A Parameter-Preadjusted Energy-Conserving Integration for Rigid Body Dynamics in Terms of Convected Base Vectors

Energy-conserving integrations are widely used in long-time simulations of mechanical systems because they exhibit excellent stability and conserve the discrete energy in conservative systems. However, in many applications, their errors of other quantities, for example, trajectory errors, are generally not bounded and increase with time. This article develops a parameter-preadjusted energy-conserving integration for rigid body dynamics in terms of convected-based vectors. To this end, we introduce the modified inertia representation into the Hamiltonian description of the rigid body dynamics, which leads to a modified formulation of Hamilton’s equations including an undetermined parameter $γ$. After that, the direct discretization of the modified Hamilton’s equations and the constraints in terms of finite increments in time gives an energy-conserving integration. Error estimation suggests that the value of $γ$ has a great influence on the discretization errors of the Hamilton’s equations. Therefore, a preadjusting stage at the beginning of the simulation is devised to optimize $γ$ for minimizing the numerical error. Numerical results demonstrate that the PECI not only preserves the energy exactly, but also presents significant higher accuracy of trajectory errors. In particular, the PECI can achieve approximately periodic trajectory errors for long-time simulations if $γ$ is well preadjusted.