We present a promising approach to reduce the difficulties associated with meshing complex curved domain boundaries for higher-order finite elements. In this work, higher-order XFEM analyses for strong discontinuity in the case of linear elasticity problems are presented. Curved implicit boundaries are approximated inside an unstructured coarse mesh by using parametric information extracted from the parametric representation (the most common in Computer Aided Design CAD). This approximation provides local graded sub-mesh (GSM) inside boundary elements (i.e. an element split by the curved boundary) which will be used for integration purpose. Sample geometries and numerical experiments illustrate the accuracy and robustness of the proposed approach.

La methode des elements finis (EF) est largement utilisee pour la simulation numerique de problemes physiques formules en terme d'equations aux derivees partielles (EDP). Une etape cruciale du processus d'analyse par cette methode est la discretisation de la geometrie du domaine afin de construire le maillage sur lequel est formule l'espace d'approximation du probleme. Cependant, la creation d'un maillage de qualite conforme aux frontieres courbes et aux aretes vives, dont depend les resultats numeriques, necessite encore un apport significatif de temps humain lors du processus globale d'analyse. L'objet de ce travail est la mise en oeuvre d'une nouvelle approche qui permet de realiser des simulations sur un objet dont la frontiere est non-conforme au maillage, tout en conservant les avantages des EF. Pour cela, on utilise une representation implicite du domaine (Level set) et la methode des elements finis etendus (XFEM). Dans un premier temps, on s'interesse a construire des objets par Level sets independamment de la discretisation spatiale (i.e. un maillage simple). Des strategies ont ete developpees afin de construire des objets implicites a partir de la representation parametrique la plus populaire en conception CAO, de preserver les aretes vives et pour pouvoir representer correctement les frontieres courbes. Dans un deuxieme temps, on s'interesse a l'adaptation de la methode XFEM afin de realiser une integration numerique correcte et de preserver la stabilite des formulations mixtes pour la gestion de la contrainte de Dirichlet. La derniere partie consiste a verifier la precision et les taux de convergence dans le cas des frontieres courbes et pour des objets entierement non-conformes au maillage

In this paper, we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis. The novel contributions of this paper are threefold: (1) describe and validate a method to represent arbitrary parametric surfaces implicitly; (2) represent arbitrary solids implicitly, including sharp features using level sets and boolean operations; (3) impose arbitrary Dirichlet and Neumann boundary conditions on the resulting implicitly defined boundaries. The methods proposed do not require local refinement of the finite element mesh in regions of high curvature, ensure the independence of the domain’s volume on the mesh, do not rely on boundary regularization, and are well suited to methods based on fixed grids such as the extended finite element method (XFEM). Numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries. This approach is one step in the desired direction of tying numerical simulations to computer aided design (CAD), similarly to the isogeometric analysis paradigm.

The goal of this chapter is to review recent avenues of investigation to alleviate meshing difficulties in computational mechanics and give a few exemplar applications.

Various numerical methods are used in engineering analysis, among which the finite element method (FEM) is the most prominent. This method relies heavily on a mesh structure, and the generation of this mesh generation accounts for about 80% of a typical analysis time for practical engineering problems. Various ideas have been proposed in the literature to avoid these problems by either simplifying this mesh generation process, or relaxing some of the constraints associated with the very presence of a mesh. This paper reviews recent advances in this direction by focusing on what the authors consider the most versatile and prominent approaches to overcome the mesh burden in computational science, namely: * Meshfree methods (MMs) somewhat reduce the constraints posed by the mesh by generalizing the concept of elements, simplifying. * Isogeometric analysis whose focus is to closely tie the geometry, i.e. computer aided design data to the analysis, e.g. FEM by using the same functions for both. If alterations to the geometry are required, the FE model is automatically modified, without changing the mesh, which greatly simplifies the design iterations. * The extended/generalized FEM (X/GFEM) where one of the aims is to increase the independence between the problem solved and the mesh. These methods are particularly attractive as they afford the modelling of crack propagation without remeshing. Inclusions and holes as well as the domain boundary can also be treated independently of the mesh. * Geometry independent methods, implicit meshing, immersed FEM, immersed boundary, fictitious domain/fixed grid FEM are alternatives to the FEM where the geometrically complex domain is implicitly embedded in a much simpler domain which is easily meshed using a regular, Cartesian grid. Geometry independent techniques have been used both in the context of finite volume method (FVM) and FEM. The literature on this type of technique goes back, according to, to the 1960s in the Russian language publication by Saulev and have subsequently been applied in different fields. Among other references in each discipline, we can mention applications in acoustics, fluid dynamics and fluid structure interaction, biomechanics, convection diffusion, optimization, etc. * Strain smoothing in finite elements allows decreasing the negative effects of mesh distortion. These methods have been used to solve a variety of mechanics problems including plates, shells, cracking, and three-dimensional viscoelastic deformation. The main goal of these methods is to rely on simplex meshes (tetrahedral, triangles) which are easier to generate than the more robust and accurate hexahedral meshes, and to do so without sacrificing accuracy. The relative merits and shortcomings of these methods are critically reviewed and a few examples are given for illustrative purposes. Other methods which are not covered here include the boundary element method and the scaled boundary finite element method.

L’objet de ce travail est la mise en œuvre d’une nouvelle approche qui evite certaines difficultes liees a la generation de maillage, pour but de liberer le maillage du respect scrupuleux des surfaces representant la structure. L’approche envisagee s’appuie sur la methode des elements finis etendus XFEM basee sur le concept de partition de l’unite. L’idee principale extraite de cette methode est effectivement d'utiliser une discretisation de l'espace ambiant comme support des fonctions de forme et de realiser des calculs de structures par elements finis a l’aide de la technique de representation de surfaces par des fonctions implicites (Level sets).