Multi-scale Dynamic Design Optimization Based On Integrated Design Of Structure And Material

This dissertation aims at extending structural optimization techniques to the integrated design of structures and materials used in dynamic environments. In order to achieve this, methods are proposed to realize the multi-scale dynamic design optimization of structures made of promising cellular material and composite material. The ultralight microstructure in the microscale and structural configuration in the macroscale with optimum dynamic performances are obtained simultaneously by the multi-scale optimization. The main works of this dissertation are as follows:1. For materials with periodic microstructures, equivalent analysis methods are proposed and improved in order to obtain effective properties of the material and realize multiscale analysis of the macrostructure made of the periodic material. First, effective elastic stiffness and initial yield strength and elastic buckling limits are derived in analytical forms for the periodic honeycomb structures with orthotropic isosceles triangle and Kagome cells. The dependence of the effective stiffness on the shape anisotropic ratio and relative density is discussed. Second, based on the numerical homogenization method, the elasto-plastic behavior is equivalently analyzed for the three dimensional (3D) heterogeneous structure made of truss-like material with periodic microstructures. The comparison with results of discrete modeling in time consumption and precision verifies the advantages of the effective multiscale analysis. Finally, methods in literatures to realize the micropolar equivalent continuum modeling of cellular materials are summarized. Different effective micropolar constants are observed in different papers for the same microstructure. A new approach is proposed to formulate the equivalent micropolar constitutive relation of 2-D periodic cellular materials. The new approach takes both displacement compatibility on the boundary and equilibrium inside the cell into account in the micromechanical analysis of a cell structure. The solutions from the classical Cauchy-type continuum and three kinds of micropolar continuum modelings using different effective micropolar constants by different authors are compared with the exact discrete simulations of the same structure under various conditions. It is found that the micropolar constants developed in this dissertation give satisfying results of equivalent analysis for square, triangular and hexagonal cells. 2. For the well-known minimum compliance problem with the constraint of given material amount, the satisfactions of the Karush-Kuhn-Tucker necessary conditions for optimality are checked for different optimum solutions obtained from different optimization algorithms. Furthermore, using the extended optimality and continuation technique, the problem of minimizing the product of the dynamic compliance and the total cost of material is considered for vibrating bi-material plate structures subjected to time-harmonic external mechanical loading with a given band of external excitation frequencies. This objective appears to facilitate reduction of vibration by driving the resonance frequencies of the structure as far away as possible from the prescribed range of excitation frequencies while simultaneously minimizing the amount of material to be used for the structure.3. Based on the integrated design of structure and material, the problem of vibro-acoustic optimization of laminated composite plates is considered. The vibration of the laminated plate is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude, and the design objective is to minimize the total sound power radiated from the surface of the laminated plate to the surrounding acoustic medium. Instead of solving the Helmholtz equation for evaluation of the sound power, advantage is taken of the fact that the surface of the laminated plate is flat, which implies that Rayleigh's integral approximation can be used to evaluate the sound power radiated from the surface of the plate. The Discrete Material Optimization (DMO) formulation has been applied to achieve the design optimization of fiber angles, stacking sequence and selection of material for laminated composite plates.4. A two-scale optimization method is developed to realize the integrated design of macro-structure and micro-structure of cellular material for maximizing structural fundamental eigenfrequency. In this method macro and micro densities are introduced as independent design variables for the macrostructure and microstructure respectively: Optimizations at two scales are integrated into one system through homogenization theory and base material is distributed between the two scales automatically with the optimization model. Microstructure of materials is assumed to be homogeneous at the macro scale to meet today's manufacture practice and reduce manufacturing cost. Plane structure with homogeneous cellular material and perforated plate are studied. Numerical experiments validate the proposed method and computational model.The research of this dissertation was partially supported by the National Natural Science Foundation of China (Grant No.90816025,10332010), and National Basic Research Program of China (Grant No.2006CB601205). This support is gratefully acknowledged by the author.