The Computations of Large Rotation Through an Index Two Nilpotent Equation
Chein-Shan Liu

Source CMES: Computer Modeling in Engineering & Sciences, Vol. 16, No. 3, pp. 157-176, 2006
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Keywords Large rotation, Nilpotent matrix, Singularity-free, Lie algebra, Noncanonical orthogonal matrix.
Abstract To characterize largely deformed spin-free reference configuration of materials, we have to construct an orthogonal transformation tensor${\bf Q}$ relative to the fixed frame, such that the tensorial equation$\mathaccentV {dot}05F{\bf Q}={\bf W}{\bf Q}$ holds for a given spin history${\bf W}$. 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${\bf Q}={\bf F}{\bf Q}_1$ is first derived here, which is amenable to a simpler treatment of${\bf Q}_1$ than${\bf Q}$, and the numerical calculation of${\bf Q}_1$ is obtained by transforming the governing equations in a space of${\mathbb R}P^3$, whose dimensions are two, and the singularity-free interval is largely extended. Then, we develop a novel method to express${\bf Q}_1$ 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${\bf Q}$ from the theoretical and computational aspects, and conclude that the new methods can be applied to the calculations of large rotations.
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