Transformations for Piola-mapped elements
Transformations for Piola-mapped elements
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DOI：
10.5802/smai-jcm.91
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
Francis R. A. Aznaran;R. Kirby;P. Farrell
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文献类型：
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作者：
Francis R. A. Aznaran;R. Kirby;P. Farrell
Arnold-Winther 单元成功地离散化了线弹性的 Hellinger-Reissner 变分公式；它的发展是有限元外微积分早期的关键突破之一。尽管它很有用，但它在标准有限元软件中不可用，因为它的自由度在标准 Piola 前推下没有保留。在这项工作中，我们应用 Kirby 最近开发的新颖变换理论 [SMAI-JCM, 4:197-224, 2018] 设计正确的映射，将参考单元的基础变换为通用物理三角形。这使得能够在广泛使用的 Firedrake 有限元软件中使用 Arnold-Winther 元素（无论是一致的还是非一致的），并结合其先进的符号代码生成和几何多重网格功能。类似的结果也使得 Mardal-Tai-Winther 单元能够正确转换为不可压缩流体流动。我们给出了这两个元素的数值结果，验证了我们理论的正确性。
The Arnold-Winther element successfully discretizes the Hellinger-Reissner variational formulation of linear elasticity; its development was one of the key early breakthroughs of the finite element exterior calculus. Despite its great utility, it is not available in standard finite element software, because its degrees of freedom are not preserved under the standard Piola push-forward. In this work we apply the novel transformation theory recently developed by Kirby [SMAI-JCM, 4:197-224, 2018] to devise the correct map for transforming the basis on a reference cell to a generic physical triangle. This enables the use of the Arnold-Winther elements, both conforming and nonconforming, in the widely-used Firedrake finite element software, composing with its advanced symbolic code generation and geometric multigrid functionality. Similar results also enable the correct transformation of the Mardal-Tai-Winther element for incompressible fluid flow. We present numerical results for both elements, verifying the correctness of our theory.