Topological shape optimization of microstructures of materials using level set methods

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Topology optimization has been regarded as a most promising approach in the conceptual stage of structural design. It has experienced rapid development over the past two decades and has been applied to a wide range of engineering problems. This thesis will focus on the level-set based topology optimization method, which was originally developed by Osher and Sethian in 1988 and have been successfully incorporated into the structural optimization. With the implicit representation scheme, the level set methods can be easily applied to handle the complex shape and topology changes of the structural design. This work is divided into three parts. The first part is the necessary background for understanding the main focuses of the thesis. It includes the first four chapters: Chapter 1 provides the background the topology optimization, and overview of the current topology optimization methods, as well as application in the material design fields. Chapter 2 gives a description of the numerical homogenization method. Chapter 3 introduces the theory of the conventional level-set methods, and Chapter 4 provides details of for a parameterized level set method with numerical examples. The second part of this thesis is about the design of mechanical/elastic metamaterials, which is contented in Chapter 5. In this part, we integrate the parameterized level set method with the numerical homogenization method for the design problems of metamaterials. Meanwhile, a multiphase level-set based scheme for designing metamaterials is proposed. In the parameterized level set method, a set of compactly supported radial basis functions (CSRBF) is employed to interpolate each implicit level set function, which transfer the most difficult topology optimization problem into an easiest “size” optimization problem in the area of structural optimization. This method will be free of the Courant–Friedrichs–Lewy (CFL) condition and the re-initialization scheme. The propagation of the level set function can be driven by other well-developed optimization that involves gradient information. Moreover, this method can freely create new holes inside the material regions of the two-dimensional (2D) design domain. The optimal designs for mechanical metamaterials with extreme and prescribed properties are presented in this chapter as well (e.g. negative thermal expansion and negative Poisson’s ratios). In the third part of this thesis, Chapter 6, we applied the parametric level set method and the numerical homogenization method for designing three-dimensional (3D) scaffolds for the tissue engineering. Numerical examples are used to demonstrate the effectiveness of the optimization method in designing the scaffold with a range of multifunctional properties. The efficiency, convergence and accuracy of the present methods are also highlighted. Finally conclusions are given in Chapter 7.
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