Level set-based topology optimization for graded acoustic metasurfaces using two-scale homogenization

Highlights

• We propose a topology optimization method for two-layered acoustic metasurfaces.

• A two-scale homogenization method is introduced to represent metasurface's system.

• A sensitivity analysis is performed based on the topological derivative.

• Level set-based topology optimization is conducted to obtain unit cell designs.

• The obtained design shows negative refraction, a typical property of metasurfaces.

Abstract

This paper proposes a level set-based topology optimization method for acoustic metasurfaces consisting of multiple types of unit cells. As a type of acoustic metasurface, we focus on a two-layered metasurface, where each layer contains various types of unit cell designs. To handle this metasurface, we first propose a two-scale homogenization method based on previous studies targeting metasurfaces composed of a single type of unit cell structure. As a result of homogenization, each layer of the array of unit cells is replaced with an interface with a nonlocal transmission condition. Next, an optimization problem is formulated. We set an objective function to achieve negative refraction, and macroscopic responses obtained by the homogenization method are used to define the objective function. The material distributions in each unit cell of the metasurface are set as the design variables. Based on these settings, we perform a sensitivity analysis based on the concept of the topological derivative. A level set-based topology optimization method is introduced to solve the optimization problem, and two-dimensional numerical examples are provided to demonstrate the validity of the proposed method. An optimal design of a two-layered acoustic metasurface that induces negative refraction is proposed, and a discussion of its mechanism is provided.

Introduction

Acoustic waves are closely related to daily life as human voices, music, noise, and so on. Noise reduction is necessary to create a comfortable environment, and acoustic structures, such as sound barrier walls and sound-absorbing materials, are utilized to solve this problem. Ultrasonic waves, or acoustic waves that are beyond audible frequencies, are also widely used, such as in imaging for medical applications and non-destructive testing. Therefore, efficient control of acoustic waves is critical in many fields.

Acoustic metamaterials have been of interest due to their unique properties compared to natural homogeneous materials. In a pioneering study, Liu et al. [1] proposed locally resonant sonic material (LRSM) consisting of a periodic arrangement of unit cells filled with different types of elastic media. This structured material induces local resonance in each unit cell, resulting in the formation of a bandgap that prevents transmission of acoustic waves. The local resonance phenomenon in LRSM enables subwavelength control of acoustic waves, which is beneficial from an engineering perspective. A variety of acoustic metamaterials have been proposed that exhibit unusual acoustic properties, such as negative bulk modulus [2], negative mass density [3,4], and negative refractive index [5,6]. These properties improve the efficiency of acoustic wave control and have potential for use not only in existing acoustic devices, but also in novel devices, such as an acoustic cloaking device [7], acoustic hyperlens [8], and acoustic diode [9,10].

Recent studies have focused on planar metamaterials, called acoustic metasurfaces. For the realization of unusual properties, more efficient designs of metasurfaces are required compared with bulk metamaterials because they are based on a two-dimensional array of unit cells with finite thickness, yielding narrow regions for wave manipulation. Ma et al. [11] proposed a metasurface that has membrane resonator structures that absorb acoustic energy by hybrid resonance. In addition, Xie et al. [12] proposed a metasurface consisting of tapered labyrinthine unit cells. In their research, a two-layered metasurface was proposed, where each unit cell was designed to have the desired phase change. Negative refraction was observed due to the graded array of unit cells in each layer. In another study, Zhai et al. [13] proposed ultrathin planar acoustic metasurfaces that can control transmitted waves, where each unit cell contains two elastic membranes and an air-filled cavity. They proposed metasurfaces for wave focusing and wave motion conversion from spherical waves to plane waves and from propagating waves to surface waves.

To achieve effective control of acoustic waves by an acoustic metasurface, the unit cell design of the metasurface is essential, as the above-mentioned properties of metasurfaces are strongly dependent on the unit cell structure. Topology optimization [14] is one method of designing metasurfaces. It has the highest design freedom among structural optimization methods and has been applied in various fields not only for static problems represented by the stiffness maximization problem of a linear elastic body, but also for dynamic problems, including acoustic wave propagation problems. For example, Wadbro and Berggren optimized an acoustic horn by topology optimization [15], while Du and Olhoff proposed a design method for minimizing sound radiation by optimizing a bi-material structure [16]. Dühring et al. tackled indoor and outdoor noise reduction problems, and optimized the distribution of reflecting material and sound barriers [17]. Christiansen and Fernandez-Grande [18] conducted topology optimization for acoustic focusing devices and offered experimental validations for the optimized designs using their three-dimensional prints.

