Model Reduction And Topology Optimization Of Periodic Beam And Plate Structure Based On Homogenization

Periodic beam and plate structures are widely used in engineering, and have broad application in aircrafts, satellites, launch vehicles and high speed trains. With the rapid development of technology, the structure with the high specific stiffness and strength, lightweight and multifunctional performance is demanded strongly. Various new composite beam and plate structures have emerged, and the demand of design optimization for advanced composite beam and plate structures is also more and more demand.The present dissertation focuses on periodic composite beam and plate structures. By the comprehensive utilization of asymptotic homogenization method, topology optimization and multi-scale analysis technology, the effective properties of periodic beam, design optimization of microstructure for specified or extreme stiffness and macro-micro integration optimal design of periodic plate are studied and implemented with integration of the commercial finite element software as a black box, which enables the high performance computing for homogenization and design optimization.The main contents of this dissertation are as follows:(1) Based on one-dimensional periodic asymptotic homogenization theory, the finite element formulation of the asymptotic homogenization of periodic beam structure is developed, and the new implementation of the asymptotic homogenization method based on commercial software is extended to the periodic beam structures. With the new method, the heterogeneous beam structure is reduced to a classical Euler-Bernoulli beam with the fast prediction of effective stiffness matrix. The new method can be implemented easily using a commercial software as a black box, and dealing with the unit cell using various kinds of elements and modeling techniques available in the commercial software. As a result, the model scale and the workload of programming are reduced significantly, thus the computational efficiency is improved. With the new asymptotic homogenization method of beam structures established, the 3D (2D) periodic continuum, the plate and shell structures with periodicity in-plane and the 1D periodic beam structures with periodicity along the axis are all unified in the same framework of the new method. The new method is compared with the bending energy method and the asymptotic homogenization based on 3D periodic medium, and the size effect in thickness direction is studied. The results indicate that the bending energy method overestimates the effective bending stiffness of the beam, because the three-dimensional stress state in the beam is neglected based on the plane cross-section assumption. The results of 3D asymptotic homogenization method, which assumes infinite periodicity in thickness direction, have large error when the number of the unit cell is small, and tend to be consistent to the results of the present method when the number is large enough.(2) When volume preserving Heaviside function nonlinear density filter is used for topology optimization, the objective function oscillates seriously as the nonlinear parameter increases. As fundamental study of microstructure optimal design, the reason of this phenomenon is studied. The accurate sensitivity expressions of volume preserving nonlinear density filter is derived, and the accuracy is verified by finite difference method. The comparison of the different sensitivity formulas is given by the numerical examples. The results indicate that the sensitivity result of the original formula deviates from the result by the finite difference method when the nonlinear parameter increases. However, the result by the accurate sensitivity formula keeps consistent with the finite difference method, which verifies the accurate sensitivity formula. In addition, when nonlinear parameter keeps constant, the original sensitivity formula is still correct. This work provides a strong support for the microstructure topology optimization.(3) Based on the new asymptotic homogenization method of one-dimensional periodic beam, the microstructure design problem of periodic beam structure aiming at extreme or specified effective stiffness is studied under certain material volume constraint. The analytical sensitivity calculation approach based on new method is developed by constructing the complex displacement fields and extracting the corresponding mutual strain energy term. The optimization procedure is implemented with close integration of commercial software to solve the objective function and sensitivity, thus improves the efficiency of the optimization. The optimization method is extended to the one-dimensional periodic truss beam structure, and the optimization of the coupling stiffness is achieved. The topology optimization of the short beam with a few of periodic unit cells is discussed based on the variable mapping, and the effect of the transverse shear can be taken into account for this optimization.(4) The stiffness design of honeycomb structure is studied. The honeycomb-like unit cell is simulated with design variable mapping along the normal direction, and the analytical sensitivity analysis approach based on the new asymptotic homogenization method is extended to the periodic plate structure. We compared this method with the method based on 2D homogenization and integration along the normal direction, which is based on Kirchhoff-love assumption, and found that the length to height ratio has an influence to the final topology result, and the Kirchhoff-love assumption based method cannot distinguish this difference. Based on this work, a two-scale topology optimization method is studied for the purpose of minimizing the compliance of the macro structure and taking into account both topology distribution in macro and unit cell configurations. The objective function and the corresponding sensitivity to the macro and micro design variables are solved based on the commercial software, in order to achieve the efficient calculation of the two-scale topology optimization problem.(5) As a practical application of beam and plate model reduction method in engineering, a template is developed to implement the fast transformation of ProE model and multilevel projectile structure analysis based on MSC.SimXpert software platform. The feature naming approach is presented to transforms the information between CAD and CAE, and the external XML file database is used to identify and create properties of the model automatically, and attribute them to corresponding components. The establishment of MSC.SimXpert template platform promotes the accurate identification from CAD model and fast transformation to FEM model. Based on model reduction technology, such as homogenization method, the complex solid model can be reduced to the simplified model and the 1D beam model, and can be replaced and assembled freely through the template platform. Therefore, the establishment of multi-level finite element model reduces the calculation cost effectively. In addition, the quality inspection file and result file can be generated automatically through the template, which can improve the speed of product development.