A new Leapfrog scheme for rotational motion in 3D

Summary Discrete element codes use complex geometric solid particles, so it is necessary to integrate three-dimensional rigid-body rotation correctly with external torque. This article presents an interpretation of the Leapfrog scheme. We begin with some rotation formulae before presenting an efficient and high-order recurrent Taylor series method for rotation. Integrating this method with Leapfrog interpretation provides a complete scheme for rotational motion with external torque. This new Leapfrog scheme has been integrated into the CeaMka3D Discrete Element code, and we present some verifications and simulations to illustrate the capabilities of this scheme. Copyright © 2015 John Wiley & Sons, Ltd.

[1]  David Fincham,et al.  Leapfrog Rotational Algorithms , 1992 .

[2]  Elena Celledoni,et al.  The Exact Computation of the Free Rigid Body Motion and Its Use in Splitting Methods , 2008, SIAM J. Sci. Comput..

[3]  A. Saccon,et al.  A numerical test of long‐time stability for rigid body integrators , 2012 .

[4]  P. Cundall,et al.  A bonded-particle model for rock , 2004 .

[5]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[6]  C. Mariotti,et al.  Lamb's problem with the lattice model Mka3D , 2007 .

[7]  Scott M. Johnson,et al.  Quaternion‐based rigid body rotation integration algorithms for use in particle methods , 2008 .

[8]  S. Buss.  Accurate and efficient simulation of rigid-body rotations , 2000 .

[9]  C.G.J. Jacobi.  Sur la rotation d'un corps. Extrait d'une lettre adressée à l'académie des sciences de Paris. , 1850 .

[10]  A. Müller.  Approximation of finite rigid body motions from velocity fields , 2010 .

[11]  C. Mariotti,et al.  An energy-preserving Discrete Element Method for elastodynamics , 2009, 0907.2202.

[12]  P. Cundall,et al.  A discrete numerical model for granular assemblies , 1979 .

[13]  Anatoly B Kolomeisky,et al.  Recursive Taylor Series Expansion Method for Rigid-Body Molecular Dynamics. , 2011, Journal of chemical theory and computation.

[15]  I. Omelyan.  A New Leapfrog Integrator of Rotational Motion. The Revised Angular-Momentum Approach , 1999, physics/9901025.

[16]  R. Olness,et al.  Two‐dimensional computer studies of crystal stability and fluid viscosity , 1974 .

[17]  L. Verlet.  Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules , 1967 .