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

科学和工程中的许多问题都需要微积分来解决，而对于其中的大多数问题，人们必须依靠计算机。为此，几十年来，微积分的技术已经被近似，以使它们适合在计算机算法中使用。微积分的更高级的技术和数学结构是被称为外微积分的数学领域的一部分。这种高级形式的微积分使得微积分的技术可以很容易地应用于空间弯曲的情况，例如地球表面。外部微积分框架还允许在比我们熟悉的三维世界更多的维度上使用微积分。这方面的例子出现在物理学、工程学和数据科学的许多问题中。离散外积分(DEC)是一种适用于计算机程序的外微积分计算框架。首席调查员(PI)将通过创建对许多应用程序至关重要但在DEC中缺失的数学对象来丰富DEC框架。具体地说，PI将开发所需的数学技术和算法，以创建称为对称张量的对象的近似，以及与这些对象相关的微积分运算。为了检验这些构造的有效性，这些近似将与求解用于模拟弹性固体的方程一起开发。虽然外部微积分通常需要研究生的数学培训，但DEC的一个好处是最终产品的简单性。得到的对象和运算可以用基本的方式解释，这将被用来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，如应力张量和微分算子，如卷曲，Hessian和对称梯度。PI将通过使用几何中的工具Bernstein-Gelfand-Gelfand(BGG)构造来执行对称张量和相关微分算子的离散化。作为该项目的一部分，将开发进行这一建设所需的新的DEC空间和操作员。BGG结构可用于组合微分复形，并在此过程中创建对称张量和高阶微分算子。PI将使用平面内的双调和方程和线性化弹性作为模型问题来衡量DEC的BGG建设的成功。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.