算法XXXX：有限元自由度变换矩阵的计算
Algorithm XXXX: Computation of finite element degree-of-freedom transformation matrices
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中文总结 AI 辅助
研究如何高效构建任意次数有限元的自由度映射，提出新算法，仅依据参考单元定义和属性计算，在Basix库实现，简化代码、便于新单元实现及支持用户自定义单元，无需输入变换信息。
中文摘要 AI 辅助
计算有限元算子的算法的算术强度随着多项式次数的增加而增加。这使得高阶方法在现代CPU和GPU架构上特别有吸引力，因为在这些架构上，低阶时的性能受到可用内存带宽的严重限制，处理器的浮点能力只有很小一部分被使用。高阶方法可以利用现代架构中更大比例的可用计算能力。然而，虽然计算高阶有限元基的稳定方法已经成熟，但对于任意次数单元，还没有通用且自动化的算法来高效构建自由度映射。我们用一种新算法解决了这个问题，该算法可用于仅使用参考单元的定义和属性来计算任意Ciarlet型有限元的自由度映射，而无需为每个单元进行特定实现。此方法在FEniCSx库的组件Basix库中实现。该算法不仅极大简化了代码库的部分内容，还能轻松实现新单元，并支持用户在运行时创建自定义单元，而无需用户输入构建自由度映射所需的任何变换信息。
英文摘要
The arithmetic intensity of algorithms for computing finite element operators increases with increasing polynomial degree. This has made high degree methods particularly attractive on modern CPU and GPU architectures, since performance at low degree is limited by memory bandwidth and only a small fraction of the floating point capacity of the processor is used. Higher degree methods can exploit a significantly greater fraction of the available compute power. While stable methods for computing high-degree finite element bases are well-established, there is no universal and automated algorithm for the efficient construction of the degree-of-freedom map for arbitrary degree elements. We introduce a new algorithm that can be used in computing degree-of-freedom maps for arbitrary Ciarlet-type finite elements using only the element's definition and properties of the reference cell, without requiring a specific implementation for each element. This method is implemented in the library Basix, a component of the FEniCSx libraries. As well as allowing vast simplifications of parts of a codebase, the algorithm has allowed us to support user-defined custom elements that a user can create at runtime without requiring the user to input any information about transformations required to construct a degree-of-freedom map.
发表机构
- University College London（伦敦大学学院）
- University of Cambridge（剑桥大学）
机构由 AI 辅助整理，请以论文原文为准。
