Structured and Parameter-Dependent Eigensolvers for Simulation-Based Design of Resonant MEMS

This dissertation is about computational tools to aid in the design of resonant Micro-Electro-Mechanical Systems (MEMS), tiny vibrating devices built by processeslike those used to make integrated circuits. Vibrating MEMS are used in accelerom-eters and gyroscopes, in sensors to detect chemicals and to measure pressure, and in communication devices such as cell phones. MEMS engineers can use computersimulations to design devices using fewer costly and time-consuming prototype tests, but these simulations are only as useful as the models on which they are built. In this work, we contribute new mathematical models, numerical methods, and software tools to simulate resonant MEMS, and apply these tools to analyze specific devices. We describe physical models of damped vibrations of MEMS, including anchor loss and thermoelastic effects which are widely recognized as important, but not modeled ingenerality by existing tools. Though the resulting systems of equations are large and non-Hermitian, and depend nonlinearly on frequency, we use the equation structureto develop efficient structured Krylov subspace projection methods for computing free vibrations and reduced-order models. We also provide efficient continuation methods for re-computing eigendecompositions under changes to design parameters or operating conditions. Our models and analysis methods are integrated into HiQLab, a newfinite element tool with a particularly flexible architecture which we have designed. Using HiQLab, we simulate example resonator designs, and compare our results tolaboratory measurements. Our simulations reveal a previously-unknown mode interference phenomenon, subsequently observed in experiments, which dramaticallyaffects the amount of damping near certain critical values of geometric parameters.