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Trimmed Spline Surfaces with Accurate Boundary Control

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Geometric Challenges in Isogeometric Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 49))

Abstract

We introduce trimmed NURBS surfaces with accurate boundary control, briefly called ABC-surfaces, as a solution to the notorious problem of constructing watertight or smooth (\(G^1\) and \(G^2)\) multi-patch surfaces within the function range of standard CAD/CAM systems and the associated file exchange formats. Our construction is based on the appropriate blend of a base surface, which traces out the intended global shape, and a series of reparametrized ribbons, which dominate the shape near the boundary.

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Notes

  1. 1.

    Basically, the only case which does not lead to exploding complexity is the special situation where \(\gamma \) parametrizes a straight line parallel to one of the coordinate axes.

  2. 2.

    This observation is a blatant contradiction to the concepts of differentiable geometry, which rely on the independence of geometric properties of the chosen parametrization.

  3. 3.

    Customary tolerances for traditional production techniques have to be decreased even further for modern 3d printing systems with their incredible attention to detail.

  4. 4.

    Advanced methods like discontinuous Galerkin may offer a remedy.

  5. 5.

    It does appear in a similar form in [12] for the special case of straight boundaries and polynomial ribbons.

  6. 6.

    The special case of straight boundaries, which admits rational constructions, was discussed in [8, 10,11,12].

  7. 7.

    The geometry shown here was kindly provided by Malcolm Sabin, who modeled a similar composite \(G^2\)-surface using a completely different, hitherto unpublished technique.

References

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Acknowledgements

We would like to thank Malcolm Sabin for fruitful discussions and comments.

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Authors and Affiliations

  1. Darmstadt University of Technology, Darmstadt, Germany

    Florian Martin & Ulrich Reif

Authors
  1. Florian Martin
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  2. Ulrich Reif
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Corresponding author

Correspondence to Ulrich Reif .

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Editors and Affiliations

  1. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Carla Manni

  2. Department of Mathematics, University of Rome Tor Vergata, Rome, Italy

    Hendrik Speleers

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Martin, F., Reif, U. (2022). Trimmed Spline Surfaces with Accurate Boundary Control. In: Manni, C., Speleers, H. (eds) Geometric Challenges in Isogeometric Analysis. Springer INdAM Series, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-92313-6_6

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