Page 222 - The Combined Finite-Discrete Element Method

ALTERNATIVE EXPLICIT TIME INTEGRATION SCHEMES 205

Table 5.1 Values of α i
PC-3 PC4 PC5
th
rd
th
(3 Order) (4 Order) (5 Order)
1/6 19/120 3/16
α 0
5/6 3/4 251/360
α 1
1 1 1
α 2
1/3 1/2 11/18
α 3
– 1/12 1/6
α 4
α 5 – – 1/60
h 2 h 2 ¨xh 2
¨ x t+ t =¨x t+h,p + α 2
2! 2! 2!
... h 3 ... h 3 ¨xh 2
x t+h = x t+h,p + α 3
3! 3! 2!
h 4 iv h 4 ¨xh 2
iv
x t+h = x t+h,p + α 4
4! 4! 2!
h 5 v h 5 ¨xh 2
v
x = x
t+h t+h,p + α 5
5! 5! 2!
The PC-3 scheme uses only first, second and third derivatives. PC-4 also uses a fourth
derivative, and PC5 uses all five derivatives – the coefficients α for all three schemes are
given in Table 5.1.

5.3.3 CHIN integration scheme

The recursive formula for the CHIN integration scheme is given by

h f t
v 1 = v t + (5.111)
6 m
h
x 1 = x t + v 1
2

2
2 f 1 h
v 2 = v 1 + h + G 1
3 m 48
h
x t+h = x 1 + v 2
2
h f t+h
v t+h = v 2 +
6 m
where v t is the velocity at time t, f t is the force evaluated at x t ,x t is the position at time
t, x t+h is the position at time (t + h), v t+h is the velocity at time (t + h), f t+h is the force
evaluated at x t+h and m is the mass. The rest of the variables are auxiliary variables used
to reach the solution at time (t + h).