Tidal Bladed Knowledge Base

The consistency of acceleration with velocity and displacement for fixed-step integrated simulations
Category: Integrators

Problem

When integrating dynamic states with fixed-step integrators, such as the "Explicit Newmark Beta" or "Generalised alpha", the acceleration displays greater sensitivity to variations in the time-step, as compared to the displacement and velocity. This can give the impression that the acceleration is not consistent with the velocity, as illustrated in the graphs below.

Velocity

Acceleration

These graphs display the x-component of the angular velocity and acceleration of the interface node for an offshore turbine with a Superelement support structure.

Solution

This is generally not considered to be a problem. The time-step required to accurately capture the accelerations is usually less than the time-step necessary for the velocity and displacement to converge. When using larger time-steps, the acceleration overshoots and has a phase-shift with respect to the converged acceleration profile. The numerical equation for the velocity takes the form


The Newmark-Beta integrator uses not only the current acceleration but also the velocity and displacements from the previous time-step to calculate the current velocity. For the simulation with the 0.02 s time-step, the contribution of the acceleration to the current velocity (in the equation above) is of order 0.01. This explains why both acceleration profiles result in a similar velocity profile, even though they vary noticeably from one another.





Keywords Explicit Integrator Acceleration Velocity Displacement Inconsistent