Periodic elastic waveguides, such as rods, beams, and shafts, exhibit frequency bands where wave reflections at impedance discontinuities cause strong wave attenuation by Bragg scattering. Such frequency bands are known as stop bands or band gaps. This work presents a shape optimization technique for one-dimensional periodic structures. The proposed approach, which aims to maximize the width of the first band gap, uses as tuning parameters the spatial Fourier coefficients that describe the shape of the cell cross-section variation along its length. Since the optimization problem is formulated in terms of Fourier coefficients, it can be directly applied to the Plane Wave Expansion (PWE) method, commonly used to obtain the dispersion diagrams, which indicate the presence of band gaps. The proposed technique is used to optimize the shape of a straight bar with both solid and hollow circular cross-sections. First, the optimization is performed using the elementary rod, the Euler-Bernoulli and Timoshenko beam, and the shaft theoretical models in an independent way. Then, the optimization is conducted to obtain a complete band gap in the dispersion diagrams, which includes the three wave types, i.e., longitudinal, bending, and torsional. All numerical results provided feasible shapes that generate wide stop bands in the dispersion diagrams. The proposed technique can be extended to two- and three-dimensional periodic frame structures, and can also be adapted for different classes of cost functions.

诸如棒，梁和轴之类的周期性弹性波导会出现频带，其中阻抗不连续处的波反射会因布拉格散射而引起强烈的波衰减。这样的频带被称为阻带或带隙。这项工作提出了一种用于一维周期结构的形状优化技术。所提出的方法旨在最大化第一带隙的宽度，它使用描述单元横截面沿其长度变化的形状的空间傅里叶系数作为调整参数。由于优化问题是根据傅立叶系数来表示的，因此可以将其直接应用于通常用于获得色散图的平面波扩展（PWE）方法，该色散图表明存在带隙。所提出的技术用于优化具有实心和空心圆形横截面的直杆的形状。首先，使用基本杆，Euler-Bernoulli和Timoshenko梁以及轴理论模型以独立的方式进行优化。然后，进行优化以在色散图中获得完整的带隙，该带隙包括三种波类型，即纵向，弯曲和扭转。所有数值结果均提供了可行的形状，这些形状在色散图中生成了较宽的阻带。所提出的技术可以扩展到二维和三维周期性框架结构，并且还可以适用于不同类别的成本函数。Euler-Bernoulli和Timoshenko梁，以及独立的轴理论模型。然后，进行优化以在色散图中获得完整的带隙，其中包括三个波类型，即纵向波，弯曲波和扭转波。所有数值结果均提供了可行的形状，这些形状在色散图中生成了较宽的阻带。所提出的技术可以扩展到二维和三维周期性框架结构，并且还可以适用于不同类别的成本函数。Euler-Bernoulli和Timoshenko梁，以及独立的轴理论模型。然后，进行优化以在色散图中获得完整的带隙，其中包括三个波类型，即纵向波，弯曲波和扭转波。所有数值结果均提供了可行的形状，这些形状在色散图中生成了较宽的阻带。所提出的技术可以扩展到二维和三维周期性框架结构，并且还可以适用于不同类别的成本函数。所有数值结果均提供了可行的形状，这些形状在色散图中生成了较宽的阻带。所提出的技术可以扩展到二维和三维周期性框架结构，并且还可以适用于不同类别的成本函数。所有数值结果均提供了可行的形状，这些形状在色散图中生成了较宽的阻带。所提出的技术可以扩展到二维和三维周期性框架结构，并且还可以适用于不同类别的成本函数。