Bandgap Tuning of Phononic Crystals Based on Grading-fractal Structural Scheme and Topology Optimization Design
Phononic crystals(PnCs)are a kind of flinctional spatial periodicity artificial structures/materials which have the unique properties of elastic absolute band gaps(ABGs),within which the propagation of elastic waves is prevented.Due to this special physical characteristic,phononic crystals have a vast application prospect in multi-flmctional fields,particularly in acoustic fields and reduction of vibration and noise.Then how to design PnCs to obtain expected ABGs is of great significance for its application.In order to break through the limitations of artificial empirical designs,combined with natural fractal structures and bidirectional evolutionary structural optimization algorithm,the structural design of PnCs for its bandgap properties are deeply studied.The main contents and conclusions include:(1)Grading PnCs are proposed by introducing the circular porous vertically distributed in the unit cells.The effects of the grading geometry parameters on the band structure in porous phononic crystals are discussed.Combined with vibration modes of the unit cell corresponding to the absolute band gap(ABG)edges,the mechanism of the bandgap generation is investigated.The numerical results show that the grading structure greatly decreases the critical porosity for the opening of the ABGs,along with the increase of the porosity,wider ABGs could be obtained at lower frequencies.The band structure can be tuned by changing the geometry parameters of the grading porous.(2)Fractal porous phononic crystals are proposed by introducing Sierpinski triangle fractal structure,strictly defined Sierpinski fractal triangle PnCs and fractal grading PnCs are designed respectively.The influence of the porosity and fractal hierarchy on the elastic wave dispersion in PnCs are investigated.Vibration modes of unit cells at the edges of ABGs are analyzed to manifest formation mechanism of bandgaps.The results show that for the strictly defined Sierpinski fractal triangle PnCs,the increase of the hierarchy at the initial stage increases the sensitivity of bandgap central frequency to the porosity,but this sensitivity will be weakened by rnrther increase of the fractal hierarchy.On the same hierarchy,Sierpinski equilateral triangle fractal PnCs is favorite to open wider ABGs;whilst lower ABGs at lower porosity are easier to be obtained in Sierpinski right-angled isosceles fractal PnCs.For fractal grading PnCs,the distribution trend of the band gap in the frequency domain and the distribution trend of the hole in the space have a certain degree of similarity,higher fractal hierarchy will result more discrete distribution of pores and band gaps in the space and frequency domain respectively.(3)Based on bidirectional evolutionary structural optimization(BESO)algorithm,the optimization models of the two-dimensional solid-solid phononic crystals to maximize the ABG are established.The effects of different matrix/scatter materials,material volume fraction and initial topological design on the optimized PnC unit cells and band structures are discussed.PnC unit cells possessing wide bandgap width between two special adjacent bands are designed.The results show that the proposed BESO strategy can be used to seek the optimal design of PnCs in an effective and efficient way.The optimization algorithm has less dependence on the initial structure,relatively wide ABGs are obtained under different objective functions and constraint functions.All optimized PnCs show some similar topology characteristics.For the in-plane model,PnCs are made of some mass blocks evenly periodically distributed in the matrix.For the out-of－plane model,the total number of the scatters in one unit cell equals the serial number of the ABGs.(4)Topology optimization design of phononic crystals with reserved inclusions are established.For a given structure containing reserved inclusions,by introducing isolation stop and design areas,the topology optimization designs of PnCs are conducted in the limited space under the given structural schemes,large bandgap between two special adjacent bands are obtained.By introducing different isolation strip widths and different materials in the inclusions and isolation strip,the influence of material and topological parameters on the optimization results is discussed in detail.The results show that for the PnCs with reserved inclusions,there was a material gravitation phenomenon of the inclusions in optimization process.Fill materials tend to be distributed around the inclusions,this phenomenon will be more obvious with higher inclusion density and narrower isolation strip.This work will provide theoretical guidance for the design of phononic crystals in multifunctional applications.