| Lattice materials have excellent mechanical properties such as high specific strength and high ratio of rigidity,and are widely used in various fields of modern industry.However,due to the large number of micro-structures contained in the structure of lattice materials,the traditional finite element calculations have great difficulty in modeling and analysis.The Extended Multiscale Finite Element Method(EMs FEM)provides an effective way to deal with these difficulties.This method also makes it possible to obtain a unit cell structure with better performance through structural optimization design.Based on the extended multi-scale finite element method,this paper studies the theory and method of numerical analysis and optimization design for macroscopic structures composed of lattice materials.Graded structures are widely found in natural materials,such as the stems of plants,the beak of birds,etc.They contain a large number of micro-structures that continuously change in gradient to meet the functional requirements of different locations of the structure.Based on the work of the lattice structure,this article will take the graded lattice structure as the research object,the multi-scale optimization design of lattice micro-structures and their distribution on the macro-scale is carried out to obtain better lattice materials and structural systems with better performance.In this paper,graded lattice structures and multi-phase material lattice structures have been optimized by hierarchical optimization and concurrent optimization.Firstly,based on the EMsFEM,this paper establishes a hierarchical optimization design model of macro structure/micro material for graded lattice structure.With the minimum compliance as the objective function and the material usage as the constraint,the graded/layered distribution of the structure was achieved on the macro scale using the Layer-wise SIMP method.Based on the macro topological configuration,the micro-scale optimization problem is decomposed into several sub-optimization problems,and the equivalent performance of periodically distributed lattice micro-structures is analyzed using EMs FEM.In each suboptimization problem,the topological configuration of microcells is taken as the design variable,and the sum of the strain energy of the element in the corresponding regions is used as the objective function to optimize the topology configuration of microcells.This method realizes the decoupling between macro and micro scales,so macro/micro two-scale optimization is divided into two different optimization sub-problems.The effectiveness of the proposed method is verified by using a planar cantilever beam under mechanical loading.The effect of material usage on the macro/micro optimization results is discussed.The size effect is discussed by changing the geometry of the lattice microstructure.The effect of the optimization results was compared with the microscopic single-scale optimization design and the method,and the advantages of structural/material integration level optimization were described.Secondly,the multi-scale concurrent optimization model of graded lattice structure is established by the scale-coupled(macro-microscopic geometry inseparable)multi-scale concurrent optimization method.In the case of a given amount of matrix material,the macrocell relative density is introduced as a design variable at the structural scale,the micro structure topological configuration is used as the design variable at the material scale,and a sequential quadratic programming algorithm is used to minimize the compliance of graded lattice structure.Numerical examples verify the superiority of multi-scale concurrent optimization of structures/materials relative to microscopic single-scale optimization of materials.The influence of the geometrical size of the lattice microstructure on the optimization results of the gradient lattice structure is discussed.The influence of the microscopic volume fraction and the amount of the matrix material on the multi-scale optimization results of the gradient lattice structure is investigated.Afterwards,with the constraint of the strength and stability of the lattice micro-bars,a multi-scale concurrent light-weight design model of graded lattice structure was established.The new aggregation function is used to aggregate the constraints of the strength and stability of all micro-elements into overall constraints,avoiding the problems of excessive constraints and the difficulty of “secondary peaks”.The influence of lattice microcell geometry on the lightweight optimization results is discussed.It provides a new theoretical basis and implementation technology for the multi-scale analysis and design of graded lattice structure.Finally,for lattice structure composed of multiphase materials(two real materials and one empty material),an interpolation format based on the volume-conservative nonlinear density filter function is proposed,and a multiphase material concurrent optimization model is established of lattice structure.On the structural scale,the traditional SIMP method is used to obtain the topological configuration.Two types of topological variables are introduced on the material scale(one used to characterize the presence or absence of a rod,and the other is used to characterize which material is used).In order to avoid the intermediate values of the topological variables,the nonlinear density filter function is used to filter the topological variables so that they finally exhibit a distribution of 0 or 1,and a clear topological configuration is obtained.The effect of material dosage on macro/micro optimization results was discussed and the effectiveness of the above method was verified. |
