Efficient numerical integration of rigid body dynamics

¹ Department of Mechanical Engineering, Yale University, New Haven, Connecticut 06520, USA
² School of Natural and Built Environment, Queen’s University Belfast, David Keir Building, Stranmillis Road, BT9 5AG Belfast, United Kingdom
³ Institute for Multiscale Simulation, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 3, 91058 Erlangen, Germany
⁴ Departamento de Física, Universidad Nacional de Colombia, Carrera 45 No. 26-85, Edificio Uriel Gutiérrez, Bogotá D.C., Colombia

^* e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Published online: 1 December 2025

Abstract

Integrating equations of motion is a crucial aspect of discrete element method (DEM) simulations. However, this integration can be particularly challenging when dealing with rigid body dynamics. In this work, we review Spiral, a third-order integration algorithm designed for the rotational motion of extended bodies. Spiral offers stability and precision, surpassing commonly used algorithms. Furthermore, Spiral addresses many challenges associated with rotation dynamics in leading DEM codes, such as YADE, MERCURY DPM, LIGGGHTS, and PFC, without compromising performance. This algorithm eliminates the need for quaternion normalization at each time step, requires only one force calculation per time step, and is compatible with both leapfrog and synchronous integration methods.