Page 75 - The Combined Finite-Discrete Element Method

58 PROCESSING OF CONTACT INTERACTION

Since individual discrete elements are discretised into finite elements, each discrete
element can be represented as the union of its finite elements:

(2.42)
β c = β c 1 ∪ β c 2 .. . ∪ β c i ... ∪ β c n
β t = β t 1 ∪ β t 2 .. . ∪ β t j ... ∪ β t m
where β c and β t are contactor and target discrete elements, respectively, while m and
n are the total number of finite elements the contactor and target discrete elements are
discretised into. The potentials ϕ c and ϕ t are therefore a sum of potentials associated with
individual finite elements:

ϕ c = ϕ c 1 + ϕ c 2 ··· + ϕ c i ·· · + ϕ c n
(2.43)
ϕ t = ϕ t 1 + ϕ t 2 ··· + ϕ t i ·· · + ϕ t m
Integration over the overlapping area may therefore be represented by summation over
finite elements:
n m

f = [gradϕ c i − gradϕ t j ]dV (2.44)
i=1 j=1 β c i ∩β t j
By replacing integration over finite elements by equivalent integration over finite element
boundaries (2.25), the following equation for contact force is obtained:
n m

f c = n (ϕ c i − ϕ t j )dS (2.45)
i=1 j=1 S β c i ∩β t j

where integration over finite element boundaries may be written as summation of integra-
tion over surfaces of finite elements. In other words, the contact force between overlapping
discrete elements is calculated by summation over the surfaces of corresponding finite
elements that overlap.
In this context, the following solution strategies are available:
• Consider contact of discrete elements with disregard for finite element discretisation.
In this case, a special data structure is needed to keep track of boundary representation.
• Consider contact of discretised domains of discrete elements with disregard to the
relationship between individual discrete elements. This is the simplest path to follow.
It may lead to unnecessary integration of contact forces over the inner boundaries of
discrete elements, slowing down execution of the algorithm.
• Consider integration of contact forces only over boundaries (surfaces) of finite elements
that are also boundaries of discrete elements. This approach requires the detection and
marking of global boundaries. In return, CPU requirements are reduced.

2.6.2 Computational aspects

The numerical procedure for integration of contact forces makes use of the fact that
discrete elements are discretised into tetrahedron shaped finite elements. Thus, the total
contact force is the sum of contact forces between individual tetrahedrons.