Topology optimization, as a scientific and efficient intelligence algorithm, can be used to distribute the structures or materials with periodic units within the design domain to obtain the desired properties or functions. To induce interface states with high transmission and localization at the specified objective frequency, we propose a topology optimization design of the mirror-symmetric periodic waveguides based on the combination of the multiple population genetic algorithm and the finite element method. In addition to the traditional Bragg gaps, which are generated by the same low-order mode resonances, interface states can also be induced through the proposed topology optimization method in the non-Bragg gaps formed by the resonances of the more complex modes. The non-Bragg interface states possess higher localization due to the involvement of high-order modes in the non-Bragg resonances. This topology optimization method is expected to realize the wavefield manipulation of sound wave localization from a new perspective and provide a novel method for the design of devices for intense localization and high transmission.

Localization and propagation of waves at the interface of two waveguides is a topic of continuing interest.1,2 Interface states were originally investigated in quantum systems3,4 as a localization phenomenon at an interface: the methodology was later extended to classical wave systems, including optical,5,6 acoustic,7,8 and elastic9,10 waves. The localization property provides interface states with a wide range of applications, including phononic crystal sensors,11 acoustic energy harvesters,12 lasers,13 sound switches,14 acoustic energy amplifiers,15 unidirectional transmission filters,16 and organic solar cells.17 In addition to the heterostructures analogous to semiconductor heterojunctions,18 interface states can be induced in the mirror-symmetric heterostructures in one-dimensional (1D) periodic systems. Wang et al. fabricated a narrow band filter at 1550 nm based on the interface state induced by a mirror-symmetric heterostructure composed of the alternating high and low refractive index dielectric materials.19 Multiple topological interface states were achieved by Zhang et al. in the mirror-symmetric 1D phononic crystal slabs by using a zone-folding strategy.20 Zhukovsky demonstrated that the perfect transmission with strong localization is explicitly related to mirror-symmetry structures.21 

Interface states in the mentioned 1D mirror-symmetric periodic systems all exhibit the localization characteristic in the field distributions. More specifically, the localized modes of interface states exhibit the low-order mode characteristic. Numerous research studies have demonstrated that the localized mode characteristics of interface states highly depend on the gaps in which they appear.22–24 Almost all relevant research focuses on the interface states in Bragg gaps25,26 and, consequently, omit those in non-Bragg gaps, which have been known in recent years.23,27 The resonant interaction between the low- and high-order transverse modes can lead to the splitting spectra and form the non-Bragg gaps.28 The involvement of the high-order modes enhances the localization of interface states but increases the difficulty of the field analysis.

With the continuous development of computer technology, intelligent algorithms are used to design the engineering structures with low-cost and high-performance. As one of the intelligent algorithms, topology optimization can be used to optimize the distribution of materials in a design domain according to constraints and design objectives.29 For instance, the combination of the multiobjective genetic algorithm and the finite element method (FEM) was used in the topology optimization to reduce the weight of structures.30 Liao et al. proposed an efficient and robust topology optimization method by hybridizing the multiple population genetic algorithm (MPGA) and the bi-directional evolutionary structural optimization (BESO) method.31 Based on the genetic algorithm (GA) and FEM, the topology optimization of two-dimensional phononic crystals was performed to maximize the gap width.32 At present, the topology optimization has been widely employed in various acoustic structures, such as acoustic black hole plates,33 ventilation barriers,34 diodes,35 and filters.36 The periodic structures designed by the topology optimization method show better performance and applicability compared to those designed by the traditional manual experience adjustment method.37 

In this paper, the strategy of combining MPGA with the FEM is employed for the topology optimization. Interface states are induced by the topology optimization in the tubular mirror-symmetric waveguides with periodically corrugated inner walls. Even in the non-Bragg gap with the complex multimode interaction, the proposed topology optimization is effective. A concise overview of each component of the paper is provided below. The parameters of mirror-symmetry periodic waveguides are calculated with the method in Sec. II. Section III presents the optimization model design and the topology optimization problem, which is described with a fitness function. Then, the specific steps of the MPGA are explained. Section IV discusses the topology optimization results and compares the localization characteristics between the Bragg and non-Bragg interface states. The main conclusions are presented in the last section of the paper.

