The Optimization of Structural Topology (OST) is a breakthrough in product design because it can optimize size, shape and topology synchronously under different physical constraints. It has promising applications in industry ranging from automobile and aerospace engineering to micro electromechanical system.; This dissertation first substitutes the nonlinear diffusion method for filter process in the optimization of structural topology. Filtering has been a major technique used in a homogenization-based method for topology optimization of structures. It plays a key role in regularizing the basic problem into a well-behaved setting. But it has a drawback of smoothing effect around the boundary of material domain. A diffusion technique is presented here as a variational approach to the regularization of the topology optimization problem. A nonlinear or anisotropic diffusion process not only leads to a suitable problem regularization but also exhibits strong "edge"-preserving characteristics. Thus, it shows that the use of the nonlinear diffusions brings desirable effects of boundary preservation and even enhancement of lower-dimensional features such as flow-like structures. The proposed diffusion techniques have a close relationship with the diffusion methods and the phase-field methods of the fields of materials and digital image processing.; Then this dissertation introduces a gradient flow in the norm of H-1 for the problem of multi-material structural topology optimization in 2/3D with a generalized Cahn-Hilliard (C-H) model with elasticity. Unlike the traditional C-H model applied to spinodal separation which only has bulk energy and interface energy, the generalized model couples the macroscopic elastic energy (mean compliance) into the total free energy. As a result, the grain morphology is not random islands or zigzag web-like objects but regular truss or bar structure. Although disturbed by elastic energy, the C-H system still keeps its two most important properties: mass conservation and energy dissipation. Therefore, it is unnecessary to compute the Lagrange multipliers for the volume constraints and make extra effort to minimize the mean compliance (elastic energy) for the optimization of structural topology. On the other hand, when pure phases separate from disordered original state, their boundaries will merge and split resulting in natural and flexible topology variation. Such aforementioned properties make the C-H model especially suitable for the problem of optimization of multi-material structural topology.; The fourth-order nonlinear parabolic C-H equations with elasticity are solved by a powerful nonlinear implicit mutigrid algorithm. To validate its correctness and efficiency, I first use it for the quadternary C-H equations without elasticity and get good results. To my best knowledge, it is the first simulation for such C-H models composed of more than three phases both in 2D and 3D.; This dissertation also extends the famous Solid Isotropic Material with Penalization (SIMP) model from 2D to 3D for topology optimization of the structure with single material. A short 177-line Matlab code including 3D Finite Element Method (FEM), filter technique, Optimality Criteria (OC) algorithm and bisection method is listed in appendix A for clear understanding of this model in 3D.; All proposed methods are demonstrated by several 2D and 3D examples which have been extensively studied in the recent literature of topology optimization.
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