Page 44 - The Combined Finite-Discrete Element Method

COMBINED CONTINUA-DISCONTINUA PROBLEMS 27

shape distribution, size of the box, etc., are the key parameters governing the behaviour
of the systems.
In this context, discontinua-based formulations are not just an alternative to continua-
based formulations. On the contrary, discontinua-based formulations are the only available
solution for problems where the continuum hypothesis is not valid. This fact has been
recognised by researchers working in different disciplines from nano-scale and atomic-
scale to terrestrial bodies.

1.4 COMBINED CONTINUA-DISCONTINUA PROBLEMS

Consider the container problem with particles being made of very soft rubber or jelly so
that, in addition to interacting with each other, they deform as well. In addition, the walls
of the container also deform. This problem is called the flexible container problem.Even
in the case of less deformable particles, the deformation of the container and deformation
of individual particles significantly influences the way particles move inside the container.
Thus, the total mass of the particles deposited into the container is also influenced by
deformability (elastic properties) of both particles and the container.
Each individual particle deforms under external forces and interaction with other parti-
cles already in the container, and also under interaction with container walls. Changes in
the shape and size of individual particles is in essence a problem of finite strain elasticity
(since finite rotations at least are present). The deformability of individual particles is
therefore well represented by a hypothetical continuum-based model. Interaction among
individual particles and interaction between particles and the container is best represented
by discontinua-based model. The flexible container problem involves aspects of both
continua and discontinua.
Problems such as the flexible container problem are a combination of continua and dis-
continua, and are therefore termed combined continua-discontinua problems. The deforma-
bility of individual particles is best described using the hypothetical continua formulation.
Interaction and the motion of individual particles is best described using discontinua
formulation. The set of governing equations obtained describes both the deformability of
individual particles and interaction with container walls, and also between particles. The
number of equations is a function of the total number of particles in the container. Ana-
lytical solutions of the governing equations obtained are rarely available, and numerical
approaches have to be employed. These include DDA and DEM with added features to
capture deformability.
However, the most advanced approach is to use the sate of the art method (the finite
element method) to model continuum-based phenomena (in this case deformability) and
the sate of the art method (the discrete element method) to model discontinuum-based
phenomena (interaction and motion of individual particles). The new method is therefore
a combination of both the finite element method and the discrete element method, and is
termed the combined finite–discrete element method (FEM/DEM).
In the combined finite-discrete element method, each particle (body) is represented
by a single discrete element that interacts with discrete elements that are close to it. In
addition, each discrete element is discretised into finite elements. Each discrete element
has its own finite element mesh. The total number of finite element meshes employed
is equal to the total number of discrete elements. Each finite element mesh employed