Topology optimization of magnetorheological smart materials included PnCs for tunable wide bandgap design

1. Introduction

Wave propagation control in artificial periodic materials and structures has been widely investigated in the last decades. As typical artificial acoustic metamaterials, phononic crystals (PnCs) have received renewed attention recently due to their bandgap characteristics. To develop more effective devices of acoustic/elastic wave control, it would be necessary to introduce tunable phononic properties in the designs of PnCs. There are two main mechanisms to achieve frequency tunability of PnCs: geometric changes and material properties changes. For the former one, the sizes and shapes of PnC unit cells are transformed with deformation or buckling by applying external stress [1-5]; for the latter mechanism, the elastic characteristics of the constitutive materials in PnCs are changed through the application of additional fields, including magnetic fields [6,7], electric fields [8], and temperature fields [9-11].

Many studies regarding tunable PnC materials and devices have been devoted to the use of intelligent materials, including magnetoelastic materials [12], shape-memory materials [13], electroactive dielectric elastomers [14-16], electrorheological (ER) materials [17-19], magneto-rheological (MR) materials [20,21] Herein, Huang et al. [22] studied the tuning and control acoustic band structures of the piezoelectric inclusions including PnCs using the biasing field. Karami Mohammadi et al. [23] investigated the bandgaps tunability of soft magnetoactive periodic laminates under remotely applied magnetic field. For the tunable PnCs design with the ER and MR materials, Zhou and Chen [17] investigated the tunability of bandgaps of ER locally resonant PnCs by using the externally applied electric field. Bayat et al. [20] studied the tunable PnCs consisting of porous hyperelastic MR elastomers for generating new tunable bandgaps upon the application of the magnetic field. Zhang and Gao [21] studied the bandgap control of locally resonant PnCs composed of ER and MR elastomers by simultaneously manipulating the external electric and magnetic fields. Although the tunable bandgaps of PnCs with smart material components could be opened with heuristic-based design strategies, the systematic optimization design methods are still in urgent need for opening wider PnC bandgaps and enlarging the tunable bandgap range.

Topology optimization [24,25], as a powerful conceptual design tool, has been applied to provide wave propagation functions [26,27], such as complete bandgaps [28], directional bandgaps [29-31], waveguides [32,33], negative reflection [34], self-collimation [32], topological insulators [35,36], which have been studied and designed with different topology optimization methods. Herein, Zhang et al. [29] proposed a gradient-free topological optimization method for full and directional bandgap design basing on a material-field series expansion framework [37]. Dong et al. [34] proposed a systematic design strategy for designing the broadband double-negative PnC structures with the genetic algorithm-based topology optimization method. Chen et al. [38] investigated the new topology optimization method for obtaining frequency-specified bandgaps of viscoelastic PnCs by maximizing the decay of evanescent waves.

Designing tunable bandgaps of PnCs with smart material components with topology optimization is an attractive and challenging research topic. Different types of smart materials have been adopted to design wide and tunable bandgaps by optimizing the material distributions inside the PnC unitcells [39-41]. For instance, Vatanabe et al. [42] proposed the solid isotropic material with penalization (SIMP) basing topology optimization to maximize the width of elastic wave bandgaps in piezo-composite materials. Hedayatrasa et al. [43] studied genetic algorithm basing topology optimization of nonlinear elastic deformation PnC plates by using various tunability targets to tune complete bandgaps. Bortot et al. [44] proposed a genetic algorithm basing topology optimization method of dielectric elastomer PnC composites for widening bandgaps and improving the bandgap tunability. Although different optimization methods based on smart materials have been proposed to achieve tunable bandgaps, the topology design optimization of MR elastomers including PnCs has yet to be explored.

