The successive node snapping scheme for an evolving branched curve in 2D and 3D

Abstract

We introduce a novel method to construct conforming meshes for an evolving branched curve immersed in a given background mesh in 2D and 3D. The proposed method is built on the successive node snapping (SNS) scheme proposed by Wan and Shen (2016) and Wan et al. (2019) which is able to generate a conforming mesh for a simple evolving curve in 2D and 3D. The essence of the SNS scheme is to snap chosen nodes to the curve, and to simultaneously relax the nodes within the neighborhood of the curve. Following the curve, each portion of the curve’s vicinity is adjusted to obtain a mesh conforming to the entire curve. The core of the generalization of the SNS scheme for a branched curve is to assign each branch a matching node, from which we start to adjust the mesh for the branch via a procedure similar to that for a simple curve. The approach to determine the matching node is based on minimizing mesh distortion. The proposed approach inherits the main advantage of the SNS scheme, including unaltered nodal connectivities, small adjustment of nodes, no a priori conformity requirements on the background mesh, and permitting obtuse-angled triangular or tetrahedral background mesh as well as acute-angled ones. As all the operations on the mesh are local, the proposed method is especially suitable for evolving branched curve problems in which the curve updates a little bit at each time/ load step. Numerical examples in 2D and 3D are provided.