112 research outputs found
A numerical study of a two-layer model for the growth of granular matter in a silo
The problem of filling a silo of given bounded cross-section with granular
matter can be described by the two-layer model of Hadeler and Kuttler [8]. In
this paper we discuss how similarity quasi-static solutions for this model can
be numerically characterized by the direct finite element solution of a
semidefinite elliptic Neumann problem. We also discuss a finite difference
scheme for the dynamical model through which we can show that the growing
profiles of the heaps in the silo evolve in finite time towards such similarity
solutions.Comment: Submitted to Proceedings of the MASCOT2015 - IMACS/ISGG Workshop,
Rome, Ital
Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks
Adding sand grains at a single site in Abelian sandpile models produces
beautiful but complex patterns. We study the effect of sink sites on such
patterns. Sinks change the scaling of the diameter of the pattern with the
number of sand grains added. For example, in two dimensions, in presence of
a sink site, the diameter of the pattern grows as for large
, whereas it grows as if there are no sink sites. In presence of
a line of sink sites, this rate reduces to . We determine the growth
rates for these sink geometries along with the case when there are two lines of
sink sites forming a wedge, and its generalization to higher dimensions. We
characterize one such asymptotic patterns on the two-dimensional F-lattice with
a single source adjacent to a line of sink sites, in terms of position of
different spatial features in the pattern. For this lattice, we also provide an
exact characterization of the pattern with two sources, when the line joining
them is along one of the axes.Comment: 27 pages, 17 figures. Figures with better resolution is available at
http://www.theory.tifr.res.in/~tridib/pss.htm
A differential model for growing sandpiles on networks
We consider a system of differential equations of Monge-Kantorovich type
which describes the equilibrium configurations of granular material poured by a
constant source on a network. Relying on the definition of viscosity solution
for Hamilton-Jacobi equations on networks, recently introduced by P.-L. Lions
and P. E. Souganidis, we prove existence and uniqueness of the solution of the
system and we discuss its numerical approximation. Some numerical experiments
are carried out
An existence result for the sandpile problem on flat tables with walls
We derive an existence result for solutions of a differential system which
characterizes the equilibria of a particular model in granular matter theory,
the so-called partially open table problem for growing sandpiles. Such result
generalizes a recent theorem of Cannarsa and Cardaliaguet established for the
totally open table problem. Here, due to the presence of walls at the boundary,
the surface flow density at the equilibrium may result no more continuous nor
bounded, and its explicit mathematical characterization is obtained by domain
decomposition techniques. At the same time we show how these solutions can be
numerically computed as stationary solutions of a dynamical two-layer model for
growing sandpiles and we present the results of some simulations.Comment: 15 pages, 11 figure
Variational Inequalities in Critical-State Problems
Similar evolutionary variational inequalities appear as convenient
formulations for continuous quasistationary models for sandpile growth,
formation of a network of lakes and rivers, magnetization of type-II
superconductors, and elastoplastic deformations. We outline the main steps of
such models derivation and try to clarify the origin of this similarity. New
dual variational formulations, analogous to mixed variational inequalities in
plasticity, are derived for sandpiles and superconductors.Comment: Submitted for publicatio
A numerical study of a two-layer model for the growth of granular matter in a silo
The problem of filling a silo of given bounded cross-section with granular
matter can be described by the two-layer model of Hadeler and Kuttler. In
this paper we discuss how similarity quasi-static solutions for this model can
be numerically characterized by the direct finite element solution of a
semidefinite elliptic Neumann problem. We also discuss a finite difference
scheme for the dynamical model through which we can show that the growing
profiles of the heaps in the silo evolve in finite time towards such similarity
solutions
Smoothing of sandpile surfaces after intermittent and continuous avalanches: three models in search of an experiment
We present and analyse in this paper three models of coupled continuum
equations all united by a common theme: the intuitive notion that sandpile
surfaces are left smoother by the propagation of avalanches across them. Two of
these concern smoothing at the `bare' interface, appropriate to intermittent
avalanche flow, while one of them models smoothing at the effective surface
defined by a cloud of flowing grains across the `bare' interface, which is
appropriate to the regime where avalanches flow continuously across the
sandpile.Comment: 17 pages and 26 figures. Submitted to Physical Review
Approximation properties of the -sine bases
For the eigenfunctions of the non-linear eigenvalue problem
associated to the one-dimensional -Laplacian are known to form a Riesz basis
of . We examine in this paper the approximation properties of this
family of functions and its dual, in order to establish non-orthogonal spectral
methods for the -Poisson boundary value problem and its corresponding
parabolic time evolution initial value problem. The principal objective of our
analysis is the determination of optimal values of for which the best
approximation is achieved for a given problem.Comment: 20 pages, 11 figures and 2 tables. We have fixed a number of typos
and added references. Changed the title to better reflect the conten
Continuum theory of partially fluidized granular flows
A continuum theory of partially fluidized granular flows is developed. The
theory is based on a combination of the equations for the flow velocity and
shear stresses coupled with the order parameter equation which describes the
transition between flowing and static components of the granular system. We
apply this theory to several important granular problems: avalanche flow in
deep and shallow inclined layers, rotating drums and shear granular flows
between two plates. We carry out quantitative comparisons between the theory
and experiment.Comment: 28 pages, 23 figures, submitted to Phys. Rev.
- …