In this paper, we propose a systematic topology optimization method of the MR materials including PnCs for opening tunable and wide bandgaps. The bandgap of the PnCs is tuned by changing the MR material properties by various applied magnetic fields. Here, the bandgap and frequency response analysis is implemented with the finite element (FE) method. To achieve the tunable and wide bandgap of PnCs, we propose an aggregated bandgap index for enveloping the extreme values of bandgap width and tunable bandgap range with the Kreisselmeier-Steinhauser (KS) function [45] as the objective function. In the optimization model, the distribution of MR and epoxy material candidates inside the smart PnCs are represented with pseudo-densities of all the FE elements, and an artificial MR interpolation model with penalization is adopted to eliminate the “gray” elements that may be generated during the optimization process. The tunable bandgap design problem is solved with the gradient-based mathematical programming method, which can provide a relatively stable convergence process in complicated dynamics optimization problems. To this end, the sensitivity analysis scheme for the aggregated bandgap index is also derived. With the proposed optimization method of PnCs with MR materials, the designs of wide bandgap PnCs under continuously changing magnetic fields are obtained, and the numerical examples prove that the optimized design can achieve a wider tunable bandgap frequency range and still have relatively stable bandgap widths under different specified magnetic fields.

This paper is organized as follows. In Sect. 2, the band analysis method of PnCs is introduced. In Sect. 3, the topology optimization formulation of PnCs considering changing magnetic field intensities is presented, and its sensitivity analysis is derived. Numerical examples and results discussion are given in Sect. 4. Section 5 concludes and discusses the whole paper.

2. Dispersion analysis of wave propagation in PnCs

For the wave propagation in three-dimensional PnCs, we ignore the damping effect of the structure, the governing equation of PnCs can be expressed as

where ρ is the material density, $r\left(x,y\right)$ represents the position vector, $U=\left(u_x,u_y,u_z\right)$ is the displacement vector along the coordinate directions x, y, and z, $\lambda$ and $\mu$ are the Lame coefficients. In this study, the wave field assumed to be independent of $z$, and the governing equation only considering the out-of-plane mode gives

For the bandgap calculation in PnCs based structures, the dispersion curve can be obtained by analyzing the unit cell with the Floquet-Bloch boundary condition. Here, the displacement vector can be given as

where $k$ is the wave vector, $\tilde{U}$ is the translation-periodic wave displacement field, $R$ is the translation vector of the square unit cell, and $\omega$ is the angular frequency ($\omega =2\text{π}f$, $f$ is frequency). The symbol $\mathrm{Im}$ represents the imaginary part of a complex number.

By substituting Eq. (4) into the governing equation (2), the band structure of out-of-plane elastic waves can be determined by solving the generalized complex eigenvalue problem for the specified wave vector $k$ as

where $K\left(k\right)$ and $M\left(k\right)$ are the stiffness matrix and mass matrix of the PnC unit cell, respectively. For any specified wave vector $k$, the corresponding characteristic frequency $\omega$ can be calculated by solving the Eq. (5). By sweeping all the boundary curves of the irreducible Brillouin zone (for the unit cell design has four-fold symmetry), the whole dispersion relations $\omega \left(k\right)$ of the PnCs can be obtained.

3. Topology optimization model

In this study, the PnCs composed of MR (matrix) and epoxy (inclusion) materials are optimized to open a tunable bandgap. Herein, the mechanism of bandgap tuning is implemented by changing the physical properties of the MR material by adjusting the intensity of the applied magnetic field.

3.1 MR material and interpolation model

Considering the bandgap characters of the out-of-plane mode PnCs mainly depend on the mass densities and shear modulus of the component materials, adjusting the shear modulus of the MR material is the key step to achieve the tunable bandgap. For the MR material, the shear modulus is composed of storage modulus $G_{\text{s}}$ and loss modulus $G_{\text{l}}$. In this paper, we ignore the relationship between shear modulus and excitation frequency, and the shear modulus of the MR elastomer can be given as [46,47]

where $B$ represents the magnetic field intensity. For instance, the shear modulus of the MR elastomer $G_{\text{MR}}$ = 1.34 MPa under a weak magnetic field (100 G) and $G_{\text{MR}}$ = 2.72 MPa under a strong magnetic field (800 G).

