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.
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