In real mechanical systems, the uncertainties in design variables or system parameters are often large enough to disrupt a given design especially a deterministic optimal design to such a degree that it can not serve original objective. Probability based optimal design is very suitable for the purpose, but it requires statistical information of uncertain variables, which is usually very difficult to obtain. Dimensional uncertainties in MEMS structures due to manufacturing errors are known to be large compared to the errors in usual structures and can significantly influence the performance. The concept of robust design can be described in relation to the sensitivities of performance functions with respect to the variations in uncertain parameters. The mathematical approach of robust design depends on how the sensitivities are defined. The main purpose of this thesis is to suggest mathematical formulations for the robustness and study their characteristics by applying to several MEMS structures. Results of the existing minimum sensitivity method and those from deterministic approach are compared with ours. Probabilities of failure calculated by the Monte Carlo simulation are also compared.
The proposed measure of robustness is based on integrated value of sensitivities over some fixed uncertain variable interval. The first one suggested is integration of the square of sensitivity of the performance function. The new objective function of the optimal robust design formulation is the maximum over the number of random variables. The second is based on taking integration of the square of the difference between the performance function and its value at current design point. The influence of the size of the interval is also studied, but its selection need be based on the size of variation of the random parameters if known. In the formulation of the optimal robust design the robustness measure just defined is taken as the objective function to be minimized while a goal for the original objective function is set and then treated as a constraint such that the functional value does not exceed the goal. To treat robustness for constraints, a penalty term method is used. Considering the importance of the dimensional uncertainties in MEMS, in our examples, only the design variables for the dimensions are taken as uncertain parameters.
It is observed that we need to choose an appropriate formulation depending on the nonlinearity of the performance function. When it is weak, the minimum sensitivity method is efficient, but in the case of large nonlinearity, the proposed integral based formulations give better results. An ideal robust optimum is possible when the optima obtained are almost the same even when the integration interval is changed. The present formulation has some arbitrariness in setting the goal for the performance function and in defining the integration interval. More experience with different problems and sets of uncertain variables is required to further identify performance characteristics of the proposed formulations.
