The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods
Real and complex analysis, 3rd ed.
Real and complex analysis, 3rd ed.
On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach
Computer Methods in Applied Mechanics and Engineering
A new look at finite elements in time: a variational interpretation of Runge-Kutta methods
Applied Numerical Mathematics
Animating rotation with quaternion curves
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics
Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics
Quaternion-based dynamics of geometrically nonlinear spatial beams using the Runge-Kutta method
Finite Elements in Analysis and Design
Dynamics of spatial beams in quaternion description based on the Newmark integration scheme
Computational Mechanics
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The integration of the rotation from a given angular velocity is often required in practice. The present paper explores how the choice of the parametrization of rotation, when employed in conjuction with different numerical time-integration schemes, effects the accuracy and the computational efficiency. Three rotation parametrizations - the rotational vector, the Argyris tangential vector and the rotational quaternion - are combined with three different numerical time-integration schemes, including classical explicit Runge-Kutta method and the novel midpoint rule proposed here. The key result of the study is the assessment of the integration errors of various parametrization-integration method combinations. In order to assess the errors, we choose a time-dependent function corresponding to a rotational vector, and derive the related exact time-dependent angular velocity. This is then employed in the numerical solution as the data. The resulting numerically integrated approximate rotations are compared with the analytical solution. A novel global solution error norm for discrete solutions given by a set of values at chosen time-points is employed. Several characteristic angular velocity functions, resulting in small, finite and fast oscillating rotations are studied.