Page 221 - The Combined Finite-Discrete Element Method

204 TEMPORAL DISCRETISATION

• The D-1/12 Time Integration Scheme (D-1/12): The recursive formula for D-1/12 is
as follows:

f t 2
x t − x t−h + v t h + h
2 m 1 df
˜
x t+h = x t−h + where b t = (5.106)
1 m dx
˜
1 − b t t 2 t
12
h
v t+h = v t + (8f t + 5f t+h − f t−h ) (5.107)
12 m
5.3.2 Gear’s predictor-corrector time integration schemes (PC-3,
PC-4, and PC-5)

The third order scheme is referred to as PC-3, while the fourth and fifth order schemes
are referred to as PC-4 and PC-5, respectively. All three schemes use the same recursive
formula, which comprises three stages:
(a) Prediction: in the prediction stage, positions are calculated at time (t + ∆t) by means
of a Taylor series based on positions and their derivatives:
h 2 ... h 3 iv t 4 v h 5
x t+h,p = x t +˙x t h +¨x t + x t + x t + x t (5.108)
2! 3! 4! 5!
... h 2 iv h 3 v h 4
˙ x t+h,p =˙x t +¨x t h + x t + x t + x t
2! 3! 4!
... iv x 2 v x 3
¨ x t+h,p =¨x t + x t h + x + x
t t
2! 3!
... ... iv v h 2
x t+h,p = x t + x h + x t
t
2!
v
iv
iv
x t+h,p = x + x h
t
t
x v t+ t,p = x t v
(b) Evaluation: in the evaluation stage, the force f t+h at time (t + h) is evaluated using
the x t+h,p position from the prediction stage. From this force, acceleration ¨x t+h at time
(t + h) is calculated, and the discrepancy between this acceleration and the predicted
acceleration ¨x t+h,p is evaluated, i.e.
¨x = (¨x t+h −¨x t+h,p ) (5.109)

(c) Correction: in the correction stage, the predicted positions and time derivatives are
corrected using the following formulae:

¨xh 2
x t+h = x t+h,p + α 0 (5.110)
2!
¨xh 2
˙ x t+h h =˙x t+h,p h + α 1
2!