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

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.