Page 193 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

184 8 Spontaneous Crack Generation Problems in Large-Scale Geological Systems

and rotational moment exerted on every particle in the simulation are calculated.
(4) Using the central finite difference method, the equations of motion expressed
by Eqs. (8.1), (8.2) and (8.3) are solved for each of the particles in the simulation
system, so that new displacements can be determined and the position of every par-
ticle can be updated at the end of the calculation time step. The above-mentioned
solution loop is repeated for each calculation time step until the final stage of the
simulation is reached.

8.2 Some Numerical Simulation Issues Associated
with the Particle Simulation Method

Although the particle simulation method such as the distinct element method was
developed more than two decades ago (Cundall and Strack 1979), some numeri-
cal issues associated with it may need to be further addressed. These include: (1) an
issue caused by the difference between an element used in the finite element method
and a particle used in the distinct element method; (2) an issue resulting from using
the explicit dynamic relaxation method to solve a quasi-static problem; and (3) an
issue stemming from an inappropriate loading procedure used in the particle simula-
tion method. Although some aspects of these numerical simulation issues have been
briefly discussed (Cundall and Strack 1979, Cundall 2001, Itasca Consulting Group,
inc. 1999, Potyondy and Cundall 2004), we will discuss them in greater detail so
that their impact on the particle simulation results of spontaneous crack generation
problems within large-scale quasi-static systems can be thoroughly understood.

8.2.1 Numerical Simulation Issue Caused by the Difference
between an Element and a Particle

In order to investigate the numerical simulation issue associated with the differ-
ence between an element used in the finite element method and a particle used in
the distinct element method, we need to understand how an element and a par-
ticle interact with their neighbors. Figure 8.4 shows the comparison of a typical
four-node element used in the finite element method with a typical particle used in
the distinct element method. In the finite element method, the degree-of-freedom
is represented by the nodal points of the element, while in the distinct element
one, the degree-of-freedom is represented by the mass center of the particle. In this
regard, the particle may be viewed as a rigid element of only one node. With a two-
dimensional material deformation problem taken as an example, the displacement
along the common side between two elements is continuous, implying that there is
no overlap between any two elements in the conventional finite element method.
However, particle overlap is allowed in the distinct element method. Since the ele-
ment is deformable, the mechanical properties calculated at the nodal point of an ele-
ment are dependent on the macroscopic mechanical properties of the element so that