拓樸絕緣體以量子霍爾效應、量子能谷霍爾效應以及量子自旋霍爾效應等理論為基礎，其強大的能量傳播能力受到矚目，在電磁波、聲波、彈性波等領域都有許多學者進行研究。本文研究具十二重旋轉對稱性之準晶體，由於準晶體缺少平移對稱性，所以必須先探討分析的單位晶格尺寸，以滿足布洛赫定理，本文成功模擬準聲子晶體之色散曲線及穿透率，並利用準聲子晶體之特性設計共振耦合之無缺陷波導。另外本文以具十二重旋轉對稱性之準晶體的幾何作為基礎，基於量子自旋霍爾效應，提出三角晶格排列之聲學拓樸絕緣體，其優點為不需外加強磁場以及在極低溫之環境下即可實現，利用改變圓柱半徑討論其能帶結構，觀察能帶反轉的現象，進而產生受拓樸保護之邊緣模態，最後利用全波模擬兩種不同型式之介面，驗證其波傳能抑制後向散射及不受路徑轉彎的影響。由於電子的自旋方向有兩種，在本文中成功激發了單一方向的波傳，進而可以設計出高穿透率且可抑制的後向散射的聲學元件。

The researches of topological insulators (TIs) have recently emerged due to the interest in robustly transport based on quantum Hall effect, quantum valley Hall effect, and quantum spin Hall effect (QSHE). These topics have been involved in distinct fields, such as electromagnetic wave, sound waves, and elastic wave, etc. In this thesis, the quasiperiodic phononic crystals with 12-fold symmetry have been investigated. Owing to the lack of translational symmetry, we discuss the size of unit cell and regard it as the periodic structure. We prove the accuracy of the dispersion relations by full wave simulation. We also design the defect-free coupled-resonator waveguides. Based on QSHE, we present acoustic topological insulators (ATIs) of the quasiperiodic phononic crystals with 12-fold symmetry. It neither need to apply magnetic field nor need to realize in extremely low temperature condition. By modulating the diameter of the cylinder, band inversion is realized. This phenomenon gives rise to a pair of counter-propagating topologically protected edge states. We demonstrate the wave propagation is immune to backscattering, disorder, and sharp bends in the full wave simulation. We also successfully excite pseudospin-dependent one-way transport.

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