Figure 1a shows a schematic PnCs design composed of epoxy and MR materials. Here, the circular epoxy inclusion with a radius of 0.0075 m is surrounded by MR material matrix in the PnC unit cell (the size of the square unit cell is $a_c=0.02\text{ m}$ and is surrounded by a negligible rubber layer). The shear modulus of the MR material under different applied magnetic field intensities is shown in Fig. 1b. Comparing Fig. 1c and d, a notable difference in dispersion curve can be found by us for the same PnC design. As shown in Fig. 1c, there are two bandgaps at the second band (f = 1.31 kHz to 1.51 kHz) and fifth band (f = 1.86 kHz to 2.02 kHz) under a weak magnetic field (B = 100 G). When the magnetic field intensity rises to B = 800 G, the bandgaps of the PnCs become higher ( f = 1.87 kHz to 2.15 kHz for the second band, and f = 2.65 kHz to 2.87 kHz for the fifth bandgap) and a bit wider (about 40% wider than the case of B = 100 G for the second bandgap). Therefore, designing the topological distribution of the MR-included PnCs under various magnetic field intensities is a feasible strategy to achieve tunable wide bandgaps.a The schematic design of PnCs, the triangle in the unit cell represents the irreducible Brillouin zone; b dependence of storage modulus of MR on applied magnetic field; c dispersion curves of PnCs under weak magnetic field (B = 100 G); d dispersion curves of PnCs under strong magnetic field (B = 800 G).

As seen in Eq. (6) and Fig. 1b, when the applied magnetic field changes from 0 to 1000 G, the shear modulus $G_{\text{MR}}$ of the material has a maximum value when the applied magnetic field reaches B = 741.67 G. When the magnetic field strength becomes higher than this extreme point, there must be a smaller magnetic field that can provide the same shear modulus of the MR material. This will make the optimization processes produce the duplicate optimized design and bandgap characteristics under two equivalent external magnetic fields. Hence, the upper bound value of the magnetic field strength variation range is approximately taken as 800 G in the numerical implementations.

To describe the bi-material distribution of the PnC unit cell with pseudo-densities based topology optimization framework, an interpolation scheme for MR and epoxy material candidates is needed by us. In this paper, an artificial MR interpolation model with penalization [46], which is similar to the rational approximation of material properties (RAMP) [48], is adopted to eliminate the “gray” elements that may be generated during the optimization process. With the artificial MR interpolation model, the shear modulus and density of element $e$ can be expressed as

where $x_e$ is the elemental density of the $e$th element for representing the material distribution inside the PnC unit cell. If $x_e=0$, it means the $e$th element fills with material 1 (MR), and for $x_e=1$, it means the $e$th element fills with material 2 (epoxy). Here, $G_1$, $G_2$ and ${\rho }_1$, ${\rho }_2$ are the shear modulus and density of material 1 and material 2, respectively. The symbol $p$ is the penalty factor, which is taken as $p=3$ in this paper to make the design variables have an obvious 0/1 distribution in the optimized solution, and the filtering technology [49] based on Heaviside function is adopted to obtain the clear topological boundary.

3.2 Objective function

For obtaining a tunable and broad bandgap of MR-included PnCs, we need to enlarge the bandgap at any considered applied magnetic field intensity and widen the frequency gap between the higher bands under the largest applied magnetic field intensities and the lower band under the smallest applied magnetic field intensities. To this end, the objective function consists of two parts: maximizing the bandgap and maximizing the tunable range of the bandgaps under various magnetic fields.

To open the wide bandgap under the specified magnetic field $B_{\xi }$, it is necessary for us to maximize the bandgap between two adjacent band orders $j$ and $j+1$. The objective function is thus written as

where $n$ is the number of wave vectors considered, and ${\omega }_j\left(k_i\right)$ is the $j\text{th}$ eigenvalue corresponding to the wave vector $k_i$. Because the objective function is composed of the maximum value of the underlying band and the minimum value of the overlying frequency, the corresponding wave vector point for the maximum value may switch during the optimization process. This will lead to the non-differentiability of the objective function, which is difficult to handle with the gradient-based optimization algorithm. In this paper, the KS aggregated function [45] is selected to make a continuous envelope of the discrete eigenvalues. Hence, the objective function can be expressed as

where ${\alpha }_1$ and ${\alpha }_2$ are the parameters of the KS function. We do not directly deal with the critical frequency corresponding to the wave vector but aggregate a single value close to the minimum or maximum. The wave vector $k$ is restricted to the boundary of the irreducible Brillouin zone (the Γ-X-M triangle shown in Fig. 1a) because the unit cell design has the one-eighth symmetry.

