Optimization Design Of Elastic Wave Band-Gap Material/Structures

Elastic wave band-gap materials, also called phononic crystalspossessing periodic distributed elastic constants and densities can attenuate elastic wave propagation within certain frequency bands, which have great application potentials in vibration and noise reduction, wave guide, and new acoustic devices. Thus, it is of great theoretical and engineering values to find and investigate the band-gap properties of the present materials and the influence from the geometric parameters, to establish material design methodology for specific band-gap properties, and to explore new high performance acoustic devices based on the band-gap characteristics such as single direction wave guide and adaptive filters.In this dissertation, firstly, the elastic wave band-gap properties and the influence from the configurations of microstructures were analyzed for advanced functional2D lattices. Secondly, the optimization model and solving methods for continuum materials and layered materials to achieve specific elastic wave band-gap characteristics were established. Finally, the design method for new elastic wave filters by utilizing nonlinear band-gap properties. The detailed works are listed as follows.(1) Analysis and configuration design for2D band-gap lattices. According to the Floquet-Bloch theory, the band-gap behaviors have been analyzed for seven typical2D lattices includingtriangle, ground, square, hexagon, reverse hexagon, kagome and diamond shapes, and influence principle of the material structure configuration on the band-gap properties were obtained. Then, a configuration selection model for2D lattices was established with the aim to achieve the wide width band-gap in low frequency range. The simulation results show that the latticematerial with reverse hexagon configuration microstructure obtains the best band-gap property in low frequency range. It can also be concluded that it will be much easier for out-of-plane waves to achieve band-gap than that of in-plane elastic waves at low frequency range. Furthermore, one useful principle can be obtained and numerically validated that the upper and lower bounds of the low-frequency band-gaps are directly proportional to the cross section length of the microstructures of the material, and inversely proportional to the square of the beam length, with which, the multi-functional2D lattice with specific band-gap character can be designed by varying the geometric parameters of the microstructure configuration.(2) Topology optimization of multi hierarchical unit cell for unidirectional band-gap materials. To obtain innovational structural configurations to improve the properties of unidirectional band-gap materials, considering the in-plane coupled elastic wave, the topology optimization model for unidirectional band-gap materials was established with the goal of maximizing the band gap width in a certain direction by extending the design domain from one dimension to two dimensions and by using the Pseudo-density to depict the material distributions in the wave propagating direction and its perpendicular direction. Hence, the periodic hierarchical structures were obtained which can be regarded not only as layered structures with alternate appeared layered structures in the wave propagating direction, but also as the composite materials with alternate distributions of two phase materials in its perpendicular direction. The numerical results show that the proposed periodic hierarchical two phase materials exhibit wider band gaps than that of the multilayered materials optimized by a one-dimensional design domain. Due to the variation of the tensile modulus and the shear modulus ofthe anisotropic combined layers, the overlap frequency ranges between the shear band gap and transverse band gap was expanded, thus widening the coupled wave band-gap.(3)New optimization model for maximized attenuation coefficients in specified frequency ranges.Firstly, the relation between cosine function of the wave number and the attenuation coefficient is analyzed. Secondly, the band gap characteristic is depicted by the integration of the cosine function of the wave number within given frequency ranges. Thirdly, with the idea of maximizing the proposed target by maximizing attenuation coefficient in a given frequency range, the optimization formulation for achieving band gap in the two-phase periodic layered material in specified frequency ranges is proposed, with which the size constraint microstructure configuration of the band-gap material was obtained. Importantly, when the given frequency range corresponds to the pass band, the solving difficulty induced by the zero sensitivity of the objective function can be effectively avoidedby the newly proposed model taking the cosine function of the wave number as the objective function instead of the attenuation coefficient or the imaginary part of the wave number. With the proposed optimization formulation, the non-band-gap materials can be transferred to band-gap materials effectively.(4)Integrated design optimization of the cell size and the configuration for layered band-gap material. Considering the elastic wave attenuation in finite-dimensional structures, the objective function is defined as maximized attenuation per unit length at a target frequency or within desired frequency ranges. By using the cosine function of the wave number to smooth and represent the attenuation coefficient, the size and the configurations of the unit cell of the periodic layered two-phase material were integrated designed by integrating the ’two-step optimization strategy’ and the analytical sensitivity based optimization method. Comparing with the reference results, the newly proposed optimization model and the solving method can significantly improve the optimization efficiency for more accurate cell parameters. Additionally, the distribution of the elastic wave in the frequency domain can be involved in the optimization model to achieve the maximum elastic wave attenuation for specific probability density distribution at given frequency ranges.(5) Design of amplitude dependent band-gap filters. In nonlinear systems, the band-gap characteristics are dependent upon not only the properties of microstructures itself, but also the vibration amplitudes. Considering the influence of the nonlinearity on the elastic wave dispersion, an amplitude dependent band-gap filter was design by using the nonlinear band-gap materials. By introducing the parallel connected positive and negative springs to simplify the proposed nonlinear cell structure. The propagation properties of elastic wave in periodic nonlinear systems are investigated by the modified L-P method. The dependent relations among the structure nonlinearity, vibrating amplitude and the band-gap frequency range were theoretically analyzed, and the influence of the vibration amplitude on the filter properties of the periodic structures was discussed. The investigation of such nonlinear band-gap properties offers a theoretical guidance for the design of new ultra-low frequency vibration isolators and amplitude dependent band-gap acoustic filters.