2d mesh adaptationThe generation of quasiuniform meshes with respect to a metric tensor field is an efficient manner of improving the accuracy as well as of reducing the computational complexity of a numerical simulation. We have investigated the generation of highly anisotropic meshes using a Delaunaybased approach based on local geometrical and topological mesh modifications. A geometric error estimate based on the interpolation error provides the desired metric field to equidistribute the error over the adapted mesh. This approach has been successfully to computational fluid dynamics applications where elements with high aspect ratio are often needed to fully resolve the flow features.
Surface adaptation (with H. Borouchaki)Surface meshes are adapted to a geometric metric based on the intrinsic properties of the discrete (triangulated) surface. Mesh decimation and mesh enrichment procedures produce quasiuniform meshes, with respect to the riemannian tensor field, wellsuited for finite elementvolume calculations.
3d anisotropic mesh adaptation (with C. Dobrzynski)Mesh adaptation in three dimensions is the natural extension of 2d anisotropic mesh adaptation. A quasiuniform anisotropic Delaunaybased mesh is generated based on a given metric tensor field. Local mesh modifications (edge flipping, node insertion, vertex deletion, node relocation) are used to generate the adapted mesh.

Geometric approximation (with V. Ducrot)We have proposed a general purpose method to control the geometric approximation of a manifold of codimension 1 associated with an isovalue of a scalar function u. To this end, we rely on the generation and the adaptation of an anisotropic triangulation to a metric tensor field related to the intrinsic properties of the manifold. Given a triangulation Th of the domain, we consider as approximation space of the function u the space of affine Lagrange finite elements P1. Under these assumptions, our Theorem provides an upper bound on the approximation error uuh in terms of the distance between the isovalue u=0 and the relevant discretisation.
Curve and surface reconstruction (with A. Claisse)We deal with the problem of constructing a smooth curve passing at best through all points of a given dataset. We define it as the isovalue {u(x) = 0} of an implicitly defined function u. To this end, we formulate this problem using the levelset method, that has been originally designed to propagate interfaces at a velocity related to the local curvatures of the manifolds. We consider that the desired curve is the result of the PDEbased advection of an initial smooth curve. In addition, we are able to show that this solution is also solution of a minimization problem and leads to the desired minimal curve. The equation we propose is composed of two terms, namely a regularization term and an attraction term.


Follow this link to download software.