Compressed symmetric graphs for the simulation of super carbon nanotubes

In this paper, we present an extremely space-saving, yet parallelizable, data structure called Compressed Symmetric Graphs (CSGs) for the simulation of Super Carbon Nanotubes (SCNTs) modeled by a graph algebra. CSGs can drastically reduce the amount of data to represent a SCNT by exploiting inherent symmetry and hierarchy to dynamically reconstruct symmetric parts from base elements as needed. This new graph structure is integrated in an existing matrix-free solving approach for simulating the mechanical behavior of SCNTs. We extend previous investigations on structural symmetry in SCNTs to now simultaneously exploit translational and rotational symmetry, thus multiplying their effects. As a result, we can reach compression ratios of over 100 for some SCNT configurations. The memory demand is further reduced by replacing the m-tuples identifying the nodes in the graph model via a structure-related compression by a serial index that can be unfolded on demand. Finally, we use still available RAM as a software-controlled cache for storing intermediate values, reducing recomputations. In this fashion, our code can represent very large configurations, but makes optimal use of the hardware at hand. We investigate for order 0 and 1 SCNTs the impact of the data access-time in CSGs on the total runtime and a suitable OpenMP parallelization strategy for minimizing this influence. We demonstrate that as a result, the new CSGs approach significantly reduces the runtime, by a factor between 1.3 and 12. While the graph algebra underlying this work was designed for the representation of SCNTs, we believe that the algorithmic principles with respect to the exploitation of structure and efficient software design are relevant to other graph settings, where hierarchy and replication play an important role in the graph design. In these cases, CSGs can help to overcome the per node/core memory-capacity limitation of current HPC systems.