In this work, a topology optimization algorithm has been developed to design bi-material lattice structures showing a band gap around a target frequency, using just one unit cell through the application of Bloch-Floquet theorem. The Bidirectional Evolutionary Structural optimization (BESO) method has been employed, based on bi-material interpolation. A new objective function has been defined, which uses only the natural frequencies closest to the target one, regardless of their position with respect to the fundamental natural frequency. This reduces the computational cost by limiting the number of frequencies considered, and improves the robustness of the optimization process, as these frequencies adapt to changes in the distribution of materials within the domain, constantly encompassing the target frequency. In addition, a novel approach has been implemented to determine the optimal volume fraction of the materials forming the structure, a parameter typically predefined in other works before starting the optimization process. Consequently, the algorithm can autonomously identify the volume that produces the widest band gap around the target frequency. The algorithm has been evaluated for different cases of lattice structures formed by the periodic repetition of a unit cell in both 1D (1D-CR) and 2D (2D-CR), comparing some results with those obtained in other works through different approaches.

中文翻译：

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通过 Bloch-Floquet 理论对具有最佳体积分数的目标带隙的晶格结构进行拓扑优化


在这项工作中，开发了一种拓扑优化算法，通过应用 Bloch-Floquet 定理，仅使用一个晶胞来设计双材料晶格结构，显示目标频率周围的带隙。采用了基于双材料插值的双向进化结构优化 （BESO） 方法。定义了一个新的目标函数，它只使用最接近目标频率的固有频率，而不考虑它们相对于基本固有频率的位置。这通过限制考虑的频率数量来降低计算成本，并提高优化过程的鲁棒性，因为这些频率会适应域内材料分布的变化，并始终包含目标频率。此外，还实施了一种新颖的方法来确定形成结构的材料的最佳体积分数，该参数通常在其他工作中在开始优化过程之前预先预定义。因此，该算法可以自主识别在目标频率周围产生最宽带隙的体积。该算法已经针对在一维 （1D-CR） 和二维 （2D-CR） 中由晶胞周期性重复形成的晶格结构的不同情况进行了评估，并将一些结果与通过不同方法在其他工作中获得的结果进行比较。