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Automatic domain partitioning for quadrilateral meshing with line constraints

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Abstract

In this paper, we present an algorithm for partitioning any given 2D domain into regions suitable for quadrilateral meshing. It is able to preserve the symmetry of the domain if any, and can deal with inner boundaries and multidomain geometries. Moreover, this method keeps the number of singularities at the junctions of the regions to a minimum. Although each part of the domain, being four-sided, can be easily meshed using a structured method, we provide a meshing process that guarantees near perfect quality for most quadrilaterals of the resulting mesh. The partitioning stage is achieved by solving a PDE-constrained equation based on the geometric properties of the domain boundaries. An analysis of the generated mesh quality is provided at the end, showcasing that the meshes obtained through our algorithm are especially suitable for finite element methods.

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Notes

  1. Note that results are independent of this choice.

  2. The representation vector used at a \(C^0\) corner of \(\partial \Omega\) is the average of the representation vectors of the two geometrical edges connected to this corner.

  3. We consider a point of \(\partial T_h\) to be a corner if the two incident edges form an angle smaller than \(\frac{3 \pi }{4}\) or greater than \(\frac{5 \pi }{4}\).

  4. A non-isolated singularity actually leads to an infinity of singularities which in turns leads to a much higher energy. As our resolution minimized it, it automatically removed such configurations.

  5. We remind the reader here that this means that the representation vector field of \(F\) is continuous, and not that any of the \({\mathbf {v}}_i\) field is continuous.

  6. This can be understood easily: if there were a majority of negative singularities, at the minimum there were only negative singularities, and one of them could merge with this positive singularity we added and disappear. It is to be noted, though, in this case we can be sure that our algorithm would not reach the minimum, as it would not merge this singularity with another, because it would distort the representation vector field too much. If the majority of singularities were positive, on the contrary, we added one to the minimum.

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Authors and Affiliations

  1. CEA, DAM, DIF, 91297, Arpajon, France

    Nicolas Kowalski & Franck Ledoux

  2. UPMC Univ Paris 06, UMR 7598, Laboratoire J.-L. Lions, 75005, Paris, France

    Nicolas Kowalski & Pascal Frey

Authors
  1. Nicolas Kowalski
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  2. Franck Ledoux
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  3. Pascal Frey
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Correspondence to Nicolas Kowalski.

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Kowalski, N., Ledoux, F. & Frey, P. Automatic domain partitioning for quadrilateral meshing with line constraints. Engineering with Computers 31, 405–421 (2015). https://doi.org/10.1007/s00366-014-0387-5

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  • DOI: https://doi.org/10.1007/s00366-014-0387-5

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