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Translating TPMS models to STEP files

Yaonaiming Zhao, Qiang Zou

TL;DR

Addresses translating TPMS from F-rep to STEP to enable CAD/CAM/CAE workflows; proposes an error-controlled, $C^2$-continuous translation based on an error-driven sampling scheme and CPIA-based NURBS approximation; proves convergence for the Gyroid-based CPIA and validates with Gyroid, Diamond, Schwarz_P across nine examples, including STEP compatibility tests; demonstrates practical potential to integrate TPMS design directly into commercial CAD pipelines.

Abstract

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 $2ε$ on the deviation can be ensured if two conditions called $ε$-density and $ε$-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.

Translating TPMS models to STEP files

TL;DR

Addresses translating TPMS from F-rep to STEP to enable CAD/CAM/CAE workflows; proposes an error-controlled, -continuous translation based on an error-driven sampling scheme and CPIA-based NURBS approximation; proves convergence for the Gyroid-based CPIA and validates with Gyroid, Diamond, Schwarz_P across nine examples, including STEP compatibility tests; demonstrates practical potential to integrate TPMS design directly into commercial CAD pipelines.

Abstract

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 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 on the deviation can be ensured if two conditions called -density and -approximation are satisfied. The constrained-PIA algorithm enforces 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.
Paper Structure (19 sections, 1 theorem, 40 equations, 15 figures, 4 tables, 1 algorithm)

This paper contains 19 sections, 1 theorem, 40 equations, 15 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. TPMS modeling
  4. NURBS approximation
  5. Methods
  6. TPMS and NURBS preliminaries
  7. The TPMS2STEP pipeline
  8. Error-controlled TPMS sampling
  9. $\epsilon$-density
  10. $\epsilon$-approximation
  11. $C^2$ continuity-preserving TPMS approximation
  12. $C^2$ continuity constraints
  13. The CPIA method
  14. CPIA convergence proof
  15. STEP file generation
  16. ...and 4 more sections

Key Result

Proposition 1

The CPIA method for Gyroid is convergent and the limit surface is the least-square approximation outcome of the initial data $\mathrm{\{\mathbf{Q}_{ij}\}^{m_1,m_2}_{i=0,j=0}}$.

Figures (15)

  • Figure 1: Illustration of TPMS solid models for Gyroid (left), Diamond (middle), and Schwarz_P (right).
  • Figure 2: The proposed TPMS-to-STEP translation pipeline.
  • Figure 3: 2D illustration of the virtual intermediate surface method. The original surface and approximation surfaces are constrained within the error band (the dashed lines), and therefore a $2\epsilon$ error bound on their deviation.
  • Figure 4: 2D illustration of $\epsilon$-density.
  • Figure 5: Illustration of the $C^2$ continuity-preserving method: (a) Gyroid; (b) Diamond; (c) Schwarz_P. The edges of the surfaces are divided into two, two, and four types with different colors for Gyroid, Diamond, and Schwarz_P TPMS respectively. Only the edges with the same color can be spliced together during assembly.
  • ...and 10 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof