Page 231 - The Combined Finite-Discrete Element Method

214 TEMPORAL DISCRETISATION

Convergence of the mode i to the zero energy state (state of rest) is given by
ω 1
− ω i t
e ω i (5.141)
The time needed to reach a static solution is therefore proportional to

const
ω 1
ω i t s = const ⇒ t s = (5.142)
ω i ω 1
This means that all frequencies (modes) reach a static solution at the same time. Since
the critical time step is proportional to 1/ω n , the total number of time steps required to
obtain a static solution is given by

t s const 1 const ω n
n = = = ω n = const (5.143)
h ω 1 h ω 1 ω 1
The high frequencies are damped relatively slowly. Their presence in the system converg-
ing to the state of rest can, for instance, produces undesirable results. Material that is in
a static solution in compression can suddenly, due to the dynamic relaxation, be found to
be in tension. In Figure 5.22, 2 ω 1 M damping is employed to solve a system comprising
100 different frequencies, with the highest frequency being 1000 rad/s and the smallest
frequency being 1 rad/s. It is evident from the results shown in the figure that high fre-
quencies oscillate for a long time. These oscillations can, for instance, produce unphysical
behaviour such as brittle fracture in tension, when in fact the equivalent static problem
comprises material in compression only. This is due to stress reversal (oscillations), as
shown in the figure.
This phenomenon is called overshooting. In this particular scheme, it is present with
all modes. In addition, estimation of the lowest frequency can be difficult. The highest
frequency is much easier to estimate (from the rigidity of the stiffest finite element) than
the lowest frequency. It is also worth mentioning that damping proportional to mass is

1
0.8 Legend:
0.6 Frequency = 1
Frequency = 10
0.4 Frequency = 50
0.2
Residual −0.2 0

−0.4
−0.6
−0.8
−1
0 500 1000 1500 2000
Time (s)

Figure 5.22 Overshooting with mass proportional damping matrix.