Qualitative and numerical aspects of a motion of a family of interacting curves in space

TL;DR

This paper studies the geometric evolution of interacting 3D curves driven by curvature, proving mathematical properties and developing numerical methods, with applications in modeling complex curve interactions.

Contribution

It introduces a Lagrangian approach and analytical framework for evolving interacting curves, along with an efficient finite volume numerical scheme.

Findings

Proved local existence and uniqueness of smooth solutions.

Developed a finite volume numerical scheme for the system.

Conducted computational experiments demonstrating flow behaviors.

Abstract

In this article we investigate a system of geometric evolution equations describing a curvature driven motion of a family of 3D curves in the normal and binormal directions. Evolving curves may be subject of mutual interactions having both local or nonlocal character where the entire curve may influence evolution of other curves. Such an evolution and interaction can be found in applications. We explore the direct Lagrangian approach for treating the geometric flow of such interacting curves. Using the abstract theory of nonlinear analytic semi-flows, we are able to prove local existence, uniqueness and continuation of classical H\"older smooth solutions to the governing system of nonlinear parabolic equations. Using the finite volume method, we construct an efficient numerical scheme solving the governing system of nonlinear parabolic equations. Additionally, a nontrivial tangential velocity is considered allowing for redistribution of discretization nodes. We also present several computational studies of the flow combining the normal and binormal velocity and considering nonlocal interactions.