In Eq. (10), the choice of the total number $n$ of the wave vector $k$ needs to balance the envelope accuracy and the analysis cost in the numerical implementation. To the authors’ numerical analysis experiences, when more than ten wave vectors are selected on each boundary, the influence of the wave vector numbers on optimization results is very weak. In this study, 34 wave vectors are selected on the boundary of the irreducible Brillouin zone (10 points on the Γ-X and X-M boundary, 14 points on the M-Γ boundary).

Although the objective function in Eq. (10) is applicable to open the bandgap of PnCs under a specified magnetic field intensity and the tunable bandgap can be realized by changing the applied magnetic field, the tunable range of the bandgaps under various magnetic fields is still not considered and optimized. For widening the tunable range of the designed MR-included PnCs, we also propose the frequency gap between the specified higher band under the largest magnetic field intensity and the lower band under the smallest magnetic field intensity as another part of the objective function as

The objective function in Eq. (11) is also a KS type aggregated function. Herein, the superscripts H and L in ${\omega }_{j+1}^{\text{H}}\left(k_i\right)$ and ${\omega }_{j+1}^{\text{L}}\left(k_i\right)$ represent the eigenvalue of the PnC unit cell under the largest and smallest magnetic field intensities, respectively.

With the two parts of the proposed objective functions (10) and (11), the objective function of the optimization problems for opening wide and tunable bandgaps of MR-included PnCs gives

where m is the number of considered applied magnetic field intensities, and the positive weight coefficient $w$ of the objective function is introduced for balancing the requirements of the bandgap width and the bandgap tunable range.

3.3 Topology optimization formulation and sensitivity analysis

By using the mentioned objective functions, we propose a topology optimization formulation of PnCs to find the layout material distribution with a wide bandgap under various magnetic field intensities in the determined material ratio as

where $N_e$ is the total number of FEs, $V_e$ represents the element volume, $V_{\text{FULL}}$ represents the total volume of the unit cell, and $f_v$ is the volume proportion of material 2. It needs to point out that although the volume constraint is not mandatory for solving the proposed optimization problem, it is still necessary to control the weight of the PnC unit cells and the energy consumption of the applied magnetic field over the MR material.

In this paper, the globally convergent method of moving asymptotes (GCMMA) [48] is used to analyze the optimization problem. The GCMMA method is a gradient-based mathematical induction algorithm that requires sensitivity analysis. The sensitivity of the objective function $\widetilde{f_{B_{\xi }}}$ to the eth element gives

For the objective function $\widetilde{f_{\text{t}}}$, its sensitivity can be expressed as

in the sensitivity expressions (14) and (15), the partial differential of characteristic frequency ${\omega }_j\left(k_i\right)$ relative to design variable $x_e$ is

where $q_j$ is the normalized eigenvector corresponding to the eigenvalue ${\omega }_j$ in the generalized eigenvalue problem (15), and$q_j^{\text{T}}M{q}_j=1$. More details of the sensitivity analysis for the repeated eigenfrequencies of the PnCs can be found in Ref. [50].

Then, the sensitivity of the objective function $f_{\text{D}}$ can be written as

4. Numerical examples

In this section, some numerical examples are given to verify the correctness of the proposed topology optimization model, and the calculation results are also discussed. The whole optimization scheme is implemented on the MATLAB platform. When the relative difference between two adjacent objective functions is less than the specified value, it is considered that the objective function value converges, and the optimization process ends. To verify the reliability of our analysis and optimization results, we used the same material and geometric parameters to compare with the bandgap analysis of the existing literature [51,52] and obtained very close analysis results (the bandgap analysis error is less than 1%). Further, we also verified the bandgap performance of our optimized results basing on the FE software platform COMSOL and acquired consistent analysis results.

