Molecular mechanics in the context of the finite element method

Abstract

SUMMARYIn molecular mechanics, the formalism of the finite element method can be exploited in order to analyze the behavior of atomic structures in a computationally efficient way. Based on the atom-related consideration of the atomic interactions, a direct correlation between the type of the underlying interatomic potential and the design of the related finite element is established. Each type of potential is represented by a specific finite element. A general formulation that unifies the various finite elements is proposed. Arbitrary diagonal-and cross-terms dependent on bond length, valence angle, dihedral angle, improper dihedral angle and inversion angle can also be considered. The finite elements are formulated in a geometrically exact setting; the related formulas are stated in detail. The mesh generation can be performed using wellknown procedures typically used in molecular dynamics. Although adjacent elements overlap, a double counting of the element contributions (as a result of the assembly process) cannot occur a priori. As a consequence, the assembly process can be performed efficiently line by line. The presented formulation can easily be implemented in standard finite element codes; thus, already existing features (e.g. equation solver, visualization of the numerical results) can be employed. The formulation is applied to various interatomic potentials that are frequently used to describe the mechanical behavior of carbon nanotubes. The effectiveness and robustness of this method are demonstrated by means of several numerical examples.

The Buckling Behavior of Boron Nitride Nanotubes under Bending: An Atomistic Study

“…Force field parameters are taken from the work by Sevik, C., et al [26] for B-N interactions. While DFT (Density Functional Theory) calculations and MD (Molecular Dynamics) simulations are time-consuming, MDFEMs (Molecular Dynamic Finite Element Methods), sometime known as atomic-scale finite element methods or atomistic finite element methods, have been developed to analyze nanostructured materials in a computationally efficient way [27,28]. To achieve the atomic positions of the BN-NT under specific boundary conditions, MDFEM is here adopted.…”

Buckling Analysis of Carbon Nanotubes – a molecular mechanics approach using the finite element framework

“…Due to the local character of the interatomic potential the global residuum vector and the global stiffness matrix can be composed of lower-dimensional elemental contributions. Therefore a universal approach to build an atom-related finite element was proposed by J. Wackerfuss [1]. This approach leads to a finite element for the Tersoff-Brenner potential given in Figure 1 consisting of a reference atom and its nine neighbouring atoms.…”

“…Therefore multiple approaches were done to describe the behaviour of the atom interactions by either bond-related or atom-related finite elements (see e.g. [1] and references therein). With such a description in hand a classical finite element framework and its algorithms can be reused.…”

Structural Analysis on Nanoscale

“…Already existing strategies to do this can be categorized into two groups: The bond-related approaches, where each individual term of E is considered separately by a single unit, and the atom-related approaches, where all those terms of E, where a reference atom is involved, are considered as one unit. For a detailed comparison, see [1]. The limitations of the existing approaches are: a) restricted to a specific type of atomic interactions, b) mostly restricted to small deformation scenarios, c) double-counting of energy contributions increases the computational effort.…”

“…The derivatives of E e i , with respect to the position vectors, can be derived analytically, as stated in [1] in detail. The presented numerical method can be embedded in existing FE-codes by using standard user-element interfaces.…”