Christian Schröppel

Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

We propose a novel finite element formulation that significantly reduces the number of degrees of freedom necessary to obtain reasonably accurate approximations of the low-frequency component of the deformation in boundary-value problems. In contrast to the standard Ritz–Galerkin approach, the shape functions are defined on a Lie algebra—the logarithmic space—of the deformation function. We construct a deformation function based on an interpolation of transformations at the nodes of the finite element. In the case of the geometrically exact planar Bernoulli beam element presented in this work, these transformation functions at the nodes are given as rotations. However, due to an intrinsic coupling between rotational and translational components of the deformation function, the formulation provides for a good approximation of the deflection of the beam, as well as of the resultant forces and moments. As both the translational and the rotational components of the deformation function are defined on the logarithmic space, we propose to refer to the novel approach as the “Logarithmic finite element method”, or “LogFE” method.

A Unified and Memory Efficient Framework for Simulating Mechanical Behavior of Carbon Nanotubes

Abstract Carbon nanotubes possess many interesting properties, which make them a promising material for a variety of applications. In this paper, we present a unified framework for the simulation of the mechanical behavior of carbon nanotubes. It allows the creation, simulation and visualization of these structures, extending previous work by the research group “MISMO” at TU Darmstadt. In particular, we develop and integrate a new matrix-free iterative solving procedure, employing the conjugate gradient method, that drastically reduces the memory consumption in comparison to the existing approaches. The increase in operations for the memory saving approach is partially offset by a well scaling shared-memory parallelization. In addition the hotspots in the code have been vectorized. Altogether, the resulting simulation framework enables the simulation of complex carbon nanotubes on commodity multicore desktop computers.

Exploiting Structural Properties During Carbon Nanotube Simulation

In this paper, we present a novel matrix-free algorithm for the simulation of the mechanical behavior of carbon nanotubes (CNTs). For small deformations, this algorithm is capable of exploiting the inherent symmetry within CNT structures. The symmetry information is encoded with a graph algebra (GA) construction process and preserved within a tuple based atom-indexing. The exploitation of symmetry leads to a reduction of the needed calculations by a factor of more than 100 in the case of larger CNTs. Combining the usage of symmetry information with a new potential caching mechanism, our software is able to store even large tubes in a compressed way with only a few megabytes of data. Altogether, our implementation allows a matrix-free, resource-aware simulation of CNTs. For larger cases it is only about the factor 1.45 - 1.6 slower than the reference solution with a fully assembled stiffness matrix, but consumes twelve times less memory. Also first results of the parallelization of our new algorithm are presented.

Methods to Model and Simulate Super Carbon Nanotubes of Higher Order

Super carbon nanotubes (SCNTs) are of interest in material design because of their strength and weight characteristics. In this paper, we present a graph algebra‐based approach to model and construct SCNTs of arbitrary order. The SCNTs are represented by directed graphs with Y junctions as basic modeling element. A new data structure to store these graphs is proposed that capitalizes on the hierarchy within SCNTs and allows efficient queries for nodes and edges. Symmetry considerations for SCNTs are conducted and related to the graph algebra‐based modeling. We present an extended and improved algorithm for simulating the mechanical behavior of SCNTs. Compared with our previous work on level 0 SCNTs, the performance is improved by a factor higher than 2 when running in serial and a factor up to 4.4 when running in parallel on a 16‐core symmetric multiprocessing system. A new pre‐processing step exploiting structural symmetry and an improved proximity‐aware matrix‐vector‐multiplication routine make this performance improvement possible while only consuming little additional memory. We also now consider SCNTs of order 1 and 2. Experimental results show that our new solver is up to 1.4 times faster than a compressed‐row‐storage based reference solver, on order 0, 1, and 2 SCNTs, with and without deformations, while requiring only half the memory. Because memory is the limiting factor for the feasibility of such simulations, our new approach significantly expands the realm of feasibility for such simulations. Copyright

An Improved Algorithm for Simulating the Mechanical Behavior of Super Carbon Nanotubes

In this paper we present an extended and improved algorithm based on our previous work for simulating the mechanical behavior of super carbon nanotubes (SCNTs). On previously considered level 0 SCNTs the performance is improved by a factor higher than 2 when running in serial and a factor up to 4.4 when running in parallel on a 16 core SMP system. A new pre-processing step exploiting structural symmetry and an improved proximity-aware Matrix-Vector-Multiplication routine make this performance improvement possible while only consuming few additional memory. We also extend our symmetry considerations to SCNTs of order 1 and give an insight into the graph algebra based construction of these structures. Experimental results show that our new solver outperforms a compressed-row-storage based reference solver, on order 0 and 1 SCNTs, with and without deformations, while requiring only half the memory.