Structural Topology Optimization Considering Natural Frequencies

The tendency of the modern equipment to be lightweight and high-precise has made the dynamic performance of the structure become more and more important.As one of the most important characteristics,the natural frequency can largely determine the dynamic response,and its optimization may improve the performance of the whole structure.Using topology optimization to find the material distribution of the structure to achieve the best natural frequencies is a scientific,systematic and efficient way for structural design,which can not only fully utilize the material,but also reduce the dependence on the engineer's experience.The solving of the generalized eigenvalue problem dominates the computing cost of the frequency topology optimization.Generally speaking,an iteration method should be applied in order to obtain natural frequencies and corresponding modes with enough accuracy.Because the implementation of the topology optimization is also an iteration process,it will lead to huge calculation when the scale of the finite element model increased.This paper proposed a method using successive iteration of analysis and design(SIAD)for topology optimization considering natural frequency.By using the Rayleigh quotients as approximations of the natural frequencies and achieving sequential approximation of the eigenpairs along with the topological evolution of the structure,the method avoids solving the time-consuming eigenvalue problem in each design iteration.When the maximum change of design variables is restricted in each iteration,this method can achieve the convergence of the approximate eigenpairs and material distribution simultaneously.Besides,based on the SIAD method,a multi-step relay method is proposed in this paper,which can further reduce the amount of computation required to update the approximate modes.This method divides the optimization process into several steps,and gradually refines the mesh of each step with the process of the optimization.Projection procedures are implemented to keep the continue change of the approximate modes and the material distribution.It can be seen from numerical examples that the method presented in this paper can effectively reduce the amount of computation required in structural optimization,while high quality results can still be obtained.This has a positive effect on reducing design cost and promoting the use of optimization results in engineering application.