Triply periodic minimal surface (TPMS) is emerging as an important way of designing microstructures. However, there has been limited use of commercial CAD/CAM/CAE software packages for TPMS design and manufacturing. This is mainly because 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. One possible solution to this gap is translating TPMS to STEP, which is the standard data exchange format of CAD/CAM/CAE. Following this direction, this paper proposes a new translation method with error-controlling and $C^2$ continuity-preserving features. It is based on an approximation error-driven TPMS sampling algorithm and a constrained-PIA algorithm. The sampling algorithm controls the deviation between the original and translated models. With it, an error bound of $2\epsilon$ on the deviation can be ensured if two conditions called $\epsilon$-density and $\epsilon$-approximation are satisfied. The constrained-PIA algorithm enforces $C^2$ continuity constraints during TPMS approximation, and meanwhile attaining high efficiency. A theoretical convergence proof of this algorithm is also given. The effectiveness of the translation method has been demonstrated by a series of examples and comparisons.

翻译：三周期极小曲面（TPMS）正逐渐成为设计微结构的重要方式。然而，商业CAD/CAM/CAE软件包在TPMS设计与制造中的应用十分有限，这主要是因为TPMS始终以函数表示（F-rep）格式描述，而现代CAD/CAM/CAE工具基于边界表示（B-rep）格式构建。弥合这一差距的潜在方案是将TPMS转换为STEP格式（CAD/CAM/CAE的标准数据交换格式）。沿此方向，本文提出了一种具有误差控制与C2连续性保持特性的新型转换方法。该方法基于近似误差驱动的TPMS采样算法与约束-PIA算法：采样算法通过控制原始模型与转换模型之间的偏差，在满足ε-密度与ε-近似两个条件时，可确保偏差的误差界为2ε；约束-PIA算法在TPMS近似过程中施加C2连续性约束，同时实现高效率。本文还给出了该算法的理论收敛性证明。通过一系列实例与对比验证了该转换方法的有效性。