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Simulation of Granular Flow using the Material Point Method
Sachith Dunatunga, Ken Kamrin
Massachusetts Institute of Technology
Cambridge, Massachusetts, USA
[email protected]
While granular flows are common in everyday life, several factors complicate the
numerical simulation of such a flow. Viewed as a bulk continuum, the elastic stress-strain
relation is nonlinear and the total deformation requires plasticity to properly describe.
Traditional solvers for solids (e.g. finite element) do not work well with the large plastic
deformations involved in flow due to mesh distortion issues, whereas the elastic part of the
constitutive relation presents a challenge for fluid solvers (e.g. finite volume, finite
difference). The discrete element method -- which is preferred for granular flow -- treats
grains as individual particles and considers interactions between them[1], but is too
computationally intensive to be practical for even medium-scale problems.
The material point method (MPM) is a recent development[2] in mesh-free computational
methods aimed at solids that seeks to combine the strengths of DEM and FEM. To this
end, MPM uses mobile ``material points'' to capture the state of the system. These are not
grains, but rather a representation of the continuum at certain points. At every time
increment, data from these material points is projected onto a background computational
domain. The equations of motion are solved on this domain, and the information is then
projected back onto the material points, which updates other material quantities such as
stress and strain through a constitutive relation.
We have implemented a basic MPM solver for 2D problems. Using first order elements
and a local model[3] for granular flow, simulations such as collapse of a sand pile and
discharge of a silo produce the physically expected behavior even when the strains
become extremely large. Particularly, Beverloo scaling is recovered in the silo. A linearized
version of the local model is used in the solution of a nonlocal model, which has been
confirmed to reproduce analytical results in the vertical chute geometry. We are currently
investigating implementation of the non-linearized nonlocal model, which involves solving
a stiff diffusion-like set of equations coupled with the original equations of motion.
References:
[1] P. A. Cundall and O. D. L. Strack, “A Discrete Numerical Model for Granular
Assemblies,” Geotechnique, vol. 30, pp. 335-336, 1980 1980.
[2] D. Sulsky, et al., “A Particle Method for History-Dependent Materials,” Computer
Methods in Applied Mechanics and Engineering, vol. 118, pp. 179-196, Sep 1994.
[3] P. Jop, et al., “A Constitutive Law for Dense Granular Flows”, Nature, vol. 441, pp.
727-730, Jun 2006.
[4] D. Henann and K. Kamrin, “A predictive, size-dependent continuum model for dense
granular flows”, PNAS, vol. 110, pp. 6730-6735, 2013.