Persistent homology-based explicit topological control for 2D topology optimization with MMA

TL;DR

This work proposes an explicit and differentiable topology-control framework by integrating persistent homology into the classical minimum-compliance topology optimization problem and demonstrates that the proposed approach enables explicit control over structural connectivity and the number of holes, thereby providing a systematic and mathematically grounded strategy for topology regulation.

Abstract

Controlling structural complexity, particularly the number of holes, remains a fundamental challenge in topology optimization, with significant implications for both theoretical analysis and manufacturability. Most existing approaches rely on indirect strategies, such as filtering techniques, minimum length-scale control, or specific level-set initializations, which influence topology only implicitly and do not allow precise regulation of topological features. In this work, we propose an explicit and differentiable topology-control framework by integrating persistent homology into the classical minimum-compliance topology optimization problem. The design domain and density field are represented using non-uniform rational B-splines (NURBS), while persistence diagrams are employed to rigorously and quantitatively characterize topological features. Given a prescribed number of holes, a differentiable topology-aware objective is constructed from the persistence pairs and incorporated into the compliance objective, leading to a unified optimization formulation. The resulting problem is efficiently solved using the method of moving asymptotes (MMA).Numerical experiments demonstrate that the proposed approach enables explicit control over structural connectivity and the number of holes, thereby providing a systematic and mathematically grounded strategy for topology regulation.

Figures (16)

• Figure 1: Cubical complex. (a)-(d) Examples of $k$-dimensional cubes. (e) An example of cubical complex represented by dark part, which is a collection of cubes. (f) The underlying space constructed from (e) by merging all cubes as a topological space on the plane.
• Figure 2: Filtration on a cubical complex. (a) is a cubical data with height function values on its grid cells. (d)$-$(g) show the corresponding filtration of (a) and each item of this filtration is a cubical complex which represents the sublevel set of original cubical data. The topological features can be extracted from those expanding cubical complexes with threshold value increasing. Using persistent homology, one can obtain the persistence diagram (b) and persistence barcode (c) of filtration (d)–(g).
• Figure 3: A two-dimensional persistence diagram. Regions I, II, III, and IV are segmented by lines $y=-x$, $x=-\bar{\rho}$ and $y=-\bar{\rho}$. The persistence pairs in regions I and II correspond to the isolated holes within the binarized structure. The blue arrows indicate the direction of the optimization objective.
• Figure 4: (a) The short beam model. (b) Optimization result without topology control. (c) Binary structure corresponding to the density field in (b).
• Figure 5: Optimization results of the short beam for $\bar{N}_{1}=1,2,3,4,5,6$, respectively.