A partition‐of‐unity‐based finite element method for level sets

Abstract

SUMMARYLevel 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.

The improvement of crack propagation modelling in triangular 2D structures using the extended finite element method

“…Meanwhile, the enrichment functions are the key of the XFEM. In the previous studies, the most common way to describe the cracks was the level set method . For the description of crack surface, the level set method considers the crack as the level set function; the crack separates the mesh into two parts in a proper range of the mesh in which one part is considered as the upper surface of the crack, and the other part is considered as the lower surface of the crack.…”

“…In the previous studies, the most common way to describe the cracks was the level set method. [22][23][24][25][26] For the description of crack surface, the level set method considers the crack as the level set function; the crack separates the mesh into two parts in a proper range of the mesh in which one part is considered as the upper surface of the crack, and the other part is considered as the lower surface of the crack. For the description of the crack tips, the location of the crack tips must be obtained to determine the nodes that are enriched by the crack tip enrichment.…”

Electrostatic simulation using XFEM for conductor and dielectric interfaces

“…We must therefore conclude that the shape functions proposed in [14] are not suitable for electro-mechanical modeling in the vicinity of conductors. It was shown in [28] that arbitrary Dirichlet boundary conditions can be imposed along an arbitrary contour using a regularization. In the following section, a different extended field is build to circumvent this shortcoming.…”

Étude mécanique d’un changement de phase allotropique à l’échelle mésoscopique

“…Une méthode de résolution est proposée par les auteurs dans [18]. La première équation du système 9 est résolue par l'utilisation d'une formulation faible du problème linéarisé alors que la condition aux limites est respectée par l'utilisation de fonctions de forme enrichies respectant la partition de l'unité [19].…”