Uncertainty-Based Design Optimization of MEMS/NEMS

  • Holger Neubert


In designing micro-electromechanical systems (MEMS), model-based design and design optimization is inevitable. This is due to the complexity of the working principles and the manufacturing technology associated with high initial costs for experimental investigations. Uncertainty-based design optimization enables stochastic variables, such as scattering material properties, manufacturing tolerances, stochastic time dependent loads, aging, or wear, to be considered in the design. In this chapter, the fundamental concept of reliability or failure probability is introduced and probabilistic approximation and simulation methods are described. These methods are used for robust design optimization (RDO) and reliability-based design optimization (RBDO) as well. Applications and examples for of both approaches are also provided.


Monte Carlo Simulation Failure Probability Response Surface Method Epistemic Uncertainty Limit State Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Adams, B.M., Eldred, M.S., Wittwer, J.W.: Reliability-based design optimization for shape design of compliant micro-electro-mechanical systems. In: 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia 6–8 Sep 2006Google Scholar
  2. 2.
    Agarwal, N., Aluru, N.R.: Stochastic analysis of electrostatic MEMS subjected to parameter variations. J. Microelectromech. Syst. 18(6), 1454–1468 (2009)CrossRefGoogle Scholar
  3. 3.
    Allen, M., Raulli, M., Maute, K., Frangopol, D.M.: Reliability-based analysis and design optimization of electrostatically actuated MEMS. Comput. Struct. 82, 1007–1020 (2004)CrossRefGoogle Scholar
  4. 4.
    Ang, A.H.S., Tang, W.: Probability Concepts in Engineering Planning and Design, vol. II Decision, Risk, and Reliability. Wiley, New York (1984)Google Scholar
  5. 5.
    Beyer, H.G., Sendhoff, B.: Robust optimization—a comprehensive survey. Comput. Method Appl. M. 196, 3190–3218 (2007)Google Scholar
  6. 6.
    Bucher, C.: Computational Analysis of Randomness in Structural Mechanics, Structures and Infrastructures Book Series, vol. 3. Taylor& Francis Ltd, Leiden (2009)Google Scholar
  7. 7.
    Bucher, C.G.: Adaptive sampling—an iterative fast Monte Carlo procedure. Struct. Saf. 5(2), 119–126 (1988)CrossRefGoogle Scholar
  8. 8.
    Canumalla, S.: Robust design of third level packaging in portable electronics: solder joint reliability under dynamic mechanical loading. In: 58th Electronic Components and Technology Conference ECTC 2008, May 27–30 2008Google Scholar
  9. 9.
    Chase, K.W., Parkinson, A.R.: A survey of research in the application of tolerance analysis to the design of mechanical assemblies. Res. Eng. Des. 3, 23–37 (1991)CrossRefGoogle Scholar
  10. 10.
    Cornell, C.A.: A probability-based structural code. J. Am. Concr. Inst. 66, 974–985 (1969)Google Scholar
  11. 11.
    Dehnad, K. (ed.): Quality Control, Robust Design, and the Taguchi Method. Wadsworth& Broos/Cole Advanced Books& Software Pacific Grove, California (1989)Google Scholar
  12. 12.
    Eldred, M.S., Adams, B.M., Copps, K.D., Carnes, B., Notz, P.K., Hopkins, M.M., Wittwer, J.W.: Solution-verified reliability analysis and design of compliant micro-electro-mechanical systems. In: 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, April 23–26 2007Google Scholar
  13. 13.
    Fiessler, B., Rackwitz, R., Neumann, H.J.: Quadratic limit states in structural reliability. J. Eng. Mech. Div. 105(4), 661–676 (1979)Google Scholar
  14. 14.
    Gurav, S., Kasyap, A., Sheplak, M., Cattafesta, L., Haftka, R., Goosen, J., van Keulen, F.: Uncertainty-based design optimization of a micro piezoelectric composite energy reclamation device. In: 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, Aug 30–1, 2004Google Scholar
  15. 15.
