Asymptotic Homogenization Of Periodic Plate And Micro-structural Optimization

Plate and shell structures are widely used in various applications, such as aeroplanes and ships, etc. To obtain higher stiffness and lighter weight, these structures often have stiffen-ers, ribs, and other complicated microstructures, such as corrugated plate, honeycomb plate, and sandwich plate with truss-lattice cores. Because of these microstructures, analysis of the macrostructures composed of them is often time consuming. To reduce computing time, this type of structure can be simplified as macroscopically homogeneous and heterogeneous plate and shell structure. The key step in this kind of algorithms is to obtain effective stiffness of the plate and shell structures at macro scale. This paper studied the asymptotic homogenization method for predicting the effective properties and optimization of periodic plate structures with microstructures, which will be called periodic plate structures in the following for simplicity.Representative volume element (RVE) method and asymptotic homogenization (AH) method are two numerical methods to predict effective properties of periodic materials. The RVE method has clear physical meaning and is easy to implement, but without rigorous mathematical foun-dation. It can only provide approximate prediction of effective properties. The AH method has rigorous mathematical foundation, and has complete theory and algorithm on calculating effec-tive properties of3D and2D periodic materials with microstructures. Periodic plate structures have microstructures periodic in mid-plane, but not in the thickness direction, which becomes difficulty in calculating effective stiffness of the periodic plate structures. Kalamkarov devel-oped mathematical theory of asymptotic homogenization of periodic plate and shell structures. However, the complex derivation and expressions of this theory made it difficult for finite ele-ment realization. Based on this theory, this paper presented a finite element formulation of the homogenization method of periodic plate structures using solid element and shell element, and used it to analyze periodic plate structures with complicated microstructures. This method pro-vides a benchmark platform for evaluating other methods to calculate effective stiffness, such as the RVE method.Based on above work, this paper presented a new implementation approach of the AH method. The new implementation firstly realizes the AH method explicitly in3D and2D pe-riodic materials with microstructures, so it has rigorous mathematical foundation. Meanwhile, it is as simple to implement as the RVE method. This method uses commercial software as a black box, and can use various element types and modeling techniques in commercial software to model unit cells with complicated microstructures. so that the finite element model can be kept relatively small. Furthermore, the new implementation approach for3D and2D periodic materials can be easily extended to the AH method of plate and shell structures without complex mathematical derivation, which realized homogenization of material with periodic microstruc-tures based on rigorous theory, and at the same time, reduced three dimensional structures into two dimensional plate/shell structures. Several examples demonstrate that the new implementa-tion is simple and effective. Based on this method, we can obtain sensitivity analysis of effective properties of periodic plate structures with respect to parameters of their microstructures. The proposed method provides a way for inverse homogenization on the microstructure of the plate structures.The widespread application of honeycomb plates require prediction of their effective stiff-ness with different unit cell sizes. Existing analytical and approximate formulas all have their limitations of application. In this paper, we used the new implementation approach to study ef-fective stiffness of the hexagonal honeycomb plate with wall thickness and cell height in large ranges. We provided a series of approximate formulas of the effective stiffness of honeycomb plate, and verified their accuracy. We also discussed about the accuracy of existing approximate formulas in literature, the conditions when the classical laminate theory stands, and the influence of the Poisson’s ratio.Based on the homogenization method of periodic plate structures, we can optimize effective properties of periodic plate structures by designing one unit cell, including topology optimiza-tion. To overcome some existing problems in topology optimization, and helps topology opti-mization of microstructures of periodic plate structures, we studied manufacturability in topology optimization. Although much work has been done on this problem, it is still an unsolved difficult problem. Based on parameterized Heaviside density filter proposed by Shengli Xu, the author of this paper, and Gengdong Cheng, and also inspired by the work on manufacturability and robust design in literature, we used a simplified minimax optimization formula, which is solved by ag-gregate function. To achieve manufacturability in topology optimization, we used some heuristic techniques in our algorithm. Assuming the erosion, intermediate, dilation designs have the same topology, we got an approximate relation between the filter radius, threshold in parameterized Heaviside function, and minimum size, which provide a way to select proper filter radius and threshold according expected minimum size. We used a relaxation technique, which can make the erosion, intermediate, dilation designs having the same topology to a large extent. Several examples, including compliance minimization, compliant mechanism design, heat conduction design, and topology optimization of periodic plate structures with microstructures, verified the effectiveness of our algorithm.At last, we studied sizing and topology optimization of effective properties and critical buckling load of honeycomb plate. We used homogenization method to calculate effective stiff-ness of the periodic honeycomb plate, then calculate buckling load using these effective stiffness. In this way, we can analyze just one unit cell, not a much refined finite element model of the macrostructure, thus achieving a significant reduction in computing time. In the sizing opti-mization, we used optimization method based on Kriging surrogate model. Through optimizing sizes of the honeycomb, such as inclination angle and respect ratio, we realized optimization of buckling load of the honeycomb plate. In topology optimization, we used the sensitivity analysis formulas of effective properties of periodic plate structures with respect to their microstructural parameters, and inverse homogenization method to realize optimization of effective properties and critical buckling load of honeycomb plate by designing microstructures of the plate. We considered honeycomb-like plate, whose cell wall is perpendicular to plate mid-plane. To get honeycomb-like results, we linked all elements through the plate thickness as one design vari-able.