Critical time step for DEM simulations of dynamic systems using a Hertzian contact model

Burns, Shane J.; Piiroinen, Petri T.; Hanley, Kevin J.

doi:10.1002/nme.6056

Cited by 59 publications

(17 citation statements)

References 23 publications

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“…Even though Equation (48) contains k n , the shear direction is usually the one which limits the critical time step as implied by Equations (22)-(24). This is consistent with the findings of Tu and Andrade 7 and Burns et al, 12 among others. is considerably more conservative than = √ 2 found by Burns et al 12 for the same two-sphere configuration.…”

Section: Amplification Matrix Methodssupporting

confidence: 93%

“…For this two‐particle case, we have made a similar assumption that the maximum relative translational acceleration at the contact, that is, the maximum magnitude of

{\ddot{x}}_{c}

, determines the critical time step.

{\ddot{x}}_{c}

appears in the differential equation describing the system's dynamics at the contact:

M {\ddot{x}}_{c} + C {\dot{x}}_{c} + K x_{c} = F,

where

M

C

, and

K

refer to the particle mass, damping, and contact stiffness matrices, respectively 12 . Imposing contact models and integrators, an amplification matrix can be obtained.…”

Section: Maximum Relative Translational Acceleration At the Contactmentioning

confidence: 99%

“…where M, C, and K refer to the particle mass, damping, and contact stiffness matrices, respectively. 12 Imposing contact models and integrators, an amplification matrix can be obtained.…”

Section: Maximum Relative Translational Acceleration At the Contactmentioning

confidence: 99%

“…Tavarez and Plesha 11 constructed mass and stiffness matrices for a DEM unit cell of monosized spheres to obtain an upper‐bound estimate of the critical time step. Burns et al 12 developed a framework for selecting a stable time step for both linear and nonlinear interactions of spheres by analyzing the equations of motion as a nonlinear map. The most common approach to estimate a critical time step for nonlinear contact between spheres is fundamentally based on the propagation of Rayleigh waves through a particle assembly 13,14 .…”

Section: Introductionmentioning

confidence: 99%

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Critical time step for discrete element method simulations of convex particles with central symmetry

Peng

Burns

Hanley

2020

Numerical Meth Engineering

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A time step must be selected for any explicit discrete element method (DEM) simulation. This time step must be small enough to ensure a stable simulation but should not be overly conservative for computational efficiency. There are established methods to estimate critical time steps for simulations of spherical particles. However, there is a comparable lack of guidance on choosing time steps for DEM simulations involving nonspherical particles: a fact which is increasingly problematic as simulations of nonspherical particles become more commonplace. In this article, the eigenvalues of the amplification matrix are used to develop an explicit formula for the critical time step for a range of shapes including ellipsoids, convex superquadrics and convex, central symmetric polyhedra. This derivation is based on a linear analysis and applies to both underdamped and overdamped systems. The dependence on the particle mass and contact stiffness expected for a system of spheres is recovered. For a fixed particle mass, as particle shape becomes increasingly nonspherical, the critical time step decreases nonlinearly. Thus, estimating a critical time step by assuming a sphere of equivalent volume may not always be conservative.

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Section: Amplification Matrix Methodssupporting

confidence: 93%

“…For this two‐particle case, we have made a similar assumption that the maximum relative translational acceleration at the contact, that is, the maximum magnitude of

{\ddot{x}}_{c}

, determines the critical time step.

