Heavy Top Dynamic Problem

For this benchmark problem, we use the heavy top to evaluate the convergence rate of the Kynema Lie-group generalized-alpha time integrator. The problem consists of a rotating body, fixed to the ground by a spherical joint, as detailed in [@Bruls-etal:2012]. The formulation of the heavy top problem as implemented in Kynema is described in Heavy top constrained-rigid-body example.

Our benchmark solution is computed using a Kynema simulation with a time step size of \(10^{-6}\) seconds; we examine the vertical displacement \(u_3\) at the end of two seconds. The following figure shows the convergence of the relative error in the vertical displacement as a function of time step size, where we observe the expected second-order convergence of the method.

7 Second-order convergence of the relative error in vertical displacement \(u_3\) at \(t = 2.0\) s for the heavy top problem. The benchmark solution is computed with \(\Delta t = 10^{-6}\) s.

Note

This benchmark is included as a regression test in the Kynema code base. The test is implemented in:

https://github.com/kynema/kynema/blob/main/tests/regression_tests/regression/verification/dynamic_heavy_top.cpp

Brüls, O., A. Cardona, and M. Arnold. 2012. “Lie Group Generalized-\(\alpha\) Time Integration For Constrained Flexible Multibody Systems.” Mechanism and Machine Theory 48: 121–137.