Application Of Generalized Multipole Technique In Band Structure Calculation Of Two-dimensional Phononic Crystals

Generalized multipole technique (GMT) is a kind of stable and flexible numerical methods which can handle various complex boundary value problems easily. Since the method was proposed in1980s, it has been widely used in calculation of electromagnetic wave scattering, acoustic/elastic wave scattering, photonic crystals and photonic crystal waveguide. However, due to the complexity of the acoustic/elastic wave propagation in periodic structures, no application of GMT in phononic crystals (PCs) has been reported. PCs are a kind of axoustic functional composite materials which have periodic structures and exhibit elastic wave band gaps. The existence of band gaps provides broad prospects for PCs in engineering application such as sound insulation, vibration damping and the design of acoustic devices. So, calculatting of band structures and analysis of band become the key topic in the research field of PCs. Although several numerical methods have been developed to calculate the band structures, each of them has its disadvantages. In order to make up the deficiencies of the existing methods and rich the application range of GMT, new numerical methods based on the GMT are proposed to calculate the band structures of various two-dimensional (2D) PCs system in this thesis. The main contents include:(1) Two numerical methods, i.e. multiple multipole (MMP) method and multiple monopole (MMoP) method, are proposed based on the GMT to calculate the band structures of scalar waves in2D PCs. The detailed calculation of the band structures of the anti-plane shear wave mode in solid/solid systems and the acoustic modes in fluid/fluid systems were performed by MMoP method, and the results were compared with those of existing methods. The validity and accuracy of the MMoP method and the influence of model parameters to the results were discussed.(2) Taking into account the fluid-solid interface conditions strictly, the MMoP method was extended to calculate the band structure of2D solid/fluid systems. The validity and applicability of the method were discussed by calculating the typical solid/fluid systems with various acoustic impedance ratios.(3) The MMoP method was extended to calculate the band structures of the plane mixed wave modes in2D solid/solid, hole/solid and fluid/solid systems.(4) To verify the validity and applicability of the MMoP method for the system with imperfect interfaces and to wxpand its applications, the MMoP method was extended to calculate the band structures of2D nano-sized PCs with circular or non-circular scatterers by considering the surface/interface effect based on the basis of surface/interface elasticity theory.The results show:(1) Both MMP and MMoP methods can yield accurate results for the scalar waves modes in2D PCs. The latter one is much simpler and faster than the former one and exhibits some advantages such as good stability, fast convergence, etc. The MMoP method is able to calculate the systems with arbitrary shaped scatterers, breaking the limitation of the traditional multiple expansion method (such as multiple scattering theory method and Dirichlet-Neumann map method) which can only deal with circular (or spherical) scatterers. In addition, this method can calculate the band structures at any frequency intervals, without calculating from zero frequency.(2) By establishing the strict interface conditions and defining an appropriate eigenvalues searching function, the MMoP method can calculate the band structures of various solid/solid, hole/solid, solid/fluid and fluid/solid PCs accurately and effectively, exhibiting the high adaptability. For the fluid-solid coupling system with small acoustic impedance ratio between the scatterer and matrix (such as rubber/Mercury and air/Aluminum systems), the MMoP method can calculate not only the bands corresponding to the Bragg scattering modes but also the physical flat bands corresponding to the locally resonant modes.(3) For a certain kind system (solid/solid, hole/solid, solid/fluid or fluid/solid), all computing model parameters in the MMoP method only depend on the geometry of the system under consideration and have nothing to do with the material properties. This makes it more convenient to deal with the system having a large acoustic mismatch.(4) The MMoP method can calculate the band structures of nano-sized PCs with circular or non-circular scatterers effectively and accurately by taking into account the surface/interface effect. Furthermore, the calculation process of the method is the same whether the surface/interface effect is considered or not, except the boundary conditions on the surface/interface. This provides a great convenience for programming, and avoids the trouble in the finite element method different special surface/interface elements should be developed for various surface/interface conditions.