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.
The Argyris isogeometric space on unstructured multi-patch planar domains
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉登计算与应用数学研究所）；Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia（帕维亚大学F.卡索拉蒂数学系）；Istituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR)（意大利国家研究委员会E.马格内斯应用数学与信息技术研究所）；Institute of Applied Geometry, Johannes Kepler University Linz（林茨约翰·开普勒大学应用几何研究所）
Space of $C^2$-smooth geometrically continuous isogeometric functions on planar multi-patch geometries: Dimension and numerical experiments
Università degli Studi di Pavia（帕维亚大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉顿计算与应用数学研究所）；University of Primorska（普里莫斯卡大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）
