Section: New Results

High order mesh generation and mesh adaptation

Participants : Luca Arpaia, Cécile Dobrzynski [Corresponding member] , Ghina El Jannoun, Mario Ricchiuto.

This year several new algorithmic improvements have been obtained which will allow to enhance our meshing tools:

• We have enhanced our work on $r$-adaptation techniques for time dependent equations. These techniques are based on mesh deformations obtained by solving continuous differential equations for the local displacements. These equations are controlled by an error monitor. Several improvements have been made. We have studied in depth the formulation of the coupling of the mesh movement with the flow solver. We have found that for both finite volume and residual distribution methods, a coupling of mesh and solution evolution (by means of an ALE method) provides accuracy enhancements, and is to be preferred to a simpler adapt-project-evolve approach. The method has been fully tested in two space dimensions. The adaptation library has been extended to three dimensions, and benchmarking is under way. We have improved the definition of the error monitor, and we are now able to prescribe directly the local mesh size. For problems with source terms, and in particular problems admitting some important physical invariants as the shallow water equations, we have solved the conflict between the conservation of either mass or the invariant, allowing for the conservation of both quantities up to machine accuracy;

• We upgrade our technique for generating high order curved meshes: starting from a straight mesh with a curved boundary, a new smoothing and untangling approach is proposed to ensure a final valid mesh. The untangling algorithm is a hybrid technique that gathers a local mesh optimization applied on the surface of the domain and a linear elasticity analogy applied in its volume. On the one hand, the local topological optimization consists in simultaneously relocating the vertices and control points of a local patch around the invalid element in order to optimize the quality and validity of the elements inside the patch. The elements' validity problem is formulated as an unconstrained optimization problem using a log-barriers that is solved progressively using the conjugate gradient method. On the other hand, the linear elasticity analogy permits the propagation of the curvature to the volume of the domain hence untangling volume mesh elements.