Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction

Given a tetrahedral mesh and objective functionals measuring the mesh quality which take into account the shape, size, and orientation of the mesh elements, our aim is to improve the mesh quality as much as possible. In this paper, we combine the moving mesh smoothing, based on the integration of an ordinary differential equation coming from a given functional, with the lazy flip technique, a reversible edge removal algorithm to modify the mesh connectivity. Moreover, we utilize radial basis function (RBF) surface reconstruction to improve tetrahedral meshes with curved boundary surfaces. Numerical tests show that the combination of these techniques into a mesh improvement framework achieves results which are comparable and even better than the previously reported ones.Comment: Revised and improved versio

A curvature-adapted anisotropic surface remeshing method

We present a new method for remeshing surfaces that respect the intrinsic anisotropy of the surfaces. In particular, we use the normal informations of the surfaces, and embed the surfaces into a higher dimensional space (here we use 6d). This allow us to form an isotropic mesh optimization problem in this embedded space. Starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by combining common local modifications operations, i.e., edge flip, edge contraction, vertex smoothing, and vertex insertion. All operations are applied directly on the 3d surface mesh. This method results a curvature-adapted mesh of the surface. This method can be easily adapted to mesh multi-patches surfaces, i.e., containing corner singularities and sharp features. We present examples of remeshed surfaces from implicit functions and CAD models

an anisoptropic surface remeshing strategy combining higher dimensional embedding with radial basis functions

Abstract Many applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy

A novel surface remeshing scheme via higher dimensional embedding and radial basis functions

Many applications heavily rely on piecewise triangular meshes to describe complex surface geometries. High-quality meshes significantly improve numerical simulations. In practice, however, one often has to deal with several challenges. Some regions in the initial mesh may be overrefined, others too coarse. Additionally, the triangles may be too thin or not properly oriented. We present a novel mesh adaptation procedure which greatly improves the problematic input mesh and overcomes all of these drawbacks. By coupling surface reconstruction via radial basis functions with the higher dimensional embedding surface remeshing technique, we can automatically generate anisotropic meshes. Moreover, we are not only able to fill or coarsen certain mesh regions but also align the triangles according to the curvature of the reconstructed surface. This yields an acceptable trade-off between computational complexity and accuracy

Tetrahedral Mesh Improvement Using Moving Mesh Smoothing and Lazy Searching Flips

In this paper we combine two new smoothing and flipping techniques. The moving mesh smoothing is based on the integration of an ordinary differential coming from a given functional. The lazy flip technique is a reversible edge removal algorithm to automatically search flips for local quality improvement. On itself, these strategies already provide good mesh improvement, but their combination achieves astonishing results which have not been reported so far. Provided numerical examples show that we can obtain final tetrahedral meshes with dihedral angles between 40 and 123 degrees. We compare the new method with other publicly available mesh improving codes

Anisotropic Finite Element Mesh Adaptation via Higher Dimensional Embedding

In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1], [2], [3], [4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in R3. In the context of adaptive finite element simulation, the solution (which is an unknown function f : Ω ⊂ d → ) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φf (x):= (x1, …, xd, s f (x1, …, xd), s ▿ f (x1, …, xd))t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function f itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function f. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φf (x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG – a metric-based adaptive mesh generator. The errors measured in the L2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been perfomed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.Comment: 21 pages, 10 tables, 4 figure