Multi-scale Analysis And Optimization Design Of Periodic Lattice Cylindrical Shell Structures

Due to the excellent mechanical properties,periodic lattice cylindrical shell structures are widely used in industrial fields.However,the traditional finite element calculation of the structure containing a large number of periodic unit cells is quite complex in modeling,expensive in computational cost,even difficult to calculate and so on.To solve the problems,new implementation of asymptotic homogenization method(NIAH)is used to predict the mechanical properties of lattice cylindrical shell structures.The static deformation responses,the fundamental frequency responses,the size optimizations of lattice cylindrical shell structures are studied and the structure / material concurrent optimization designs of planar lattice structures are studied in this paper by using NIAH.For the three-dimensional lattice cylindrical shell structure,the expression of the initial displacement field function of the unit cell is deduced,NIAH is used to calculate the equivalent elastic matrices of the unit cells.The homogeneous equivalent cylindrical shell models are established to solve the displacement results under tensile,bending and torsion conditions.The relative errors between the equivalent model results and the accurate finite element method(FEM)models results are very small,and the computing costs of equivalent model are smaller.Therefore,NIAH is efficient and accurate for the calculation of the lattice cylindrical shell structures.And the size effect of the lattice cylindrical shell structure is studied.It is found that the relative error between the accurate model results and the equivalent model results decreases significantly with the increase of the number of unit cells at first,then it approaches stability.Based on NIAH,the natural frequencies of planar lattice structures and lattice cylindrical shell structure are studied.It is found that the relative errors between the natural frequencies of accurate FEM models and equivalent models are within 1% when the overall vibration occurs.However,the relative errors between the natural frequencies of accurate FEM models and equivalent models are relatively obvious but also within 9% when radial breathing vibration occurs.The relative errors are relatively obvious for the reason that,along the axial direction,unit cells do not strictly meet the periodic boundary conditions at the free end.For strength analysis problems,the first order approximation stress of the local micro bars in the planar lattice structure is studied,and the relative errors between the stress results of accurate FEM models and equivalent models are tiny.It shows that calculating the stress of planar lattice structure by NIAH is valid.The sensitivity analysis is very important to improve the computational efficiency of the optimization design of the lattice cylindrical shell structure.To calculate sensitivities of the compliance of lattice cylindrical shell structures to cross section sizes of beam element,an implementation method using business simulation software has been derived based on NIAH.Taking the widths of unit cell walls as the design variables,considering the volume constraint,the minimum compliance designs of the lattice cylindrical shell structure under tensile,bending and torsion conditions are carried out through the implementation method and sequential quadratic programming algorithm(SQP),and the optimal size of the lattice material is obtained.On the basis of the above research,taking the widths of unit cell walls as the design variables,considering the volume constraint,the maximum fundamental frequency design is carried out,in the example,sensitivities are obtained through difference method and SQP is used.For the periodic planar lattice structures,to calculate sensitivities of the compliance of planar lattice structures to macro and micro design variables,an implementation method using business simulation software has been derived based on NIAH.At the macro scale,porous anisotropic material with penalty(PAMP)is used,at micro scale,NIAH is used to connect the macro scale and micro scale,to get clear topologies at macro scale,the modified Heaviside function is used to filter the macroscopic density.Then,the structure / material concurrent designs of planar lattice structures are carried out.The optimal configurations at macro scale and the optimal wall bar areas of unit cell are obtained.It is proved that the structure / material concurrent design is effective with the compliance reducing significantly after optimization.