The propagation of sound waves in an ideal fluid can be described by means of the Helmholtz equation. The relationship between the sound pressure amplitude p and the angular frequency ω satisfies
(1)
where c is the speed of sound. For an acoustic cylindrical waveguide with periodically corrugated inner walls Γwall, the rigid boundary condition is
(2)
where n represents the unit vector perpendicular to the wall outward. According to the Bloch–Floquet theory, the sound wave amplitude distribution in the periodic structure can be expressed with the relationship of the wave vector k and position vector r,38,
(3)
where k z ( − π / Λ ≤ k z ≤ π / Λ ) is the propagation constant along the z axis. Λ is the corrugated wall period length of the cylindrical waveguide. Analogous to the earlier theoretical work,38,39 this equation is substituted into Eq. (1) to find the dispersion relation,
(4)
where k t , m and k l , m represent the transverse and longitudinal wavenumbers for the mth mode, respectively. The interactions between multiple modes can be illustrated along the following lines:39 
(5)
where r0 is the mean radius. As the zero of the first-order Bessel function, k r , m ( = k t , m r 0 ) represents the transverse wavenumber of the mth mode corresponding to the nth space harmonic waves (n = ±1, ±2, ±3 … ), i.e., { k r , m , m = 0 , 1 , 2 , … } = { 0 , 3.8317 , 7.0156. … } . kz is equal to k l , m − 2 n π Λ⁠.

Finite element simulations of the sound propagation in a cylindrical waveguide with periodically corrugated walls were performed using the COMSOL Multiphysics. The propagation velocity of the sound in the air of the waveguide is set to 343 m/s, and the density of the air is 1.25 kg/m3. The boundary conditions for the inlet and outlet of the waveguide were set to the plane wave radiation.

The center frequency of the interface state is highly related to the corresponding frequency range of the gap in the waveguide spectrum. To demonstrate the universality of the method, the Bragg and non-Bragg interface states are set at the same objective frequency of 3000 Hz. Consequently, the resonant frequencies of the Bragg and non-Bragg waveguides are both set to the objective frequency. For distinctly describing the mode interaction, the mode reference lines of the two waveguides are obtained by formula (5) and shown in Fig. 1. Each of the lines is marked by a pair of indices l(m,n), denoting the curve for the mth mode of the nth harmonic. The first zero of the first-order Bessel function is zero, so the low-order mode has a vanishing transverse wavenumber, which is the approximate plane wave. The low- and high-order modes are indicated by the blue and red lines, respectively. The frequency position of the green dashed line indicates the resonant frequency of 3000 Hz. For the Bragg waveguide in Fig. 1(a), the intersection of the blue lines l(0, 0) and l(0, 1) at 3000 Hz represents that two low-order modes participate in the resonance. Figure 1(b) illustrates the resonance mode interactions of the non-Bragg waveguide. A blue line l(0, −1) and a red line l(1, 0) intersect at 3000 Hz, indicating that the low-order mode and the high-order mode resonate. Under the simplified condition of r0 = Λ and the resonant frequency of 3000 Hz, the corresponding k r , m and l(m, n) are substituted in the formula (5), and the parameters of the Bragg waveguide r0 = Λ = 57 mm and the non-Bragg waveguide r0 = Λ = 78 mm are obtained.

FIG. 1.
Reference lines of the multi-mode interactions. (a) Reference lines of the Bragg waveguides with the Bragg resonant frequency of 3000 Hz. (b) Reference lines of the non-Bragg waveguides with the non-Bragg resonant frequency of 3000 Hz. The blue and red lines represent the low- and high-order modes, respectively. The green dashed line indicates the frequency of 3000 Hz.
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Reference lines of the multi-mode interactions. (a) Reference lines of the Bragg waveguides with the Bragg resonant frequency of 3000 Hz. (b) Reference lines of the non-Bragg waveguides with the non-Bragg resonant frequency of 3000 Hz. The blue and red lines represent the low- and high-order modes, respectively. The green dashed line indicates the frequency of 3000 Hz.

