Energy dissipative numerical scheme for gradient flows of planar curves using discrete partial derivatives and B-spline curves

TL;DR

This paper introduces an energy dissipative numerical scheme for planar curve gradient flows, combining discrete partial derivatives with B-spline curves to handle higher order derivatives and complex evolutions.

Contribution

It presents a novel framework integrating structure-preserving discretization with B-spline approximation for efficient simulation of planar curve flows.

Findings

Successfully simulates topology-changing elastic flows

Demonstrates stability and accuracy of the scheme

Provides visualizations of complex curve evolutions

Abstract

In this paper, we develop an energy dissipative numerical scheme for gradient flows of planar curves, such as the curvature flow and the elastic flow. Our study presents a general framework for solving such equations. To discretize time, we use a similar approach to the discrete partial derivative method, which is a structure-preserving method for the gradient flows of graphs. For the approximation of curves, we use B-spline curves. Owing to the smoothness of B-spline functions, we can directly address higher order derivatives. In the last part of the paper, we consider some numerical examples of the elastic flow, which exhibit topology-changing solutions and more complicated evolution. Videos illustrating our method are available on YouTube.