Research On Several Problems Of Structural Topology Optimization Under Harmonic Excitations

Structural dynamic response widely exists in high-end equipment,precision instruments,civil engineering,aerospace and other industries.On the other hand,researchers often obtain the best performance of structure design by means of structural optimization methods.This makes structural topology optimization under harmonic excitations become a subject under attention.In this work,a systematic study on this issue is carried out.It is found that the existing model of structural topology optimization under harmonic excitations is very time consuming when it is applied to deal with large-scale problems.This may greatly increases the period of structure design.Further research on this model shows that the accuracy of the method for analyzing structure harmonic responses is not enough that makes the optimized results less reliable and controllable.It will restrict the development of this research field.Focusing on the above key issues,the main contents of this paper are as follows:1.The existing topological optimization model of large-scale structure harmonic response is studied,i.e.structural topology optimization based on modal acceleration method.It is found that optimized results often need to iterate hundreds of steps.In the topology optimization procedure,one of the most time-consuming part is computing structural harmonic response by modal acceleration method.This work introduces mature structural reanalysis methods.It can reduce the number of stiffness matrix decomposition during the topology optimization procedure and improve computational efficiency.Structural reanalysis method make full use of the information obtained by calculating the harmonic response of the reference structure,including the decomposed stiffness matrix,the pseudo-static response and the results of the modal analysis to obtain the harmonic response of the updated structure approximately used by modal acceleration method.The sensitivity analysis based on the adjoint variable method is derived.Numerical examples demonstrate the applicability of structural reanalysis methods in such topology optimization problems.Both the objective functions of dynamic compliance and displacement amplitude of a certain degree of freedom have good approximations.2.Further study features of the modal acceleration method to solve harmonic responses of structures.It is found that the error of the solution increases as excitation frequency increases.On the other hand,modal acceleration method cannot adaptively determine the number of lower modes required for modal superposition,which makes the CPU time of modal analysis beyond controllable.Therefore,this work introduces the combined method of structural harmonic response analysis,i.e.modal superposition combined with a model order reduction method,and applies the method to the structural topology optimization under harmonic excitations.By using the Sturm sequence,the combined method can self-adaptively determine the number of lower modes for modal superposition,as modes re-distribute due to updating of design variables.This step effectively controls the CPU time required for computing modes.The effect of truncation modes on harmonic responses of structures is approximated by a model reduction method.The reduction bases are the search vectors obtained by applying precondition conjugate gradient method to solve the response of the highest point in the frequency range.The Galerkin projection is used to derive responses of all the points in the frequency range.The sensitivities of objective functions are obtained by using the adjoint variable method.The numerical examples show that the combined method is superior to the modal acceleration method in terms of efficiency and accuracy,and sensitivity results are more accurate.Therefore,results of structural topology optimization under harmonic excitations based on the combined method is more reliable than the modal acceleration method.3.A new method to calculate harmonic responses of structures with proportional damping is proposed,which is named hybrid iteration method.Moreover,it is applied in the structural topology optimization under harmonic excitations.The hybrid iteration method includes modal superposition and a new proposed iteration algorithm.As the combined method,the Sturm sequence is applied to self-adaptively determine the number of lower modes required for modal superposition before modal analysis.The contribution of the higher modes is approximated by a partial sum of power series obtained from a new proposed iteration algorithm.The power series are obtained by a new proposed iteration format solving the harmonic response of the largest excitation frequency.In this way,harmonic responses of all the desired frequency points in the frequency range arrive at a given accuracy.This method has characteristics of self-adaptability,stable precision and theoretical convergence.The results of sensitivity analysis are obtained by using the adjoint variable method.In numerical examples,harmonic response analysis based on the hybrid iteration method are compared with the results obtained by full method to verify the accuracy and efficiency of the proposed method and its applicability in topology optimization.