    Gurav, S.P., Goosen, J.F.L., vanKeulen, F.: Bounded-but-unknown uncertainty optimization using design sensitivities and parallel computing: application to MEMS. Comput. Struct. 83, 1134–1149 (2005)Google Scholar
  16. 16.
    Haldar, A., Mahadevan, S. (eds.): Probability, Reliability and Statistical Methods in Engineering Design. John Wiley& Sons, New York (2000)Google Scholar
  17. 17.
    Hansen, L.P.: Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054 (1982)CrossRefGoogle Scholar
  18. 18.
    Harik, V.M., Luo, L.S. (eds.): Micromechanics and Nanoscale Effects. Kluwer Academic Publishers, Amsterdam (2004)Google Scholar
  19. 19.
    Hsiung, K.L.: Design of microelectromechanical systems for variability via chance-constrained optimization. J. Phys. Conf. Ser. 34, 162–167 (2006). International MEMS Conference 2006Google Scholar
  20. 20.
    Huang, B., Du, X.: Analytical robustness assessment for robust design. Struct. Multidiscip. Optim. 34, 123–137 (2007)Google Scholar
  21. 21.
    Jung, H.S., Cho, S.: Reliability-based topology optimization of geometrically nonlinear structures with loading and material uncertainties. Finite Elem. Anal. Des. 41(3), 311–331 (2004)CrossRefGoogle Scholar
  22. 22.
    Jurecka, F.: Robust design optimization based on metamodeling techniques. Ph.D. thesis, Fakultät für Bauingenieur und Vermessungswesen der Technischen Universität München (2007)Google Scholar
  23. 23.
    Karpat, F., Ekwaro-Osire, S., Khandaker, M.P.H.: Probabilistic analysis of MEMS asymmetric gear tooth. J. Mech. Des.130, 042306-1–042306-6 (2008)Google Scholar
  24. 24.
    Katzan, H.j.: Managing Uncertainty: A Pragmatic Approach. Van Nostrand Reinhold, New York (1992)Google Scholar
  25. 25.
    Khandaker, M., Ekwaro-Osire, S.: Sensitivity based optimum design process for MEMS devices. In: 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, Aug. 30–1, 2004Google Scholar
  26. 26.
    Kim, C., Choi, K.K.: Reliability-based design optimization using response surface method with prediction interval estimation. J. Mech. Des. 130, 121401–121412 (2008)Google Scholar
  27. 27.
    Kim, C., Wang, S., Hwang, I., Lee, J.: Parallel computed reliability-based topology optimization using response surface method. In: 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, 30 May–03 June 2005Google Scholar
  28. 28.
    Kim, C., Wang, S., Hwang, I., Lee, J.: Application of reliability-based topology optimization for microelectromechanical systems. AIAA J. 45(12), 2926–2934 (2007)CrossRefGoogle Scholar
  29. 29.
    Kleppmann, W.: Taschenbuch Versuchsplanung, 6th edn. Hanser Verlag München Wien (2009)Google Scholar
  30. 30.
    Kong, J.S., Frangopol, D.M., Raulli, M., Maute, K., Saravanan, R.A., Liew, L.A., Raj, R.: A methodology for analyzing the variability in the performance of a MEMS actuator made from a novel ceramic. Sens. Actuators, A 116, 336–344 (2004)Google Scholar
  31. 31.
    Lee, O.S., Park, Y.C., Kim, D.H.: Reliability estimation of solder joints under thermal fatigue with varying parameters by using form and mcs. J. Mech. Sci. Technol. 22, 683–688 (2008)CrossRefGoogle Scholar
  32. 32.
    Leondes, C.T. (ed.): MEMS/NEMS Handbook Techniques and Applications, vol. 1 Design Methods. Springer Science+Business Media, New York (2006)Google Scholar
  33. 33.
