Page 220 - The Combined Finite-Discrete Element Method

ALTERNATIVE EXPLICIT TIME INTEGRATION SCHEMES 203

convincing representation of this highly complex motion. It stands up to sensory percep-
tion of collisions between spinning and bouncing fragments of rock and their container
walls and floor. The predicted rocking and pirouetting behaviour of such irregular shaped
pebbles is extremely realistic.

5.3 ALTERNATIVE EXPLICIT TIME INTEGRATION SCHEMES

5.3.1 The Central Difference time integration scheme (CD)

CD as described in detail in this chapter is a second order time integration scheme,
originally developed in the context of structural dynamics. Apart from CD, a whole range
of explicit direct time integration schemes is available for the combined finite-discrete
element simulations. Recursive formulae for some of the schemes are as follows.

• The Leap Frog or Position Verlet Time Integration Scheme (PV): PV is very similar to
CD. It appears to be in essence the same as CD, except that velocities and positions
are shifted by h/2. The recursive formulae are given by

(5.101)
x t+h/2 = x t−h/2 + hv t
h
v t+h = v t + f t+h/2
m
• The T-1/12 Time Integration Scheme (T-1/12): T-1/12 is a third order scheme. The
expressions for the position and the velocity at time (t + h) are as follows:

1 2
v t h + (7f t − f t−h )h
12m
x t+h = x t + (5.102)
1
˜
1 − b t h 2
12

1 df
˜
where b t =
m dx t
h
v t+h = v t + (8f t + 5f t+h − f t−h ) (5.103)
12m
• The T-1/6 Time Integration Scheme (T-1/6): T-1/6 is a third order algorithm. The posi-
tion and the velocity at time t + h are given by
1
f t 2 3
˜
x t+h = x t + v t h + h + b t v t h (5.104)
2 m 6
1
f t 2
˜
v t + h + b t v t h
m 3
v t+h = where (5.105)
1
1 − b t+h h 2
˜
6
1 df 1 df

˜
˜
b t = and b t+h =
m dx m dx
t t+h