本研究基於組合雙向演進式結構最佳化法及有限元素法的優化算法，對由兩種不同的固體材料組成之正方形晶格排列的二維聲子晶體進行拓樸最佳化，建立了完整的聲子晶體分析和優化流程，分別針對出平面波 (out-of-plane waves)和平面波(in-plane waves)計算頻帶結構，透過最佳化流程找出兩相鄰能帶之間具有寬頻且低頻能帶的週期結構，再將結果與文獻做比對。先驗證程式正確性，再進行參數分析。
在拓樸最佳化方面，考慮由環氧樹脂-金組成的聲子晶體對出平面波及平面波進行能帶最大化分析，且迭代歷程最後會收斂。並分析在最佳化時，出平面波及平面波分析會遇到的問題，及解決辦法。
在參數分析方面，利用不同材料組合進行最佳化，找到出平面波及平面波各頻帶間具有較大目標函數之參考拓樸形狀。並透過改變材料性質分析能帶大小及中心頻率的改變，使用不銹鋼作為材料2，材料1之性質為材料2的倍數比例，且材料1之密度及剛性皆小於材料2，如當固定材料2性質，調整材料1之密度時，可使能帶寬度及中心頻率變大，調整材料1之剛性時，可使能帶寬度及中心頻率變小，而同時調整材料1之密度及剛性時，則會使能帶寬度變大且中心頻率變低，進而達到調控能帶大小及中心頻率高低的能力。

This study was based on an optimization algorithm that combined the bidirectional evolutionary structural optimization (BESO) method and finite element method. It aimed to topologically optimize a two-dimensional phononic crystal with a square lattice arrangement composed of two distinct solid materials. A comprehensive analysis and optimization process for phononic crystals were established. Band structures were separately calculated for out-of-plane waves and in-plane waves. Through the optimization process, periodic structures with wide and low-frequency bands between adjacent energy bands were identified. The results were then compared with existing literature. The correctness of the program was validated first, followed by parameter analysis.
In terms of topological optimization, band gap maximize analysis of out-of-plane waves and in-plane waves considering a phononic crystal composed of epoxy and gold, with the iterative process eventually converging. The study also addressed the problems encountered during the optimization of these wave types and provided solutions.
Regarding parameter analysis, different material combinations were optimized to find reference topological shapes with larger objective functions among various frequency bands for out-of-plane waves and in-plane waves. And analyze the change of energy band size and center frequency by changing the material properties. Using steel as material 2, the property of material 1 is a multiple ratio of material 2, and the density and stiffness of material 1 are smaller than material 2, such as fixing material 2's properties and adjusting material 1's density, the band width and central frequency could be increased. Similarly, changing material 1's rigidity reduced both. Simultaneously adjusting both density and rigidity of material 1 increased the band width and lowered the central frequency, thereby achieving the ability to control band size and central frequency.

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