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.

中文翻译：

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在多边形和多面体网格上构造任意阶的有限元自由度图

我们开发了一种方法，可以为任何像元形状的任意阶数有限元空间生成自由度图。该方法基于单元子实体的排列和变换的组成。生成针对任意顺序问题的自由度图的当前方法通常依赖于单元实体的一致方向，该方向允许在共享边和面上定义公共局部坐标系。但是，尽管网格的定向对于单纯形单元很简单并且是局部操作，但对于四边形单元来说并不是严格的局部操作，对于六面体单元，并非所有网格都可以定向。排列和变换方法是针对一系列元素类型（包括拉格朗日，散度和卷曲符合元素）开发的，以及一系列的单元格形状。该方法是局部的，可以应用于任何形状的单元，包括普通的多面体和具有混合单元类型的网格。给出了许多示例，并且已在开源有限元库中实现了开发的方法。