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Acoustic anisotropic metamaterials are known for its potential of applications to various acoustic devices, such as hyperlens and waveguides. Anisotropic metamaterial typically has a periodic structure constituted of unit cells, and exhibits highly anisotropic wave propagation based on local resonance induced by its periodic medium. In previous works, design methods for anisotropic metamaterials which shows unusual wave propagation properties have been proposed, though since metamaterials comprised of only a single type of unit cell were in interest, the macroscopic wave propagation properties that could be obtained were limited. Therefore, in the present work, we propose a design method for acoustic structures which are composed of several unit cells, in order to realize waveguides that navigate acoustic waves and achieve complex wave propagation overall. First, we introduce the high-frequency homogenization method, which is capable of evaluating the macroscopic property of periodic structures based on local resonance. We construct a multiscale optimal design method by formulating the macro-scale optimization problem for obtaining the properties required for different unit cells in order to realize the waveguide, along with the micro-scale optimization problem for designing each unit cell based on topology optimization. Numerical examples are introduced to demonstrate the validity of our proposed method.

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A multi-material structure that is composed of several different material properties is promising for achieving an ideal functionality that can outperform a single material structure. In the course of automotive design, the combination of lightweight and stiff materials can reduce the weight of a car body without sacrificing its performance. This paper proposes a multi-material topology optimization (MMTO) framework for the eigenfrequency maximization problem based on the Multi-material level set (MMLS) based topology optimization. The key idea of MMLS is to use M level set functions to represent M material regions and one void region without overlap. To demonstrate the proposed method, first, we formulate an MMTO problem for maximizing the eigenfrequency based on the shape representation by the MMLS method. Next, we derive the topological derivatives of multiple materials in the eigenfrequency problem and construct an optimization algorithm in which the level set functions are evolved by solving a reaction-diffusion equation (RDE) based on the topological derivatives. Several numerical examples are provided to validate the proposed methodology.