The purpose of this work is the multiscale modeling of a single-layered graphene sheet. The model is divided into three parts. One is an atomistic domain which is simulated with the atomic-scale finite element method (AFEM). Another is a continuum domain. In this domain, the mechanical properties are investigated by using a finite element based on a nonlocal continuum shell model with a high order strain gradient. To be exact, it is a 4-node 60-generalized degree of freedom (DOF) Mindlin–Reissner finite shell 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 ...

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 ...

Mesh generation is an important step inmany numerical methods.We present the “HierarchicalGraphMeshing” (HGM)method as a novel approach to mesh generation, based on algebraic graph theory.The HGM method can be used to systematically construct configurations exhibiting multiple hierarchies and complex symmetry characteristics. The hierarchical description of structures provided by the HGM method can be exploited to increase the efficiency of multiscale and multigrid methods. In this paper, the HGM method is employed ...