FIG. 1.
Reference lines of the multi-mode interactions. (a) Reference lines of the Bragg waveguides with the Bragg resonant frequency of 3000 Hz. (b) Reference lines of the non-Bragg waveguides with the non-Bragg resonant frequency of 3000 Hz. The blue and red lines represent the low- and high-order modes, respectively. The green dashed line indicates the frequency of 3000 Hz.
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Reference lines of the multi-mode interactions. (a) Reference lines of the Bragg waveguides with the Bragg resonant frequency of 3000 Hz. (b) Reference lines of the non-Bragg waveguides with the non-Bragg resonant frequency of 3000 Hz. The blue and red lines represent the low- and high-order modes, respectively. The green dashed line indicates the frequency of 3000 Hz.

Close modal

The schematic diagram of the cylindrical waveguides corresponding to the topology optimization model depicted in Fig. 2(a) is shown in Fig. 2(b). The design unit near the waveguide wall of the topology optimization model is meshed into N1 × N2 small squares. The material distribution in the design unit can be arranged by a structure matrix with corresponding dimensions. When the value of the element in the structure matrix is set to 1, the corresponding small square in the design unit indicated by the gray area is filled with a rigid material. The value of structure matrix 0 represents that the filling material is the air, which corresponds to the white area. The left domain denoted by the red dashed lines is the design domain, and the right domain is the mirror-symmetric domain of the design domain along the interface. The four-period design domain forms the left half of the topology optimization model, while the right half is the mirror-symmetric structure of the left half.

FIG. 2.
(a) Periodic unit structure of the topology optimization model [the magnified view of the blue dashed box in (b)]. (b) Cylindrical mirror-symmetric waveguide corresponding to the topology optimization model. The orientation of the blue arrow indicates the direction of sound wave incidence. The interface of the mirror-symmetric waveguide is marked with a yellow dashed line.
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(a) Periodic unit structure of the topology optimization model [the magnified view of the blue dashed box in (b)]. (b) Cylindrical mirror-symmetric waveguide corresponding to the topology optimization model. The orientation of the blue arrow indicates the direction of sound wave incidence. The interface of the mirror-symmetric waveguide is marked with a yellow dashed line.

FIG. 2.
(a) Periodic unit structure of the topology optimization model [the magnified view of the blue dashed box in (b)]. (b) Cylindrical mirror-symmetric waveguide corresponding to the topology optimization model. The orientation of the blue arrow indicates the direction of sound wave incidence. The interface of the mirror-symmetric waveguide is marked with a yellow dashed line.
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(a) Periodic unit structure of the topology optimization model [the magnified view of the blue dashed box in (b)]. (b) Cylindrical mirror-symmetric waveguide corresponding to the topology optimization model. The orientation of the blue arrow indicates the direction of sound wave incidence. The interface of the mirror-symmetric waveguide is marked with a yellow dashed line.

Close modal

Although the localization of the field distributions is a significant feature of the interface states, it cannot be used to describe the topology optimization problem, as the field distributions of interface states show different localization characteristics in the gaps with different resonance mechanisms. Another prominent feature of interface states is the transmission peak phenomenon in the gap. Here, the transmission is the integral ratio of the acoustic amplitude in the outlet and inlet of the waveguide. The transmission spectrum feature of the transmission peak can be quantitatively characterized so that the fitness function can be constructed. This constructing method of fitness function is fit for the topology optimization of the interface states in various gaps, including the Bragg and non-Bragg gaps.

The optimization objective for interface states is to narrow the gap between the transmission spectrum of the individual to be optimized and the objective transmission spectrum. The objective transmission spectrum can be constructed with a rectangular function through the feature analysis of the transmission spectrum of interface states. In order to ensure the maximal transmission of the interface state, the height of the rectangular function is set to 1. In the case of the objective frequency of 3000 Hz, the transmission spectrum that still needs to be optimized is plotted in Fig. 3. The yellow and blue colors represent the magnitude of transmission. The black lines show that the weight factor W(f) value varies from 1 to 2.5 with the frequency. Introducing W(f) into the objective function facilitates the interface state induced at the exact frequency. Thus, the objective function can be formulated as
(6)
where g(f)–o(f) is the area difference between the transmission spectrum needed to be optimized and the objective transmission spectrum in the range of f1–f2.
FIG. 3.
Transmission spectrum of the individual to be optimized. The black line represents the distribution of the weight factor values.
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Transmission spectrum of the individual to be optimized. The black line represents the distribution of the weight factor values.

