Finite Element-Based Level Set Methods for Higher Order Flows
Finite Element-Based Level Set Methods for Higher Order Flows
复制标题
高阶流的基于有限元的水平集方法
DOI：
10.1007/s10915-008-9204-x
复制
发表时间：
2008
影响因子：
2.5
通讯作者：
A. Voigt
中科院分区：
文献类型：
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作者：
M. Burger;C. Stöcker;A. Voigt
在本文中，我们将讨论水平集方法对几何流的数值模拟。考虑的主要例子是高阶流，例如表面扩散和威尔莫尔流以及它们具有更复杂表面能的变体。此类问题有多种应用，例如材料科学（薄膜生长、晶界运动）、生物物理学（膜形状）和计算机图形学（表面平滑和恢复）。我们将使用基于局部变分原理的有限元方法和半隐式时间步进的空间离散化，这允许通过离散化保持流动的耗散特性。为了补偿缺失的最大值原理（这确实是水平集方法应用于高阶流的主要障碍），我们频繁地重新调整水平集函数的距离。最后，我们还讨论了每个时间步中出现的离散线性系统的解决方案以及有限元方法的一些特殊优点，例如允许有效处理高阶和各种各向异性的变分公式以及零水平集周围局部自适应的可能性。
In this paper we shall discuss the numerical simulation of geometric flows by level set methods. Main examples under considerations are higher order flows, such as surface diffusion and Willmore flow as well as variants of them with more complicated surface energies. Such problems find various applications, e.g. in materials science (thin film growth, grain boundary motion), biophysics (membrane shapes), and computer graphics (surface smoothing and restoration).We shall use spatial discretizations by finite element methods and semi-implicit time stepping based on local variational principles, which allows to maintain dissipation properties of the flows by the discretization. In order to compensate for the missing maximum principle, which is indeed a major hurdle for the application of level set methods to higher order flows, we employ frequent redistancing of the level set function.Finally we also discuss the solution of the arising discretized linear systems in each time step and some particular advantages of the finite element approach such as the variational formulation which allows to handle the higher order and various anisotropies efficiently and the possibility of local adaptivity around the zero level set.
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