Page 202 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8.2 Some Numerical Simulation Issues Associated with the Particle Simulation Method 193

It is noted that the total number of time steps to produce the total solution S is
calculated using the following formula:

M

N total = N i . (8.35)
i=1
Similarly, at the end of each loading step, there exist the following relationships:

j

P j = ΔP i ( j = 1, 2,..., M), (8.36)
i=1

j

S j = ΔS i ( j = 1, 2,..., M), (8.37)
i=1

where P j is the total applied loading force at the end of loading step j; S j is the total
solution at the end of loading step j. This indicates that at the end of a loading step,
a point (P j , S j ) has been obtained in the load-solution space. Therefore, at the end
of loading step M, we have obtained M points so that it is possible to link all these M
points together to obtain a load-solution path in the load-solution space. Clearly, if
P and S represent the applied loading force and resulting displacement, then a force-
displacement curve is obtained in the force-displacement space. Alternatively, if P
and S stand for the applied loading stress and resulting strain respectively, then a
stress-strain curve is obtained in the stress-strain space.
With the biaxial compression test of a particle simulation model taken as an
example, the general steps of using the proposed loading algorithm can be sum-
marized as follows. (1) For a given load increment, which needs to be applied to
the boundary of a particle model, the servo-control technique (Itasca Consulting
Group, inc. 1999) is used to apply the equivalent velocity to the boundary of the
particle model. This means that an equivalent velocity is applied to the boundary of
the particle model for a few time steps during the particle simulation. The number
of time steps, during which the equivalent velocity needs to be applied to the bound-
ary of the particle model, depends on the desired load increment that is applied to
the boundary of the particle model. This step is called the loading step and the cor-
responding loading time or the number of related time steps is called the loading
period. It is noted that in theory, a loading period should be as small as possible, so
that any unphysical damage/crack can be prevented from occurring during the load-
ing period. (2) After a loading period, the applied equivalent velocity is set to be zero
so that the loading boundary of the particle model becomes frozen (i.e. fixed). This
step is called the frozen step. During a frozen step, the particle simulation model
is kept running until an equilibrium state is reached. Therefore, the duration of a
frozen period depends on how quickly the particle simulation model can reach its
corresponding equilibrium state. (3) When the particle simulation model reaches an
equilibrium state, an applied load increment such as a stress increment on the load-