The aim of this project is to develop a way to generate efficient code for parallel architectures for the simulation of carbon nanotube structures. This should be realized without actually setting up the full stiffness matrix A. The way to reach this is generating the corresponding code directly from the graph algebra description of the nanotubes, taking advantage of the special strucutral properties of super carbon nanotubes (SCNTs). The generated code shall in addition exploit the current parallel architectures. These offer a rich amount of variants like multicore processors in different configurations (SMP-, NUMA-Systems) or accelerators such as AMD/NVidia GPUs, Intel’s Xeon Phi coprocessors and reconfigurable devices like FPGAs. Figure 4 shows some recent devices and their properties. Better usage of the hardware resources will also lead to a more power efficient calculation.

# Efficient Carbon Nanotube Simulation

Carbon based materials present a big variety of forms and physical and chemical properties. One of such forms are carbon nanotubes. They can be imagined as flat graphene sheets which are rolled up. Because of there characteristics it is very likely that nanotubes will play an important role in nanotechnology. Figure 1 shows a graphene sheet and the corresponding tube.

With the help of Y-formed junctions it is possible to compose tubes to structures called super carbon nanotubes (SCNT). These structures are highly hierarchical and symmetric, what you can see in figure 2. This process can be repeated recursively as often as it is necessary.

These symmetries and hierarchies can formally be described by a graph algebra introduced in [3]. Every carbon atom can be identified by a tuple of n values which encodes its position within the whole tube. Figure 3 shows these tuples for a flat graphene sheet as a pre-stage of a 3D tube. This is a compact way to deal with these structures which preserve the hierarchy and symmetry properties for later calculations. Current simulations based on standard models ignore this information and simply produce a large sparse matrix system Ax = b which must be solved by standard methods [4] [5].

# Construction of different tubes

The already mentioned graph algebra allows to create arbitrary tubes. To determine the shape of a level 0 SCNT only two parameters need to be set. The first one is its diameter D0 and the second one the length L0 of the tube. We notate the tube configuration as a pair (D0, L0).

As the two figures indicate this process allows to create a high diversity of shapes. Both shown tubes consist of 8192 atoms, but while the first one might remind you of a drinking straw the second one looks much more like a ring.

# Literature

[1] *Geometry of Multi-Tube Carbon Clusters and Electronic Transmission in Nanotube Contacts*, S. Ferrer et al, 1999

[2] *Geometric and electronic structure of carbon nanotube networks: ‚super‘-carbon nanotubes*, V. R. Coluci et al, 2006

[3] *Modelling super carbon nanotubes as hierarchically symmetric graphs*, C. Schröppel and J. Wackerfuß, 2013

[4] *The atomic-scale finite element method*, B. Liu et al, 2004

[5] *Molecular mechanics in the context of the finite element method*, J. Wackerfuß, 2008

[6] http://www.amd.com/de/products/workstation/graphics/ati-firepro-3d/w9000/Pages/w9000.aspx

http://www.intel.de/content/www/de/de/processors/xeon/xeon-processor-5000-sequence.html

http://www.xilinx.com/products/silicon-devices/fpga/virtex-7.html