Dynamic Insertion with Pre-Stretch

To minimize the nodal oscillations resulting from blind insertion, we employ a pre-stretch to cohesive nodes during insertion. With this pre-stretch the cohesive node is inserted in a state most closely approximating the state it would be in had the insertion occurred at the beginning of the simulation. Using again the reference problem described at the beginning of this chapter, we pre-stretch nodes $\char93 12$ through $\char93 20$ during dynamic insertion at the $5000th$ and $10000th$ time step. The velocity profiles for node $\char93 15$ are presented in in Figures 3.17 and 3.18, for each of the insertions.

Comparing these velocity profiles to the blind insertion profiles in Figures 3.13 and 3.14, we can see that pre-stretching is able to minimize the oscillations quite well. This method is also able to avoid the minor stabilization period that is required with damping. Furthermore, Figures 3.19 and 3.20 plot the separation distance for two cohesive halves of node $\char93 15$, for dynamic pre-stretched insertion at the $10000th$ time step. From these two figures we can clearly see how oscillatory the separations cohesive node are, as well as how significantly pre-stretching minimizes them.

Figure 3.17: Velocity profile of node $\char93 15$ resulting from insertion with pre-stretching at the $5000th$ time step ($12.5$ $s$).
Figure 3.18: Velocity profile of node $\char93 15$ resulting from insertion with pre-stretching at the $10000th$ time step ($25.0$ $s$).

Figure 3.19: Cohesive separation for node $\char93 15$ resulting from blind insertion at $10000th$ time step ($25.0$ $s$)
Figure 3.20: Cohesive separation for node $\char93 15$ resulting from insertion with pre-stretching at $10000th$ time step ($25.0$ $s$)

In order to obtain timing information we have increased the size of the reference problem to one having $300$ segments of length $1.0$ $m$, as seen in Figure 3.21, The middle $180$ nodes are made cohesive at the $110,000th$ time step or $2750$ $s$ for a critical time step reduced by $1/40th$ to $\Delta t= 0.025$ $s$ The total simulation is run for $200,000$ time steps ($5000$ $s$).

Figure 3.21: Test case for dynamic cohesive node insertion with pre-stretching.

Table 3.2 gives the timing results for the cohesive and internal force calculations as well as the total simulation time. We gain a $58$ $\%$ time savings in the cohesive calculations which is on par with the approximate time of insertion, $110,000th$ time step out of $200,000$, or $55$ $\%$ through the simulation. The internal force calculations are increased slightly because the number of nodes has increased and hence the number of internal force calculations has also increased.

Table 3.2: Timing results for dynamic cohesive node insertion with pre-stretching. CPU time saving (in %) is given in parentheses.

Subroutine	Reference Case [s]	Dynamic Case [s]
$R^\textrm{co}$	9.17	3.82 ( 58%)
$R^\textrm{in}$	6.74	6.88 ( 0%)
$Total$	60.91	51.34 ( 16%)