Meshing Highly Regular Structures: The Case of Super Carbon Nanotubes of Arbitrary Order

Schröppel, Christian; Wackerfuß, Jens

doi:10.1155/2015/736943

Cited by 6 publications

(2 citation statements)

References 38 publications

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“…Burger et al [89,90] address compressing graphs used to model Super Carbon Nanotubes (SCNTs). Example such graphs are Hierarchically Symmetric Graphs (HSG) [403]. They can model hierarchical structure of SCNTs.…”

Section: Chemistry Networkmentioning

confidence: 99%

Survey and Taxonomy of Lossless Graph Compression and Space-Efficient Graph Representations

Besta,

Hoefler

2018

Preprint

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Various graphs such as web or social networks may contain up to trillions of edges. Compressing such datasets can accelerate graph processing by reducing the amount of I/O accesses and the pressure on the memory subsystem. Yet, selecting a proper compression method is challenging as there exist a plethora of techniques, algorithms, domains, and approaches in compressing graphs. To facilitate this, we present a survey and taxonomy on lossless graph compression that is the first, to the best of our knowledge, to exhaustively analyze this domain. Moreover, our survey does not only categorize existing schemes, but also explains key ideas, discusses formal underpinning in selected works, and describes the space of the existing compression schemes using three dimensions: areas of research (e.g., compressing web graphs), techniques (e.g., gap encoding), and features (e.g., whether or not a given scheme targets dynamic graphs). Our survey can be used as a guide to select the best lossless compression scheme in a given setting.

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Section: Chemistry Networkmentioning

confidence: 99%

Survey and Taxonomy of Lossless Graph Compression and Space-Efficient Graph Representations

Besta,

Hoefler

2018

Preprint

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“…Super carbon nanotubes are large graphene-based structures that exhibit complex hierarchies and symmetries [4]. To compute the mechanical response of such structures to different loads, we have set up an atomistic finite element formulation on the fine grid [5] and a LogFE formulation for the coarse grid correction.…”

Section: Formulationmentioning

confidence: 99%

The Logarithmic finite element method in a multigrid setting

Schröppel

Wackerfuß

2016

Proc Appl Math and Mech

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The Logarithmic finite element ("LogFE") method is a novel finite element approach for solving boundary-value problems proposed in [1]. In contrast to the standard Ritz-Galerkin formulation, the shape functions are given on the logarithmic space of the deformation function, which is obtained by the exponentiation of a linear combination of the shape functions given by the degrees of freedom.Unlike many existing multigrid formulations, the LogFE method allows for a very smooth interpolation between nodal values on the coarse grid. It can thus avoid problems with regard to locking and convergence that appear in multigrid applications using only linear interpolation, especially for larger coarsening factors.We illustrate the use of the LogFE method as a coarse grid algorithm, in conjunction with an atomistic finite element method on the fine grid, for calculating the mechanical response of super carbon nanotubes. MotivationMany multigrid models do not take account of the actual geometry of a structure. For algebraic multigrid methods, the explicit aim is to formulate an algorithm that can be applied independently of the geometry of the structure under consideration. While this significantly simplifies the generation of the coarse meshes, one of its drawbacks is that rather simple prolongation functions, such as linear interpolation, are generally being used to identify the coarse grid correction on the fine grid. This procedure often introduces spurious high frequency deformations (such as simple kinks in piecewise linear functions) that lead to a deterioration of the convergence characteristics on the fine grid. The errors on the fine grid then may lead to deficient estimates of the derivatives with regard to the objective function, which in turn impede on the convergence of the coarse grid or even lead to a breakdown of the optimization algorithm.In this respect, it is important to recall that, generally, a smoothing algorithm, i.e. not a global solver, is being employed on the fine grid. As a result, if the coarse grid experiences locking, that effects cannot be mitigated by the fine grid algorithm, which generally exhibits very slow convergence for low-frequency spatial residuals. As a result, the use of linear interpolation on the coarse grid may lead to locking effects in the global calculation, even for small coarsening factors. While higher order interpolation functions can avoid kinks, they can also introduce spurious high frequencies, e.g. oscillations, on the fine grid.The Logarithmic finite element ("LogFE") method [1-3] provides a smooth interpolation of the field variables on the coarse grid. It thus minimizes spurious high-frequency components on the fine grid and can avoid locking phenomena associated with the use of linear interpolation. As a result, it allows for an increase in the coarsening factor, leading to a significant reduction of the relative number of degrees of freedom on the coarse grid, compared to the fine grid. This reduction of the number of degrees of freedom on the coarse grid is...

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