论文标题
结构保存的数值方法，用于具有显式切向速度的平面封闭曲线的约束梯度流量
Structure-preserving numerical methods for constrained gradient flows of planar closed curves with explicit tangential velocities
论文作者
论文摘要
在本文中，我们考虑了平面封闭曲线的约束梯度流的数值近似，包括Willmore和Helfrich流。这些方程式具有耗能，由于限制，后者具有保护特性。我们将为这些方程式开发结构性的方法，以保留耗散和约束。为了保留能量结构，我们根据离散梯度方法介绍了离散的梯度版本，并适当地确定Lagrange乘数。我们通过使用与B-Spline曲线的Galerkin方法直接解决高阶导数来离散曲线。此外，我们将通过增加切向速度来考虑对方案的稳定。我们引入了一个新的Lagrange乘数，以获得能量结构和稳定性。提出了几个数值示例，以验证所提出的方案是否保留了控制点良好分布的能量结构。
In this paper, we consider numerical approximation of constrained gradient flows of planar closed curves, including the Willmore and the Helfrich flows. These equations have energy dissipation and the latter has conservation properties due to the constraints. We will develop structure-preserving methods for these equations that preserve both the dissipation and the constraints. To preserve the energy structures, we introduce the discrete version of gradients according to the discrete gradient method and determine the Lagrange multipliers appropriately. We directly address higher order derivatives by using the Galerkin method with B-spline curves to discretize curves. Moreover, we will consider stabilization of the schemes by adding tangential velocities. We introduce a new Lagrange multiplier to obtain both the energy structures and the stability. Several numerical examples are presented to verify that the proposed schemes preserve the energy structures with good distribution of control points.
