Page 19 - The Combined Finite-Discrete Element Method

2 INTRODUCTION

These include the finite difference method, finite volume method, finite element method,
mesh-less finite element methods, etc.
The most advanced and the most often used method is the finite element method.
The finite element method is based on discretisation of the domain into finite sub-
domains, also called finite elements. Finite elements share nodes, edges and surfaces,
all of which comprise a finite element mesh. The solution over individual finite elements
is sought in an approximate form using shape (base) functions. Balanced principles are
imposed in averaged (integral or weak) form. These usually yield algebraic equations, for
instance equilibrium of nodal forces, thus effectively replacing governing partial differ-
ential equations with a system of simultaneous algebraic equations, the solution of which
gives results (e.g. displacements) at the nodes of finite elements.

1.2 GENERAL FORMULATION OF DISCONTINUUM PROBLEMS

Taking into account that the mean free path of molecules for most engineering materials is
very small in comparison to the characteristic length of most of the engineering problems,
one may arrive at the conclusion that most engineering materials are well represented by
a hypothetical continuum model. That this is not the case is easily demonstrated by the
following problem:

A glass container of square base is filled with particles of varying shape and size, as
shown in Figure 1.1. The particulate is left to fall from the given height. During the fall
under gravity, the particles interact with each other and with the walls of the container.
In this process energy is dissipated, and finally, all the particles find state of rest. The

Figure 1.1 Letting particles of different shape and size pack under gravity inside the glass con-
tainer. The question is how much space will they occupy?