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.