We propose an asymptotic concentration approach combined with isogeometric analysis (IGA) for the topology optimization of two-dimensional (2D) linear elasticity structures under the commonly-used framework of the solid isotropic materials and penalty (SIMP) model. Based on the SIMP framework, the B-splines are used as basis functions to describe geometric model in structural finite element analysis, which closely combines geometric modeling with structural analysis. Isogeometric analysis is utilized to define the geometric characteristics of the 2D linear elasticity structures, which can greatly improve the calculation accuracy. In addition, to eliminate the gray-scale intervals usually caused by the intermediate density in the SIMP approach, we utilize the asymptotic concentration method to concentrate the intermediate density values on either 0 or 1 gradually. Consequently, the intermediate densities representing gray-scale intervals in topology optimization results are sufficiently eliminated by virtue of the asymptotic concentration method. The effectiveness and applicability of the proposed method are illustrated by several typical examples.

We propose a parameterized level set method (PLSM) for structural topology optimization based on reaction diffusion equation (RDE) and fuzzy PID control algorithm. By using the proposed method, the structural compliance minimization problem under volume constraints is studied. In this work, the RDE is used as the evolution equation of level set function, and the topological derivative of the material domain is used as the reaction term of the RDE to drive the evolution of level set function, which has little dependence on the initial design domain, and can generate holes in the material domain; the compactly supported radial basis function (CS-RBF) is used to interpolate the level set function and modify the RDE, which can improve the computational efficiency, and keep the boundary smooth in the optimization process. Meanwhile, the fuzzy PID control algorithm is used to deal with the volume constraints, so that the convergence process of the structure volume is relatively stable. Furthermore, the proposed method is applied to 3D structural topology optimization. Several typical numerical examples are provided to demonstrate the feasibility and effectiveness of this method.

We propose a polygonal topology optimization method combined with the alternating active-phase algorithm to address the multi-material problems. During the process of topology optimization, the polygonal elements generated by signed distance functions are utilized to discretize the structural design domain. The volume fraction of each material is considered as a design variable and mapped to its corresponding element variable through a filtering matrix. This method is used to solve a multi-material structural topology optimization problem of minimizing compliance, in which a descriptive model is established by using the alternating active-phase algorithm and the solid isotropic microstructure with penalty theory. This method can accomplish the topology optimization of multi-material structures with complex curve boundaries, eliminate the phenomena of checkerboard patterns and a one-node connection, and avoid sensitivity filtering. In addition, this method possesses fine numerical stability and high calculation accuracy compared to the topology optimization methods that use quadrilateral elements or triangle elements. The effectiveness and feasibility of this method are demonstrated through several commonly used and representative numerical examples.