Generalised primal-dual grids for unstructured co-volume schemes

Highlights: • New approach for the construction of staggered unstructured computational grids. • Use of Regular-Power tessellations to define optimal staggered orthogonal meshes. • Development of new methods for the optimisation of Regular-Power structures. • Results confirm Regular-Power approach is superior to Delaunay–Voronoi methods. • Open-source implementation provided in the JIGSAW meshing library. The generation of high-quality staggered unstructured grids is considered, leading to the development of a new optimisation-based strategy designed to construct weighted 'Regular-Power' tessellations appropriate for co-volume type numerical discretisation techniques. This new framework aims to extend the conventional Delaunay–Voronoi primal-dual structure; seeking to assemble generalised orthogonal tessellations with enhanced geometric quality. The construction of these grids is motivated by the desire to improve the performance and accuracy of numerical methods based on unstructured co-volume type schemes, including various staggered grid techniques for the simulation of fluid dynamics and hyperbolic transport. In this study, a new hybrid optimisation strategy is proposed; seeking to optimise the geometry, topology and weights associated with general, two-dimensional Regular-Power tessellations using a combination of gradient-ascent and energy-based techniques. The performance of this new method is tested experimentally, with a range of complex, multi-resolution primal-dual grids generated for various coastal and regional ocean modelling applications.