An algorithm to mesh interconnected surfaces via the Voronoi interface

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.

Geometric modeling of complex anatomical structures based on implicit functions

“…From each homogeneous region a set of points P = {p 1 Figure 3). In this process an appropriate element size needs to be fitted on the boundary and inside the implicit model [7], [16], [13] (see Section 4).…”

Topology-controlled reconstruction of multi-labelled domains from cross-sections

“…Typical representations of multi-labeled domains include (regional or global) implicit functions [Feng et al 2010;Kim 2010;Losasso et al 2006;Mitchell et al 2015;Saye 2015;Yuan et al 2012;Zhao et al 1996;Zheng et al 2006] and volume fractions [Ahn and Shashkov 2007;Anderson et al 2008Anderson et al , 2010Bonnell et al 2003]. While implicit function representations are often based on level sets, current works typically utilize a single level as the underlying function evolves (e.g., during a simulation), which creates a univariate family of domains.…”

Multiphase surface tracking with explicit contouring

“…Bronson et al [2013] introduced a new algorithm for generating tetrahedral meshes that conform to volumetric domains of multiple materials. Saye and Sethian [2012] extracted the multiphase interfaces with piecewise linear interpolation and further quality-improved [Saye 2013] under the framework of VIIM, which requires the continuous piecewise interpolation of every distance function and mesh abstraction and chopping. Anderson et al [2008;2010] addressed the Material Interface Reconstruction problem in standard VOF method.…”