Memory-Efficient and Parallel Simulation of Super Carbon Nanotubes

Carbon nanotubes (CNTs) received much attention since their description in Nature in 1991. In principle, a carbon nanotube is a rolled up sheet of graphene, which can be imagined as a honeycomb grid of carbon atoms. This allotrope of carbon has many interesting properties like high tensile strength at very low weight or its high temperature resistance. This motivates the application of CNTs in material science to create new carbon nanotube enforced materials. They also possess interesting electronic properties since CNTs show either metallic or semiconducting behavior, depending on their configuration. The synthesis of branched carbon nanotubes allows the connection of straight CNTs to carbon nanotubes networks with branched tubes employed as junction elements. One of these networks are the so-called super carbon nanotubes (SCNTs) that were proposed in 2006. In that case, each carbon-carbon bond within the honeycomb grid is replaced by a CNT of equal size and each carbon atom by a Y-branched tube with three arms of equal length and a regular angle of 120° between the arms. This results in a structure that originates from tubes and regains the outer shape of a tube. It is also possible to repeat this process, replacing carbon-carbon bonds not with CNTs but with SCNTs, leading to very regular and self-similar structures of increasingly higher orders. Simulations demonstrate that the SCNTs also exhibit very interesting mechanical properties. They are even more flexible than CNTs and thus are good candidates for high strength com- posites or actuators with very low weight. Other applications arise again in microelectronics because of their configurable electronic behavior and in biology due to the biocompatibility of SCNTs. Despite progress in synthesizing processes for straight and branched CNTs, the production of SCNTs is still beyond current technological capabilities. In addition, real experiments at nanoscale are expensive and complex and hence, simulations are important to predict properties of SCNTs and to guide the experimental research. The atomic-scale finite element method (AFEM) already provides a well-established approach for simulations of CNTs at the atomic level. However, the model size of SCNTs grows very fast for larger tubes and the arising n-body and linear equation systems quickly exceed the memory capacity of available computer systems. This renders infeasible the simulation of large SCNTs on an atomic level, unless the regular structure of SCNTs can be taken into account to reduce the memory footprint. This thesis presents ways to exploit the symmetry and hierarchy within SCNTs enabling the simulation of higher order SCNTs. We develop structure-tailored and memory-saving data struc- tures which allow the storage of very large SCNTs models up to several billions of atoms while providing fast data access. We realize this with a novel graph data structure called Compressed Symmetric Graphs which is able to dynamically recompute large parts of structural information for tubes instead of storing them. We also present a new structure-aware and SMP-parallelized matrix-free solver for the linear equation systems involving the stiffness matrix, which employs an efficient caching mechanism for the data during the sparse matrix-vector multiplication. The matrix-free solver is twice as fast as a compressed row storage format-based reference solver, requiring only half the memory while caching all contributions of the matrix employed. We demonstrate that this solver, in combination with the Compressed Symmetric Graphs, is able to instantiate equation systems with matrices of an order higher than 5∗10^7 on a single compute node, while still fully caching all matrix data.