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

含参数的非线性方程是许多科学和工程问题的基础。数值分支分析的主要方法是基于定义良好的算子方程的解的延拓。这样的计算结果使我们对解的性态、稳定性、多重性和分叉有了更深入的了解。该项目致力于开发用于大型非线性系统分叉分析的新算法和软件，以及在微电子机械系统(MEMS)中的应用。典型的MEMS器件导致了多尺度-多物理问题。这些装置包括耦合的机械、电气、热力等部件。除了像梁屈曲这样的分叉涉及单一类型的物理外，MEMS还由于不同能域之间的相互作用而表现出分叉。分叉在MEMS中扮演着多种角色：(I)分叉作为一种失效模式(导致系统失效)，(Ii)分叉作为设计特征(即双稳系统)，(Iii)分叉提取材料属性。从汽车发动机到医院手术室，微电子机械(MEMS)产品正以越来越快的速度应用于我们生活的各个方面。GASH是美国加州大学伯克利分校用MatLab开发的一种新的三维MEMS系统级仿真程序。它已经在学术界和工业界得到了广泛的应用，目前正在积极开发中，这是MEMS设计革命的一部分。Suge将在源代码和网络服务中免费提供。用于MEMS分叉分析的新算法将在MatLab中实现，然后产生的新代码将被整合到由Utrecht大学开发的MatLab延续工具箱中，以增强其对大系统分叉分析的能力。升级后的MatLab延续工具箱将被集成到GASE中，并将成为MEMS设计循环的一部分，该循环包括设计、模拟、制造、实验室测量以及模拟与测量的比较。

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.