We present a new P 2 extension of the P 1 cavity operator used as the basis for topological modification in 3D metric based mesh adaptation, with notable success in strongly anisotropic industrial cases of CFD. The P 2 operator inherits the P 1 cavity operator's robustness -mesh validity is guaranteed at all times -and manages to recover a metric field's inherent curvature through a Riemannian edge length optimization algorithm. This generic approach allows it to tackle a variety of problems, which are defined only by the input metric field such as the classic problem of surface approximation -through a geometric error surface metric propagated to the volume -or unit mesh construction such as for interpolation error minimization through high-order L p error estimates.Consistence with the log-euclidian metric interpolation scheme used in P 1 adaptation is obtained by a rigorous formulation of the optimization problem. This guarantees full compliance of the operator with the general adaptation process, by accurately measuring Riemannian edge lengths.Particular stress was put on the performance of the operator because of its central role in anisotropic mesh adaptation. All curving operations are carried out locally: this is in opposition with global approaches, be they optimization or PDE based. The optimization itself is carried out by an inhouse solver tailored to the problem at hand. As a result, the added cost is strictly linear.Numerical results illustrating the P 2 cavity operator's ability to recover curvature, be it surface curvature extended to boundary layers or metric field induced curvature of the volume, will be presented through cases representative of real-world geometries encountered in CFD. Finally, the operator's ability to handle rather large cases (10M elements) in minutes will be demonstrated.

This paper describes developments on the P 2 cavity operator stemming from a new Bézier untangling algorithm. Both surface and volume are adapted to an anisotropic solution field with the cavity operator as the low-level driver handling all topological changes to the mesh. The P 2 extension of the cavity operator handles curvature through Riemannian curved edge length minimization in the volume and geometry projection on the surface. In particular, the anisotropy conserving log-euclidean metric interpolation scheme was extended to high-order elements to facilitate differentiating edge length in the metric field. As a step forward from previous iterations of the P 2 cavity operator, validity is now enforced through optimization of Jacobian coefficients using the simplex algorithm for linear programs. This is made possible by the fact that Jacobian control coefficients are linear with regards to each control point and enables the global optimization of the minimum of all control coefficients surrounding an edge at once. Numerical results illustrate the ability of metric-induced curving to relatively quickly curve 3D meshes with complex geometries involved in Computational Fluid Dynamics (CFD) using only local schemes. This framework allow us to curve highly anisotropic meshes with around 10 million elements within minutes.

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