Page 234 - The Combined Finite-Discrete Element Method

THE COMBINED FINITE-DISCRETE ELEMENT SIMULATION 217

The high frequencies are damped relatively quickly. The lowest frequency reaches the
steady state last. The damping does not introduce any artificial external forces to the
system. The convergence to the state of rest is natural. However, the total number of steps
can be too large. Overshooting associated with highest frequencies is mostly eliminated.
However, overshooting associated with small frequencies can still be a problem. This can
be easily avoided by simply supplying the load gradually over what is a relatively long
time that the system takes to reach the state of rest.
In most combined finite-discrete element simulations, stiffness proportional damping
is preferred option. It comes naturally with the deformability of discrete elements, where
viscous forces calculated using the rate of deformation tensor act as an energy dissipa-
tion mechanism, which is not artificial but is the physical property of the material of
discrete elements.
• Stiffness proportional damping: if damping is introduced in the form

2
C = K (5.159)
ω 1
then equation (5.126) can be written as follows:

2
Kx + M¨x + K˙x = p (5.160)
ω 1
which, after applying modal analysis, yields the equation for mode i

2
ω i u i +¨u i + 2 ω i ω i ˙u i = q i where (5.161)
ω 1
ω i
ξ i = is damping ratio
ω 1

The damping ratio for the lowest frequency is equal to 1, i.e. the damping of lowest
frequency is critical damping. The damping ratio for all other frequencies is greater than
1, which means that all other frequencies are overdamped.
Convergence of the lowest frequency mode to the zero energy state (state of rest) is
given by
ω i ω i t ω 1 ω 1 t
e − ω 1 = e − ω 1 = e −ω 1 t (5.162)

The highest frequency mode is highly overdamped, and its convergence to the state of
rest is therefore given by
ω n ω n ω 1
− t − t − t
e 2ξ n = e 2ω n /ω 1 = e 2 (5.163)
The time needed to reach a static solution is therefore proportional to

const
ω 1 t s = const ⇒ t s = (5.164)
ω 1