4.1 Topology optimization of PnCs considering single magnetic field intensity

We first optimize the bandgap between the first and second bands of PnCs under a specified magnetic field intensity. In all examples, the whole square PnC unit cell is selected as the design domain of the topology optimization. The unit cell has the size of $a_c=0.02\text{ m}$ (surrounded by a negligible rubber layer) and is uniformly discretized with 80 × 80 square elements. The typical design of PnCs with epoxy inclusions (the circular radius is 0.002 m) surrounded by the MR material matrix is selected as the initial design of the optimization, as shown in Fig. 2a. The shear modulus of the MR material is given in Eq. (6), and the shear modulus of the epoxy material is $G_2=1.59\text{ GPa}$, the mass densities of the two materials are ${\rho }_1=3500{\text{ kg m}}^{-3}$ and ${\rho }_2=1180{\text{ kg m}}^{-3}$, respectively. For the initial design, the dispersion curves under two representative external magnetic fields (B = 100 G and B = 800 G) are given in Fig. 2b and c. It is observed from Fig. 2b and c that the expected bandgap characteristics cannot be achieved by either magnetic field.Initial design and its dispersion curves: a initial design of PnC with nine unit cells; b dispersion curves under weak magnetic field (B = 100 G); c dispersion curve under strong magnetic field (B = 800 G).

In the first optimization example, we only consider the objective function in Eq. (10) to open a wide bandgap of the MR-include PnCs at specified applied magnetic field. In the optimization model, the volume proportion constraint $f_v$ for the epoxy material is set as $f_v=0.3$. The aggregation parameters in the objective function are initially set to ${\alpha }_1=-10$ and ${\alpha }_2=10$, and their absolute value increases by 0.5 in each iteration before reaching 20. The optimization process will be terminated when the relative difference of the objective function values between two adjacent iteration steps satisfies the convergence criterion $1\times {10}^{-4}$.

Two specified applied magnetic field intensities (B = 100 G and B = 800 G) are first considered, with the proposed optimization method, the optimized designs, and corresponding dispersion curves are shown in Fig. 3. Notable differences in the optimization solutions of Fig. 3a and d under different magnetic field intensities are observed, which also shows that the magnetic field intensity plays an important role in the optimization of MR-included PnCs. By comparing the dispersion curves of the two optimized designs under different magnetic field intensities in Fig. 3b, c, e, and f, we can find that the optimized designs show better bandgap characteristics under respective magnetic field intensity conditions. Specifically, that the optimized design obtained under B = 100 G in Fig. 3a has bandgaps of 0.34 kHz (f = 1.11-1.45 kHz) and 0.45 kHz (f = 1.57-2.02 kHz) under the magnetic fields B = 100 G and B = 800 G, respectively, while the optimized design in Fig. 3d presents a 0.56 kHz (f = 1.58-2.14 kHz) bandgap under B = 800 G but only has a 0.28 kHz (f = 1.08-1.36 kHz) bandgap under B = 100 G. The bandgaps of the two optimized designs under various applied magnetic fields are also shown in Table 1. It is obvious that the optimized solution obtained under a specified magnetic field intensity does not guarantee good bandgap characteristics under other magnetic field intensities. Optimized designs obtained under single magnetic field intensity and corresponding dispersion curves: a optimized design obtained under weak magnetic field (B = 100 G); b dispersion curve for the optimized design in a at B = 100 G; c dispersion curve for the optimized design in a at B = 800 G; d optimized design obtained under strong magnetic field (B = 800 G); e dispersion curve for the optimized design in d at B = 100 G; f dispersion curve for the optimized design in d at B = 800 G.Bandgaps of optimization results under different magnetic fields

4.2 Tunable bandgap design of PnCs under different magnetic fields

In this section, we consider the tunable bandgap design of MR-included PnCs simultaneously considering several magnetic field intensities (B = 100, 400, and 800 G). Here, the objective function in Eq. (12) is considered to open a tunable and wide bandgap at all representative magnetic field intensities. In this part, the weight coefficient $w$ of the objective function is taken as $w=1$. With the same convergence criterion as the previous example, the optimization process converged after 69 iterations, the optimized designs, and the iteration histories are given in Fig. 4a and b. It is shown that the optimized solution (Fig. 4a) for the tunable bandgap design has notable differences with the optimized designs (Fig. 3a and d) obtained with the single magnetic fields. Figure 4c and d show the dispersion curves of the optimized design under the magnetic field of B = 100 G and B = 800 G, respectively. For the tunable bandgap design, it has a bandgap of 0.33 kHz (f = 1.13-1.46 kHz) under the magnetic field B = 100 G. Although its bandgap is narrower than that of the optimized design (Fig. 3a) under the single specified magnetic field B = 100 G, it still has a considerable bandgap. When B = 800 G, the tunable bandgap design has the bandgap of 0.48 kHz (f = 1.57-2.05 kHz). Figure 4e shows the variation of bandgap start and stop bandgap frequencies with various magnetic fields in the optimized design. It is worth mentioning that the tunable bandgap optimized design present larger and more uniform bandgap characteristics under different magnetic field intensities. The whole bandgap range for the optimized design is 0.92 kHz (f = 1.13-2.05 kHz).Optimized results of PnCs considering various magnetic fields: a optimized design; b iteration histories; c dispersion curves at B = 100 G; d dispersion curves at B = 800 G; e tunability of bandgap with various magnetic fields.

