Topology Optimization Of Vibrating Structures And Phononic Crystals With Specific Band Gaps

The eigenfrequency and eigenmode are the inherent nature of the structure.Through reasonable design and adjust their value and mode,could achieve vibration and noise reduction and waveguide wave absorption of the structure,which has important application value in aerospace,submarine stealth and MEMS devices.By adjusting the natural frequency of the structure to keep it away from the specific external load frequency range,could avoid structural resonance,reduce the amplitude of the structure,and improve the life of the structure.That is,the structure needs to have a specific natural frequency band gap.Topology optimization is an important tool of structural innovation design to improve the specific properties of structures by designing the optimal spatial distribution of material.In this thesis,the vibration structure and phononic crystal design with the specific band gap are studied and the corresponding topology optimization method are proposed.The main research works are as follows:(1)A continuous and differentiable mathematical representation model of frequency band constraints is established to achieve topology optimization design with specific frequency bands.When the existing topology optimization methods use frequency band constraints,only the maximum optimization design of two adjacent frequency is considered,and the order of natural frequency needs to be determined in advance.Therefore,the automatic imposition of specific band gap constraints cannot be realized.According to this problem,a topology optimization design method of vibration structure with specific frequency band is proposed in this thesis.Based on the modified Heaviside function,a continuous and differentiable mathematical formulation of frequency band constraints is established,which could integrate into the topology optimization model to maximize the fundamental frequency with volume constraint and frequency band constraint.Moreover,this method can deal with multi-frequency band constraints and obtain a design scheme with arbitrary band constraints.The effectiveness of the proposed method is verified by several typical examples.The natural frequency of the designed structure can avoid the working frequency and prevent resonance.(2)A topology optimization method of phononic crystals with specific band gaps is established.Phononic crystals could block the propagation of elastic wave/acoustic wave in a specific frequency range through periodic arrangement of unit cells,which widely used in acoustic functional devices,vibration and noise reduction and other fields.However,the existing research on phononic crystal band gap is still limited to the optimization model considering the maximum adjacent two-order band gaps,which is difficult to adapt to the practical requirements of engineering structures and devices with the specified frequency band gaps.Compared with the application of structural band constraints,the difficulty of specific band gap design of phononic crystals is that the stiffness matrix changes with the change of wave vector,which is a typical multi-condition eigenvalue problem.The direct application of frequency band constraints established in Study 1 often meets difficulties such as convergence difficulties.Therefore,based on the frequency band constraint,this thesis proposes an isolateral frequency constraint to ensure the convergence stability of the optimization problem,and realizes the optimization design of specific band gap of phononic crystal structure.The accuracy of the design scheme is verified by frequency sweep analysis.(3)A nonlinear eigenvalue topology optimization method is established for structures with frequency-dependent materials.In practical engineering structure applications,the elastic modulus of materials often has a nonlinear relationship with frequency,such as viscoelastic materials and fluid-structure coupling problems.However,most of the existing dynamic topology optimization assumes that the elastic modulus of materials is independent of the frequency,and the resulting structural eigenfrequencies are obtained by solving a linear eigenvalue problem.However,the frequency-dependent materials makes modal analysis of structures a nonlinear eigenvalue problem,which brings challenges to topology optimization design.Firstly,based on homotopy algorithm and perturbation expansion technique,the continuous asymptotic numerical method is used to establish the solving strategy of nonlinear eigenvalue equation.Several schemes are proposed to improve the computational accuracy,applicability,and robustness of the method for the application in topology optimization,including Faà di Bruno’s theorem,bisection method,and inverse iteration based eigenvector modification method.Numerical examples verify the effectiveness of the proposed method,and show that the frequency-dependent material properties have a large influence on the results of topology optimization.