Bounding Box Insertion

The bounding box method relies on a changing boundary to select cohesive edges. At periodic intervals during the simulation, the bounding box method calculates the farthest extents of all failing cohesive elements within the domain. Where a failing element is defined as one whose strength parameter is below the initial value, but has not yet reached zero. The initial bounding box is the increased slightly and all non-cohesive edges within this new box are made cohesive.

As a test of the bounding box method, we have selected the mode I crack problem presented in Figure 4.15. We apply an applied velocity of $0.25$ $m/s$ along the left and right boundaries. The bulk material is PMMA with a Young's modulus $E = 3.24$ $GPa$, Poisson's Ratio $\nu = 0.35$, and density $\rho = 1190$ $kg/m^3$. The cohesive elements have a maximum stress $\sigma_{max} = 32.4$ $MPa$, initial strength parameter $S_{init} = 0.995$ and a normal and tangential critical separations of $\Delta_{crit,N} = \Delta_{crit,T} = 2.2 \times 10^{-5}$ $m$. The domain is meshed into $4043$ nodes, $11857$ edges, and $7815$ volumetric elements. The resulting critical time step for the problem is reduced to $\Delta t = 2.8 \times 10^{-9}$ $s$.

Figure 4.15: Schematic of a mode I crack problem.

Using a Pentium III, 600 MHz, 750Mb RAM processor, running Mandrake Linux 7.2, the simulations were run for $72000$ time steps - or approximately $0.0002$ seconds.For the reference simulation, cohesive elements are present everywhere in the domain from time $t = 0$. The bounding box simulation requires only a few initial cohesive elements in the vicinity of the crack tip, so that failure can begin. The selection test occurs every $500$ time steps at which time the bounding box is scaled up by $4$ characteristic lengths in each of the principal directions. Both the selection interval and scaling factors can be selected to best fit a given problem. Decreasing the selection interval increases the frequency of the bounding box selections, which results in an increased computational time as well as an increase in cohesive insertions. The size of the scaling factor is directly proportional to the number of new cohesive elements that will be inserted during each bounding box selection cycle. As a result, if this scaling factor is large, the domain can become quickly filled with cohesive elements.

The timing results for the bounding box selection case are presented in Table 4.5 for the cohesive and internal force subroutines, the main solution code and the total simulation. From this table we can see that the bounding box method saves nearly $65\%$ of the time needed to solve the mode I crack problem. A major portion of the savings is due to the decreased number of internal force calculations, which saved approximately $42\%$ of the total time. This can be seen in Figure 4.16, where the number of cohesive elements present in the domain was very limited for most of the simulation. The major influx of new cohesive elements occurred at approximately the same time as the main crack began to form. These new cohesive elements extended the cohesive failure zone allowing for the formation and propagation of this crack. Figure 4.17 shows the growth in the crack length occurring at nearly three quarters of the way through the simulation - immediately after the cohesive element insertions. This figure also shows how well the bounding box selection method was able to track the crack tip distance over time.

The final solutions, at time $t = t_o + 72000 \Delta t$, for both the reference case as well as the bounding box selection case are presented in Figures 4.18 and 4.19. In both figures, the non-cohesive edges are represented by thin lines, while the cohesive edges or elements are dark. Any failing cohesive elements are represented by a dashed bold line, and completely failed ones are bold. In the bounding box selection case, we can see that the part of the upper region is free of cohesive elements. In addition, the final solution appears to have many more failing cohesive elements, both near the crack itself, as well as in fringe regions. This is most likely the result of the oscillations that occur as a part of the insertion. Since the bounding box method selects new cohesive elements based on any neighboring cohesive elements, certain regions may be under high stress although have no existing cohesive elements in their vicinity. As a result, when a cohesive element is finally inserted into one of these regions, even pre-stretching is not able to compensate well for the high stresses already present.

Table 4.4: Mode I case timing results, in seconds, for the reference and bounding box insertion cases.

Subroutine	Reference Case [s]	Bounding Box Case
$R^\textrm{co}$	3013.84	619.30
$R^\textrm{in}$	1035.90	794.97
$Main$	1645.03	561.36
$Total$	5749.17	2040.19
$\%$ $Total$ $Savings$		65%

Figure 4.16: Number of cohesive elements present in the domain over time for bounding box insertion.
Figure 4.17: Mode I case crack tip distance versus time.

Figure 4.18: Mode I reference case with cohesive elements present from the beginning of the simulation (10x exaggeration).
Figure 4.19: Mode I bounding box solution (10x exaggeration) (edge key: thin = normal edge, dark = cohesive element, dashed = failing cohesive element, bold = failed cohesive element).