多重耦合有限元素軟體模擬非均質鐵電材料之相場法應用

Abstract

本文主要使用COMSOL Multiphysics有限元素軟體來進行相場架構模擬，並建立一套標準的計算流程，有助於日後建立相場模型上更快速有效率。有別於過去的快速傅立葉演算法在理論相場法中，須假設均質情況來求解，COMSOL解決了此問題，成功地在非均質假設下模擬出晶域排列。此外COMSOL在數學模型建構上簡單明瞭，大幅降低建立相場模型時間。 本團隊依據Eshelby等效理論及Hashin-Shtrikman變分理論推導出另一種能量表示式，進而建構出等效型相場架構。在本文中，首先簡化等效相場架構，在不考慮極化情形下模擬麻田散鐵來驗證此架構正確性，發現微結構排列確實符合諧和條件與許多實驗上的成果。接著延伸到非均質鐵電材料上來模擬正方晶與菱方晶，等效型確實可以演化出正確晶域排列並且成功算出材料等效係數，提供了更精準的材料性質，也證實使用有限元素軟體COMSOL來建立等效相場架構與計算材料等效係數是可行的。

The thesis aims to develop numerical algorithms implemented in COMSOL finite element software for phase-field simulations. The conventional phase-field simulations adopt algorithms based on the fast Fourier transform. However, such a method is not suitable for the case with material inhomogeneity. Instead, the algorithms developed here are suitable for both cases with material homogeneity and inhomogeneity. In addition, the algorithms based on COMSOL are much simpler than those implemented in programming languages. The framework is based on the equivalent phase-field models developed by the previous team efforts. It is established by involving the spirit of Eshelby equivalent principle and Hashin-Shtrikman variational formulation. The results are first validated by simulating microstructure patterns in martensitic thin films. Next, the developed algorithms are extended to the case of tetragonal and rhombohedral ferroelectric domain simulations. The results show that many more electromechanical self-accommodation patterns as well as engineered domains are simulated and are consistent with experimental observations. Finally, the effective constants of several engineered domains are calculated and compared with those based on micromechanics calculations.