Structural a fixed number can be chosen. Topology optimization

Structural optimization seeks the optimal material layout in a given design space. It is believed to be a powerful tool for reducing the component weight in manufacturing industry. The general form of structural optimization problem is minimizing/maximizing objective function f(x) by changing the design variable x subject to a certain constraints. The design variable x usually represents some geometrical features. According to the difference in these geometrical features, the structural optimization problems of continuum structures can be divided into three categories, size optimization, shape optimization and topology optimization. In size optimization, the design variable x usually represents the thickness of some features, such as the thickness of sheet. For shape optimization, the design variable x represents one of the pieces consisting the boundary, such as control points of spline curve. For topology optimization, the design variable x can be the material property (such as density or Young’s modulus) of one of the pieces consisting the whole structure. It usually takes a value from 0 to a predefined maximum. For some hard-kill topology optimization methods, only 0 and a fixed number can be chosen. Topology optimization is more generally used in practice for the optimization of continuum structure because it’s able to create any shape within the design space while shape and size optimization work on the given configurations.(Christensen and Klarbring 2008) There are varying methods solving topology optimization problem. Some methods are cell based, including Solid Isotropic Material with Penalisation (SIMP) and Bidirectional Evolutionary Structural Optimization (BESO) method. The SIMP method was developed by Bendsøe (Bendsøe 1989) and Zhou and Rozvany (Zhou and Rozvany 1991) independently. The term “SIMP” was first used by Rozvany et al (Rozvany and Sobieszczanski-Sobieski 1992). SIMP method can be classified into density-based methods, where the design space consists of finite elements consist of solid material or void. This actually forms an large-scale integer programming problem which was challenging to solve. (Deaton and Grandhi 2014) SIMP method replaces the discrete variables with continuous variables, and then converts the elements with intermediate material property into solid or void elements. BESO method belongs to the hard-kill methods, where the structure is optimized by gradually removing/adding a finite volume of material. The initial version of BESO is Evolutionary Structural Optimization (ESO), which was firstly developed by Xie and Steven (Xie and Steven 1993) (Xie and Steven 1997)