FIG. 3.
Transmission spectrum of the individual to be optimized. The black line represents the distribution of the weight factor values.
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Transmission spectrum of the individual to be optimized. The black line represents the distribution of the weight factor values.

Close modal
The objective of topology optimization is to minimize the objective function F(x) within the expected frequency range. Consequently, our optimization model can be written as follows:
(7)
where the number of the matrix elements ei is N1 × N2 in the design domain. The topology optimization problem is essentially to maximize the fitness function 1/F(ei). f band ( f 1 ≤ f band ≤ f 2 ) contains the frequency range apart from the window part of the rectangular function in the gap, corresponding to the black dashed lines in Fig. 3. The transmission T(fband) is set to less than 0.5 to form the gap for inducing the interface state. The value of T(fband) can also be adjusted according to the application requirements.

Since the design variables in the objective function are multi-modal, the optimization algorithms based on the local search can easily become trapped at a local optimum and lose the best achievable performance. As an adaptive and globally optimizing probabilistic search method, GAs can search the global optimal solutions by directly using the objective function and avoid falling into trap of a local optimum. The MPGA, an advanced GA, overcomes the limitation of relying on a single population for genetic and evolutionary operations in the standard genetic algorithm (SGA) by the synergistic optimization of multiple populations, thus preventing the premature convergence of the SGA.

The steps of the MPGA are illustrated in Fig. 4. The initial step involves the generation and initialization of multiple populations, which are then evaluated in terms of their respective fitness values. Next, different control parameters are assigned to each population, which evolves separately and performs the steps of the SGA. The fitness value of each individual in every population is recalculated. After completing the steps of the SGA in each population, immigration operators introduce the optimal individual during evolution to other populations. The optimal individual in each population is collected to form an elite population by an elite reserved strategy. The convergence of the MPGA is judged by the continuous unchanging iteration number of the maximum fitness value in the elite population or the maximum number of optimization iterations. If the results are judged to be convergent, the algorithm terminates. Otherwise, it returns to perform the steps of the SGA.

FIG. 4.
Flowchart of the MPGA.
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Flowchart of the MPGA.

FIG. 4.
Flowchart of the MPGA.
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Flowchart of the MPGA.

Close modal

The topology optimization of mirror-symmetric waveguides is performed by using the method described in Sec. III. As a result, an interface state is generated at the preset objective frequency of 3000 Hz in both Bragg and non-Bragg gaps. The unit length N1 and width N2 of the optimization model are set to 60 and 6, respectively. MPGA related parameters are listed below. Four populations of 15 individuals each are initialized, for a total of 60 individuals. The maximum number of iterations, denoted as Nm, is limited to 200. The crossover probability Pc corresponding to four different populations can be randomly selected with the range from 0.7 to 0.9, and the mutation probability Pm varies from 0.001 to 0.05. As a classic case, the topology optimization process of the interface state in the Bragg gap is shown in Fig. 5. The variation in the fitness function value of the optimal individual with the iterations is represented by a blue solid line. The fitness function value of the optimal individual rapidly approaches the final value after only five iterations, which exhibits the high efficiency of the optimization algorithm. The fitness function value reaches the maximum at the 80th iteration and remains a constant in successive another 50 iterations, and the optimization algorithm is judged to be convergent. Although the preset Nm is 200, the optimization algorithm converges after 130 iterations. In addition, three typical optimized periodic units are shown in Fig. 5. They are the unit structure in the initial mirror-symmetric waveguide, the mirror-symmetric waveguide after four iterations, and the mirror-symmetric waveguide after 85 iterations, respectively.

FIG. 5.
The variation in the fitness function value in the topology optimization process of the interface state in the Bragg gap. The illustrations show the periodic unit structures in the initial mirror-symmetric waveguide, the mirror-symmetric waveguide after four iterations, and the mirror-symmetric waveguide after 85 iterations, respectively.
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The variation in the fitness function value in the topology optimization process of the interface state in the Bragg gap. The illustrations show the periodic unit structures in the initial mirror-symmetric waveguide, the mirror-symmetric waveguide after four iterations, and the mirror-symmetric waveguide after 85 iterations, respectively.

