Page 235 - The Combined Finite-Discrete Element Method

218 TEMPORAL DISCRETISATION

Table 5.3 Comparison of dynamic relaxation schemes
M C Overshoot Time steps Momentum balance
0 M NO (ω n /ω 1 ) 2 NO
M 2 ω 1 M YES (ω n /ω 1 ) NO
M 2 ω n M NO (ω n /ω 1 ) 2 NO
M (1/ω 1 )K NO (ω n /ω 1 ) 2 YES
M (1/ω n )K YES (ω n /ω 1 ) 2 YES

2
Since the critical time step for such an overdamped system is proportional to ω 1 /ω n ,the
total number of time steps required to obtain a static solution is given by

t s const ω 2 n
ω n 2
n = = = const (5.165)
h ω 1 ω 1 ω 1
This damping does not introduce any artificial external forces to the system, while the
convergence to the state of rest is natural.

A summary of the alternative dynamic relaxation schemes listed above is given in
Table 5.3: In the first column, the schemes are separated with regard to the mass matrix
The second column presents the method of damping. The third column indicates whether
overshooting is present or not. The theoretical number of time steps to obtain a steady
state solution is given in column 4. An important factor in both dynamic relaxation and
transient dynamics is momentum balance. Mass proportional damping introduces external
viscous forces into the system. This results in a significant momentum imbalance, which
can result in inaccurate stress and strain fields. For nonlinear systems this can lead to
a wrong prediction of the response of the system. For instance, some discrete element
systems, if subjected to a heavy mass proportional damping, can lead to a very distorted
picture of the final static state (state of rest), i.e. the dynamic relaxation scheme may
converge to a non-physical static solution.