Paul-Emile Bernard

High-order discontinuous Galerkin schemes on general 2D manifolds applied to the shallow water equations

An innovating approach is proposed to solve vectorial conservation laws on curved manifolds using the discontinuous Galerkin method. This new approach combines the advantages of the usual approaches described in the literature. The vectorial fields are expressed in a unit non-orthogonal local tangent basis derived from the polynomial mapping of curvilinear triangle elements, while the convective flux functions are written is the usual 3D Cartesian coordinate system. The number of vectorial components is therefore minimum and the tangency constraint is naturally ensured, while the method remains robust and general since not relying on a particular parametrization of the manifold. The discontinuous Galerkin method is particularly well suited for this approach since there is no continuity requirement between elements for the tangent basis definition. The possible discontinuities of this basis are then taken into account in the Riemann solver on inter-element interfaces. The approach is validated on the sphere, using the shallow water equations for computing standard atmospheric benchmarks. In particular, the Williamson test cases are used to analyze the impact of the geometry on the convergence rates for discretization error. The propagation of gravity waves is eventually computed on non-conventional irregular curved manifolds to illustrate the robustness and generality of the method.

Frame field smoothness-based approach for hex-dominant meshing

An indirect approach for building hex-dominant meshes is proposed: a tetrahedral mesh is constructed at first and is recombined to create a maximum amount of hexahedra. The efficiency of the recombination process is known to significantly depend on the quality of the sampling of the vertices. A good vertex sampling depends itself on the quality of the underlying frame field that has been used to locate the vertices. An iterative procedure to obtain a high quality three-dimensional frame field is presented. Then, a new point insertion algorithm based on a frame field smoothness is developed. Points are inserted in priority in smooth frame field regions. The new approach is tested and compared with simpler strategies on various geometries. The new method leads to hex-dominant meshes exhibiting either an equivalent or a larger volume ratio of hexahedra (up to 20%) compared to the frontal point insertion approach. A frame field smoothness-based algorithm for bulk point insertion is proposed.An iterative procedure for smoothing the frame field is proposed.The impact of geometric singularities on the final mesh is reduced.Volumic ratio of hexahedra is increased up to twenty percents.

Handling Localization in Damage Models With the Thick Level Set Approach (TLS)

In this paper, we discuss a new way to model damage growth in solids. A level set is used to separate the undamaged zone from the damaged zone. In the damaged zone, the damage variable is an explicit function of the level set. This function is a parameter of the model. Beyond a critical length, it is assumed that the material is totally damaged, thus allowing a straightforward transition to fracture. The damage growth is expressed as a level set propagation. The configurational force driving the damage front is non local in the sense that it averages information over the thickness in the wake of the front. Three important theoretical advantages of the proposed approach are as follows: (a) The zone for which the materials is fully damaged is located inside a clearly identified domain (given by an iso-level set). (b) The non-locality steps in gradually in the model. At initiation the model is fully local. At initiation, micro-cracks being absent no length scale should prevail. (c) It is straightforward to prove that dissipation is positive. A numerical experiment of the cracking of a multiply perforated plate is discussed.Copyright