Gradient Index Based Robust Optimal Design Method for MEMS Structures

구배 지수에 근거한 MEMS 구조물의 강건 최적 설계 기법

  • 한정삼 (한국과학기술원 기계기술연구소) ;
  • 곽병만 (한국과학기술원 기계공학과)
  • Published : 2003.07.01


In this paper we present a simple and efficient robust optimal design formulation for MEMS structures and its application to a resonant-type micro probe. The basic idea is to use the gradient index (GI) to improve robustness of the objective and constraint functions. In the robust optimal design procedure, a deterministic optimization for performance of MEMS structures is followed by design sensitivity analysis with respect to uncertainties such as fabrication errors and change of operating conditions. During the process of deterministic optimization and sensitivity analysis, dominant performance and uncertain variables are identified to define GI. The GI is incorporated as a term of objective and constraint functions in the robust optimal design formulation to make both performance and robustness improved. While most previous approaches for robust optimal design require statistical information on design variations, the proposed GI based method needs no such information and therefore is cost-effective and easily applicable to early design stages. For the micro probe example, robust optimums are obtained to satisfy the targets for the measurement sensitivity and they are compared in terms of robustness and production yield with the deterministic optimums through the Monte Carlo simulation. This method, although shown for MEMS structures, may as well be easily applied to conventional mechanical structures where information on uncertainties is lacking but robustness is highly important.


Gradient Index;Robust Optimal Design;Resonant-Type Micro Probe


  1. Muchow, J., Kretschmann, A., Henn, R., Skrobanek, K., Finkbeiner, S. and Krauss, H.R., 2002, 'Influence of Process Variation on the Functionality of a High Pressure Sensor,' DTIP of MEMS/MOEMS, Proc. SPIE, Vol. 4755, pp. 143-148
  2. Germer, C., Hansen, U., Franke, H.J. and Buttgenbach, S., 2002, 'Tolerance Analysis and Synthesis in Micro Systems,' DTIP of MEMS/MOEMS, Proc. SPIE, Vol. 4755, pp. 355-364
  3. Vanderplaats, G.N., 1984, 'Numerical Optimization Techniques for Engineering Design with Applications,' McGraw-Hill, New York
  4. DOT Users Manual version 4.20, 1995, VR&D
  5. Lebrasseur, E., Bourouina, T, Pourciel, J., Ozaki, M., Masuzawa, T and Fujita, H., 2000, 'Resonant-Type Micro-Probe for Vertical Profiler,' 2000 International Conference on Modeling and Simulation of Microsystems, pp. 285-288
  6. Han, J.S. and Kwak, B.M., 2002, 'Robust Design in Terms of Minimization of Sensitivity to Uncertainty and Its Application to Design of Micro Gyroscopes,' Transactions of the KSME, A, Vol. 26, No. 9, pp. 1931-1942
  7. Hwang, K.H., Lee, K.H., Park, G.J., Lee, B.L., Cho, Y.C. and Lee, S.H., 2002, 'Robust Design of the Vibratory Gyroscope with Unbalanced Inner Torsion Gimbal Using Axiomatic Design,' Transactions of the KSME, A, Vol. 26, No. 5, pp. 914-923
  8. Delauche, F., Affour, B. and Dufaza, C., 2002, 'Parametric Yield Optimization of MEMS,' DTIP of MEMS/MOEMS, Proc. SPIE, Vol. 4755, pp. 126-135
  9. Ko, J.S., Lee, M.L., Lee, D.S., Choi, C.A. and Kim, YT, 2002, 'Development and Application of a Laterally Driven Electromagnetic Microactuator,' Appl. Phys. Lett., Vol. 81, pp. 547-549
  10. Chiao, M. and Lin, L., 2000, 'Self-Buckling of Micromachined Beams under Resistive Heating,' J. Microelectromech. Syst., Vol. 9, No. 1, pp. 146-151
  11. Han, J.S. and Kwak, B.M., 2001, 'Robust Optimal Design of a Vibratory Micro Gyroscope Considering Fabrication Errors,' J. Micromech. Microeng., Vol. 11, pp. 662-671
  12. Kwak, B.M. and Lee, T.W., 1987, 'Sensitivity Analysis for Reliability-based Optimization Using an AFOSM Method,' Computers & Structures, Vol. 27, No.3, pp. 399-406
  13. Jung, D. H. and Lee, B.C., 2000, 'Development of an Efficient Optimization Technique for Robust Design by Approximating Probability Constraints,' Transactions of the KSME, A, Vol. 24, No. 12, pp. 3053-3060
  14. Seo, H.S. and Kwak, B.M., 2002, 'Efficient Statistical Tolerance Analysis for General Distributions Using Three-Point Information,' Int. J Prod. Res., Vol. 40, No. 4, pp. 931-944
  15. Ko, J.S., Cho, Y.H., Kwak, B.M. and Park, K.H., 1996, 'Design, Fabrication and Testing of a Piezoresistive Cantilever-Beam Microaccelerometer for Automotive Airbag Applications,' Transactions of the KSME, A, Vol. 20, No. 2, pp. 408-413
  16. Parkinson, A., Sorensen, C. and Pourhassan, N., 1993, 'A General Approach for Robust Optimal Design,' J. Mech. Des., Vol. 115, pp. 74-80
  17. Wu, Y.T, Millwater, H.R. and Cruse, T.A., 1990, 'Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions,' AIAA Journal, Vol. 28, No. 9, pp. 1663-1669
  18. Hasofer, A.M. and Lind, N.C., 1974, 'Exact and Invariant Second-Moment Code Format,' ASCE Journal of the Engineering Mechanics, Vol. 100, pp. 111-121
  19. Ayyub, B.M. and Haldar, A., 1984, 'Practical Structural Reliability Techniques,' Journal of Structural Engineering, Vol. 110, No. 8, pp. 1707-1724
  20. Nikolaidis, E. and Burdisso, R., 1988, 'Reliability based Optimization: a Safety Index Approach,' Computers & Structures, Vol. 28, No. 6, pp. 781-788
  21. Shoemaker, A.C., Tsui, K.L. and Wu, C.F.J., 1991, 'Economical Experimentation Methods for Robust Parameter Design,' Technometrics, Vol. 33, pp. 415-427
  22. Chen, W., Allen, J.K., Tsui, K.L. and Mistree, F., 1996, 'A Procedure for Robust Design: Minimizing Variations Caused by Noise Factors and Control Factors,' J. Mech. Des., Vol. 118, pp. 478-485
  23. Gao, X., Low, T.S., Chen, S. and Liu, Z., 2001, 'Structural Robust Design for Torque Optimization of BLDC Spindle Motor Using Response Surface Methodology,' IEEE Transactions on Magnetics, Vol. 37, No. 4, pp. 2814-2817