  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.
