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An algorithm to mesh interconnected surfaces via the Voronoi interface

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Abstract

Many scientific and engineering problems involve interconnected surfaces meeting at junctions. For example, understanding the dynamics of a soap bubble foam can require modelling the fluid mechanics of liquid inside an intricate network of thin-film membranes. If a mesh of these surfaces is needed, the use of standard meshing algorithms often leads to voids, overlapping elements, or other artefacts near the junctions. Here, we present an algorithm to generate high-quality triangulated meshes of a set of interconnected surfaces with high surface accuracy. By capitalising on mathematical aspects of a geometric construction known as the “Voronoi interface”, the algorithm first creates a topologically consistent mesh automatically, without making heuristic or complex decisions about surface topology. In particular, elements that meet at a junction do so by sharing a common edge, leading to simplifications in finite element calculations. In the second stage of the algorithm, mesh quality is improved by applying a short sequence of force-based smoothing, projection, and edge-flipping steps. Efficiency is further enhanced by using a locally adaptive time stepping scheme that prevents inversion of mesh elements, and we also comment on how the algorithm can be parallelised. Results are shown using a variety of examples arising from multiphase curvature flow, geometrically defined objects, surface reconstruction from volumetric point clouds, and a simulation of the multiscale dynamics of a cluster of soap bubbles. In this last example, generating high-quality meshes of evolving interconnected surfaces is crucial in determining thin-film liquid dynamics via finite element methods.

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Notes

  1. For simplicity, in the following, interfaces are assumed to contain no holes. However, holes could be made by creating additional phases in the multiphase system whose purposes are to implement void regions, by using additional ϕ i functions. In a simple implementation, the meshing algorithm may then proceed unaltered and any surface meshes of the void phases may be ignored or discarded upon completion. Alternatively, to improve efficiency, one could make minor modifications to the algorithm such that surface elements of void phases are never created.

  2. Note that ϕ i  − ϕ j is a continuous piecewise linear function defined on \(\Upomega\), and so its zero level set must necessarily be piecewise planar. In particular, because ϕ i  − ϕ j is linear on each tetrahedron, it follows that its zero level set in each tetrahedron, if it exists, is either a triangle or a planar quadrilateral.

  3. With mild assumptions on the functions ϕ i , it is possible to show that the vertex snapping procedure guarantees that if two vertices on a surface are within a distance \(\epsilon\) from each other, such that \(\epsilon \ll h\) (where h is the tetrahedron length), then they are in fact the same vertex. In this work, a tolerance of \(\epsilon = 10^{-14}\) was used (corresponding to double precision and unit-length domains). Extensive tests analysing separation distance of vertices found that this tolerance correctly uniquified vertices in all cases.

  4. The scaling factor of 1.2 was determined empirically, and is the same as that used in [29]. It forces mesh vertices to spread apart across the whole surface and leads to a smoothing behaviour that performs consistently well.

  5. In the projection step, the original, unperturbed functions ϕ i are used, i.e. they have not been altered by the vertex snapping procedure used in creating the initial mesh.

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Acknowledgments

This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under contract number DE-AC02-05CH11231, by the Division of Mathematical Sciences of the National Science Foundation, and by an American Australian Association Sir Keith Murdoch Fellowship.

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  1. Department of Mathematics and Lawrence Berkeley National Laboratory, University of California, Berkeley, Berkeley, California, 94720, USA

    R. I. Saye

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Saye, R.I. An algorithm to mesh interconnected surfaces via the Voronoi interface. Engineering with Computers 31, 123–139 (2015). https://doi.org/10.1007/s00366-013-0335-9

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