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Geodesic finite elements for Cosserat rods
Author(s) -
Sander Oliver
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2814
Subject(s) - geodesic , mathematics , discretization , finite element method , interpolation (computer graphics) , mathematical analysis , manifold (fluid mechanics) , hyperelastic material , exponential map (riemannian geometry) , space (punctuation) , pure mathematics , geometry , classical mechanics , motion (physics) , computer science , sectional curvature , physics , mechanical engineering , scalar curvature , curvature , engineering , thermodynamics , operating system
We introduce geodesic finite elements as a new way to discretize the non‐linear configuration space of a geometrically exact Cosserat rod. These geodesic finite elements naturally generalize standard one‐dimensional finite elements to spaces of functions with values in a Riemannian manifold. For the special orthogonal group, our approach reproduces the interpolation formulas of Crisfield and Jelenić. Geodesic finite elements are conforming and lead to objective and path‐independent problem formulations. We introduce geodesic finite elements for general Riemannian manifolds, discuss the relationship between geodesic finite elements and coefficient vectors, and estimate the interpolation error. Then we use them to find static equilibria of hyperelastic Cosserat rods. Using the Riemannian trust‐region algorithm of Absil et al . we show numerically that the discretization error depends optimally on the mesh size. Copyright © 2009 John Wiley & Sons, Ltd.