Kuwabara-Kono numerical dissipation: a new method to simulate granular matter

• Published in: IMA journal of applied mathematics Vol. 85; no. 1; pp. 27 - 66
• Main Authors: James, Guillaume, Vorotnikov, Kirill, Brogliato, Bernard
• Format: Journal Article
• Language: English
• Published: Oxford University Press 28.02.2020; Oxford University Press (OUP)
• Subjects: Dynamical Systems; Engineering Sciences; Mathematics; Mechanics; Nonlinear Sciences; Numerical Analysis; Pattern Formation and Solitons; Physics; Solid mechanics; implicit Runge-Kutta methods; solitary waves; Newton’s cradle; modified equation; numerical dissipation; viscoelastic impact models; granular chains
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxz034

Abstract A new method is introduced for the simulation of multiple impacts in granular media using the Kuwabara-Kono (KK) contact model, a nonsmooth (not Lipschitz continuous) extension of Hertz contact that accounts for viscoelastic damping. We use the technique of modified equations to construct time-discretizations of the nondissipative Hertz law matching numerical dissipation with KK dissipation at different consistency orders. This allows us to simulate dissipative impacts with good accuracy without including the nonsmooth KK viscoelastic component in the contact force. This tailored numerical dissipation is developed in a general framework, for Newtonian dynamical systems subject to dissipative forces proportional to the time-derivative of conservative forces. Numerical tests are performed for the simulation of impacts in Newton’s cradle and on alignments of alternating large and small balls. Resulting wave phenomena (oscillator synchronization, propagation of dissipative solitary waves, oscillatory tails) are accurately captured by implicit schemes with tailored numerical dissipation, even for relatively large time steps.