{\ddot{x}}_{c}

appears in the differential equation describing the system's dynamics at the contact:

M {\ddot{x}}_{c} + C {\dot{x}}_{c} + K x_{c} = F,

where

M

C

, and

K

refer to the particle mass, damping, and contact stiffness matrices, respectively 12 . Imposing contact models and integrators, an amplification matrix can be obtained.…”

Section: Maximum Relative Translational Acceleration At the Contactmentioning

confidence: 99%

“…where M, C, and K refer to the particle mass, damping, and contact stiffness matrices, respectively. 12 Imposing contact models and integrators, an amplification matrix can be obtained.…”

Section: Maximum Relative Translational Acceleration At the Contactmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Critical time step for discrete element method simulations of convex particles with central symmetry

Peng

Burns

Hanley

2020

Numerical Meth Engineering

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“…Δt was set at 100 ns: far lower than the critical value of 325 s appropriate for a normal incident velocity around 1 m s −1 . 22 15 000 time steps, that is, 1.5 ms, were run in both the forward-and reversed-time frames: sufficient time for the particles to collide and fully separate. Figure 1 shows that much of the momentum of particle A is transferred to particle B by the collision.…”

Section: Two-particle Collisionmentioning

confidence: 99%

Time reversibility of the discrete element method

Hanley

2020

Numerical Meth Engineering

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An algorithm is presented for discrete element method simulations of energy-conserving systems of frictionless, spherical particles in a reversed-time frame. This algorithm is verified, within the limits of round-off error, through implementation in the LAMMPS code. Mechanisms for energy dissipation such as interparticle friction, damping, rotational resistance, particle crushing, or bond breakage cannot be incorporated into this algorithm without causing time irreversibility. This theoretical development is applied to critical-state soil mechanics as an exemplar. It is shown that the convergence of soil samples, which differ only in terms of their initial void ratio, to the same critical state requires the presence of shear forces and frictional dissipation within the soil system. K E Y W O R D S discrete element method, geomechanics, granular media, solids 1 INTRODUCTION Most simulations aim to model a real system with maximal fidelity, limited by practical considerations such as the finite nature of computational resources. However, there is a second category of simulations in which some nonphysical element is deliberately introduced into a simulated system in order to further our understanding of the real system. One relatively commonplace example is discrete element method (DEM) simulations of frictionless particles. These serve as a valuable limiting case for real particle systems which can vary significantly in interparticle friction but are never completely frictionless. The subject of this paper, reversed-time DEM, is within this second category of simulations. While Einstein's theory of special relativity allows for time to speed up or slow down, reversing time is not thought possible according to our current understanding of physics. An algorithm which is deterministic, such as DEM, is not necessarily time-reversible. 1 To the best of the author's knowledge, since the development of DEM by Cundall and Strack 2 more than 40 years ago, no one has explored the possibility of setting DEM within a reversed-time framework. This is somewhat surprising as the time reversibility of molecular dynamics, which is algorithmically related to DEM, has been explored in depth, beginning with Orban and Bellemans 3 in the 1960s. More recent investigations of time reversibility in molecular dynamics have involved the use of integer arithmetic to eliminate round-off errors in floating-point computations. 4 The development of symplectic, reversible integrators for molecular dynamics has been an area of significant research activity. 5-8 These integrators have been proposed for application to both Hamiltonian and non-Hamiltonian dynamical systems. 8 However, one of the simple variants of Verlet, for example, velocity-Verlet, is almost invariably chosen as the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Phase‐resolving direct numerical simulations of particle transport in liquids—From microfluidics to sediment

2022

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The article describes direct numerical simulations using an Euler–Lagrange approach with an immersed‐boundary method to resolve the geometry and trajectory of particles moving in a flow. The presentation focuses on own work of the authors and discusses elements of physical and numerical modeling in some detail, together with three areas of application: microfluidic transport of spherical and nonspherical particles in curved ducts, flows with bubbles at different void fraction ranging from single bubbles to dense particle clusters, some also subjected to electro‐magnetic forces, and bedload sediment transport with spherical and nonspherical particles. These applications with their specific requirements for numerical modeling illustrate the versatility of the approach and provide condensed information about main findings.

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Critical time step for DEM simulations of dynamic systems using a Hertzian contact model

Cited by 59 publications

References 23 publications

Critical time step for discrete element method simulations of convex particles with central symmetry

Critical time step for discrete element method simulations of convex particles with central symmetry

Time reversibility of the discrete element method

Phase‐resolving direct numerical simulations of particle transport in liquids—From microfluidics to sediment

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