Simple finite element‐based computation of distance functions in unstructured grids

Abstract

SUMMARYA distance field is a representation of the closest distance from a point to a given surface. Distance fields are widely used in applications ranging from computer vision, physics and computer graphics and have been the subject of research of many authors in the last decade. Most of the methods for computing distance fields are devoted to Cartesian grids while little attention has been paid to unstructured grids. Finite element methods are well known for their ability to deal with partial differential equations in unstructured grids. Therefore, we propose an extension of the fast marching method for computing a distance field in a finite element context employing the element interpolation to hold the Eikonal property ( ∇ = 1). A simple algorithm to develop the computations is also presented and its efficiency demonstrated through various unstructured grid examples. We observed that the presented algorithm has processing times proportional to the number of mesh nodes.

Numerical Modeling and Experimental Validation of Free Surface Flow Problems

“…Several alternatives for keeping the regularity of the level set function field can be found in the literature, e.g. [14,75,100,156]. Mut et al [156] have proposed a redistancing algorithm for the LS function for simplicial unstructured meshes that includes a local mass correction algorithm.…”

“…Improvements to such method were reported in [14], where a comparison with a PDE-based method [116] is also performed. Moreover, Elias et al [75] have developed a fast marching method for computing the distance for a finite element grid based on using the finite element interpolation to hold the property |∇φ| = 1 by element, restricting the computation to a narrow band around the zero-LS.…”

An efficient edge‐based level set finite element method for free surface flow problems

“…The method proposed by Elias, Martins and Coutinho in [36] is taken as reference. For the calculation of the distance field of the domain 0 , numerical methods have to be employed because the use of analytical solution is not trivial.…”

“…This is done using the method presented in [36] and previously commented. The evaluation of ' on the rest of the domain.…”

Analysis of an efficient finite element method for embedded interface problems

“…Some of this has been driven by the adoption of level-set concepts by the finite element community [8,19,24] and the success of fixed-grid finite difference techniques such as the Ghost Fluid Method [9]. Related work concerns the extended finite element method (X-FEM) [2,18] and the generalized finite element method (G-FEM) and their application to a wide range of interface problems [1,4,7,10].…”