Page 72 - The Combined Finite-Discrete Element Method

POTENTIAL CONTACT FORCE IN 3D 55

demonstrated, there is no need for it regardless of the shape of the interacting dis-
crete elements.
• The contact force is discretised with the same algorithm and the same piece of code,
regardless of the shape (concave, convex, hollow, etc.) of discrete elements. Thus,
algorithmic complexities are greatly reduced.

• The contact processing is in general faster than alternative solutions. It is obvious that
sphere to sphere contact is processed faster than triangle to triangle contact. However,
contact between two discrete elements of general shape is processed almost as quickly
as contact between two triangles. This is due to the fact that contact detection eliminates
all triangular finite elements from two finite element meshes, except for the couples of
finite elements that are in contact. The number of these triangles in a general shaped
body is governed by the number of contact points. However, very often two such bodies
(unless they are flat) touch at just one point, making the speed of interaction processing
almost independent of the shape of the individual discrete elements. There is contact
detection overhead, but the state of the art contact detection algorithms are so fast that
contact detection as a whole consumes very little overall CPU time.

2.6 POTENTIAL CONTACT FORCE IN 3D

The easiest way of implementing the potential contact force interaction kinematics for
discrete elements of general shape in 3D is by discretising the domain of each dis-
crete element into finite elements. In this way, the potential is applied piecewise, i.e.
elementwise.
To reduce implementation complexities and increase CPU efficiency, the simplest possi-
ble 3D solid finite element (i.e. four-noded tetrahedra) is chosen. The same finite element
and the same discretisation is best used for deformability of discrete elements, including
any non-linearity. Special procedures are often introduced to avoid phenomena such as
locking. Most of them can be found in any good finite element textbook, and are outside
of the scope of this book.
The potential contact force algorithm for 3D problems described in this section is based
on the following assumptions:
• The domain of each discrete element is discretised into finite elements. The geometry
of each element is assumed to be defined by a tetrahedron, although extension to other
geometry is possible.

• Maximum allowed penetration at any point of contact is a function of the size of the
finite element to which a particular point belongs. The logic behind this assumption is
the fact that contact constraints at points of interface are satisfied only approximately.
The same applies for field equations, for which an approximate solution is obtained
by employing the finite element discretisation. The element size at each point of the
domain is usually based on some kind of error estimator expressed, for instance, in
terms of displacements. Relative displacements between any two points of the domain
are obtained by integrating the strain field over any path connecting these two points. If
this path intersects two surfaces in contact, the error in relative displacement is altered
by penetration at the place of interface. According to the St. Venant principle, the