TY - EJOU
AU - Liu, Chein-Shan
TI - The Computations of Large Rotation Through an Index Two Nilpotent Equation
T2 - Computer Modeling in Engineering \& Sciences
PY - 2006
VL - 16
IS - 3
SN - 1526-1506
AB - To characterize largely deformed spin-free reference configuration of materials, we have to construct an orthogonal transformation tensor **Q** relative to the fixed frame, such that the tensorial equation **Q**^{˙} = **WQ** holds for a given spin history **W**. This paper addresses some interesting issues about this equation. The Euler's angles representation, and the (modified) Rodrigues parameters representation of the rotation group *SO*(3) unavoidably suffer certain singularity, and at the same time the governing equations are nonlinear three-dimensional ODEs. A decomposition **Q** = **FQ**_{1} is first derived here, which is amenable to a simpler treatment of **Q**_{1} than **Q**, and the numerical calculation of **Q**_{1} is obtained by transforming the governing equations in a space of R*P*^{3}, whose dimensions are two, and the singularity-free interval is largely extended. Then, we develop a novel method to express **Q**_{1} 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 **Q** from the theoretical and computational aspects, and conclude that the new methods can be applied to the calculations of large rotations.
KW - Large rotation
KW - Nilpotent matrix
KW - Singularity-free
KW - Lie algebra
KW - Noncanonical orthogonal matrix
DO - 10.3970/cmes.2006.016.157