Page 61 - The Combined Finite-Discrete Element Method

44 PROCESSING OF CONTACT INTERACTION

may comprise a few thousand or even a few million separate bodies. In addition, each
body is represented by a single discrete element, which in turn is discretised into one or
more finite elements. Thus, the evaluation of contact forces at each time step may involve
a large number of contacting couples of discrete elements. As each discrete element is
discretised into a finite element, each contacting couple of discrete elements is in fact
represented by a whole set of contacting couples of finite elements. Thus, making the
total number of contacting couples of finite elements even larger.
In this context, the summation as given by (2.29) usually involves a large number of
contacting couples of finite elements, and the total CPU time and overall efficiency of
the contact algorithm critically depend upon implementation of part of the interaction that
processes finite element to finite element contact.
It is therefore important to employ the simplest possible finite element and make that
element work well for both contact and deformability. Using the simplest possible finite
elements for deformability has resulted in a constant strain triangle being employed for
almost any conceivable (linear and non-linear) 2D problem. Much has been published on,
for instance, special techniques to avoid locking. All of these can be found in most good
finite elements textbooks, and is outside the scope of this book.
The simplest possible finite element in 2D is a tri-noded triangle. The edges of a trian-
gular finite element are straight, and its geometry is uniquely defined by the coordinates
of its nodes, as shown in Figure 2.5.
As explained above, the potential ϕ should be constant on the boundary of a discrete
element. This constraint is unconditionally met if the potential ϕ is defined in such a way
that it is constant on the boundaries of each finite element. Thus, for a triangular finite
element it is assumed that ϕ at point P inside the triangle is given by

ϕ(P ) = min{3A 1 /A, 3A 2 /A, 3A 3 /A} (2.30)

where A i (i = 1, 2, 3) is the area of corresponding sub-triangles, as shown in Figure 2.5.
The potential ϕ equals 1 at the centre of triangle and 0 at the edges of the triangle. For
any point P outside the triangle, the potential ϕ is zero.
According to (2.29), the problem of interaction between two triangles can be reduced
to interactions between the contactor triangle and the edges of the target triangle, coupled

0

A 1 2

P
A 2
A 3

1
Figure 2.5 Potential ϕ at any point P of a triangle shaped finite element.