In areas that require high performance components, such as the automotive, aeronautics and aerospace industries, optimization of the dynamic behavior of structures is sought through different approaches, such as the design of materials specific to the application, for instance through structural topology optimization. The bi-directional evolutionary structural optimization (BESO) method, in particular, has been used for the simultaneous design of hierarchical structures, which means that the structural domain consists not only of the macrostructure but also of the microstructural topology of the materials employed. The purpose of this work is to apply the BESO method to solve two-dimensional multiscale problems in order to minimize the response of structures subjected to forced vibrations in a given frequency range. The homogenization method is applied to integrate the different scales of the problem. In particular, the material interpolation model for two materials is used. The BESO method is applied to different cases of optimization, in macroscale, microscale, and multiscale structural domains. Numerical examples are presented to validate the optimization and demonstrate the potential of this approach. The numerical examples show that the multiscale bi-material topology optimization method implemented here is able to produce structures and microstructures for optimization of the frequency domain response, satisfying prescribed volume constraints.

在需要高性能组件的区域，例如汽车，航空和航天工业，通过不同的方法来寻求结构动态性能的优化，例如通过特定于应用的材料设计，例如通过结构拓扑优化。特别是双向进化结构优化（BESO）方法已用于同时设计层次结构，这意味着结构域不仅由宏观结构组成，而且还由所用材料的微观结构拓扑组成。这项工作的目的是应用BESO方法来解决二维多尺度问题，以使结构在给定频率范围内受到强迫振动的响应最小化。均质化方法用于整合问题的不同尺度。特别是，使用了两种材料的材料插值模型。BESO方法适用于宏观，微观和多尺度结构领域的不同优化案例。数值例子表明了该优化方法的有效性，并证明了这种方法的潜力。数值算例表明，在此实现的多尺度双材料拓扑优化方法能够产生结构和微结构，以优化频域响应，从而满足规定的体积约束。微观和多尺度结构域。数值例子表明了该优化方法的有效性，并证明了这种方法的潜力。数值算例表明，在此实现的多尺度双材料拓扑优化方法能够产生结构和微结构，以优化频域响应，从而满足规定的体积约束。微观和多尺度结构域。数值例子表明了该优化方法的有效性，并证明了这种方法的潜力。数值算例表明，在此实现的多尺度双材料拓扑优化方法能够产生结构和微结构，以优化频域响应，从而满足规定的体积约束。