Numerical results

• 2d, 3d and surface anisotropic mesh adaptation

Anisotropic mesh adaptation allows to reduce dramatically the number of mesh nodes while improving the numerical accuracy of the solution. Here, quasi-uniform meshes are created with respect to a discrete anisotropic metric tensor field based on a geometric error estimate.



2d mesh adaptation The generation of quasi-uniform 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 Delaunay-based approach based on local geometrical and topological mesh modifications. A geometric error estimate based on the interpolation error provides the desired metric field to equi-distribute 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. Anisotropic mesh adaptation for CFD computations Delaunay-admissibilité des triangulations de surfaces Mesh gradation control 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 quasi-uniform meshes, with respect to the riemannian tensor field, well-suited for finite element-volume calculations. About surface mesh generation Surface meshing using a geometric error estimate Simplification of surface mesh using Hausdorff enveloppe Generation and adaptation of computational surface meshes from discrete anatomical data 3d anisotropic mesh adaptation (with C. Dobrzynski) Mesh adaptation in three dimensions is the natural extension of 2d anisotropic mesh adaptation. A quasi-uniform anisotropic Delaunay-based 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. Fast and accurate simulations of air-cooled structures

• Moving meshes (with C. Dobrzynski)

Rigid body displacements The local mesh modification tools can be efficienty used to resolve rigid-body displacement applications. This approach is coupled with a linear elasticity solver to prescribe a suitable field of displacements inside the computational domain. Anisotropic Delaunay mesh adaptation for unsteady simulations

• Unsteady problems (with F. Alauzet)

Transient fixed-point The classical mesh adaptation scheme appears inappropriate when dealing with unsteady problems. Hence, another approach based on a new mesh adaptation algorithm and a metric intersection in time procedure, suitable for capturing and tracking such phenomena, has been developed. A specific loop is introduced in the main adaptation loop in order to solve a transient fixed point problem for the (mesh,solution) couple. Here, an anisotropic geometric error estimate based on a bound of the interpolation error is used, using the Hessian of the solution. The mesh adaptation stage is based on the generation, by global remeshing, of a quasi-uniform with respect to the prescribed metric. 3d transient fixed point mesh adaptation for time-dependent problems Transient fixed point based unstructured mesh adaptation Adaptation de maillages pour des problèmes instationnaires

• Levelset method and adaptation (with A. Claisse and V. Ducrot)

We have recently started to couple mesh adapation techniques with levelsets in different types of applications (geometry capturing, interface tracking in flow simulations). The results obtained so far confirm that this may be a winning combination to improve the accuracy and the efficiency of our simulations.

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 |u-uh| in terms of the distance between the isovalue u=0 and the relevant discretisation. Contrôle de l'approximation géométrique d'une interface par une métrique anisotrope 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 PDE-based 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. Construction d'une courbe régulière d'approximation d'un ensemble de points Level set driven smooth curve approximation from unorganized or noisy point set (submitted)

• Viscous flow (Stokes) solver (with C. Bui and B. Maury)

Modification of the right-hand side term In this Note, we present a method to solve numerically the Stokes equation for the incompressible flow between two immiscible fluids presenting very different viscosities. The resolution of the finite element systems of equations is performed using Uzawa's method. The stiffness matrix conditioning problems related to the very important viscosity ratios are circumvented using an new iterative scheme. A numerical example is proposed to show the efficiency of this approach. Méthode du second membre modifié pour la gestion de rapports de viscosité importants dans le problème de Stokes bifluide

• Elasticity problems (with M. de Buhan)

Biomedical application In this project, we like to develop a mathematial model and analyze numerical methods to address the problem of the deformation of the cerebral structures in view of surgical treatment planning.

Software

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