Construction of Arbitrary Order Finite Element Degree-of-Freedom Maps on Polygonal and Polyhedral Cell Meshes
Construction of Arbitrary Order Finite Element Degree-of-Freedom Maps on Polygonal and Polyhedral Cell Meshes
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DOI：
10.1145/3524456
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发表时间：
2021-02
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影响因子：
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通讯作者：
Matthew W. Scroggs;J. Dokken;C. Richardson;G. N. Wells
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文献类型：
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作者：
Matthew W. Scroggs;J. Dokken;C. Richardson;G. N. Wells
我们开发了一种方法生成自由度映射任意阶的Ciarlet型有限元空间的任何细胞形状。该方法是基于细胞子实体的排列和变换的组合。当前生成任意阶问题的自由度映射的方法通常依赖于单元实体的一致取向，该取向允许在共享边缘和面上定义公共局部坐标系。然而，虽然网格的定向对于单纯形单元是直接的并且是局部操作，但是对于四边形单元它不是严格的局部操作，并且在六面体单元的情况下，不是所有的网格都是可定向的。置换和变换的方法是开发的一系列的元素类型，包括任意程度的拉格朗日，意外发现，发散和卷曲协调元素，并为一系列的细胞形状。该方法是局部的，可以应用于任何形状的细胞，包括一般多面体和混合细胞类型的网格。一些例子，并已在开放源代码图书馆实施的方法。
We develop a method for generating degree-of-freedom maps for arbitrary order Ciarlet-type 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 arbitrary degree Lagrange, serendipity, 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 open-source libraries.