    Liu, N., Manoochehri, S.: Reliability-based MEMS system modeling and optimization. In: Proceedings of the 44th Annual Reliability Physics Symposium, San Jose, March 26–30 2006. pp. 403–409, ISBN: 0-7803-9498-4Google Scholar
  34. 34.
    Liu, P.L., Kiureghian, A.: Multivariate distribution models with prescribed marginals and covariances. Probab. Eng. Mech. 1, 105–112 (1986)CrossRefGoogle Scholar
  35. 35.
    Luo, R.C., Lin, C.F.: A review of mechatronics and bio-inspired mechatronics systems. J. Biomechatron. Eng. 1, 1–36 (2008)Google Scholar
  36. 36.
    Maute, K., Frangopol, D.M.: Reliability-based design of MEMS mechanisms by topology optimization. Comput. Struct. 81, 813–824 (2003)CrossRefGoogle Scholar
  37. 37.
    Melchers, R.E.: Structural Reliability Analysis and Prediction, 2nd edn. John Wiley& Sons, New York (1999)Google Scholar
  38. 38.
    Neubert, H., Fleischer, D., Kamusella, A., Pham, T.Q.: Optimization of bipolar magnetic actuators for microvalves with regard to the tolerances. In: Proceeding of the 11th International Conference and Exhibition on New Actuators and Drive Systems ACTUATOR 2008, pp. 1038–1041, Bremen, June 9–11, (2008)Google Scholar
  39. 39.
    Neubert, H., Kamusella, A., Pham, T.Q.: Uncertainly design analysis and optimization of LTCC based accelerometers in COMSOL Multiphysics with OptiY. In: Proceeding of the European COMSOL Conference, Grenoble, Oct 22–23, 2007Google Scholar
  40. 40.
    Neubert, H., Partsch, U., Fleischer, D., Gruchow, M., Kamusella, A., Pham, T.Q.: Thick film accelerometers in LTCC-technology design optimization, fabrication, and characterization. JMEP 5(4), 150–155 (2008)Google Scholar
  41. 41.
    Nikolaidis, E., Ghiocel, D.M., Singhal, S. (eds.): Engineering Design Reliability Handbook. CRC Press LLC, New York (2005)Google Scholar
  42. 42.
    Oberkampf, W.L., DeLand, S.M., Rutherford, B.M., Diegert, K.V., Alvin, K.F.: Error and uncertainty in modeling and simulation. Reliab. Eng. Syst. Safe. 75, 333–357 (2002)CrossRefGoogle Scholar
  43. 43.
    Pfingsten, T., Herrmann, J., Rasmussen, C.E.: Model-based design analysis and yield optimization. IEEE Trans. Semiconduct. M. 19, 475–486 (2006)Google Scholar
  44. 44.
    Pham, T.Q., Kamusella, A.: Zuverlässigkeitsanalyse und zuverlässigkeitsbasierte Optimierung mit probabilistischen Methoden am Beispiel eines Magnetantriebes. In: VDI-Tagung Technische Zuverlässigkeit, Leonberg, April 29-30, (2009)Google Scholar
  45. 45.
    Pham, T.Q., Neubert, H., Kamusella, A.: Design for reliability and robustness through probabilistic methods in COMSOL Multiphysics with OptiY. In: Proceedings of the 2nd European COMSOL Conference, Hannover, Nov 4–6, (2008)Google Scholar
  46. 46.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T. (eds.): Numerical Recipes, 3rd edn. Cambridge University Press, New York (2007)Google Scholar
  47. 47.
    Rackwitz, R., Fiessler, B.: Structural reliability under combined random load sequences. Comput. Struct. 9, 489–494 (1978)CrossRefGoogle Scholar
  48. 48.
