Arnoldi精簡模型於壓電元件之應用

Abstract

在研究工作中，我們採用Arnoldi精簡模型演算法應用於有限元素之壓電元件的分析。經由有限元素法的推導，得到了壓電元件的二階常微分方程式系統。利用適當的降階方法，將此二階常微分方程式系統降為一階常微分方程式系統以求得其暫態行為。而此一階常微分方程式系統可以配合Arnoldi方法得到精簡模型。此精簡模型經過證實為被動性(passive)系統。在這精簡模型流程中，我們採用了20個節點的三次元有限元素(20-node brick element)來建立模型。經過精簡流程的模型計算後可得元件上位移以及電壓的分佈。本文中所模擬的壓電元件為圓盤結構的壓電片以及Rosen-type 壓電變壓器。針對此兩種元件我們利用Arnoldi方法將之有限元素模型做降階。在不同的輸入電壓下，經過精簡的模型在位移分佈與電壓分佈依然與原有限元素模型的暫態行為表現相符，卻較有限元素法節省大量計算時間與資源，效率提升至少一千倍以上。

In this work, we present a methodology of efficient piezoelectric analysis using the Arnoldi-based model-order reduction technique. The piezoelectric numerical models, which are actually systems of ordinary differential equations (ODEs), are formulated by the finite element method. The ODEs systems can be reduced into low-order ODE systems using the Arnoldi-based model-order reduction technique. Also, it is shown that the reduced system to be passive. The finite-element formulation of 20-node Brick elements is used to create the full-meshed ODE models. Disk-type piezoelectric actuators and Rosen-type piezoelectric transformers are modeled and measured in this work. The results of displacement and voltage distributions by using the reduced models match very well with the results by the full-meshed models for different applied input voltage. The computational efficiency of the reduced models is at least 1,000 times higher than the full-meshed finite-element models.