Recently, topology optimization has also been applied for the design of acoustic metamaterials and metasurfaces. Lu et al. [19] proposed a topology optimization method for acoustic metamaterials that exhibit a negative bulk modulus, and this method was later expanded to handle acoustic-structural coupling effects [20]. Dong et al. [21] used genetic-algorithm-based topology optimization to obtain the optimal design of double-negative acoustic metamaterials that induce negative refraction. Christiansen and Sigmund [22] experimentally validated the optimal design of an acoustic hyperbolic metamaterial obtained by topology optimization and confirmed a negative refractive behavior. In Ref. [23], an acoustic metasurface was optimized so that an input longitudinal acoustic wave was converted to a transverse elastic wave.

Since topology optimization requires iterative estimation of the target system, an analysis method with low computational cost should be used for the optimization of metasurfaces. A popular method is the S-parameters-based retrieval method proposed by Smith et al. [24] in the field of electromagnetic metamaterials. The principle of this method is to identify a homogeneous material with effective material parameters that has the same S-parameters (corresponding to complex transmission and reflection coefficients) as the target metamaterial or metasurface using analytical formulas under the assumption that the system can be described by plane waves. This method was later expanded to handle acoustic metastructures by Fokin et al. [25], and has been widely used with topology optimization due to its simplicity and ease of implementation. However, it is difficult to apply this method to a general system with complex incident wave conditions due to the plane wave assumption. Recently, Pestourie et al. proposed a locally periodic approximation method that can be applied to the inverse design of metasurfaces [26]. This method is based on scalar diffraction theory, and the metasurface system is divided into unit cell problems with periodic boundary conditions, where the amplitude and phase of the transmitted fields are examined in advance and used for the entire system analysis. This method was also combined with topology optimization, and metasurfaces composed of many unit cells for electromagnetic waves were optimized [27]. Casadei et al. [28] proposed a geometric multiscale finite element method for the analysis of the elastic waves propagating in metamaterials, where the multiscale elements modeling the heterogeneities in the unit cells are utilized to reduce the computational cost.

Another method for estimating a metasurface system is a two-scale homogenization method [[29], [30], [31]]. By introducing two types of characteristic scales for the system (i.e., microscale and macroscale), asymptotic expansions of the solution are considered, leading to a homogenized equation that governs the macroscopic behavior of the solution. The classical homogenization method imposes the assumption that the system is represented by a periodic array of unit cells, and each unit cell structure is equivalently replaced with coefficients of the homogenized equation, called homogenized coefficients. However, direct application to the metasurface system is impossible because the metasurface has a finite thickness. Several homogenization methods have been proposed for a system containing a periodic array of unit cells with finite thickness. Marigo and Maurel [32,33] derived effective transmission conditions for acoustic and electromagnetic metasurfaces based on matched asymptotic expansions. In addition, Rohan and Lukeš [34,35] proposed a homogenization method for acoustic metasurfaces whose thicknesses are the same order as the period of the unit cell. With their method, the entire metasurface system is decomposed into a fictitious layer containing unit cells and external regions filled with the background acoustic medium. The fictitious layer is converted into an interface characterized by homogenized coefficients based on the periodic unfolding method [36]. Then, the acoustic wave propagation behavior can be obtained by solving the Helmholtz equation in the external regions with a nonlocal transmission condition on this interface. Shape sensitivity analysis was also performed [37,38], which can be used in shape optimization for unit cell design. The homogenization method proposed by Rohan and Lukeš was also combined with topology optimization [39]. The distribution of two types media was taken into account in the fictitious layer, and a topological derivative was derived that measures the rate of change of an objective function when an inclusion is inserted in the homogeneous material domain. Using level set-based topology optimization with the reaction–diffusion equation [40], the material distribution in a unit cell was optimized to achieve the desired performance of the metasurface at the macroscale. This optimization method can handle a single type of unit cell design; however, the performance is expected to increase with additional unit cell structures, as suggested by existing designs of graded metasurfaces [12,13].

In this study, we propose a topology optimization method for the design of acoustic metasurfaces that consist of multiple unit cells. To increase the degree of freedom of design, which will result in obtaining unique properties of the metasurface, we target a two-layered metasurface, as illustrated in Fig. 1(a). To this end, we first modify the homogenization method proposed by Rohan and Lukeš to handle a two-layered metasurface, as discussed in Section 2. Then, an optimization problem for metasurface design is formulated in Section 3. We focus on negative refraction as the unique property of the metasurface, and an objective function is defined with the macroscopic response obtained by the homogenization method. The formula for the topological derivative is modified based on the settings of multiple unit cells. Then, the level set-based topology optimization method is introduced. A discussion of the numerical implementation of the proposed optimization method is provided in Section 4, while two-dimensional numerical examples are provided in Section 5 to demonstrate the validity of the proposed method. We provide an optimal design of an acoustic metasurface where negative refraction occurs, and discuss its mechanism by the distributions of the sound intensity vector and the homogenized coefficients. Furthermore, the solution obtained by the homogenization method is compared to the solution obtained by the finite element method (FEM) without using homogenization to evaluate the proposed method. We summarize our conclusions in Section 6.