Abstract 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.
A number of approaches for the generation of high-quality primal-dual grids have previously been pursued, typically focusing on the optimisation of conventional Delaunay-Voronoi type pairs. Such methods include a range of well-known techniques, including methods based on Centroidal Voronoi Tessellation (CVT) @cite_45 @cite_24 and Optimal Delaunay Triangulation (ODT) @cite_19 @cite_38 @cite_7 that seek to reposition primal vertices to minimise an that depends on the geometry and topology of the grid . While sophisticated CVT ODT schemes have been shown to generate high quality centroidal Voronoi-type tessellations, the construction of structures that are additionally well-centred has proven to be a difficult task; requiring the generation of so-called triangulations in which all angles in the primal mesh are bounded below @math . In @cite_0 @cite_48 , have shown that a non-linear optimisation-based strategy can be used to create such non-obtuse triangulations, though the resulting primal-dual tessellations do not appear to be centroidal or necessarily of optimal quality.
@cite_38 @cite_7 @cite_48 @cite_24 @cite_19 @cite_0 @cite_45
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In this paper, several mesh optimization schemes based on Optimal Delaunay Triangulations are developed. High-quality meshes are obtained by minimizing the interpolation error in the weighted L1 norm. Our schemes are divided into classes of local and global schemes. For local schemes, several old and new schemes, known as mesh smoothing, are derived from our approach. For global schemes, a graph Laplacian is used in a modified Newton iteration to speed up the local approach. Our work provides a mathematical foundation for a number of mesh smoothing schemes often used in practice, and leads to a new global mesh optimization scheme. Numerical experiments indicate that our methods can produce well-shaped triangulations in a robust and efficient way. Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, and optimal allocation of resources. In this paper, we explore the use of CVTs in grid generation in connection with finite element approximations of partial differential equations. We being by describing these tessellations and methods for their determination. We then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations. In this paper, a novel Delaunay-based variational approach to isotropic tetrahedral meshing is presented. To achieve both robustness and efficiency, we minimize a simple mesh-dependent energy through global updates of both vertex positions and connectivity. As this energy is known to be the ∠1 distance between an isotropic quadratic function and its linear interpolation on the mesh, our minimization procedure generates well-shaped tetrahedra. Mesh design is controlled through a gradation smoothness parameter and selection of the desired number of vertices. We provide the foundations of our approach by explaining both the underlying variational principle and its geometric interpretation. We demonstrate the quality of the resulting meshes through a series of examples. We present an iterative algorithm to transform a given planar triangle mesh into a well-centered one by moving the interior vertices while keeping the connectivity fixed. A well-centered planar triangulation is one in which all angles are acute. Our approach is based on minimizing a certain energy that we propose. Well-centered meshes have the advantage of having nice orthogonal dual meshes (the dual Voronoi diagram). This may be useful in scientific computing, for example, in discrete exterior calculus, in covolume method, and in space-time meshing. For some connectivities with no well-centered configurations, we present preprocessing steps that increase the possibility of finding a well-centered configuration. We show the results of applying our energy minimization approach to small and large meshes, with and without holes and gradations. Results are generally good, but in certain cases the method might result in inverted elements. A centroidal Voronoi tessellation is a Voronoi tessellation whose generating points are the centroids (centers of mass) of the corresponding Voronoi regions. We give some applications of such tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals. We discuss methods for computing these tessellations, provide some analyses concerning both the tessellations and the methods for their determination, and, finally, present the results of some numerical experiments.