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）正逐渐成为微结构设计的重要方法。然而，商业CAD/CAM/CAE软件在TPMS设计与制造中的应用仍然有限。这主要是因为TPMS通常以函数表示（F-rep）格式进行描述，而现代CAD/CAM/CAE工具则建立在边界表示（B-rep）格式之上。解决这一差异的一种可行方案是将TPMS转换为CAD/CAM/CAE的标准数据交换格式——STEP。基于这一方向，本文提出了一种具备误差控制与保持$C^2$连续性特征的新型转换方法。该方法基于近似误差驱动的TPMS采样算法与约束渐进迭代逼近（constrained-PIA）算法。采样算法可控制原始模型与转换模型之间的偏差：若满足称为$\epsilon$-密度与$\epsilon$-近似的两个条件，则可确保偏差的误差界为$2\epsilon$。约束-PIA算法在TPMS近似过程中强制施加$C^2$连续性约束，同时实现高效计算。本文亦给出了该算法的理论收敛性证明。通过一系列实例与对比分析，验证了该转换方法的有效性。