Implicit Meshing for Finite Element Methods using Levelsets

Finite Element methods (FEM) usually require a mesh to describe the geometric domain on which the computations are occuring. These meshes must have several properties: 1) they must approximate the geometrical domain accurately, 2) they must have good numerical properties, and 3) they must be small enough so that the computations take a reasonable amount of time. These goals are somewhat contradictory and in many cases such as biomedical images - and particularly in the case of the head -, even though the geometric domains can effectively be extracted, eg from Magnetic Resonance Images (MRI), the generation of such meshes is quite difficult. This paper describes a technique that bypasses this mesh generation step going directly from a description by levelsets of the interfaces separating the various domains to the matrix associated to the FEM method. Using the levelsets description is quite convenient as it is already used by many segmentation tools. The technique is illustrated on spherical and realistic geometries for the Electroencephalography (EEG) direct problem.