A partition‐of‐unity‐based finite element method for level sets
A partition‐of‐unity‐based finite element method for level sets
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DOI：
10.1002/nme.2371
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发表时间：
2008-12
影响因子：
2.9
通讯作者：
S. Valance;R. de Borst;J. Réthoré;M. Coret
中科院分区：
文献类型：
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作者：
S. Valance;R. de Borst;J. Réthoré;M. Coret
水平集方法最近获得了很大的普及，以捕捉不连续性，包括其可能的传播。通常，水平集方法中出现的偏微分方程，特别是Hamilton-Jacobi方程，通过有限差分方法求解。然而，有限差分法不太适合不规则区域。此外，使用有限差分来捕捉不连续性似乎有点尴尬，而在随后的应力分析中通常使用有限元。为此，我们在这里提出了一个有限元方法来解决水平集方法的控制方程。在回顾了控制方程之后，将讨论水平集的初始化、有限域上的离散化以及由此产生的有限元方法的稳定性。将特别注意内部边界条件的适当处理，这是通过利用有限元形函数的单位分割特性来实现的。最后，对一维算例进行了包括精度分析在内的定量分析，对二维曲线不连续性算例进行了定性分析。版权所有© 2008约翰威利父子有限公司.
Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. Typically, the partial differential equations that arise in level set methods, in particular the Hamilton–Jacobi equation, are solved by finite difference methods. However, finite difference methods are less suited for irregular domains. Moreover, it seems slightly awkward to use finite differences for the capturing of a discontinuity, while in a subsequent stress analysis finite elements are normally used. For this reason, we here present a finite element approach to solving the governing equations of level set methods. After a review of the governing equations, the initialization of the level sets, the discretization on a finite domain, and the stabilization of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition‐of‐unity property of finite element shape functions. Finally, a quantitative analysis including accuracy analysis is given for a one‐dimensional example and a qualitative example is given for a two‐dimensional case with a curved discontinuity. Copyright © 2008 John Wiley & Sons, Ltd.