Dynamic Topology Optimization Of Structures And Phononic Crystals Considering Eigenvalues

The design of dynamic performance is considered as one of the key issues of structure dynamic optimization,which includes the eigenvalues and eigenmodes of the structures.It has important applications in the fields of vibration reduction and avoidance of resonance effects.The precondition dynamic topology optimization considering eigenvalues is to obtain the natural frequency and vibration mode of the structure.In dynamic analysis,the inverse iteration or subspace iteration method is generally used to calculate the natural frequency and mode,which usually starts from the initial trial modes and obtains the accurate natural frequencies and modes by iteration.When the scale of the structure increases,this calculation is very time-consuming.With the continuous enhancement and development of computing power and computing methods,people expect to extend the dynamic topology optimization method to large-scale structural design.When the analysis scale of the structure increases,the calculation burden of the structure analysis increases significantly.This limits the dynamic analysis and optimization of large-scale structures.Therefore,it is crucial to find an efficient method to calculate natural frequencies and modes for dynamic optimization of large-scale structures.The dynamic topology optimization considering eigenvalues can be used not only to improve engineering structures,but also to the research and manufacture of metamaterials.Phononic crystals and acoustic metamaterials are considered as one of them.Phononic crystal is a kind of artificial periodic distribution material,and it is possible to design unit cells with band gap characteristics by optimization methods.With the development of mechanisms and optimization methods,the traditional design methods can no longer meet the design requirements.Therefore,the role of topology optimization is gradually highlighted.The topology optimization can be used to systematically design the unit cell of phononic crystal and change its natural frequencies,so as to obtain a larger band gap characteristic.The design of the band gap of phononic crystals is an important dynamic optimization application field considering eigenvalues.However,there are few existing research and application fields on the partial band gap of phononic crystals,it is necessary to carry out research and exploration on the optimization and application of partial band gap.Moreover,the existing band gap design does not take into account the uncertainty,so its band gap properties will change with the random disturbance of material and geometry.It is necessary to consider the uncertainty in the optimization of the phononic crystal.Based on the above research background,this dissertation has carried out the following research work:(1)The successive iteration of analysis and design considering eigenfrequencies.In the structural analysis process,the approximate natural mode vectors are used instead of the exact ones.The approximate natural mode vectors are updated by inverse-like iteration or subspace-like iteration methods,and the updated approximate natural mode vectors are applied to the optimization process.This method enables the continuous analysis and design process of the structure,which avoids the independence of the two processes.Since the number of updating the approximate natural vectors is equal to the optimization iterations of the structure,the computational pressure of dynamic topology optimization is significantly reduced.This dissertation verifies the effectiveness of this method through several numerial examples,including maximizing the fundamental frequency of the structure,maximizing the high-order natural frequency,and maximizing the natural frequency band gap.This dissertation briefly discusses the convergence of the method,discusses the selection methods of the initial approximate vectors,and analyzes the convergence characteristics of the approximate vectors.(2)The partial band gap topolpgy optimzaiton of phononic crystals.The full band gap of phononic crystals has been widely concerned because it can prevent the propagation of elastic waves in all directions and has wide application prospects.However,due to the narrow application prospect and the complicated design methods,few related studies have been conducted on the partial band gap of phononic crystals,which can prevent specific direction propagation of elastic waves.Topology optimization for maximizing the partial band gap characteristics needs to consider elastic waves in different directions,which means that the design domain has no symmetry.Therefore,the design space is expanded and the difficulty of optimizing unit cell configuration is increased.Meanwhile,the wave vectors in different directions need to be considered separately,so that the objective function and constraint conditions of its optimization formula are more complicated,which affects the convergence of optimization.In this dissertation,a reasonable optimization formula for the maximum partial band gap characteristics is proposed.The aggregation function is used to ensure the derivability of the objective function.The Method of Moving Asymptotes is used to makeoptimization efficient and stable.The Random Morphology Description Functions are used to generate different initial designs to obtain different unit cell designs.Using the designed phononic crystals with partial band gap characteristics,this dissertation explores directional propagation characteristics of the elastic waves in the waveguide(3)The robust topology optimization of phononic crystal considering the uncertainty of material property.The deterministic design of phononic crystal has less robustness of band gap properties.When the material distribution is changed,the band gap of phononic crystal is also changed,thereby affecting its application.It is necessary to consider the influence of uncertainty in the optimization of the band gap of the phononic crystal and improve its robustness.This dissertation considers phononic crystals with uncertain material properties and assumes that their randomness satisfies the Gaussian distribution.The random field is discretized by using the Expanded Optimal Linear Estimation method,and the bandgap response of the structure is predicted by the Polynomial Chaos Expansion method.An objective function including band gap mean and standard deviation is proposed,a sensitivity calculation method for robust design considering uncertainty is derived.The in-plane and out-of-plane mode of two dimensional elastic wave are both considered.The influence of the weight coefficient of the objective function and the coefficient of variation of the random field on the optimization results are discussed.