Constraints Matrices and Convergence Proof of TPMS2STEP
Yaonaiming Zhao, Qiang Zou
TL;DR
The paper tackles translating TPMS from a functional representation (F-rep) to a STEP boundary representation (B-rep), introducing constraint matrices that guarantee $C^2$ continuity and bound the translation deviation by $2\varepsilon$. It provides detailed constraint matrices for Gyroid, Diamond, and Schwarz_P within the CPIA framework and proves convergence of the CPIA iteration for Diamond and Schwarz_P to the least-squares fit of the initial data, ensuring a well-defined limit surface. Additionally, it derives the second-order derivatives of the offset equation through a complex-parameter formulation and discusses numerical strategies to compute these derivatives. Together, these contributions enable rigorous, accurate TPMS-to-STEP translations suitable for integration into CAD/CAM/CAE workflows, with quantified continuity and error guarantees.
Abstract
TPMS is consistently described in the functional representation (F-rep) format, while modern CAD/CAM/CAE tools are built upon the boundary representation (B-rep) format. To solve this issue, translating TPMS to STEP is needed, called TPMS2STEP. This paper provides constraint matrices and convergence proof of TPMS2STEP so that $C^2$ continuity and an error bound of $2ε$ on the deviation can be ensured during the translation.