Robust Optimization: Design in MEMS - University of California ...

43figure (5.9). Clearly the average and worst designs differ significantly from the robustsolutions. Comparing the costs associated with the designs in Φ 1000 against our robustsolution is useful in verifying that the optimization was successful, but less helpful inunderstanding how the robustness improved.12010080# of occurences60402000 0.005 0.01 0.015 0.02 0.025s(x, Σ)Figure 5.10: Histogram of the sensitivity cost term, s(x, Σ U ), for the set Φ 1000 .To demonstrate the robustness of our solution we ran a monte carlo simulationto model an uncertain MEMS fabrication process. To represent the variation in theprocess we generated 400,000 gaussian random vectors with a standard deviation,σ, of 0.1 µm. We then calculated the natural frequencies of the robust design andthe average and worst design from Φ 1000 subject to these variations. The results aredisplayed in figure (5.11). The distribution for the robust design has a much tighterspread than both the average and worst-case designs from Φ 1000 . This is an excellentindication that the optimized design is much less insensitive to geometric uncertaintiesin the design variables.