Symmetric Tensors in Discrete Exterior Calculus and Linearized Elasticity in the Plane

离散外微积分中的对称张量和平面线性弹性

• 批准号: 2208581
• 负责人: Anil Hirani
• 金额: $ 23.91万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-15 至 2025-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208581&HistoricalAwards=false
• 关键词: Symmetric; Tensors; Discrete; Exterior; Calculus

Many problems in science and engineering require calculus for their solution and for most of these one must rely on a computer. For this purpose, for decades the techniques of calculus have been approximated to make them suitable for use in computer algorithms. The more advanced techniques and mathematical structures of calculus are part of a field of mathematics called exterior calculus. This advanced form of calculus allows the techniques of calculus to be easily applied in situations where the space is curved, such as the surface of earth. Exterior calculus framework also allows the use of calculus in more dimensions than the familiar three dimensional world. Examples of this appear in physics, in engineering, and in many problems in data science. Discrete exterior calculus (DEC) is a computational framework for exterior calculus suitable for computer programs. The principle investigator (PI) will enrich the DEC framework by creating mathematical objects that are critical for many applications but are missing from DEC. Specifically, the PI will develop the mathematical techniques and algorithms needed to create approximations of objects called symmetric tensors, and calculus operations related to these objects. In order to test the validity of these constructions, the approximations will be developed in conjunction with solving equations for modeling elastic solids. While exterior calculus often requires graduate training in mathematics, one benefit of DEC is the simplicity of the final product. The resulting objects and operations can be explained in an elementary manner and this will be leveraged to introduce these topics in an undergraduate computer programing course.The PI proposes to create discrete symmetric tensors such as the stress tensor and differential operators such as the curl curl, hessianand symmetric gradient in the DEC framework. The PI will carry out this discretization of symmetric tensors and related differential operators by using the Bernstein-Gelfand-Gelfand (BGG) construction, a tool from geometry. New DEC spaces and operators needed to carry out this construction will be developed as part of the project. The BGG construction can be used to combine differential complexes and in the process create symmetric tensors and higher order differential operators. The PI will use biharmonic equation and linearized elasticity in the plane as model problems to gauge the success of the BGG construction for DEC.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.

科学和工程中的许多问题都需要微积分来解决其解决方案，对于其中大多数，必须依靠计算机。为此，数十年来，微积分的技术已被近似以使其适合于计算机算法。微积分的更先进的技术和数学结构是数学领域的一部分，称为外部微积分。这种高级演算的形式允许在空间弯曲（例如地球表面）的情况下轻松应用微积分的技术。与熟悉的三维世界相比，外部演算框架还允许在更大的维度上使用演算。这在物理，工程和数据科学中的许多问题中都出现在示例中。离散的外部演算（DEC）是适合计算机程序的外部微积分的计算框架。原则研究者（PI）将通过创建对许多应用程序至关重要但从DEC丢失的数学对象来丰富DEC框架。具体而言，PI将开发创建称为对称张量的对象以及与这些对象相关的计算操作所需的数学技术和算法。为了测试这些结构的有效性，将与求解弹性固体建模方程一起开发近似值。虽然外观演算通常需要数学培训，但DEC的一个好处是最终产品的简单性。可以用基本的方式来解释所得的对象和操作，这将被利用以在本科计算机编程课程中引入这些主题。PI建议在DEC框架中创建离散的对称张量，例如应力张量和诸如curl curl curl curl curl curl curl curl curl curl curl。 PI将通过使用Bernstein-Gelfand-Gelfand（BGG）结构（一种几何图形的工具）进行对称张量和相关差异操作员的离散化。将作为该项目的一部分开发需要开发此构建的新的DEC空间和运营商。 BGG构造可用于结合差分络合物，在此过程中，可以创建对称张量和高阶差分运算符。 PI将在飞机上使用Biharmonic方程和线性化弹性作为模型问题，以评估BGG构建成功的成功。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的审查标准通过评估来通过评估来获得支持的。