FIG. 5.
The variation in the fitness function value in the topology optimization process of the interface state in the Bragg gap. The illustrations show the periodic unit structures in the initial mirror-symmetric waveguide, the mirror-symmetric waveguide after four iterations, and the mirror-symmetric waveguide after 85 iterations, respectively.
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The variation in the fitness function value in the topology optimization process of the interface state in the Bragg gap. The illustrations show the periodic unit structures in the initial mirror-symmetric waveguide, the mirror-symmetric waveguide after four iterations, and the mirror-symmetric waveguide after 85 iterations, respectively.

Close modal

To demonstrate the universality of the topology optimization method and provide an intuitive comparison, the topology optimizations of the Bragg and non-Bragg waveguides are performed at the same objective frequency of 3000 Hz. The transmission spectra of the Bragg and non-Bragg waveguides through the topology optimization are shown in Figs. 6(a) and 7(a), respectively. The transmission spectrum analysis indicates that an interface state appears from both the Bragg and non-Bragg gaps, and the center frequencies of the Bragg and non-Bragg interface states have a good agreement with the objective frequency. The frequency range of the Bragg gap is from 2634 to 3327 Hz, and the non-Bragg gap is from 2839 to 3142 Hz. The bandwidth of the Bragg and non-Bragg gaps is 693 and 303 Hz, respectively. The full width at half maximum (FWHM) of the Bragg and non-Bragg interface state is 181 and 64 Hz, respectively. The involvement of high-order modes enhances the non-Bragg resonance, which subsequently results in the narrow bandwidth and low transmission of the non-Bragg gap and non-Bragg interface state.

FIG. 6.
(a) Transmission spectrum of the Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.
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(a) Transmission spectrum of the Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.

FIG. 6.
(a) Transmission spectrum of the Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.
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(a) Transmission spectrum of the Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.

Close modal
FIG. 7.
(a) Transmission spectrum of the non-Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.
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(a) Transmission spectrum of the non-Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.

FIG. 7.
(a) Transmission spectrum of the non-Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.
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(a) Transmission spectrum of the non-Bragg waveguide by the topology optimization at the objective frequency of 3000 Hz. (b) Sound field distribution in the corresponding mirror-symmetric waveguide by the topology optimization at 3000 Hz.

Close modal

Localization characteristics of interface states can be identified from the corresponding sound field distributions. The sound field intensity distributions of the Bragg and non-Bragg waveguides at the objective frequency of 3000 Hz are presented in Figs. 6(b) and 7(b), respectively. For both the Bragg and non-Bragg interface states, the sound field presents the obvious localization distributions near the interface of the mirror-symmetry waveguides. The Bragg gap is generated with the same low-order mode resonances, and as a result, the sound pressure amplitude of the Bragg interface state has a tiny variation along the r axis, and the field distribution is similar to the plane wave field. However, the appearance of non-Bragg gaps originates from the resonances of low- and high-order modes. The localized sound pressure amplitude of the non-Bragg interface state is highly variable along the r axis, and the maximum amplitude is at the center of the waveguide. The field distribution approximates a sphere. The non-Bragg interface state localizes more than twice the maximum sound pressure amplitude of the Bragg interface state. The introduction of high-order modes enhances the non-Bragg resonance along with the localization of the non-Bragg interface state in the aspect of the density and intensity of localized sound pressure amplitudes.

In addition to 3000 Hz, the topology optimization of interface states can be achieved, if other objective frequencies are selected within the frequency range of the corresponding gap. The frequency range of the gap depends on the structure parameters of heterostructures. In order to verify the universal applicability of the proposed topology optimization method, interface states are optimized at four objective frequencies each in the Bragg and non-Bragg gaps. For the Bragg interface states, the transmission spectra of the mirror-symmetric waveguides optimized at the objective frequencies of 2900, 2950, 3050, and 3100 Hz are shown in Fig. 8(a) from the left to right in order. Since the FWHM of the non-Bragg gap is narrower than that of the Bragg gap, the topology optimizations of the non-Bragg interface states are performed at the objective frequencies of 2900, 2950, 3050, and 3100 Hz, with the corresponding transmission spectra shown in Fig. 8(b) from the left to right in order. The transmission spectra show that the center frequencies of the Bragg and non-Bragg interface states are highly consistent with the preset objective frequencies. The optimization results show an extraordinary efficiency of the proposed topology optimization method.

