Page 197 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

188 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems
p
p
where [M] is the lumped mass matrix of the particle and {¨ u} is the acceleration
vector of the particle.
A major difference between solving the global quasi-static equilibrium problem
(i.e. Eq. (4)) in the finite element analysis and solving the fictitious dynamic problem
(i.e. Eq. (8.29)) in the particle simulation is that the numerical solution to Eq. (8.26)
is unconditionally stable so that it can be solved using the Gaussian elimination
method and the like, while the numerical solution to Eq. (8.29) is conditionally
stable if an explicit solver is used. For this reason, the critical time step, which is
required to result in a stable solution for the fictitious dynamic problem, can be
expressed as follows (Itasca Consulting Group, inc. 1999):

6
m
Δt critical = , (8.30)
k
where m is the mass of a particle and k is the stiffness between two particles.
It is immediately noted that since the value of the mass of a particle is usually
much smaller than that of the stiffness of the particle, the critical time step deter-
mined from Eq. (8.30) is considerably smaller than one. This indicates that for a
slow geological process of more than a few years, it may take too long to obtain
a particle simulation solution. To overcome this difficulty, the scaled mass is often
used in the distinct element method (Itasca Consulting Group, inc. 1999) so that the
critical time step can be increased to unity or any large number if needed. How-
ever, since the scaled mass, namely the fictitious mass, is used, the time used in the
distinct element method is fictitious, rather than physical. In order to remove pos-
sible chaotic behavior that may be caused by the use of arbitrarily-scaled masses,
fictitious damping is also added to the particles used in a distinct element simulation
(Itasca Consulting Group, inc. 1999). Thus, the numerical simulation issue resulting
from using the explicit dynamic relaxation method to solve a quasi-static problem is
that the time used in the distinct element method is fictitious, rather than physical.
If the mechanical response of a quasi-static system is elastic, then the solution to
the corresponding quasi-static problem is unique and independent of the deforma-
tion path of the system. In this case, the explicit dynamic relaxation method is valid
so that the elastic equilibrium solution can be obtained from the particle simulation
using the distinct element method. However, if any failure takes place in a quasi-
static system, then the quasi-static system behaves nonlinearly so that the solution
to the post-failure quasi-static problem is not unique and therefore, becomes depen-
dent on the deformation path of the system. Due to the use of both the fictitious
time and the fictitious scaled mass, the physical deformation path of a system can-
not be simulated using the explicit dynamic relaxation method. This implies that
the post-failure particle simulation result obtained from using the explicit dynamic
relaxation method may be problematic, at least from the rigorously scientific point
of view. Nevertheless, if one is interested in the phenomenological simulation of the
post-failure behavior of a quasi-static system, a combination of the distinct element
method and the explicit dynamic relaxation method may be used to produce some
useful simulation results in the engineering and geology fields (Cundall and Strack