The Logarithmic finite element (LogFE) method extends the Ritz-Galerkin method to approximations on a non-linear finite-dimensional manifold in the infinitedimensional solution space. Formulating the interpolant on the logarithmic space allows for a novel treatment of the rotational component of the deformation, and induces a strong coupling between rotations and translations. The Logarithmic finite element method provides a transformation of the initial configuration that is not restricted to an isoparametric formulation.

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 ...

Die Logarithmische Finite-Elemente-Methode (Log-FE-Methode) bietet einen neuartigen Ansatz zur Berechnung von FE-Modellen, insbesondere im Rahmen von Multigrid-Verfahren. In Multigrid-Verfahren wird eine schnelle und robuste Konvergenz insbesondere durch die Entkopplung der Einflüsse der Freiheitsgrade auf den unterschiedlichen Längenskalen erreicht. Das Näherungsverfahren auf dem groben Netz sollte daher auf einer deutlich gröberen Skala arbeiten und (räumlich) hochfrequente Deformationsanteile vermeiden.

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 ...

An extension of the Ritz-Galerkin method, based on finding approximations on a finite-dimensional manifold of functions (i.e., not a linear subspace) in the infinite-dimensional exact configuration space, will be presented. This new approach is particularly efficient in computing geometrically exact solutions for problems involving large rotations.

Concurrent simulations of atomistic (discrete) and continuum models are one key to perform static calculations of large atomistic structures. Occurring ghost forces at the interfaces between discrete (atomistic) and continuous regions are a major challenge for the definition of such interfaces. The so called virtual projection method [1] is a class of force-based a/c schemes which reduce (or even avoid) ghost forces. This class of coupling schemes is able to deal with atomistic models consisting of bonded, multi-body ...

In dem Buch werden die Grundlagen der klassischen Statik in einer modernen, zeitgemäßen Form dargestellt. Dabei wird auf Präzision und Prägnanz in der Darstellung Wert gelegt. Spezielles Gewicht wird auf Einflussfunktionen gelegt, da diese die Basis der finiten Elemente bilden. Das Rechnen mit finiten Elementen ist, wie in dem Buch gezeigt wird, ein Rechnen mit genäherten Einflussfunktionen. Der Text geht ausführlich auf die Fragen der Modellierung von Tragwerken mit finiten Elementen und der Genauigkeit der numerischen ...

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 ...