Page 239 - The Combined Finite-Discrete Element Method
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222 SENSITIVITY TO INITIAL CONDITIONS
Figure 6.8 A beam-shaped heap impact problem.
the system or behaviour of its parts. However, it is worth mentioning that a drastic loss
of symmetry is only possible with a small system. A large system tends to compensate
for this local rounding error type bifurcation by shear numbers of such bifurcations, and
much more constrained motion of individual discrete elements.
This is demonstrated by the example shown in Figure 6.8, where a beam shaped heap of
triangular discrete elements is impacted by a large triangular discrete element moving at
constant velocity. The motion sequence is shown in Figure 6.9. Initially, complete preser-
vation of symmetry is observed despite a large number of discrete elements comprising
the system, and a large number of contacts being resolved. This is because the surrounding
discrete elements are in a way constraining the motion of discrete elements in contact.
When the density of discrete elements close to the mid-span of the beam decreases,
discrete elements in front of the impactor become less constrained. Further resolution
of contacts leads to a slight loss of symmetry. The overall behaviour of the system still
shows symmetry, although discrete elements close to the tip of the impactor do not match
the symmetry requirements. It is worth noting that, given the total number of resolved
contacts and possible bifurcations, the fact that the symmetry has been preserved to a
great extent demonstrates the robustness of the discretised distributed contact algorithm
used to resolve contacts (see Chapter 2). A loss of symmetry and sensitivity to initial
conditions or rounding errors is not only due to sharp corner contacts. It is not only
limited to 2D problems. Instead, it is almost a second nature of large scale combined
finite-discrete element simulations. This is demonstrated by a 3D contact impact problem,
shown in Figure 6.10. A rod shaped body has been placed at the centre of a rigid cube
shaped box. The rod moves with initial vertical velocity towards the top end of the box.
The simulation is run twice. The first time, the rod is placed exactly in a vertical position.
The second time, the rod is slightly inclined relative to the vertical axis of the box. The
angle of inclination is very small (10 −11 radians). In practical terms, the two problems
are identical, and one could easily state that there is no difference in initial conditions.
There is almost no difference in the results of simulation either, as shown in Figure 6.11,
where both rods have moved towards the top of the box. In Figure 6.12, the rods have
bounced off the upper side of the box and are moving towards the bottom side of the
box, as shown in Figure 6.13.
In Figure 6.14, the rods have bounced off the bottom side of the box, and are moving
towards the upper side of the box.
In Figure 6.15, the rods have bounced again from the upper side of the box, and are
moving towards the bottom of the box.
In Figure 6.16, subsequent bouncing from the bottom of the box is shown. This is
followed by bouncing from the top of the box, followed by yet another bouncing from
the bottom of the box, as shown in Figure 6.17. The rods still seem to be at same
position and parallel, i.e. the results of two numerical experiments seem to completely
much each other.