We develop an approach to generating degree-of-freedom maps for arbitrary order finite element spaces for any cell shape. The approach is based on the composition of permutations and transformations by cell sub-entity. Current approaches to generating degree-of-freedom maps for arbitrary order problems typically rely on a consistent orientation of cell entities that permits the definition of a common local coordinate system on shared edges and faces. However, while orientation of a mesh is straightforward for simplex cells and is a local operation, it is not a strictly local operation for quadrilateral cells and in the case of hexahedral cells not all meshes are orientable. The permutation and transformation approach is developed for a range of element types, including Lagrange, and divergence- and curl-conforming elements, and for a range of cell shapes. The approach is local and can be applied to cells of any shape, including general polytopes and meshes with mixed cell types. A number of examples are presented and the developed approach has been implemented in an open-source finite element library.
翻译:我们开发了一种方法,为任何细胞形状的任意顺序限定元素空间绘制自由度地图,该方法以细胞子实体的变异和变异构成为基础。目前为任意顺序问题绘制自由度地图的方法通常依赖细胞实体的一致方向,这种方向允许在共享边缘和面孔上确定共同的本地协调系统。虽然网状方向对简单x单元格来说是直截了当的,是一种局部操作,但对于四边细胞和对于六面细胞来说,这不是严格的局部操作,对于六面细胞来说,并不是所有中间细胞都是可调整的。变异和变异方法是为一系列元素类型,包括拉格朗、差异和曲线组合元素,以及一系列细胞形状而开发的。该方法是局部的,可以适用于任何形状的细胞,包括普通的多面和混合细胞类型的金属。介绍了一些实例,开发的方法已在开放源的有限元素库中实施。