摘要
利用有限元或边界元等方法分析和仿真MEMS器件,需要耗费大量的计算资源来求解大规模耦合常微分方程组。这些方法还难以满足器件的优化设计、实时反馈控制和系统级仿真的需求。寻求能准确刻画器件复杂行为特性的低自由度数简化模型,来替代原有的大规模复杂系统,用于器件的优化设计和MEMS系统级仿真,以大幅降低计算费用,是实现MEMSCAD的关键技术之一。对其它工程领域也同样具有重大的应用价值。
论文以静电驱动的MEMS器件为对象,研究适合于多物理场耦合MEMS器件的模型自由度缩减(MOR)方法及其相关的计算技术。
在系统综述了器件MOR技术研究现状之后,论文对静电—非线性结构(梁、板)耦合器件的模型自由度缩减(MOR)方法进行研究,以结构件无阻尼模态振型为基函数,结合Galerkin方法,建立器件的缩减自由度模型(ROM)。利用Simulink(?)建立静电微梁器件ROM,对器件从Pull-in到释放全过程进行仿真,探讨静电器件共有的非线性特性；以非线性板理论为基础,导出以广义模态坐标表达的静电非线性板器件的系统能量无量纲表达式,利用Rayleigh-Ritz法和系统能量辨识的方法建立器件ROM,将ROM的仿真结果与有限元或文献报道的结果比较,验证了MOR方法的正确性和ROM的有效性。
微器件的动态特性受到器件中流体环境的显著影响,建立能考察器件中流固耦合效应的ROM对微器件的精确设计和优化,提高其动态性能至关重要。本文对基于本征正交分解(POD)和Galerkin映射的MOR方法进行研究。给出了Hilbert空间下连续形式POD的理论和计算方法,利用POD从全网格全耦合非线性瞬态仿真数据集合中提取出位移和压力变量的最优基集合,将非线性梁方程和非线性可压缩Reynolds方程映射到由最优基张成的子空间上,构造出能考虑结构大变形几何非线性和挤压膜阻尼耦合特性的ROM。为提高ROM的适应性,本文提出了自适应参数化ROMs的建模新方法,利用子空间角度内插值技术,构造参数化ROMs。将ROMs的仿真结果与全网格全耦合瞬态仿真结果、文献报道的实验和仿真结果相比较,表明：自适应参数化ROMs能够替代全网格耦合非线性模型,对多种载荷输入情况进行快速仿真计算,计算效率高且具有足够的仿真精度,当器件的几何参数和材料参数发生变化时,ROMs也具有足够的仿真能力。
在论文的最后部分,给出了主要研究结果并展望了今后的研究工作。
Using the finite element methods or boundary element methods for Micro-Electro-Mechanical Systems (MEMS) devices simulation require of intensive computational resource for solving the time integration of a very high dimensional of coupled ordinary-differential equations. Even so, those methods are inadequate for MEMS devices optimization, real-time feedback control, and predicting the MEMS dynamical behaviors under system-level simulation. As a result, abstracting lower dimensional systems from these higher dimensional ones while retaining the essence of the physical phenomena as a way to make system-level simulation and optimization is gaining attention and becomes one of the key technologies for MEMS CAD realization. It is also extremely necessary for other engineering applications
The objective of this dissertation is to develop an accurate model order reduction (MOR) methods and associated numerical technology which suit for the electrostatic- mechanical coupling microdevices and the electrostatic-mechanical-fluidic coupling microdevices.
A systematic review of the state of the MOR for MEMS applications is presented in the first part of the dessertation. Following that, a MOR procedures for the electrostatic-mechanical coupling microdevices is introduced, which uses the undamped modal shape functions of the structural element as a basis function sets for Galerkin projection. The nonlinear characteristics of the electrostatically actuated micorbeam device were probed by using the reduced order model (ROM), and the Pull-in and release dynamic process of the devices were simulated in Simulink(?) environment. After that, The derivation of strain energy and kinetic energy expression based on nonlinear plate theory have been developed, and the formulation of the ROM of an electrostatically actuated micro-plate device has been obtained by using Rayleigh-Ritz method, and a MOR method based on system energy identification has been presented. The simulation results by using the ROM are compared well with those from high-fidelity finite element model under various electric loads, and thus this MOR approach has been validated.
Considered that the dynamical behavior of MEMS devices is strongly affected by viscous fluid damping effects from the surrounding, these damping effects have to be carefully accounted for the design and optimization. The ROM of microdevices should have the ability of capturing the fluidic-structural interactions of the microdevice. And the MOR method based on Galerkin projection and subspace angles interpolation technique has been developed. The commercial software ESI-CFD(?) has been used to achieve the full coupling transient results. Subsequently, the global optimal basis functions are extracted from the full-order solution by the application of proper orthogonal decomposition (POD) methods. The nonlinear Euler-Bernoulli beam equation and the nonlinear compressible Reynolds equation are projected on the subspaces spanned by those optimal basis functions, and thus the ROMs have been obtained. Since the constructed ROMs by POD/Galerkin are valid only within a limited state-space, a subspace angles interpolation strategy is introduced to widen the range of electric loads under which the ROMs are applicable. The numerical results of ROMs show good agreement with the ESI-CFD(?) data and the experimental data available in the literature. All those have demonstrated that the ROMs can be used in the place of the full meshed coupling models for various electric loads inputs and device paramenters changing with reduced computational complexity immensely.
In the final part of this dissertation, the main conclusions are summarized and the prospects for future research are suggested.
引文
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