Recommendations for Future Research

While some important methods have been developed and many encouraging results presented in this research, various issues remain to be investigated.

Although the subcycling algorithm was able to generate some time savings, its implementation requires additional memory storage requirements not present in a non-subcycled scheme. Furthermore, the stability of the solution is highly dependent on the time step ratio as well as the distribution of the time steps across the nodal regions. The solutions may be more stable if the subcycled region is given a buffer zone whose the nodes are also subcycled, so that the transitions between the critical and non-critical regions are not so severe.

The dynamic insertion algorithm can also be improved by more accurately applying the nodal separations used in pre-stretching. Under the current implementation, these separations are averaged for each cohesive node and equally distributed across each half. Future research may concentrate on applying a weighted distribution approach to further minimize the nodal oscillations. Additionally, this method can also be extended for use with six node linear strain triangles (LST) in order to generate even more accurate solutions.

As seen in the previous chapter, our code parallelization using Charm++ was not able to generate completely ideal speedups, although the results were quite promising. The primary cost of the parallelization was due to the pre-processing of input data and serial insertion portions of the code. Parallelizing these areas can greatly improve the performance of the code. In particular, the current serial insertion requires the periodic assembly and repartitioning of the entire mesh. This interval of this periodic update can be initially increased since many insertions are not required during the initial stages of the simulation, where the stresses have not yet reached their desired levels. Furthermore, extending the dynamic insertion algorithm from serial based insertion to parallel can also improve performance since no assembly and repartitioning of the mesh would be required.

Lastly, we can also generate significant computational savings by using and adaptive remeshing technique, as illustrated in Figure 5.1. Since cracks move through the systems during the simulation, the adaptive remeshing technique would continually adjust the critical region(s) to ensure that the most optimal solution can be achieved. Using the CVFE scheme in conjunction with the adaptive remeshing would require not only dynamic insertion of cohesive elements but also a dynamic removal of failed or unloaded elements.

Figure 5.1: Adaptive crack propagation.

In our initial investigations of this topic, we have applied a adaptive remeshing technique to a simple 1-D problem shown in Figure 5.2 The beam is composed of $21$ equal segments of length $1.0$ $m$.

The beam is similar to the reference 1-D beam problem presented in Chapter 3. We break the 11th segment into $7$ smaller pieces at time step $10000$ for a $30000$ time step simulation. After element break up, and cohesive node insertion, the middle of the cohesive region is represented by node $26$ for which we plot the velocity in Figure 5.3, Comparing this velocity profile to the reference profile for node $15$ if the reference case of Figure 3.6, we can see that they match quite well, although the velocity for the adaptive meshing case are a little oscillatory.

Figure 5.2: 11th element broken into 7 pieces.

Figure 5.3: Velocity profile for new node #26 after element break up at time step 10000.

Extending this investigation into 2-D requires more complex issues such as element conformity, database management and time step stability. Also, element removal presents similar, if not more difficult, issues but combined with the dynamic insertion and multi-time step nodal subcycling it can significantly decrease the computational effort for solving dynamic fracture problems.