One of the straightforward definitions of structural topology optimization is to design the optimal distribution of the holes and the detailed shape of each hole implicitly in a fixed discretized design domain. However, typical numerical instability phenomena of topology optimization, such as the checkerboard pattern and mesh dependence, all take the form of an unexpected number of holes in the optimal result in standard density-type design methods, such as SIMP and ESO. Typically, the number of holes is indirectly controlled by tuning the value of the radius of the filter operator during the optimization procedure, in which the choice of the value of the filter radius is one of the most opaque and confusing issues for a beginner unfamiliar with the structural topology optimization algorithm. Based on the soft-kill bi-directional evolutionary structural optimization (BESO) method, an optimization model is proposed in this paper in which the allowed maximal number of holes in the designed structure is explicitly specified as an additional design constraint. The digital Gauss-Bonnet formula is used to count the number of holes in the whole structure in each optimization iteration. A hole-filling method (HFM) is also proposed in this paper to control the existence of holes in the optimal structure. Several 2D numerical examples illustrate that the proposed method cannot only limit the maximum number of holes in the optimal structure throughout the whole optimization procedure but also mitigate the phenomena of the checkerboard pattern and mesh dependence. The proposed method is expected to provide designers with a new way to tangibly manage the optimization procedure and achieve better control of the topological characteristics of the optimal results.

结构拓扑优化的直接定义之一是在固定的离散化设计域中隐式设计孔的最佳分布以及每个孔的详细形状。但是，在诸如SIMP和ESO等标准密度类型设计方法的最佳结果中，拓扑优化的典型数值不稳定现象（如棋盘图案和网格依赖性）均以意外数量的孔的形式出现。通常，孔的数量是通过在优化过程中调整滤波器算子的半径值来间接控制的，其中对于不熟悉的初学者来说，滤波器半径值的选择是最不透明和令人困惑的问题之一结构拓扑优化算法。基于软杀伤双向演化结构优化（BESO）方法，提出了一种优化模型，其中明确规定了设计结构中允许的最大孔数作为附加设计约束。数字Gauss-Bonnet公式用于在每次优化迭代中计算整个结构中的孔数。本文还提出了一种填孔方法（HFM），以控制最佳结构中的孔的存在。几个二维数值算例表明，所提出的方法不仅在整个优化过程中限制了最佳结构中的最大孔数，而且还减轻了棋盘图案和网格依赖性的现象。