Frequency Analysis And Topology Optimization Of The Lattice Structures Based On The Multiscale Finite Element Method

As a new type of ultra-lightweight material with designable microstructure,lattice material has superior characteristics such as high specific rigidity,specific strength and heat insulation,and is widely used in aerospace and other fields.When lattice materials are used to form largescale aerospace components,because they contain a large number of periodically distributed microstructures,the use of accurate finite element models for structural analysis and optimization will cause huge computational costs.To this end,the study established an efficient and accurate equivalent analysis method of lattice structure frequency by introducing the Multiscale Finite Element Method(Ms FEM).On this basis,the design of maximizing the fundamental frequency of the lattice structure and the minimum flexibility design under the frequency constraint were carried out,and the size effect of the lattice structure,the microvolume fraction,the amount of matrix material,the concentrated mass size and location and other factors were carried out.The influence of optimized structure was studied.First,the research derives the expressions of the equivalent stiffness matrix and the mass matrix of the lattice structure based on the Ms FEM theory,and establishes a multiscale equivalent analysis method of the lattice structure frequency considering the modal basis functions.The comparison with the analysis results of the accurate finite element model verifies the accuracy and efficiency of the established Ms FEM method for the frequency prediction of the lattice structure.Numerical results show that the introduction of modal basis functions can effectively improve the prediction accuracy of lattice structure frequencies.And the frequency performance of the lattice structure shows a certain size effect.As the number of microscopic unit cells increases,the characteristic value of the lattice structure gradually stabilizes.Secondly,based on the PAMP method,the cross-sectional area of the rods in the microscopic unit cell and the density of the coarse grid elements are used as design variables.Based on the Ms FEM analysis framework,a macro-micro integrated optimization mathematical model for maximizing the fundamental frequency of the lattice structure is established.By comparing the optimization results of the micro-single scale and the macro-micro dual-scale,it shows that the multiscale optimization can improve the frequency performance of the structure to a greater extent.At the same time,the study found that the number of base structures(size effect),the micro-volume fraction,the amount of base material,the size and location of the concentrated mass in the coarse grid unit have a great influence on the optimization results.With the increase in the number of base structures in the microscopic unit cell,the fundamental frequency of the optimized structure gradually increases and tends to a stable value;when the amount of base material remains constant,the increase in the microvolume fraction helps to increase the fundamental frequency of the structure;With the increase of the concentrated mass,the macroscopic material will gradually distribute around the position where the concentrated mass is applied,and the symmetry of the concentrated mass will affect the symmetry of the macroscopic topology and the microscopic unit cell configuration.Finally,based on the Ms FEM analysis framework,a macro-micro dual-scale optimization mathematical model of the minimum flexibility of the lattice structure under the fundamental frequency constraint and the volume constraint is established,and the influence of the lower limit value of the fundamental frequency constraint on the optimization result is studied.Studies have shown that the introduction of frequency constraints will reduce the structural rigidity and have a great impact on the macro-topology of the structure.With the increase of the lower limit of the fundamental frequency,more materials will be distributed near the fixed end of the structure to improve the anti-vibration capability of the structure.