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A partition‐of‐unity‐based finite element method for level sets
Author(s) -
Valance Stéphane,
de Borst René,
Réthoré Julien,
Coret Michel
Publication year - 2008
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2371
Subject(s) - finite element method , extended finite element method , partition of unity , mixed finite element method , classification of discontinuities , partial differential equation , mathematics , smoothed finite element method , discontinuity (linguistics) , finite element limit analysis , finite difference , hp fem , finite difference method , boundary knot method , level set method , mathematical analysis , computer science , boundary element method , physics , segmentation , artificial intelligence , image segmentation , thermodynamics
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.