對於任何形式的波而言，是無法僅藉由單一均質性的材料所控制，必須具有如梯度分佈、材料間介面以及曲線排列等某種程度的非均質性存在，才能被加以控制。
而本文所探討的聲子晶體是由兩種或兩種以上不同的彈性材料或流體經週期排列所構成的結構，進而具有聲子能隙(禁止帶)與特殊的色散現象(通帶)。然而，於巨觀下，聲子晶體所擁有的禁止帶與通帶特性皆僅為均質的性質，因此，對於在控制彈性波或聲波上的應用十分有限。藉此，本文利用了一種名為空間變動晶格的晶格轉換法，有效地將原始均勻晶格分佈下之晶體，依所設想的形式進行轉換，如單位晶胞的旋轉角、晶格常數以及填充比等分佈，而轉換後之晶格分佈具有連續、平滑、無缺陷及最小形變量等特性，在最大限度下保持單位晶胞的尺寸與形狀，藉以於空間中分佈原始晶格所具有的現象。並基於如此非均質性的分佈下，來控制波的傳遞。
接著，本文選用背景基材為流體，散射體為彈性體所構成的二維聲子晶體，並在兩種幾何形式的散射體下，分別為矩形柱與圓形柱，利用平面波展開法計算聲子晶體之能帶結構圖與等頻圖來分析其波傳特性，並以有限元素軟體加以模擬並相互驗證。最後，針對由矩形柱週期排列而成的聲子晶體上，所對應到的自我準直現象，藉由空間變動晶格的轉換，以有效地於結構中控制自我準直聲束進行彎曲轉向傳遞；針對由圓形柱週期排列而成的聲子晶體於低頻(長波長)下，可等效成均質材料的特性，依經空間變動後的晶格分佈與所對應到的等效折射率與圓形柱幾何之關係，有效益地構成梯度聲子晶體，進以實現彎曲聲波導與聲學黑洞等聲學元件。

The present study propose using the numerical technique, which names synthesis of spatially variant lattice, to spatially vary the orientation of the unit cell, lattice spacing, filling ratio and other properties of a periodic structure throughout its volume in a way that leaves the overall lattice smooth, continuous, defect-free and minimal deformation. Based on keeping the geometry of the unit cells in a spatially variant lattice consistent with that of a uniform lattice, the ability to spatially vary the orientation of the unit cells throughout a lattice enables directional phenomena like band gaps, anisotropy and dispersion to be fully exploited to control the waves. Considering two-dimensional square lattice phononic crystals with two kinds of elastic inclusions (rectangular and circular inclusions) immersed in fluid, the plane wave expansion method is used to obtain the band structures and equi-frequency contours of the phononic crystals. And, the finite element commercial software is employed to simulate the acoustic pressure field in the structures. As a result, using spatially variant lattice to spatially vary self-collimation of the two-dimensional phononic crystals of rectangular inclusions can manipulate the self-collimation beam with abrupt bends. Besides, using spatially variant lattice to spatially vary the effective refractive index of the two-dimension phononic crystals of circular inclusions can realize the acoustic bending waveguide and acoustic black hole.

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