Design Optimization Of Periodic Lattice Plate Structures Based On Homogenization Method For Maximizing The Fundamental Frequency

As a new type of material,lattice material has the advantages of high specific strength,high specific stiffness and high porosity.But there are two problems in the design of lattice structures: the first problem is that there are many kinds of unit cells,and what kind of unit cells should be used on the micro scale? the second problem is how to arrange the unit cells on the macro scale? A method to solve these two problems is two-scale optimization.Since the lattice structure has multi-scale characteristics,if a discrete finite element model is used for analysis,it will cost a lot,and the optimization will be more difficult.Therefore,in this thesis,through the asymptotic homogenization method of the periodic plate,the lattice plate is equivalent to the Mindlin plate,and the effective properties of the lattice structure are obtained,which are used for the frequency analysis of the lattice structure.Choosing the preceding threeorder natural frequencies of the structure as the performance indicator,the equivalent accuracy of asymptotic homogenization method for three-dimensional periodic materials and asymptotic homogenization method for periodic plate structures was compared.Then,based on homogenization method,two-scale optimization of topological design of macro structure and micro structure of lattice plate structures was carried out.In this thesis,the two-scale topology optimization design of lattice plate structure is realized by maximizing the fundamental frequency of the structure and constraining the microscopic volume fraction of the unit cell and the amount of matrix materials.The main work of this thesis includes:1)Based on the asymptotic homogenization method of three-dimensional periodic materials and the asymptotic homogenization method of periodic plate structures,the equivalent calculation of the effective properties of the lattice structure is studied.The natural frequencies of lattice plate were solved using the obtained effective properties,then compare them with the frequencies calculated by the discrete finite element models.The error analysis is carried out to verify the accuracy of prediction of the effective property based on the asymptotic homogenization method of the periodic plate structure equivalent to the Mindlin plate.2)Choosing the maximization of the first-order natural frequency of the structure as the objective function,the microscopic volume fraction of the unit cell as the fixed value and the upper limit of the material usage as the constraint conditions,the optimization model of this thesis was constructed.Optimization formulation of fundamental frequency maximization with single microstructure and multiple microstructures are respectively listed.In order to ensure the smooth progress of the optimization,the accuracy of the finite element program written by myself was tested,the results calculated by the program were compared with the results calculated by the commercial finite element software.Mesh convergence analysis was done to ensure that calculations of objective function,natural frequency of the structure is accurate.The sensitivities of the first-order natural frequency of the structure,base material amount and micro volume fraction respected to the design variables are derived respectively,which are used to solve the optimization problem.Finally,the two-scale topology optimization of lattice plate structures is carried out.