In this paper, we propose adaptive moving mesh methods for the planar Willmore flow by incorporating a tangential velocity into the original geometric evolution. The tangential velocity is designed based on a monitor function constructed from the curvature and its variation, enabling dynamic mesh redistribution along the evolving interface. This adaptive redistribution enhances spatial resolution in regions of high geometric complexity while preserving mesh regularity. The resulting moving mesh formulation is discretized using the kth-order backward differentiation formula (BDFk) in time and finite difference methods in space. Moreover, a class of new relaxed Lagrange multiplier approaches is further incorporated into the adaptive moving mesh framework to construct energy-stable adaptive moving mesh methods. Furthermore, to enhance the adaptivity and flexibility of the proposed framework, we develop an adaptive strategy for selecting the monitor function and introduce an alternative redistribution approach. Finally, extensive numerical experiments demonstrate that the proposed BDFk-based adaptive scheme accurately captures the geometric evolution of the planar Willmore flow and exhibits excellent robustness and computational efficiency for problems involving complex interface geometries.
The Modified Direct Method: an Approach for Smoothing Planar and Surface Meshes
修正的直接法（Modified Direct Method，MDM）是一种用于光滑平面和曲面网格的迭代网格光滑方法，它是在Balendran [1]提出的非迭代光滑方法的基础上发展起来的。当平滑平面网格时，对于三角形和四边形网格，MDM的性能实际上与拉普拉斯平滑的性能相同;然而，MDM优于三四边形网格的拉普拉斯平滑。在光滑曲面网格时，对于三角形、四边形和四边形为主的混合网格，在光滑过程中，所有网格单元的平均质量（MQ）总是增加，均方误差（MSE）总是减小;对于三角形为主的混合网格，三角形的质量总是下降，四边形的质量总是上升。算例表明，该方法对平面和曲面三角形、四边形和四边形网格都是收敛的。
Construction of near-boundary Voronoi mesh layers for planar domains
Dorodnicyn Computing Center FRC CSC RAS（多罗德尼琴科计算中心俄罗斯科学院联邦研究中心）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Keldysh Institute of Applied Mathematics RAS（凯尔迪什应用数学研究所俄罗斯科学院）
