二維聲子晶體在複式晶格結構與時域 有限差分法分析頻溝之探討

Abstract

聲子晶體的頻帶間隙特性在表面聲波元件上有很多應用。本文研究了複式晶格聲子晶體的頻帶結構特性。應用平面波展開法（PWM）計算複式晶格聲子晶體頻帶結構，本文採用二維鎳合金/空氣體系，計算了以蜂窩格子和Kagome格子為代表的複式晶格的頻帶結構；並與正方晶格和三角晶格為代表的簡單晶格的結果作了相互比較分析，結果顯示，在某些填充率範圍內，將聲子晶體排列為複式晶格要優於簡單晶格，可以得到更寬的頻溝。此外，借助於頻溝分佈圖，分析了頻溝數目、頻溝寬度以及頻溝上下邊界頻率隨填充係數的變化特性。 文中亦應用時域有限差分方法（FDTD）來分析頻帶的特性，時域有限差分法是對偏微分波動方程的離散化處理，通過對時間和空間的離散化，將偏微分方程轉化為差分方程，繼而採用數值計算方法，求解波傳播過程中各個離散點的所有動態參數與時間的函數關係。時域有限差分方法（FDTD）不但可以計算週期結構中的頻帶結構而且可以計算有限結構的透射、反射等特性，且不受固液耦合、結構形式等因素的影響。

Frequency band gap characteristics of phononic crystals have many applications on the surface acoustic wave device. In this thesis, we investigate frequency band structure characteristics in phononic crystals of multi-composition. We apply Plane Wave Method（PWM）to calculate frequency band structure in phononic crystals of multi-composition. Two-dimensional nickel/air system for the bee nest crystal and Kagome crystal are taken for numerical calculations of the frequency band structure in multi-composition. Then we make comparison with the results for simple crystal such as square and triangle crystal. The results show that we can get wider band gap with arranging the phononic crystal into multi-composition than simple crystal. In addition, we analyze the number and width of band gap and the characteristic of top and bottom boundary frequency of band gap varying with filling coefficient. In this thesis, we also analyze characteristics of frequency band with Finite-Difference Time-Domain（FDTD）method. Finite-Difference Time-Domain method is to discrete partial differential wave equation by means of discrete to time and space. We transform partial differential equation into difference equation, then the numerical calculation method is used to solve the function relation of all dynamic parameter and time of each discrete point in the wave propagation process. Not only band structure in period structure is calculated but also characteristics of homology and reflection in finite structure is obtained. The FDTD method has the advantage that it can be applied to solve the coupled solid and liquid problem.