Research On Multi-scale Topology Optimization Of Periodic Material Structures Based On SIMP Method

Structural topology optimization is based on the material layout in the structural design domain.Under the given constraints,a certain optimization algorithm is used to find the optimal material distribution,so that some performance index of the structure can reach the optimal state.After decades of theoretical research and development,topology optimization technology has been widely used in various engineering fields.Periodic materials have been widely used in automobile,aerospace and other engineering fields due to their good designability and various excellent properties.Accurate prediction of equivalent mechanical properties of periodic materials is the basis of relevant applications.Asymptotic homogenization method is a common and effective method to solve the equivalent mechanical properties of periodic materials due to its strict mathematical theory.In this paper,the SIMP method,which is commonly used in topology optimization,is used as the theoretical framework to study the topology optimization design of periodic material structures.The main contents are as follows.Firstly,the basic idea of SIMP method for topology optimization is expounded.According to the optimization formulation of SIMP method,the sensitivity formula of objective function and the iterative formula of OC criterion method for updating design variables are deduced.The flow chart of structural topology optimization based on OC criterion method is given,which provides a theoretical solution framework for the full text.Secondly,the prediction of equivalent mechanical properties of composites is studied.The numerical calculation formulas of asymptotic homogenization method and the finite element solution formulas of a new asymptotic homogenization method(NIAH)are derived.Based on NIAH method,the corresponding APDL command stream is compiled by ANSYS commercial software,and the the calculation of macroscopic equivalent mechanical properties of different types of microscopic unit cells in two-dimensional and three-dimensional cases is realized.Then,a two-dimensional multi-scale topology optimization method is proposed for the optimization design of two-dimensional periodic material structures.This method utilizes the advantages of NIAH method and other geometric analysis in the framework of SIMP method,and couples the two methods to realize the topological optimization design of two-dimensional periodic material structure.Taking two-dimensional periodic cantilever beams and simply supported beams with different cell types and different working conditions as numerical examples,the effects of different cell materials on two-dimensional periodic material structure topology optimization are calculated to verify the advantages of composite materials and provide theoretical basis for two-dimensional periodic material structure topology optimization design.Finally,a three-dimensional multi-scale topology optimization method is proposed to optimize the three-dimensional periodic material structure.In the framework of SIMP method,this method utilizes the advantages of NIAH method and couples it with the three-dimensional structure topology optimization method to realize the three-dimensional periodic material structure topology optimization design.Taking three-dimensional periodic cantilever beams and MBB beams with different cell types and different working conditions as numerical examples,the effects of different cell materials on three-dimensional periodic material structure topology optimization are calculated,in order to provide theoretical basis for three-dimensional periodic material structure topology optimization design.