Page 207 - The Combined Finite-Discrete Element Method

190 TEMPORAL DISCRETISATION

that direct integration is necessary to obtain angular velocity and spatial orientation of the
discrete element.

5.2.7 Change in spatial orientation during a single time step

By employing equations (5.40) and (5.41), the spatial orientation of the discrete element
at time t (i.e. at the beginning of the time step) can be obtained:

( t i, t j, t k) (5.61)

In a similar way, the spatial orientation of the discrete element at time t + h (i.e. at the
end of the time step) is
( t+h i, t+h j, t+h k) (5.62)

In equations (5.61) and (5.62) h is the time step, while the unit vectors are described
using the inertial frame of reference:

˜
˜
˜
t i = t i ˜x i + t i ˜y j + t i ˜z k (5.63)
˜
˜
t j = t j ˜x i + t j ˜y j + t j ˜z k ˜
˜
˜
t k = t k ˜x i + t k ˜y j + t k ˜z k ˜
˜
˜
t+h i = t+h i ˜x i + t+h i ˜y j + t+h i ˜z k ˜ (5.64)
˜
˜
t+h j = t+h j ˜x i + t+h j ˜y j + t+h j ˜z k ˜
˜
˜
t+h k = t+h k ˜x i + t+h k ˜y j + t+h k ˜z k ˜
The difference between the two triads is due to the rotation of the discrete element. This
rotation is uniquely described by the angular velocity:
t+h

t+h i = t i + ω(t) × i(t)dt (5.65)
t
t+h
t+h j = t j + ω(t) × j(t)dt
t
t+h
t+h k = t k + ω(t) × k(t)dt
t
If the time step h is small, it is reasonable to assume that the angular velocity is constant
during the time step, so that
 
 
 ω ˜x 
ω =  ω ˜y  (5.66)
ω ˜z

The assumption of a constant angular velocity means that the discrete element rotates
about a fixed axis during this time interval. The total rotation angle is