含曲率的水平集方程在非结构四边形网格上的数值离散方法

NUMERICAL SCHEMES FOR THE LEVEL SET EQUATIONS ON UNSTRUCTURED QUADRILATERAL MESHES

在非结构四边形网格上,含曲率的水平集方程采用伽辽金等参有限元方法空间离散,时间离散采用半隐格式.离散形成的线性方程组的系数矩阵是对称的稀疏矩阵,采用共轭梯度法求解.数值算例表明,在笛卡儿网格和随机网格上,含曲率的水平集方程离散格式可达到近似二阶精度.重新初始化方程的离散格式精度可达到近似一阶精度.给出了非结构四边形网格上不光滑界面以曲率收缩的运动过程.在不采用重新初始化的情况下,收缩过程未出现不稳定现象.

Level set equations containing curvature are solved on unstructured quadrilateral meshes. We use spatial discretization by the Galerkin isoparametric finite element method, and semi-implicit time stepping. Conjugate gradient method solves the linear system of equa- tions, whose coefficient matrix is symmetric and sparse. On Cartesian meshes and random meshes, the scheme of level set equations containing curvature is nearly second order accuracy in L2 and L∝ norms. Example is given of nonsmooth level sets shortening stably without reinitialization by local curvature on unstructured quadrilateral meshes.