEM for analyzing pull-in Introduction Microsystems or Micro-Electro-Mechanical Systems. Typical dimensions 1~100 micrometers Microsystems

3. 3 Stochastic 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

4. 4 Stochastic FEM for analyzing pull-in Introduction Pull-in voltage

5. 5 Stochastic FEM for analyzing pull-in Introduction Finite element model

6. 6 Stochastic 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

7. 7 Stochastic 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

8. 8 Stochastic 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

9. 9 Stochastic 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 FEM Perturbation Stochastic FEM

10. 10 Stochastic 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

11. 11 Stochastic FEM for analyzing pull-in Static pull-in FEM model

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

13. 13 Stochastic 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

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

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

16. 16 Stochastic 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

17. 17 Stochastic FEM for analyzing pull-in Dynamic pull-in FEM model Step load of 40 VoltStep load of 41 Volt

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

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

20. 20 Stochastic FEM for analyzing pull-in Dynamic pull-in Uncertainty analysis Monte CarloPerturbation approach

21. 21 Stochastic FEM for analyzing pull-in Dynamic pull-in Reliability analysis Monte CarloPerturbation approach What is the chance that the dynamic pull-in is below a critical value of V=37 Compute critical E c, V(E c )=37 What is the chance that E<E c

22. 22 Stochastic 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

23. 23 Stochastic 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.

24. 24 Stochastic FEM for analyzing pull-in Thank you for your attention