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.

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