Collaborative Research: Transforming Serendipity Elements from Theory to Practice

合作研究：将意外元素从理论转化为实践

• 批准号: 1913094
• 负责人: Joshua Levine
• 金额: $ 13.3万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1913094&HistoricalAwards=false
• 关键词: Collaborative; Research; Transforming; Serendipity; Elements

This project aims to significantly reduce the computation time and effort required to carry out a wide variety of simulations used in modern scientific and engineering applications. In settings as varied as magnetohydrodynamics (used in studies of nuclear fusion) and electro-diffusion (used in studies of cardiac arrhythmia), a technique called the "finite element method" is a preferred computational procedure for designing and computing highly accurate descriptions of the relevant physical phenomena. While finite element methods have been in use for nearly half a century, recent mathematical insights have indicated that certain kinds of methods could be implemented in such a way that they would produce results of the same order of accuracy while requiring orders of magnitude less computational work. Such techniques are known as "serendipity methods" and serve as the focus of the proposed work. A key impact of the research will be incorporation of these methods into a widely used, open source, community-developed software package known as the Firedrake Project.From high-level method analysis to specific computational tricks, the proposed work will significantly expand understanding of the benefits and limitations of serendipity finite element methodologies. For serendipity elements, the project will make precise the common mantra of "same accuracy, less work" by simultaneously exploring the expected computational benefits via algebraic and analytical techniques, as well as actual computational benefits via implementation and numerical simulations within the Firedrake software package. On the theoretical side, the research will advance knowledge via investigation of the algebraic structure of serendipity spaces and its exploitation to allow new Helmholtz decompositions of the associated polynomial spaces, identification of efficient "short injections" of serendipity bases into standard tensor product bases, and development of sparse quadrature rules. The investigators will combine expertise on element mapping and physically-defined basis elements to create an order-preserving handling of non-affinely mapped square element geometries, a key roadblock in prior attempts at implementation of serendipity elements.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目旨在显著减少现代科学和工程应用中各种模拟所需的计算时间和工作量。 在磁流体力学（用于核聚变研究）和电扩散（用于心律失常研究）等不同的环境中，一种称为“有限元法”的技术是设计和计算相关物理现象的高度精确描述的首选计算程序。 虽然有限元方法已经使用了将近半个世纪，但最近的数学见解表明，某些类型的方法可以以这样一种方式实现，即它们将产生相同精度的结果，同时需要数量级更少的计算工作。 这种技术被称为“意外发现方法”，是拟议工作的重点。 该研究的一个关键影响将是将这些方法纳入一个广泛使用的开源社区开发的软件包，称为Firedrake Project.从高级方法分析到具体的计算技巧，拟议的工作将显着扩大对偶然性有限元方法的好处和局限性的理解。 对于偶然发现的元素，该项目将通过同时探索通过代数和分析技术的预期计算效益以及通过Firedrake软件包中的实施和数值模拟的实际计算效益来精确“相同精度，更少工作”的共同咒语。 在理论方面，该研究将通过调查偶然空间的代数结构及其开发来推进知识，以允许相关多项式空间的新亥姆霍兹分解，将偶然基有效的“短注入”识别到标准张量积基中，并开发稀疏求积规则。 研究人员将结合联合收割机在元素映射和物理定义的基础元素方面的专业知识，创建一个非亲和映射正方形元素几何形状的保序处理方法，这是以前尝试实施偶然元素的一个关键障碍。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

项目成果

Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake

将修剪的偶然性方法引入 Firedrake 的计算实践中

• DOI: 10.1145/3490485
• 发表时间: 2022
• 期刊: ACM Transactions on Mathematical Software
• 影响因子: 2.7
• 作者: Crum, Justin;Cheng, Cyrus;Ham, David A.;Mitchell, Lawrence;Kirby, Robert C.;Levine, Joshua A.;Gillette, Andrew
• 通讯作者: Gillette, Andrew