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

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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.