Bifurcation Analysis for Large Problems: Algorithms, Software Applications to Micro Electro Mechanical Systems (MEMS)

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Nonlinear equations depending on parameters are the basis for many scientific and engineering problems. The principal approach of numerical bifurcation analysis is based on continuation of solutions to well-defined operator equations. Such computational results give a deeper understanding of the solution behavior, stability, multiplicity, and bifurcations. This project is concerned with development of new algorithms and software for bifurcation analysis in large nonlinear systems and applications to MicroElectroMechanical Systems (MEMS). Typical MEMS devices lead to multiscale-multiphysics problems. These devices include coupled mechanical, electrical, thermal and other parts. In addition to the bifurcations like beam buckling which involve a single type of physics, MEMS also exhibit bifurcations due to interactions between different energy domains. Bifurcations in MEMS play a variety of roles: (i) bifurcation as a failure mode (lead systems to fail), (ii) bifurcation as a design feature (i.e. bistable systems), (iii) bifurcations to extract material properties.From the engine of an automobile to a hospital operating room, MicroElectroMechanical (MEMS) products are finding applications in every aspect of our lives at an increasing rate. SUGAR is a new system level simulation program in MATLAB for three-dimenisonal MEMS developed at the University of California Berkeley. It is already widely used in academia and industry and is currently under active development, which is a part of the revolution in the MEMS design. SUGAR will be made freely available in source and as a web service. New algorithms for bifurcation analysis in MEMS will be implemented in MATLAB, and then the resulting new code will be incorporated into a MATLAB continuation toolbox, developed at Universiteit Utrecht, to enhance its capabilities for bifurcation analysis of large systems. The upgraded MATLAB continuation toolbox will be integrated into SUGAR and will become a part of the MEMS design loop which includes design, simulation, fabrication, laboratory measurements, and comparison of simulation with measurements.