Page 223 - The Combined Finite-Discrete Element Method

206 TEMPORAL DISCRETISATION

5.3.4 OMF30 time integration scheme

The recursive formula for this scheme is as follows:

f t
v 1 = v t + λh (5.112)
m
h
x 1 = x t + v 1
2
f 1
v 2 = v 1 + (1 − 2λ)h
m
h
x t+h = x 1 + v 2
2
f t+h
v t+h = v 2 + λh
m
1 1 & √ 1 1
3
λ = − 36 + 2 326 + # √
2 12 6 3 36 + 2 326

5.3.5 OMF32 time integration scheme

This is an eleven-stage, fourth order scheme. The recursive formula for this scheme is
as follows:

f 1
v 1 = v t + ϑh
x 1 = x t + ρh v t
m
f 2 3
v 2 = v 1 + λh
x 2 = x 1 + θh v 1 + ξh G 2
m
f 3
x 3 = x 2 + (1 − 2(θ + ρ))(h/2)v 2 v 3 = v 2 + (1 − 2(λ + ϑ))h
m
f 4 3
v 4 = v 3 + λh
x 4 = x 3 + (1 − 2(θ + ρ))(h/2)v 3 + ξh G 4
m
x 5 = x 4 + θh v 4
(5.113)
f 5
v t+ t = v 4 + ϑh
m where :
r t+ t = r 5 + ρ t v t+ t
ρ = 0.6419108866816235 E − 01
θ = 0.1919807940455741 E + 00
ϑ = 0.1518179640276466 E + 00
λ = 0.2158369476787619 E + 00
ξ = 0.9628905212024874 E − 03