Explicit 2D topological control using SIMP and MMA in structural topology optimization

Structural topology can be measured on the basis of its betti numbers. A fundamental feature of structural topology optimization is that it allows the structural topology to be changed during the optimization process. However, traditional structural topology optimization methods use indirect and nonquantitative approaches to change the structural topology during the optimization procedure. Therefore, these traditional methods leave the detailed implementation of optimization with nonintuitive parameters (e.g., filter radius) to adjust the final topology of optimized results. Choosing a suitable nonintuitive parameter for beginners is not straightforward, and makes the optimization procedure complex when applying structural topology optimization methods to engineering design tasks with a preferred level of complexity (number of structural holes). A 2D structure has two betti numbers, B_0 and B_1 , where B_0 and B_1 correspond to the number of independent connected components and the number of holes in the structure, respectively. To solve the aforementioned problems, this paper explicitly quantitatively controls over the number of structural holes within the framework of the solid isotropic material with penalty (SIMP) interpolation of the design variable and the method of moving asymptotes (MMA) optimization algorithm in 2D, thus achieving direct unilateral constraint (constraining the maximum number of structural holes) over structural topology. The framework of SIMP and MMA is a powerful way because of its ability to handle more complex problems. Thus, the proposed topological control method based on SIMP and MMA is useful for structural topology optimization research field. For example, the proposed method is based on triangular meshing discretization of the initial design domain; therefore, irregular design domains can be easily processed, and adaptive meshes can be used to improve the geometric approximation of the design domains. Numerical examples show that the proposed method can effectively control the topology, the maximum number of holes (complexity) of the optimized structure.