On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach
Computer Methods in Applied Mechanics and Engineering
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
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This paper deals with Newmark time stepping methods and finite rotations for nonlinear finite element analysis of flexible beam structures. The corotational method is used to develop expressions of the internal forces and the corresponding tangent stiffness matrices. For the dynamic part, four formulations based on different parameterizations of rotations are compared. The first three are classic formulations taken from the literature with some modifications for two of them. The last one is new and uses three of the four Euler parameters (quaternion) as rotational variables. For all these approaches, theoretical derivations as well as practical implementations are given in detail. The similarities and differences between the formulations are pointed out. Six numerical examples are studied in order to compare these four formulations in terms of numerical accuracy and computational efficiency. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. The numerical results show that these four formulations have the same numerical accuracy, but that the computational efficiency depends on the choice of the tangent inertia matrix. Besides, the new formulation proposed in this paper turns out to be the fastest one.