Page 60 - The Combined Finite-Discrete Element Method

IMPLEMENTATION DETAILS FOR DISCRETISED CONTACT FORCE IN 2D 43

which can also be written as an integral over the boundary of the overlapping area

f c = n (ϕ c − ϕ t )d (2.25)

β t ∩ϕ c
where n is the outward unit normal to the boundary of the overlapping area.

2.4 DISCRETISATION OF CONTACT FORCE IN 2D

In the combined finite-discrete element method, individual discrete elements are discre-
tised into finite elements, and each discrete element can be represented as union of its
finite elements:
β c = β c 1 ∪ β c 2 .. . ∪ β c i ... ∪ β c n
(2.26)
β t = β t 1 ∪ β t 2 ... ∪ β t j .. . ∪ β t m
where β c and β t are the contactor and target discrete elements, respectively, while m
and n are the total number of finite elements into which the contactor and target discrete
elements are discretised. In this context, the potentials ϕ c and ϕ t can be written as a sum
of potentials associated with individual finite elements:

ϕ c = ϕ c 1 + ϕ c 2 ·· · + ϕ c i ·· · + ϕ c n
(2.27)
ϕ t = ϕ t 1 + ϕ t 2 ··· + ϕ t i ·· · + ϕ t m
Integration over overlapping area may therefore be represented by summation over
finite elements:
n m

f c = [gradϕ c i − gradϕ t j ]dA (2.28)
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 = (ϕ c i − ϕ t j )d (2.29)
n β c i
∩β t j
i=1 j=1 β c i ∩β t j
where integration over finite element boundaries may be written as summation of inte-
gration over the edges of finite elements.
In other words, the contact force between overlapping discrete elements is calculated
by summation over the edges of corresponding finite elements that overlap.

2.5 IMPLEMENTATION DETAILS FOR DISCRETISED CONTACT
FORCE IN 2D

As pointed out earlier, combined finite-discrete element problems involve a large number
of separate bodies that are free to move and interact with each other. Average problems