On the Spline-Based Parameterisation of Plane Graphs via Harmonic Maps

TL;DR

This paper introduces a spline-based method for conformally parameterising plane graphs using harmonic maps, enabling applications in engineering simulations and mesh generation with flexible local features.

Contribution

It develops a novel spline-based harmonic map framework for plane graph parameterisation, allowing for locally adaptable material properties and integration with numerical analysis.

Findings

Suitable for isogeometric analysis

Enables dense mesh extraction

Supports application-specific feature integration

Abstract

This paper presents a spline-based parameterisation framework for plane graphs. The plane graph is characterised by a collection of curves forming closed loops that fence-off planar faces which have to be parameterised individually. Hereby, we focus on parameterisations that are conforming across the interfaces between the faces. Parameterising each face individually allows for the imposition of locally differing material parameters which has applications in various engineering disciplines, such as elasticity and heat transfer. For the parameterisation of the individual faces, we employ the concept of harmonic maps. The plane graph's spline-based parameterisation is suitable for numerical simulation based on isogeometric analysis or can be utilised to extract arbitrarily dense classical meshes. Application-specific features can be built into the geometry's mathematical description either on the spline level or in the mesh extraction step.