P² Cavity Operator and Riemannian Curved Edge Length Optimization: a Path to High-Order Mesh Adaptation

Abstract

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.

Developments on the P^2 cavity operator and Bézier Jacobian correction using the simplex algorithm.

“…Full integration remains in progress, preventing full P 2 adaptation. Results provided in previous papers about the P 2 cavity operator [35,36] have already established the speed of the Riemannian edge length minimization algorithm as well as its ability to curve.…”

“…Applications shown in previous publications [35,36] already establish the Riemannian edge length minimization procedure's ability to curve the volume quickly, at about 40000 edges per second on a laptop. Applications shown here therefore focus on illustrating the merits of the simplex-based optimization approach and its ability to conserve a curved surface and volume edges.…”

“…In previous works [35,36], we have introduced high-order log-euclidean metric interpolation and used it to devise an efficient Riemannian edge length minimization algorithm. This was done in order to recover curvature from the metric field's variations much in the same fashion P 1 adaptation recovers anisotropy from metric values.…”

“…The original cavity operator is the single operator responsible for all topological changes in our mesh adaptation algorithm. In this section, we present the P 2 cavity operator first introduced in [36] and further developped on in [35]. The main development between the two is in the generalization of high-order log-euclidean metric interpolation since, in the earliest work, metrics on P 2 edges were derived as the P 1 log-euclidean metric interpolate in the P 1 triangle formed by the three edge control points.…”

“…The first is to curve individual new edges and surface edges or to leave them straight, as requested by the user. The curvature driver is the Riemannian edge length optimization algorith described in [35,36] in the volume and mid-point CAD projection on the surface. The second step is to correct the new cavity.…”