Page 224 - The Combined Finite-Discrete Element Method

ALTERNATIVE EXPLICIT TIME INTEGRATION SCHEMES 207

5.3.6 Forest & Ruth time integration scheme

Recursive formula for this scheme is as follows

f t
v 1 = v t + c 1 h x 1 = x t + d 1 hv 1
m
f 1
v 2 = v 1 + c 2 h x 2 = x 1 + d 2 hv 2
m
f 2 (5.114)
v 3 = v 2 + c 3 h
m
x t+h = x 2 + d 3 hv 3
f t+h
v t+h = v 3 + c 4 h
m
where

c 1 = λ + 1/2; c 2 =−λ; c 3 =−λ; c 4 = λ + 1/2 (5.115)
1
d 1 = 2λ + 1; d 2 =−4λ − 1; d 3 = 2λ + 1; λ = (2 1/3 + 2 −1/3 − 1)
6
Any of these schemes could in principle be applied in a given combined finite-discrete
element simulation. To choose the most appropriate scheme, it is necessary to compare the
schemes in terms of stability, accuracy and CPU efficiency. The easiest way to accomplish
this is to perform comparisons by considering a one degree of freedom mass-spring system

2
¨ x + ω x = 0 (5.116)
where x is the position and ω is the natural frequency

#
ω = k/m = 1 (5.117)

and initial conditions at t = 0 are given by

x = 0; v = 1 (5.118)

The analytical solution for this system is given by

x = sin ωt; v = cos ωt (5.119)

The system has been integrated numerically using different time integration schemes.
In all cases, the time step varied from 0.01 seconds to 7.00 seconds. For each time step,
the total time of the simulation t e was equal to 40 periods, i.e.

t e = 40 · 2π = 251 s (5.120)