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 many applications in engineering applications, 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.

翻译：本文提出了一种基于样条的平面图参数化框架。该平面图由一系列形成闭合环路的曲线所表征，这些曲线围出了需要单独参数化的平面面片。在此，我们重点关注在面片交界处保持保形性的参数化方法。对面片进行单独参数化允许施加局部不同的材料参数，这在许多工程应用中具有重要价值，例如弹性力学和热传导问题。对于单个面片的参数化，我们采用调和映射的概念。这种基于样条的平面图参数化方法适用于基于等几何分析的数值模拟，也可用于提取任意密集的经典网格。特定应用的功能特征可以在样条层面或网格提取步骤中融入几何的数学描述中。