    Radhakrishnan, S., Subbarayan, G., Nguyen, L., Mazotte, W.: Optimization and stochastic procedures for robust design of photonic packages with applications to a generic package. In: Proceedings of the 53rd Electronic Components and Technology Conference, pp. 720–726, May 27–30 2003Google Scholar
  49. 49.
    Raulli, M., Maute, K.: Reliability based design optimization of MEMS considering pull-in. J. Mech. Des. 131, 061014-1–10 (2009)Google Scholar
  50. 50.
    Raulli, M., Subramanyaswamy, R.P., Maute, K.: Reliability based design optimization of analog micro-mirrors using pull-in criteria. In: The 9th International Conference on Structural Safety and Reliability - ICOSSAR 2005, June 19-23, Rome, pp. 2041–2048 (2005)Google Scholar
  51. 51.
    Roos, D.: Approximation und Interpolation von Grenzzustandsfunktionen zur Sicherheitsbewertung nichtlinearer Finite-Elemente-Strukturen. Ph.D. thesis, Bauhaus-Universität Weimar (2002)Google Scholar
  52. 52.
    Roos, D.: Advanced methods of stochastic and optimization in industrial applications. In: Proceedings of the 7th international Conference and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes, Interlaken, Sept 1–5 2008Google Scholar
  53. 53.
    Roos, D., Adam, U., Bayer, V.: Design reliability analysis. In: 24th CAD-FEM Users Meeting 2006 International Congress on FEM Technology, Stuttgart, Oct 26–27, 2006Google Scholar
  54. 54.
    Roos, D., Adam, U., Bucher, C.: Robust design optimization. In: Weimarer Optimierungs- und Stochastiktage 3.0, Weimar, Nov 23–24, 2006Google Scholar
  55. 55.
    van Roosmalen, A.J., Zhang, G.Q.: Reliability challenges in the nanoelectronics era. Microelectron. Reliab. 46, 1403–1414 (2006)CrossRefGoogle Scholar
  56. 56.
    Rubinstein, R.Y., Kroese, D.P.: Simulation and the Monte Carlo Method, 2nd edn. John Wiley& Sons, New York (2008)Google Scholar
  57. 57.
    Taguchi, G.: System of Experimental Design: Engineering Methods to Optimize Quality and Minimize Costs. Kraus International, New York (1987)Google Scholar
  58. 58.
    Tsompanakis, Y., Lagaros, N., Papadrakakis, M. (eds.): Structural Design Optimization Considering Uncertainties. Taylor& Francis, London (2008)Google Scholar
  59. 59.
    Wang, G.G., Dong, Z., Aitchison, P.: Adaptive response surface method—a global optimization scheme for approximation-based design problems. Eng. Optim. 33(6), 707–734 (2001)Google Scholar
  60. 60.
    Wittwer, J.W.: Simulation-based design under uncertainty for compliant microelectromechanical systems. Ph.D. thesis, Department of Mechanical Engineering, Brigham Young University Provo (2005)Google Scholar
  61. 61.
    Wu, Y.T.: Methods for efficient probabilistic analysis of models with large numbers of randamo variables. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. paper 98–4908, St. Louis, September 1998Google Scholar
  62. 62.
    Zang, T.A., Hemsch, M.J., Hilburger, M.W., Kenny, S.P., Luckring, J.M., Maghami, P., Padula, S.L., Stroud, W.J.: Needs and opportunities for uncertainty-based multidisciplinary design methods for aerospace vehicles. Technical Report Nr. TM-2002-211462, NASA (2002)Google Scholar
  63. 63.
    Zhao, Y.G., Ang, A.H.S.: System reliability assessment by method of moments. J. Struct. Eng. 129(10), 1341–1349 (2003)CrossRefGoogle Scholar
  64. 64.
    Zhao, Y.G., Ono, T.: Moment methods for structural reliability. Struct. Saf. 23, 47–75 (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Electromechanical and Electronic DesignTechnische Universität DresdenDresdenGermany

Personalised recommendations