Dynamic Insertion Combined with Subcycling

The multi-time step nodal subcycling algorithm, presented in the beginning of this chapter, has shown to generate significant computational time savings, while maintaining a fairly accurate solution. The solution does begin to degrade though for excessive time step differences in the subcycled and non-subcycled regions, as well as for small region ratios. In this section we present the results of combining both the subcycling and dynamic insertion algorithms.

As a test case we present the mode I crack specimen shown in Figure 4.42 whose bulk material has properties of PMMA. The critical time step for this problem is $\Delta t = 2.8 \times 10^9$ $s$, which is a $1/30$ reduction of the problem's Courant condition. In order to initiate the subcycling portion of the simulation, we pre-insert cohesive elements in a small region near the crack tip. This region is therefore given a time step of $1 \Delta t$ while the larger outer region has a time step $m \Delta t$.

The simulation was run for $60000$ time steps for a reference solution having cohesive elements present everywhere and no subcycling, for a second reference solution using a $45\%$ stress insertion and for three solutions using both dynamic insertion and subcycling with parameters of $m = 4, 10$ and $14$. The final crack profiles for each of the above simulations are presented in Figures 4.44 through 4.46. From these figures we can see that the combined solutions closely match both reference solutions. Only the $m = 14$ combined solution appears to have a different final solution. Careful observation of this solution shows that there are large instabilities near the crack tip - as seen by the large number of failing cohesive elements. These instabilities as well as the effect of a large time step in the non-subcycled region have most likely compounded the errors present in the solution. In addition, at the end of the simulation, the region ratio is much lower than its initial value - as seen in Figure 4.43. This increase in subcycled nodes increases the instabilities as discussed in the subcycling section earlier in this chapter.

Although the solution loses some accuracy for the larger subcycling parameters, significant savings can still be achieved for the lower values - as presented by Table 4.8. The % savings is presented for both the savings with respect to the reference solution using cohesive elements everywhere at the beginning as well as the reference solution of only dynamic insertion at the $45\%$ stress level - the latter savings is presented in parentheses in the table. We can see that the major savings for the combined solutions is a direct results of the dynamic insertion and only small overall percentage is obtained through subcycling.

Figure 4.42: Schematic of a mode I crack problem using nodal subcycling and dynamic stress insertion of $45\%$.

Table 4.8: Timing results (in $s$) for combined dynamic stress insertion of $45\%$ with subcycling of ${\it m} = 6, 10$ and $14$. Time savings from the reference solution is given first while the savings from the dynamic insertion case is presented in parentheses.

Subroutine	Reference	Dynamic, m = 1	m = 6	m = 10	m = 14
$R^\textrm{co}$	2627.37	76.71	95.23	88.18	79.20
$R^\textrm{in}$	870.83	614.38	247.38	167.45	134.03
$Main$	1425.46	251.39	344.63	291.36	271.16
$Total$	4973.63	1005.79	739.50	617.36	528.33
$\%$ $Total$ $Savings$		79% (0%)	85% (26%)	87% (38%)	89% (47%)

Figure 4.43: Region ratio over time for the subcycling solutions with $m = 6, 10$ and $14$.

Figure 4.44: Reference solution with cohesive elements present everywhere in the domain at the beginning of the simulation. No subcycling is used (10x exaggeration).

Figure 4.45: (a) Solution having only dynamic insertion at 45% of the local stress with no subcycling. (b) Combined dynamic insertion with subcycling, $m = 6$ (10x exaggeration).
Figure 4.46: (a) Combined dynamic insertion with subcycling, $m = 10$ (b) Combined dynamic insertion with subcycling, $m = 14$ (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).