On the equivalence between the cell-based smoothed finite element method and the virtual element method

We revisit the cell-based smoothed finite element method (SFEM) for quadrilateral elements and extend it to arbitrary polygons and polyhedrons in 2D and 3D, respectively. We highlight the similarity between the SFEM and the virtual element method (VEM). Based on the VEM, we propose a new stabilization approach to the SFEM when applied to arbitrary polygons and polyhedrons. The accuracy and the convergence properties of the SFEM are studied with a few benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined with the scaled boundary finite element method to problems involving singularity within the framework of the linear elastic fracture mechanics in 2D

Optimal Voronoi Tessellations with Hessian-based Anisotropy

International audienceThis paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries

Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric

International audienceThis article presents a new method to compute a self-intersection free high-dimensional Euclidean embedding (SIFHDE) for surfaces and volumes equipped with an arbitrary Riemannian metric. It is already known that given a high-dimensional (high-d) embedding, one can easily compute an anisotropic Voronoi diagram by back-mapping it to 3D space. We show here how to solve the inverse problem, i.e., given an input metric, compute a smooth intersection-free high-d embedding of the input such that the pullback metric of the embedding matches the input metric. Our numerical solution mechanism matches the deformation gradient of the 3D → higher-d mapping with the given Riemannian metric. We demonstrate applications of the method, by being used to construct anisotropic Restricted Voronoi Diagram (RVD) and anisotropic meshing, that are otherwise extremely difficult to compute. In the SIFHDE-space constructed by our algorithm, difficult 3D anisotropic computations are replaced with simple Euclidean computations, resulting in an isotropic RVD and its dual mesh on this high-d embedding. The results are compared with the state-ofthe-art in anisotropic surface and volume meshings using several examples and evaluation metrics

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

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