The structural complexity (the number of holes) of the 2D or 3D continuum structures can be measured by their topology invariants (i.e., Euler and Betti numbers). Controlling the 2D and 3D structural complexity is significant in topology optimization design because of the various consideration, including manufacturability and necessary structural redundancy, but remains a challenging subject. In this paper, we propose a programmable Euler–Poincaré formula to efficiently calculate the Euler and Betti numbers for the 0–1 pixel-based structures. This programmable Euler–Poincaré​ formula only relates to the nodal density and nodal characteristic vector that represents the nodal neighbor relation so that it avoids manually counting the information of the vertices, edges, and planes on the surfaces of the structure. As a result, the explicit formulations between the structural complexity (the number of holes) and the discrete density design variables for 2D and 3D continuum structures can be efficiently constructed. Furthermore, the discrete variable sensitivity of the structural complexity is calculated through the programmable Euler–Poincaré formula so that the structural complexity control problem can be efficiently and mathematically solved by Sequential Approximate Integer Programming and Canonical relaxation algorithm Various 2D and complicated 3D numerical examples are presented to demonstrate the effectiveness of the method. We further believe that this study bridges the gap between structural topology optimization and mathematical topology analysis, which is much expected in the structural optimization community.

2D 或 3D 连续体结构的结构复杂性（孔的数量）可以通过它们的拓扑不变量（即 Euler 和 Betti 数）来衡量。由于各种考虑因素，包括可制造性和必要的结构冗余，控制 2D 和 3D 结构复杂性在拓扑优化设计中很重要，但仍然是一个具有挑战性的课题。在本文中，我们提出了一个可编程的 Euler-Poincaré 公式来有效地计算基于 0-1 像素的结构的 Euler 和 Betti 数。这种可编程的欧拉-庞加莱公式只涉及表示节点邻域关系的节点密度和节点特征向量，从而避免了人工计算结构表面顶点、边和平面的信息。其结果，可以有效地构建结构复杂性（孔数）与 2D 和 3D 连续结构的离散密度设计变量之间的显式公式。此外，通过可编程的欧拉-庞加莱公式计算结构复杂度的离散变量灵敏度，从而可以通过序列近似整数规划和典型松弛算法有效地数学解决结构复杂度控制问题 提供各种 2D 和复杂 3D 数值示例以证明该方法的有效性。我们进一步认为，这项研究弥合了结构拓扑优化和数学拓扑分析之间的差距，这在结构优化社区中备受期待。