Triply periodic minimal surfaces (TPMSs) play a vital role in the design of porous structures, with applications in bone tissue engineering, chemical engineering, and the creation of lightweight models. However, fabrication of TPMSs via additive manufacturing is feasible only within a specific range of relative densities, termed the effective relative density range (EDR), outside of which TPMSs exhibit unmanufacturable features. In this study, the persistent homology is applied to theoretically calculate and extend the EDRs of TPMSs. The TPMSs with extended EDRs are referred to as extended TPMSs. To achieve this, TPMSs are converted into implicit B-spline representation through fitting. By analyzing the symmetry of TPMSs, a partial fitting method is utilized to preserve the symmetry and enhance fitting precision. A topological objective function is modeled based on the understanding of topological features, resulting in extended TPMSs that possess extended EDRs while maintaining a high degree of similarity to the original TPMSs. Experimental validation confirms the effectiveness of the approach in extending the EDRs of TPMSs. Furthermore, the extended TPMSs demonstrate superior performance in porous model design and topology optimization compared to their original counterparts. The extended TPMSs with increased EDRs hold promise for replacing traditional TPMSs in applications that require porous structures with varying densities.

翻译：三周期极小曲面（TPMS）在多孔结构设计中发挥关键作用，广泛应用于骨组织工程、化学工程及轻量化模型构建。然而，通过增材制造制备TPMS仅能在特定相对密度区间内可行，该区间称为有效相对密度范围（EDR），超出此范围则TPMS会出现不可制造的特征。本研究应用持续同调理论，对TPMS的EDR进行理论计算与拓展。具有扩展EDR的TPMS被称为扩展型TPMS。为实现该目标，首先通过拟合将TPMS转化为隐式B样条表示。通过分析TPMS的对称性，采用局部拟合方法以保持对称性并提高拟合精度。基于对拓扑特征的理解构建拓扑目标函数，由此获得的扩展型TPMS在保持与原TPMS高度相似性的同时，实现了EDR的扩展。实验验证表明该方法在扩展TPMS的EDR方面具有有效性。此外，与原始TPMS相比，扩展型TPMS在多孔模型设计与拓扑优化中展现出更优性能。具有增强EDR的扩展型TPMS有望在需要变密度多孔结构的应用中替代传统TPMS。