Uncertain Numerical Analysis And Topology Optimization Of Two-dimensional Phononic Crystals

Phononic crystals are periodic composites or structures with bandgap characteristics.Within the bandgap frequency range,the propagation of acoustic or elastic waves will be partly or totally prohibited.But without the bandgap frequency range,the propagation of acoustic or elastic waves will be lossless due to the dispersion.Thus,phononic crystals can be widely applied for the reduction of vibration and noise in engineering structures.In view of this,the research on the bandgap characteristics and topology optimization of phononic crystals has great significance for reducing the structural vibration and noise.At present,the studies of the bandgap characteristics mainly rely on the determined system parameters and structural models.However,due to the environmental changes,material inhomogeneity,load variations,and manufacturing errors,uncertainties are generally existed in phononic crystals in practical engineering.These uncertainties may have a non-trivial impact on the bandgap characteristics.Therefore,it is necessary to study the uncertain numerical analysis of phononic crystals.Supported by the National Natural Science Foundation of China(No.71271078),this dissertation mainly focuses on the uncertain numerical analysis and topology optimization of phononic crystals.Firstly,an improved fast plane wave expansion method is proposed.Secondly,combined with the Chebyshev polynomial expansion method,an interval uncertain numerical analysis method is developed based on the interval model.Then the effects of uncertainties on the bandgap characteristics are obtained.After that,the hybrid discretization model is introduced for the discretization of phononic crystals.Finally,based on the genetic algorithm,a microstructural topology optimization algorithm is proposed for the interval uncertain phononic crystals.The main research work in this dissertation is as follows:(1)An improved fast plane wave expansion method is proposed for the bandgap analysis of phononic crystals.The off-line calculation and storage of Fourier series expansion about the material parameters are developed based on the displacement translation characteristics.Then the conventional plane wave expansion method has been improved,which is only applied for phononic crystals with regular shape scatterers.Thus,the bandgap analysis of solid/solid phononic crystals with any shapescatterers is realized.Furthermore,by eliminating the jump discontinuity points on the boundary of different materials,the continuity of the Fourier series expansion about the material parameters is improved.Meanwhile,as the number of wave vectors decreases,the efficiency of the fast plane wave expansion method is improved.Numerical results show that the analytical accuracy of the proposal method is equal to the finite element method,but the computational efficiency of the proposal method is much higher.Thus,the improved fast plane wave expansion method is suitable for the bandgap analysis of solid-solid phononic crystals with complex shape scatterers.(2)A sparse point sampling based interval Chebyshev polynomial expansion method is developed for the response analysis of uncertain system or structure.The interval variable is used to describe the uncertain parameter when the sample data is limited.Then an uncertain analysis model based on Chebyshev polynomial expansion is constructed by combining the sparse point sampling strategy and numerical analysis method.After that,the sampling points generated randomly are substituted into the uncertain analysis model to calculate the variation range of system response.Finally,the response characteristics of interval uncertain system are obtained.Numerical results show that the sparse point sampling based interval Chebyshev polynomial expansion method can achieve the response characteristics of interval uncertain system effectively.Compared with the interval Monte Carlo method,the computational efficiency of the proposed method is much higher.(3)The interval Chebyshev surrogate model of band structure is developed for the uncertain numerical analysis of phononic crystals.The interval model is introduced into the phononic crystals,and the interval variable is used to describe the uncertain material parameter.Then,based on the sparse point sampling strategy,combined with the improved fast plane wave expansion method,the effects of the uncertain material parameters on the band structure and bandgap,especially in the bandgap width and frequency range,are studied.Numerical results show that as the uncertainty increases,the upper bound of the bandgap decreases,but the lower bound of the bandgap increases.Thus,the bandgap width decreases gradually.(4)A hybrid discrete model is proposed to discretize the microstructure unit of phononic crystals.Based on the accumulation strategy,the existence of isolated,redundant,and cavity are found,and the continuity of structure is improved.Then the hybrid discrete model is introduced to eliminate the point connection between the design units in the phononic crystals.Finally,the continuous discretization of the microstructure unit is completed.Numerical example shows that the hybrid discretemodel solves the structural discontinuity in the topology optimization and improves the manufacture of the optimized results.(5)The optimization model of the interval uncertain phononic crystals is constructed,and its corresponding optimization algorithm is proposed.In the optimization model,the material distribution of the microstructure unit is regarded as the design variable,and the bandgap width or microstructure unit quality is defined as the objective function.In addition,the bandgap frequency range is considered as the constraint condition.Then,combined with the adaptive genetic algorithm,the reliability based topology optimization of phononic crystals is realized.Numerical results show that the interval uncertainties involved in microstructural topology optimization have some effects on the design of phononic crystals.Moreover,the reliability optimum design with interval uncertainty usually can achieve better performance than the deterministic optimum design.The research work mentioned above may provide the theoretical basis and analysis method for the bandgap analysis,uncertainty research and topology optimization.Meanwhile,it also provides the guidance and reference to the practice engineering design.Therefore,it has good engineering potential application value.