@Article{cmes.2006.016.157,
AUTHOR = {Chein-Shan Liu},
TITLE = {The Computations of Large Rotation Through an Index Two Nilpotent Equation},
JOURNAL = {Computer Modeling in Engineering \& Sciences},
VOLUME = {16},
YEAR = {2006},
NUMBER = {3},
PAGES = {157--176},
URL = {http://www.techscience.com/CMES/v16n3/26701},
ISSN = {1526-1506},
ABSTRACT = {To characterize largely deformed spin-free reference configuration of materials, we have to construct an orthogonal transformation tensor <b>Q</b> relative to the fixed frame, such that the tensorial equation <b>Q<sup style="margin-left:-6.8px">˙</sup></b> = <b>WQ</b> holds for a given spin history <b>W</b>. This paper addresses some interesting issues about this equation. The Euler's angles representation, and the (modified) Rodrigues parameters representation of the rotation group <i>SO</i>(3) unavoidably suffer certain singularity, and at the same time the governing equations are nonlinear three-dimensional ODEs. A decomposition <b>Q</b> = <b>FQ<sub>1</sub></b> is first derived here, which is amenable to a simpler treatment of <b>Q<sub>1</sub></b> than <b>Q</b>, and the numerical calculation of <b>Q<sub>1</sub></b> is obtained by transforming the governing equations in a space of R<i>P<sup>3</sup></i>, whose dimensions are two, and the singularity-free interval is largely extended. Then, we develop a novel method to express <b>Q<sub>1</sub></b> in terms of a noncanonical orthogonal matrix, the governing equation of which is a linear ODEs system with its state matrix being nilpotent with index two. We examine six methods on the computation of <b>Q</b> from the theoretical and computational aspects, and conclude that the new methods can be applied to the calculations of large rotations.},
DOI = {10.3970/cmes.2006.016.157}
}