Local bisection for conformal refinement of unstructured 4D simplicial meshes

We present a conformal bisection procedure for local refinement of 4D unstructured simplicial meshes with bounded minimum shape quality. Specifically, we propose a recursive refine-to-conformity procedure in two stages, based on marking bisection edges on different priority levels and defining specific refinement templates. Two successive applications of the first stage ensure that any 4D unstructured mesh can be conformingly refined. In the second stage, the successive refinements lead to a cycle in the number of generated similarity classes and thus, we can ensure a bound over the minimum shape quality. In the examples, we check that after successive refinement the mesh quality does not degenerate. Moreover, we refine a 4D unstructured mesh and a space-time mesh (3D + 1D) representation of a moving object

Generating admissible space-time meshes for moving domains in d+1d+1-dimensions

In this paper we present a discontinuous Galerkin finite element method for the solution of the transient Stokes equations on moving domains. For the discretization we use an interior penalty Galerkin approach in space, and an upwind technique in time. The method is based on a decomposition of the space-time cylinder into finite elements. Our focus lies on three-dimensional moving geometries, thus we need to triangulate four dimensional objects. For this we will present an algorithm to generate $d+1$-dimensional simplex space-time meshes and we show under natural assumptions that the resulting space-time meshes are admissible. Further we will show how one can generate a four-dimensional object resolving the domain movement. First numerical results for the transient Stokes equations on triangulations generated with the newly developed meshing algorithm are presented

An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems

Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem. Several variants of the new condition are obtained. Based on one of them, two metric tensors for use in anisotropic mesh generation are developed to account for DMP satisfaction and the combination of DMP satisfaction and mesh adaptivity. Numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors.Comment: 34 page

Conformal n-dimensional bisection for local refinement of unstructured simplicial meshes

[English] In n-dimensional adaptive applications, conformal simplicial meshes must be lo cally modified. One systematic local modification is to bisect the prescribed simplices while surrounding simplices are bisected to ensure conformity. Although there are many conformal bisection strategies, practitioners prefer the method known as the newest vertex bisection. This method guarantees key advantages for adaptivity when ever the mesh has a structure called reflectivity. Unfortunately, it is not known (i) how to extract a reflection structure from any unstructured conformal mesh for three or more dimensions. Fortunately, a conformal bisection method is suitable for adap tivity if it almost fulfills the newest vertex bisection advantages. These advantages are almost met by an existent multi-stage strategy in three dimensions. However, it is not known (ii) how to perform multi-stage bisection for more than three dimensions. This thesis aims to demonstrate that n-dimensional conformal bisection is possible for local refinement of unstructured conformal meshes. To this end, it proposes the following contributions. First, it proposes the first 4-dimensional two-stage method, showing that multi-stage bisection is possible beyond three dimensions. Second, fol lowing this possibility, the thesis proposes the first n-dimensional multi-stage method, and thus, it answers question (ii). Third, it guarantees the first 3-dimensional method that features the newest vertex bisection advantages, showing that these advantages are possible beyond two dimensions. Fourth, extending this possibility, the thesis guarantees the first n-dimensional marking method that extracts a reflection struc ture from any unstructured conformal mesh, and thus, it answers question (i). This answer proves that local refinement with the newest vertex bisection is possible in any dimension. Fifth, this thesis shows that the proposed multi-stage method al most fulfills the advantages of the newest vertex bisection. Finally, to visualize four dimensional meshes, it proposes a simple tool to slice pentatopic meshes. In conclusion, this thesis demonstrates that conformal bisection is possible for local refinement in two or more dimensions. To this end, it proposes two novel methods for unstructured conformal meshes, methods that will enable adaptive applications on n-dimensional complex geometry. [Español] En aplicaciones adaptativas n-dimensionales, las mallas simpliciales conformes deben modificarse localmente. Una modificación local sistemática es bisecar los símplices prescritos mientras que los símplices circundantes se bisecan para garantizar la conformidad. Aunque existen muchas estrategias conformes de bisección, en aplicaciones prácticas se prefiere el método conocido como newest vertex bisection (NVB). Este método garantiza las propiedades deseadas para la adaptatividad siempre y cuando la malla tenga una estructura llamada reflectividad. Desafortunadamente, no se sabe (i) cómo extraer una estructura de reflexión de cualquier malla conforme no estructurada para tres o más dimensiones. Afortunadamente, un método de bisección conforme es adecuado para la adaptatividad si casi cumple con las propiedades de NVB. Estas propiedades son casi satisfechas por una estrategia existente de múltiples etapas en tres dimensiones. Sin embargo, no se sabe (ii) cómo realizar una bisección en múltiples etapas para más de tres dimensiones. Esta tesis tiene como objetivo demostrar que la bisección conforme n-dimensional es posible para el refinamiento local de mallas conformes no estructuradas. Para ello propone las siguientes aportaciones. Primero, propone el primer método de dos etapas de 4 dimensiones, que muestra que la bisección de múltiples etapas es posible en más de tres dimensiones. En segundo lugar, siguiendo esta posibilidad, la tesis propone el primer método n-dimensional de múltiples etapas y, por tanto, responde a la pregunta (ii). En tercer lugar, garantiza el primer método tridimensional que presenta las propiedades NVB, lo que demuestra que estas propiedades son posibles más allá de dos dimensiones. En cuarto lugar, ampliando esta posibilidad, la tesis garantiza el primer método de marcado n-dimensional que extrae una estructura de reflexión de cualquier malla conforme no estructurada y, por lo tanto, responde a la pregunta (i). Esta respuesta demuestra que el refinamiento local con NVB es posible en cualquier dimensión. Quinto, esta tesis muestra que el método de múltiples etapas propuesto casi cumple con las propiedades de NVB. Finalmente, para visualizar mallas de cuatro dimensiones, propone una herramienta simple para cortar mallas pentatópicas. En conclusión, esta tesis demuestra que la bisección conforme es posible para el refinamiento local en dos o más dimensiones. Con este fin, propone dos métodos novedosos para mallas conformes no estructuradas, métodos que harán posible aplicaciones adaptativas en geometría compleja n-dimensionalPostprint (published version