5. Implicit Time Integration Schemes for Constrained Rotations

5.1 Description of the problem

In the computation dynamics, besides computation of displacements and rotations, one also needs to obtain velocities and accelerations at the chosen instant in the time interval of interest. We use the Newmark family of algorithms for that end; for energy conserving algorithms (Simo and Tarnow, 1994; Brank et al. 1998; Briseghella et al., 2001). Standard implementation is used for computing the translational motion components, and necessary modifications are proposed for computing the components related to the constrained rotation of the shell-director.

Considering the typical time interval between t _n and t _{n + 1} the algorithmic problem can be described as: given at time t _n displacement, ? _n, velocity, _n, and acceleration, _n , of translational motion of the shell mid-surface, and shell-director, t _n , its constrained rotation, ? _n, velocity, _n , and acceleration, _n , find such values of ? and t at time t _{n + 1} that

(91)

and update velocities and accelerations of displacements and shell director by using the corresponding Newmark approximations. The update for displacements, constrained rotation tensors and shell-director vectors, which we need when solving equation (91) iteratively by Newton solution procedure, was discussed in the previous section. In this section, we address the remaining ingredients of the problem, namely the update of velocities and accelerations.

5.2 Newmark scheme...