Structural Analysis on Nanoscale

Abstract

Atomic structures, exhibiting a physical dimension in a range of 1-100 nanometers provide a basis for novel applications in nanotechnology. In molecular mechanics the formalism of the finite element method can be exploited to analyze the behavior of atomic structures in a computational 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 has been developed. Each type of potential is represented by a specific finite element. A general formulation that unifies the various finite elements is proposed. The mesh generation can be performed using well-known procedures typically used in molecular dynamics. Although adjacent elements overlap, a double counting of the element contributions (as a result of the assembly process) can not occur a priori in this new formulation. 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 and can be applied to various interatomic potentials. As an example, the method is applied in order to model the structural response of pristine and defective carbon nanotubes. Atomic modelConsidering an atomic structure consisting of N atoms in a 3-dimensional Euclidian space, where R i and r i denote the position vectors of atom i in the reference and the current configuration. The displacement vector of atom i is given by u i = r i − R i . The total energy π of the atomic structure is given bywhere in general the internal energy E consists of a multiplicity of terms, describing different specific atomic interactions. These terms depend on the atomic kinematics: bond length r ij , valence angle θ ijk , dihedral angle ϕ ijkl , improper dihedral angle ψ ijkl and inversion angle χ ijkl . The indices of the kinematics indicate the atoms involved in the specific interactions. The kinematics depend only on the position vectors r i of these atoms. The external energy considers the (dead) loadf i acting on the atom i. The equilibrium state of the atomic structure can be obtained by minimizing the total energy. Mathematical methods widely used in molecular mechanics are the conjugate gradient method and the steep decent method, which are order-N 2 methods. In contrast, the Newton-Raphson method which is computational more efficient (order-N method) is rarely used. The letter method leads to the following equation for the unknown displacement increments ∆u jwhere K and R are the global stiffness matrix and residuum vector of the atomic structure. In the next section it will be shown how to calculate K and R in a computationally efficient way. Numerical implementationTaking into account that the atomic interactions are localized, E can be divided into small units. 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 t...