To study the influence of weight coefficient $w$ in Eq. (12) on the optimization results and bandgap characters, we take another three weight coefficients, $w=0.1,\text{ 0}\text{.5, and 5}\text{.0}$. The optimized designs are given in Fig. 5, and bandgap characters (including the bandgap widths and bandgap tunable range) are summarized in Table 2. It is seen that the optimization results have noticeable changes as the weight coefficient varies. When the weight coefficient increases, the tunable bandgap range of the PnCs becomes slightly larger.Optimized results of PnCs considering different weight coefficients: a $w=0.1$, b $w=0.5$, and c $w=5$.Bandgaps of optimization results under various magnetic fields

It is also found from Fig. 4e that, for the same optimized design, the absolute bandgap width increases by 42% when the magnetic field increases from 100 G to 800 G, while the relative bandgap widths, which equals the absolute bandgap width divided by the center frequency of the bandgap, are almost unchanged (relative bandgap width δ = 0.257 under the 100 G magnetic field, and δ = 0.258 under the 800 G magnetic field). It is seen that although the absolute bandgap has increased, its intermediate frequency has also risen at the same time (the starting and ending frequency values are both larger). The reason for this phenomenon may be that there is no significant difference in impedance coefficient between MR material and epoxy, changing the modulus of the MR material cannot have a significant effect on the relative bandgap. While our research mainly focuses on the absolute bandgap and its tunable range, whether the relative bandgap changes do not affect the realization of the target acoustic metamaterial.

To verify the effectiveness of the optimization scheme, a waveguide tube for directional propagation of elastic waves is established by us in the commercial FE analysis software COMSOL to implement the transmission spectrum analysis of finite PnC based structure. The waveguide tube is composed of 10 layers of optimized unit cells arranged along the wave propagation direction, as shown in Fig. 6. Two perfect matched layers on both ends of the analysis model are introduced to eliminate the influence of reflected waves. Periodic boundary conditions are applied to the upper and lower boundaries of the waveguide. The incident wave excitation acts on the left side of the waveguide tube, and the transmission rate $T_c=20\times {\log }_{10}\left(D_{\text{out}}/D_{\text{in}}\right)$ ($D_{\text{in}}$ and $D_{\text{out}}$ are the displacement amplitudes at the output and input ports) of the incident wave is measured on the output port.Model for transmission analysis in COMSOL Multiphysics.

Transmission curves along the $\Gamma \text{-X}$ direction under different magnetic fields in the optimized designs considering different magnetic fields are shown in Fig. 7. A tunable complete bandgap (which are painted as shaded areas in Fig. 7) can be observed from the transmission curves under different magnetic fields, and some other low transmission rate regions can also be found due to directional bandgaps. The designed MR-included PnCs have different complete bandgaps 0.33 kHz (f = 1.13-1.46 kHz), 0.43 kHz (f = 1.48-1.91 kHz), and 0.48 kHz (f = 1.57-2.05 kHz) under three selected magnetic field intensities (B = 100, 400, and 800 G), respectively. To further examine the propagation characteristics of waveguide tube under different magnetic fields, the amplitude fields at six specified typical wave frequencies (500, 1300, 700, 1500, 900, and 1800 Hz) are given by us in Fig. 8. It is seen that the amplitude fields and transmission curve both have good agreement with the bandgap calculation results.The transmission spectrum of waves propagating along $\Gamma \text{-X}$ direction for the PnC shown in Fig. 4a: a B = 100 G; b B = 400 G; c B = 800 G.The elastic wave amplitude fields of the optimized design in Fig. 4a under different magnetic fields: a B = 100 G; b B = 400 G; c B = 800 G.

