Regge Finite Elements with Applications in Solid Mechanics and Relativity

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Regge Finite Elements with Applications in Solid Mechanics and Relativity

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Li, Lizao

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2018-05

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Abstract

This thesis proposes a new family of finite elements, called generalized Regge finite elements, for discretizing symmetric matrix-valued functions and symmetric 2-tensor fields. We demonstrate its effectiveness for applications in computational geometry, mathematical physics, and solid mechanics. Generalized Regge finite elements are inspired by Tullio Regge’s pioneering work on discretizing Einstein’s theory of general relativity. We analyze why current discretization schemes based on Regge’s original ideas fail and point out new directions which combine Regge’s geometric insight with the successful framework of finite element analysis. In particular, we derive well-posed linear model problems from general relativity and propose discretizations based on generalized Regge finite elements. While the first part of the thesis generalizes Regge’s initial proposal and enlarges its scope to many other applications outside relativity, the second part of this thesis represents the initial steps towards a stable structure-preserving discretization of the Einstein’s field equation.

Keywords

Geodesics

Regge Calculus

Regge finite elements

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University of Minnesota Ph.D. dissertation. May 2018. Major: Mathematics. Advisor: Douglas Arnold. 1 computer file (PDF); ix, 183 pages.

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Li, Lizao. (2018). Regge Finite Elements with Applications in Solid Mechanics and Relativity. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/199048.

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