A complete mesh generation or mesh adaptation process usually requires a large number of operators : Delaunay insertion, edge-face-element point insertion, edge collapse, point smoothing, face/edge swaps, etc. Independently of the complexity of the geometry, the more operators are involved in a remeshing process, the less robust the process may become. In addition, deriving a parallel version of the process may involve a large number of modifications for each operator. Consequently, the multiplication of operators implies additional difficulties in maintaining, improving and parallelizing a code. The scope of this paper is to address these issues by introducing a unique cavity-based operator. As it embeds all aforementioned operators, it can be used as the unique operator at each step of the process from surface and volume remeshing to boundary layer extrusion. In addition, we show that a coarse grain parallelization is possible by using a surface-constrained version of the operator.