- 學位論文
分段連續線性化降階演算法在非線性雙穩態結構之應用
Trajectory Piecewise-linear Model Order Reduction Technique for Nonlinear Bistable Mechanism
摘要
本篇論文探討片段線性模型降階法（trajectory piecewise-linear model order reduction, TPWLMOR）在非線性雙穩態結構模型（bistable mechanisms）的應用。本研究中所分析的雙穩態結構應用於微機電系統中的（micro-electro-mechanical systems, MEMS）光開關構造，此雙穩態結構由double curved-beam mechanisms（DCBM）所組成。當施加超過某臨界值的外力，這個雙穩態結構便會由原本的穩態住置跳躍（snap-through）至另一穩態位置。本篇論文著重於探討這種雙穩態結構的靜態及暫態反應之特性並加以分析。 我們首先利用有限元素法（finite element method, FEM）軟體ABAQUS建構出此雙穩態結構的3-D實體模型，模擬其穩態及暫態分析，並發展出一個三次元非線性有限元素法的動態方程式；之後，我們使用片段線性模型降階法將FEM產生的數值模型降階（model order reduction, MOR）成為精簡模型。TPWLMOR是從片段線性簡化及Arnoldi降階演算法的觀念而來：Arnoldi演算法所產生的線性精簡模型將用來找出線性展開的平衡點，根據這些平衡點所展開的線性模型將依權重而重新組合成為片段線性精簡模型。應用片段線性模型降階法技巧來分析此非線性雙穩態結構，相較於使用有限元素法軟體ABAQUS，我們可以有效地提高計算效率，節省所需的計算時間。 我們利用微機電SOI製程來製作實體元件，實驗量測部份，我們架設了一組Doppler雷射干涉儀系統來觀察此雙穩態結構受力時的位移量變化，並與ABAQUS的模擬值以及TPWLMOR演算法對此結構的模擬結果作比較及討論。模擬結果顯示，當施加較小的力時，DCBM的位移變化量是準確的，並且計算效率可達到ABAQUS的200倍。然而，當施力超過臨界值時，DCBM並沒有如期地發生跳躍（snap-through）的現象。模擬結果不如預期的原因可能由許多因素所影響，這個問題還有待後續研究的探討。
並列摘要
In this work, we investigate the applications of the trajectory piecewise-linear model order reduction (TPWLMOR) technique on bistable mechanisms. The bistable mechanism, which is composed of a double curved-beam mechanism (DCBM), is employed in an MEMS (micro-electro-mechanical systems) optical switch. While applying an external force which is larger than a certain critical value, the DCBM quickly snaps through from a stable state to another stable state. We focused on analyzing the properties of the static and transient behaviors of the DCBM. First, we used the finite element method (FEM) software, ABAQUS, to construct a 3-D solid model of the DCBM and to simulate both the static and the transient analyses. Subsequently, we developed a 3-D dynamic nonlinear FEM numerical model. Then, the TPWLMOR is used to reduce the full-mesh FEM models into low-order models. The idea of TPWLMOR comes from the concept of piecewise-linear approximation and an Arnoldi-based model order reduction (MOR) algorithm. Reduced models are generated using the Arnoldi algorithm at appropriate linearization points, and then are superposed into a compact model (i.e., the trajectory piecewise-linear model) using weighted sum. Compared to traditional FEM modeling, the TPWLMOR models increases computational efficiency. The bistable device was realized using a simple SOI MEMS process with one photo-mask. Then we set up a Doppler laser interferometer system in order to measure the dynamics of the DCBM. We then compared the empirical data with the simulated dataset obtained by ABAQUS , FEM, and the TPWLMOR algorithm. The simulated DCBM displacements of the reduced models agree with the results of the ABAQUS, while the computational performance reaches 200 times of that of the ABAQUS. However, when the force is larger than the critical value, the DCBM does not snap-through as expected. The inaccurate results might be caused by many factors, which are subject to further study.