On error controlled atomistic-to-continuum modeling with application to graphene monolayers

Abstract: In order to consider virtual materials, i.e. materials that are not manufactured yet, it can be advantageous to model physical phenomena on a lower length scale than the macroscopic one. In such a case, homogenization (or coarse-graining) can be a useful tool. When atomistic systems are analyzed, homogenization can be used to derive continuum properties whereby the need for empirical continuum models is avoided. The macroscopic stress-strain response is then obtained from a Representative Volume Element (RVE), also called Representative Unit Lattice (RUL) in the case of a discrete lattice. Since lattice defects may play an important role, the RUL should be very large in general. In this thesis, we study the coupling between the atomistic and continuum behavior in graphene using an adaptive version of the quasi-continuum (QC) method. In conjunction with atomistic-to-continuum homogenization, a strategy based on goal-oriented a posteriori error estimation is designed for the adaptive error control of a given quantity of interest, enabling automatic resolution of the critical parts in the lattice, and thus, retaining accuracy in the output of interest within reasonable computational cost. The adaptive algorithm is used to study crystallographic defects in the graphene lattice and the model is evaluated under various loading conditions. The thesis consists of two appended papers: In Paper A, an efficient strategy for goal-oriented error control of QC-approximations is introduced. Special attention is paid to major characteristics of the quasi-continuum method, interpolation discretization and summation quadrature via clusters of atoms. In Paper B, focus is placed on developing strategies to obtain a macroscopically relaxed RUL, which defines a stress-free initial configuration. The task is thus to find a deformation map that corresponds to a zero stress state. It is then possible to use the procedure developed in Paper A to analyze both the relaxation procedure as well as the subsequent deformation due to external loading.