Structural Topology Optimization Methods Solving Problems With Minimum Compliance As The Object

Structural topology optimization can provide optimum material distribution within the design domain in the initial stage of product design under constraints such as objective function, boundary conditions, and volume constraints and so on. With great development potential and prospects, structural topology optimization is one of the research hotspots of structural optimization. Structural topology optimization is based on multi-disciplinary knowledge, including computational mathematics, structural mechanics and computer technology. Topology optimization method is not yet fully mature at present. There are many problems waiting to be solved in this area.The background and significance of topology optimization are firstly discussed in this paper. Then the history of topology optimization, some research achievements and several typical topology optimization methods are briefly introduced. During this process, the development history of meshless method and some research achievements about topology optimization methods based on meshless method are presented. Then topology optimization methods with minimum compliance as the object are studied, including methods solving single-material problems and multi-material problems.In the field of single-material topology optimization, the method based on the level set function is studied, which is one of the most popular methods. Methods in which the level set function is updated by solving the so called Hamilton-Jacobi partial differential equation have two major drawbacks. One is the level set function needs to be re-initialized during the optimization procedure and the other is holes cannot be automatically generated in the design domain. Aiming to improve methods based on level set functions, the reaction diffusion equation method is introduced to avoid re-initialization process. Pareto optimization methods and the algorithm proposed by Vivien J. Challis are combined to present a new method with Pareto optimization method generating holes. Numerical examples show that the new method is feasible and effective. The method presented by Takayuki Yamada can control the geometric complexity of the topology optimization results. In this paper some improvements are proposed to make this method more stable and can get better optimal results.In the field of multi-material topology optimization, the method proposed by Tavakoli and Mohseni is combined with a general-purpose mesh generator for polygonal elements presented by Cameron Talischi to solve problems with curved boundaries. Then the active-phase algorithm for multi-material topology optimization problems is extended to solve three-dimensional problems by MATLAB program. Further, some improvements of the method are also proposed in this paper. Meshless method can deal with problems with large deformation and cracks, which cannot be solved exactly by finite element method. Another advantage of topology optimization methods based on meshless method is mesh dependent problems and checkerboard can be eliminated or alleviated. A meshless multi-material topology optimization method based on active-phase algorithm is presented in this paper. Element-free Galerkin method(EFG) is used to analyze structure in this method. And the nodal relative density obtained using Shepard interpolation is taken as design variable. Numerical examples show topology optimization methods based on EFG can address multi-material problems well. This paper concludes with a multi-material topology optimization method based on the level set function. This method is extended form the method proposed by Takayuki Yamada using a so-called Multi-Material Level Set(MM-LS) topology description model. And the effective and feasible of the method in solving multi-material topology optimization problems is proved through several typical numerical examples. This method has inherited the advantage of the original algorithm that is geometric complexity of optimal results can be easily controlled via the parameter.