Date: 06 Oct 2013

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.