Citation: | K. Liang, J. He, Z. Jia, and X. Zhang,Topology optimization of magnetorheological smart materials included PnCs for tunable wide bandgap design. Acta Mech. Sin., 2022, 38, |
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
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 unit
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.
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,
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
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
where
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.
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
where
Figure 1a shows a schematic PnCs design composed of epoxy and MR materials. Here, the circular epoxy inclusion with a radius of
As seen in Eq. (6) and Fig. 1b, when the applied magnetic field changes from 0 to 1000 G, the shear modulus
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
where
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
where
where
In Eq. (10), the choice of the total number
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
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
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
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
For the objective function
in the sensitivity expressions (14) and (15), the partial differential of characteristic frequency
where
Then, the sensitivity of the objective function
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.
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
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
Two specified applied magnetic field intensities (B =
Magnetic fieldintensity (G) | Bandgap of weakmagnetic field (kHz) | Bandgap of strongmagnetic field (kHz) |
100 | 0.347 | 0.279 |
200 | 0.378 | 0.362 |
300 | 0.408 | 0.437 |
400 | 0.429 | 0.494 |
500 | 0.443 | 0.530 |
600 | 0.452 | 0.553 |
700 | 0.456 | 0.563 |
800 | 0.456 | 0.564 |
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
To study the influence of weight coefficient
Weight factor | Bandgap range (kHz) | Weak magnetic field (kHz) | Strong magnetic field (kHz) | |||||
1st band | 2nd band | bandgap | 1st band | 2nd band | bandgap | |||
0.1 | 0.905 | 1.121 | 1.460 | 0.339 | 1.569 | 2.026 | 0.457 | |
0.5 | 0.912 | 1.127 | 1.458 | 0.331 | 1.565 | 2.039 | 0.474 | |
1.0 | 0.918 | 1.132 | 1.462 | 0.330 | 1.574 | 2.050 | 0.476 | |
5.0 | 0.942 | 1.137 | 1.485 | 0.348 | 1.598 | 2.079 | 0.481 |
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
Transmission curves along the
In this subsection, we consider optimization design for opening tunable bandgaps by considering higher band orders (the weight coefficient is also taken as
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
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.
This work was supported by the National Natural Science Foundation of China (Grant No. 12102079).Executive Editor: Zishun Liu
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