Adaptive tetrahedral mesh generation by constrained centroidal voronoi‐delaunay tessellations for finite element methods

Abstract

This article presents a tetrahedral mesh adaptivity algorithm for three-dimensional elliptic partial differential equations (PDEs) using finite element methods. The main issues involved are the mesh size and mesh quality, which have great influence on the accuracy of the numerical solution and computational cost. The first issue is addressed by a posteriori error estimator based on superconvergent gradient recovery. The second issue is solved by constrained centroidal Voronoi-Delaunay tessellations (CCVDT), which guarantees good quality tetrahedrons over a large class of mesh domains even, if the grid size varies a lot at any particular refinement level. The CCVDT enjoys the energy equidistribution property so that the errors are very well equidistributed with properly chosen sizing field (density function). And with this good property, a new refinement criteria is raised which is different from the traditional bisection refinement.

The effect of Knudsen diffusion and adsorption on shale transport in nanopores

“…The computational domain is illustrated in Figure 5. High-quality unstructured mesh (centroidal Voronoi Delaunay triangulation 27,28 ) is used to partition the computational domain. The temperature is fixed at 388.15 K. For different cases, the pore radius (characteristic length) varies from 5 to 50 nm and the characteristic pressure varies from 1:0 3 10 5 Pa to 1:0 3 10 7 Pa.…”

Three‐dimensional superconvergent gradient recovery on tetrahedral meshes

“…Thus, the a posteriori error estimators will determine the local mesh size, which can be used to generate the adaptive CVDT meshes. We have constructed an efficient and robust adaptive finite element method based on MSPR recovery method in three dimensions .…”

Adaptive mixed-hybrid and penalty discontinuous Galerkin method for two-phase flow in heterogeneous media

“…We first introduce an a posterior error estimator based on superconvergent patch recovery (SPR) technique [23] and then adaptive mesh generation algorithm based on constrained centroidal Voronoi-Delaunay tessellations (CCVDT) is illustrated. Although there is no definition of interface in the two-phase Darcy model for two-phase flow in porous media, high resolution is needed where the variation of saturation is large.…”

“…According to this simple observation, we recover the gradient saturation G h S k,m w from PDG approximations S k,m w using the SPR method. The computational aspects can be found in [23]. Then The recovery-type local error estimator η T associated with the element K ∈ T h is given by…”

“…modified CCVDT[23]). Given a domain Ω, a density function ρ(x) defined on Ω and an initial constrained Delaunay tetrahedral mesh, 1. predetermine a subset of generating points on the boundary via a lower dimensional CVT construction based on the pre-defined density function ρ(x);2. construct the Voronoi regions for all interior points that are allowed to change their positions and compute the mass centers of the Voronoi regions based on the given density function ρ(x) and use the computed mass centers as generating points;3. retriangulation the domain Ω using constrained Delaunay tessellation with generating points; the resulting triangulation is the new T h ;…”