FIG. 8.
Transmission spectra and sound field distribution of mirror-symmetric waveguides by topology optimization at different objective frequencies. (a) Transmission spectra of the Bragg interface states at the objective frequency of 2900, 2950, 3050, and 3100 Hz. (b) Transmission spectra of the non-Bragg interface states at the objective frequency of 2950, 2975, 3025, and 3050 Hz. (c) and (d) show the periodic unit structures and the sound field distributions corresponding to (a) and (b).
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Transmission spectra and sound field distribution of mirror-symmetric waveguides by topology optimization at different objective frequencies. (a) Transmission spectra of the Bragg interface states at the objective frequency of 2900, 2950, 3050, and 3100 Hz. (b) Transmission spectra of the non-Bragg interface states at the objective frequency of 2950, 2975, 3025, and 3050 Hz. (c) and (d) show the periodic unit structures and the sound field distributions corresponding to (a) and (b).

FIG. 8.
Transmission spectra and sound field distribution of mirror-symmetric waveguides by topology optimization at different objective frequencies. (a) Transmission spectra of the Bragg interface states at the objective frequency of 2900, 2950, 3050, and 3100 Hz. (b) Transmission spectra of the non-Bragg interface states at the objective frequency of 2950, 2975, 3025, and 3050 Hz. (c) and (d) show the periodic unit structures and the sound field distributions corresponding to (a) and (b).
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Transmission spectra and sound field distribution of mirror-symmetric waveguides by topology optimization at different objective frequencies. (a) Transmission spectra of the Bragg interface states at the objective frequency of 2900, 2950, 3050, and 3100 Hz. (b) Transmission spectra of the non-Bragg interface states at the objective frequency of 2950, 2975, 3025, and 3050 Hz. (c) and (d) show the periodic unit structures and the sound field distributions corresponding to (a) and (b).

Close modal

To verify the localization characteristics of all the topologically optimized interface states, the sound field distributions of the Bragg and non-Bragg interface states are investigated and presented in Figs. 8(c) and 8(d), respectively. The periodic unit structures corresponding to the sound field distributions are illustrated at the top of the figures. It is evident that all the interface states exhibit an obvious localization characteristic. For the Bragg interface states, the maximum sound pressure amplitude is concentrated near the interface of the mirror-symmetry waveguides, with a gradual attenuation toward the ends of the mirror-symmetry waveguides. In contrast, the sound field distributions of non-Bragg interface states show a rapid attenuation of the maximum sound pressure amplitude toward the ends of the mirror-symmetry waveguides. The introduction of high-order modes leads to an intense localization along the z and r axes, which accordingly further enhances the localization of interface states.

In summary, the proposed topology optimization is designed for interface states at objective frequencies based on the combination of the MPGA and the FEM. The design of the objective frequency can be carried out according to a variety of practical applications. The topology optimization results show that high transmission interface states can be induced in both the Bragg and non-Bragg gaps. The sound field distribution analyses of the Bragg and non-Bragg interface states show that there is an obvious localization of the sound wave amplitude appearing near the interface of the mirror-symmetry waveguides. The participation of high-order modes enhances the non-Bragg resonances and the localization of the non-Bragg interface states, leading to the different sound field distributions between the Bragg and non-Bragg interface states. The topology optimization results of multiple objective frequencies demonstrate a high degree of consistency between the optimized frequencies of the interface state and the objective ones, confirming the broad applicability of the proposed topology optimization method. This research is expected to contribute to the optimization design of mode control engineering and various functional devices, which include multimode filters, sensors, and couplers. The proposed topology optimization method can facilitate the design of interface states in classical wave systems.

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 12304520, the National Science Found for Distinguished Young Scholars under Grant No. 62125104, and the China Postdoctoral Science Foundation under Grant No. 2023M740888.

The authors have no conflicts to disclose.

Ting Liu: Investigation (equal); Supervision (equal); Writing – original draft (equal). Linge Wang: Investigation (equal); Methodology (lead); Software (lead); Validation (lead); Visualization (lead). Hongwei Liu: Formal analysis (lead); Writing – review & editing (equal). Jingwei Yin: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of the study are available from the corresponding author upon reasonable request.

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