15 research outputs found
Finite element approximation of the Levi-Civita connection and its curvature in two dimensions
We construct finite element approximations of the Levi-Civita connection and
its curvature on triangulations of oriented two-dimensional manifolds. Our
construction relies on the Regge finite elements, which are piecewise
polynomial symmetric (0,2)-tensor fields possessing single-valued
tangential-tangential components along element interfaces. When used to
discretize the Riemannian metric tensor, these piecewise polynomial tensor
fields do not possess enough regularity to define connections and curvature in
the classical sense, but we show how to make sense of these quantities in a
distributional sense. We then show that these distributional quantities
converge in certain dual Sobolev norms to their smooth counterparts under
refinement of the triangulation. We also discuss projections of the
distributional curvature and distributional connection onto piecewise
polynomial finite element spaces. We show that the relevant projection
operators commute with certain linearized differential operators, yielding a
commutative diagram of differential complexes.Comment: v2: Several revisions throughout, including major revisions to
Section 2.4, Section 5, and the beginning of Section
A serendipity fully discrete div-div complex on polygonal meshes
In this work we address the reduction of face degrees of freedom (DOFs) for
discrete elasticity complexes. Specifically, using serendipity techniques, we
develop a reduced version of a recently introduced two-dimensional complex
arising from traces of the three-dimensional elasticity complex. The keystone
of the reduction process is a new estimate of symmetric tensor-valued
polynomial fields in terms of boundary values, completed with suitable
projections of internal values for higher degrees. We prove an extensive set of
new results for the original complex and show that the reduced complex has the
same homological and analytical properties as the original one. This paper also
contains an appendix with proofs of general Poincar\'e--Korn-type inequalities
for hybrid fields
A finite element elasticity complex in three dimensions
A finite element elasticity complex on tetrahedral meshes is devised. The
conforming finite element is the smooth finite element developed by
Neilan for the velocity field in a discrete Stokes complex. The symmetric
div-conforming finite element is the Hu-Zhang element for stress tensors. The
construction of an -conforming finite element for symmetric
tensors is the main focus of this paper. The key tools of the construction are
the decomposition of polynomial tensor spaces and the characterization of the
trace of the operator. The polynomial elasticity complex and
Koszul elasticity complex are created to derive the decomposition of polynomial
tensor spaces. The trace of the operator is induced from a
Green's identity. Trace complexes and bubble complexes are also derived to
facilitate the construction. Our construction appears to be the first
-conforming finite elements on tetrahedral meshes without
further splits.Comment: 23 page
Regge Finite Elements with Applications in Solid Mechanics and Relativity
University of Minnesota Ph.D. dissertation. May 2018. Major: Mathematics. Advisor: Douglas Arnold. 1 computer file (PDF); ix, 183 pages.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