On Topology of Three-dimensional Continua with Singular Points

TL;DR

This paper models the topology of 3D continua with singular points using Yin sets, providing a unique boundary representation and characterizing their topology, which has applications in modeling and simulations involving complex topologies.

Contribution

It introduces Yin sets as a new model for 3D continua with singular points and offers a unique boundary representation based on glued surfaces.

Findings

Characterization of local and global topology of Yin sets

Unique boundary representation via glued surfaces

Applicability to complex 3D continua in scientific fields

Abstract

We propose to model the topology of three-dimensional (3D) continua by Yin sets, regular open semianalytic sets with bounded boundary. Our model differs from manifold-based models in that singular points of a 3D continuum, i.e., boundary points where the tangent plane is not uniquely defined, are treated not as anomalies but as a central subject of our theoretical investigation. We characterize the local and global topology of Yin sets. Then we give a unique boundary representation of Yin sets based on the notion of a glued surface, a quotient space of an orientable compact 2-manifold along a one-dimensional CW complex. Our results apply to 3D continua with arbitrarily complex topology and may be useful in a number of scientific and engineering applications such as solid modeling, computer-aided design, and numerical simulations of multiphase flows with topological changes.