Pierre Alliez Institut National de Recherche en Informatique Automatique (INRIA) Hexahedral meshes generated from polycube mapping often exhibit a low number of singularities but also poor quality elements located near the surface. Improving the quality may be obtained via global padding, which pushes the singularities inside by adding an extra layer of hexahedra on the entire domain boundary. Such a global padding operation suffers from a large increase of complexity, with unnecessary hexahedra added. In addition, the quality of elements near the boundary may decrease. We propose a novel optimization method which inserts sheets of hexahedra so as to perform selective padding, where it is most needed for improving the mesh quality. A sheet can pad part of the domain boundary, traverse the domain and form singularities. Our global formulation, based on solving a binary problem, enables us to control the balance between quality improvement, increase of complexity and number of singularities.