In the operation of high frequency resonators in micro electromechanical systems (MEMS) there is a strong need to be able to accurately determine the energy loss rates or alternatively the quality of the resonance. The resonance quality is directly related to a designer’s ability to assemble high fidelity system response for signal filtering, for example. This has implications on robustness and quality of electronic communication and also strongly influences overall rates of power consumption in such devices – i.e. battery life. Past design work was highly focused on the design of single resonators; this arena of work has now given way to active efforts at the design and construction of arrays of coupled resonators. The behavior of such systems in the laboratory shows un-necessarily large spread in operational characteristics, which are thought to be the result of manufacturing variations. However, such statements are difficult to prove due to a lack of available methods for predicting resonator damping – even the single resonator problem is difficult. The physical problem requires the modeling of the behavior of a resonant structure (or set of structures) supported by an elastic half-space. The half-space (chip) serves as a physical support for the structure but also as a path for energy loss. Other loss mechanisms can of course be important but in the regime of interest for us, loss of energy through the anchoring support of the structure to the chip is the dominant effect.
The construction of a basic discretized model of such a system leads to a system of equations with complex-symmetric (not Hermitian) structure. The complex-symmetry arises from the introduction of a radiation boundary conditions to handle the semi-infinite character of the half-space region. Requirements of physical accuracy dictate rather fine discretization and, thusly, large systems of equations. The core to the extraction of relevant physical performance parameters is dependent upon the underlying modeling framework. Notwithstanding, in two-dimensions it is possible to utilize eignenvalue methods with direct solvers to effectively determine quantities of interest. However, in three dimensional settings of practical interest, such systems are too large to be handled directly and must be solved iteratively; alternate time-stepping methods in conjuction with harmonic inversion thus start to look attractive. In this talk, I will cover the physical background of the problem class of interest, how such systems can be modeled, and then solved. Particular interest will be paid to the radiation boundary conditions (perfectly matched layers versus higher order absorbing boundary conditions), issues associated with frequency domain versus time domain methods, and how these choices interact with iterative solver technologies in sometimes unexpected ways.