Page 229 - The Combined Finite-Discrete Element Method

212 TEMPORAL DISCRETISATION

explicit schemes is available, however as explained before, the central difference time inte-
gration scheme appears to be the optimal one for many problems of practical importance.
The most important advantage of dynamic relaxation in comparison to iterative methods
in general is probably the physical meaning, which can be attached to the convergence
process itself through gradual motion of the system toward the steady state, which can be
expressed in terms of inertia forces. This is very useful in problems with slow monotonic
loading and nonlinear problems, in which non-unique solutions may exist. However, the
path through which a steady state is reached and the speed at which it is reached are
heavily dependent on both the C and M matrices.
The recursive formula for dynamic relaxation using the system (5.127) is as follows:

0) n = 0; x n = 0; ˙ x n = 0; (5.128)
1) n = n + 1;
−1
2) ˙ x n = C (p − Kx n−1 )
3) x n = x n−1 + ˙x n h

4) if the state of rest is not reached go to 1

The upper limit of the time step for scheme (5.128) to be stable is given by the follow-
ing inequality:
1
h< (5.129)
ω n 2

where
T
2 v i Kv i
ω = T (5.130)
i
v i Cv i
is the frequency of mode i, the shape of which is defined by the eigenvector v i .Using
modal analysis, it can be proven that each mode ω i converges to the steady state with factor
2
s = e −tω i (5.131)
2
which means that a steady state is practically reached at a time proportional to 1/ω .The
i
longest time will be needed for the lowest frequency ω 1 to reach a steady state, thus the
total time needed for the system as a whole to reach a steady state is proportional to
2
1/ω 1 , which leads to the conclusion that the total number of time steps is proportional
2
2
to ω n /ω 1 .
For problem (5.126), the recursive formula for dynamic relaxation is given by
0) n = 0; x n = 0; ˙ x n = 0; (5.132)
1) n = n + 1;
−1
2) ¨ x n = M (p − Kx n−1 − C˙x n−1 )
3) ˙ x n = ˙x n−1 + ¨x n−1 h

4) x n = x n−1 + ˙x n h
5) if the state of rest is not reached go to 1