4.3 Tunable bandgap optimization design for higher band orders

In this subsection, we consider optimization design for opening tunable bandgaps by considering higher band orders (the weight coefficient is also taken as $w=1$). By optimizing the second, third, and fifth bandgaps under various magnetic fields, the optimized designs and corresponding bandgaps are shown in Figs. 9-11. The optimized material distributions of the PnCs in Figs. 9a, 10a, and 11a for second, third, and fifth bandgaps show clear differences, and the corresponding bandgap ranges (as shown in Figs. 9b-c, 10b-c, and 11b-c) are also different that has the potential to open wider and tunable bandgaps. It is also noted that as the increase of bandgap order, the start and stop bandgap frequencies of the optimized designs are also increasing, but the bandgap width is decreasing. This may be because more local modes of the PnCs make the bandgap difficult to be opened as the bandgap order increases. By changing the magnetic field, the bandgap with different start and stop frequencies can be obtained by us to wide-range tunable bandgaps. It should be noted that the 4th bandgap is difficult to be opened with the current material composition and geometric dimensions, because the bandgap topological design and optimization of the PnCs are greatly affected by the initial designs, and a simple initial bandgap configuration (starting design) is sometimes hard to be given, which makes it difficult for gradient-based optimization methods to find a suitable optimized design.Tunable optimized design for opening second bandgap under different magnetic fields: a optimized design; b dispersion curves at B = 100 G; c tunability of bandgap with various magnetic fields.Tunable optimized design for opening third bandgap under different magnetic fields: a optimized design; b dispersion curves at B = 100 G; c tunability of bandgap with various magnetic fields.Tunable optimized design for opening fifth bandgap under different magnetic fields: a optimized design; b dispersion curves at B = 100 G; c tunability of bandgap with various magnetic fields.

To open a wider tunable bandgap, we construct a conceptual MR-included PnCs based device by assembling multiple optimized unit cells (Figs. 4a, 9a, 10a, and 11a) as shown in Fig. 12. The transmission curves in the $\Gamma \text{-}\text{X}$ direction under different magnetic fields are shown in Fig. 13. It can be seen that the conceptual device has a wider tunable complete bandgap frequency ranges (f = 1.13-2.05 kHz) than any PnC composed of single unit cell designs (Figs. 4a, 9a, 10a, and 11a), which proves that the superposition of tunable bandgaps is possible by assembling single optimized designs. Figure 14 shows the amplitude field of the device under the magnetic field B = 800 G, for the incident waves at frequencies of 1800, 2200, 2450, and 2650 Hz the wave propagation is blocked, while at frequencies of 1000 Hz and 3000 Hz, the wave can pass through the device almost without attenuation.The MR-included PnCs based device consisting of optimized designs.Transmission curves of MR-included PnCs based device under different magnetic fields: a B = 100 G; b B = 400 G; c B = 800 G.Amplitude field of MR-included PnCs based device under the magnetic field B = 800 G at different frequencies.

5. Conclusion

This paper presents an optimization formulation and numerical techniques for the topological design of MR included smart PnCs to obtain a tunable and broad bandgap. To this end, a KS aggregated function based bandgap index for balancing the tunable bandgap range and the bandgap widths at different magnetic field intensities are proposed as the objective function of the optimization problem. Here, the bandgap analysis is implemented with the FE method, and the elemental pseudo-densities indicating MR material distribution are taken as the design variables. An artificial MR model with penalization is adopted for interpolating the shear stiffness of the two material candidates. The optimization problem is solved by using the gradient-based mathematical programming method with the derived sensitivity information. The numerical examples confirm that the tunable wide bandgap can be achieved by the optimized designs under continuously changing magnetic field intensities. In particular, the optimized PnC designs can maintain the gradually increasing wide bandgaps as the applied magnetic field intensity increases, which may be good bandgap characters and has application potential in the design of wave propagation devices. Also, the optimization solutions for high-order bandgap optimization of the MR included PnCs show the possibility of tuning the bandgap of the PnCs in a wider frequency range. Based on the optimized designs under different target bandgap orders, a conceptual MR-included PnCs based tunable and wide bandgap device is reported. The proposed optimization model and solution methods can also be extended to the topological design optimization of smart PnCs with ER fluids.

Executive Editor: Zishun Liu