Dirichlet-to-Neumann映射法應用於多層結構柱之二維與三維聲子晶體頻帶結構之計算

Band structure calculations with Dirichlet-to-Neumann map method for multi-shell-rods-in-fluid-background phononic crystals

王大為(Da-wei Wang)

指導教授 ： 欒丕綱

國立中央大學/理學院/光電科學與工程學系/碩士(2015年)

摘要

本篇論文使用Dirichlet to Neumann法來計算彈性波在由多殼層介質柱子與流體背景構成的聲子晶體之頻帶結構。傳統常用之平面波展開法 (plane wave expension) 在計算聲子晶體之頻帶時，若遇到固體與流體共存之系統，會由於固體區與流體區之波函數分量數量不同，而遭遇無法執行的困難。本論文使用之Dirichlet-to-Neumann法能夠輕易的藉由連接邊界條件來解決上述困難，並且大幅縮小計算資源。第二章首先介紹二維基本結構以及彈性波之基本理論，然後推導含有剪力波之彈性固體柱在在流體背景中之聲子晶體頻帶計算細節，以盡可能模擬真實情況。在第三章中，我們將原本的Dirichlet to Neumann法推廣至三維，並提出了可能的執行方法，以計算三維聲子晶體之頻帶。最後探討在二維結構中使用不同的殼層材料與厚度搭配對於聲波在週期介質中傳播情形之影響。

關鍵字

多層結構 ； 聲子晶體 ； DtN映射法 ； 頻帶結構

並列摘要

In this thesis the Dirichlet- to-Neumann map method is applied to the calculation of the band structure of the phononic crystal which consists of multi-shell rods and fluid background. Due to the inconsistency of the total number of dynamical wave variables in the solid and fluid regions of the phononic crystal, the conventional plane wave expansion method is not directly applicable. We thus turn to use the Dirichlet-to-Neumann map method to overcome such a problem. In chapter 2, we first introduce the typical structure of the phononic crystal and the basic theory of elastic waves, and then explain how to derive the wave fields in every region in a unit cell and give the calculation details for the band structures of the phononic crystals in the text and in the appendix. In order to mimic the real acoustic/elastic systems, in most calculations the shear waves are taken into account. In chapter 3, we then generalize the original Dirichlet-to-Neumann map method to 3D and propose a possible procedure for its implementation to the band structure calculations of 3D phononic crystals. Finally, we discuss the effects of the materials used in the phononic crystal and the thicknesses of the shells on the resultant band structures.

並列關鍵字

Dirichlet-to-Neumann map method ； band structure ； multi-shell-rods

參考文獻

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