Modeling Curved Fibers by Fitting R-vine Copulas to their Frenet Representations

Abstract

Abstract In the present paper, we propose a novel single-fiber model which exploits a description of fibers as a sequence of bond and torsion angles. Using the Frenet–Serret formulas, this representation can be translated into three-dimensional (3D) space and vice-versa. While the precise locations of points along a fiber do not directly convey information about the inner material properties of the fiber, the distribution of bond, and torsion angles may be related to various material characteristics and, thus, our model may form a direct link between inner material properties and emerging microstructure properties. More precisely, we model curved fibers in the 3D Euclidean space R3 as polygonal tracks that we represent by their local curvature and torsion at each sampling point. The 2D sequences of curvatures and torsions obtained in this way are then considered as realizations of a Markov chain with finite memory which takes its values in R2. The transition kernel of this Markov chain is given by a family of conditional multivariate probability distributions. They are constructed using so-called R-vine copulas, which are fitted and validated by means of experimental data.