Topology optimization of magnetorheological smart materials included PnCs for tunable wide bandgap design
doi: 10.1007/s10409-021-09076-5
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摘要: 由于可谐调声子晶体具有有效操纵声波和弹性波的能力, 因此其设计和应用受到越来越多的关注. 本文研究了由磁流变材料构成的声子晶体的拓扑优化设计, 以打开声子晶体可调谐的宽禁带. 其中, 声子晶体的带隙可谐调性是通过磁流变材料在不断改变的外加磁场下剪切模量的变化来实现的. 代表声子晶体内双材料分布的伪单元密度被视为设计变量, 并采用人工磁流变惩罚模型进行插值, 提出了一种包络磁流变智能声子晶体带隙宽度和带隙可调范围极值的凝聚函数作为目标函数. 在此背景下, 对目标函数的灵敏度进行了分析, 并用基于梯度的数学规划方法解决了优化问题. 数值算例表明了该优化方法的有效性, 优化结果在不同磁场下具有可谐调、稳定的宽带隙特性. 本文还探讨了基于可谐调优化声子晶体的器件, 该器件可以提供更宽的可谐调带隙范围.Abstract: Design and application of tunable phononic crystals (PnCs) are attracting increasing interest due to their promising capabilities to manipulate acoustic and elastic waves effectively. This paper investigates topology optimization of the magnetorheological (MR) materials including PnCs for opening the tunable and wide bandgaps. Therein, the bandgap tunability of the PnCs is achieved by shear modulus variation of MR materials under a continuously changing applied magnetic field. The pseudo elemental densities representing the bi-material distribution inside the PnC unit cell are taken as design variables and interpolated with an artificial MR penalization model. An aggregated bandgap index for enveloping the extreme values of bandgap width and tunable range of the MR included smart PnCs is proposed as the objective function. In this context, the sensitivity analysis scheme is derived, and the optimization problem is solved with the gradient-based mathematical programming method. The effectiveness of the proposed optimization method is demonstrated by numerical examples, where the optimized solutions present tunable and stably wide bandgap characteristics under different magnetic fields. The tunable optimized PnCs based device that can provide a wider tunable bandgap range is also explored.
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Key words:
- Topology optimization /
- Tunable bandgap /
- Phononic crystals /
- Magnetorheological material
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1. 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). 
2. 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).
3. 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.
4. 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.
5. Optimized results of PnCs considering different weight coefficients: a
, b , and c . 
7. The transmission spectrum of waves propagating along
direction for the PnC shown in Fig. 4a: a B = 100 G; b B = 400 G; c B = 800 G. 
8. 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.
9. 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.
10. 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.
11. 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.
13. Transmission curves of MR-included PnCs based device under different magnetic fields: a B = 100 G; b B = 400 G; c B = 800 G.
14. Amplitude field of MR-included PnCs based device under the magnetic field B = 800 G at different frequencies.
Table 1. Bandgaps of optimization results under different magnetic fields
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 
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Table 2. Bandgaps of optimization results under various magnetic fields
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
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