Effective discontinuous interface coupled models for atomistic energy minimisation

In the field of multiscale modelling of materials, a class of significant problems involves the atomistic-to-continuum coupling in crystals. Continuum models frequently fail to produce accurate predictions near singularities and defects and hence coupled atomistic/continuum methods have become popular. The ad-hoc coupling of atomistic and continuum energies results in numerical artifacts on the interface between the continuum and atomistic regions, known as ghost forces. The design and analysis of atomistic/continuum coupling methods that are ghost-force free is important in computational and mathematical modelling of materials and one of the very few well defined problems in multi-scale algorithm design for nonlinear phenomena. In this thesis we developed a discontinuous ghost-force free bond volume based method in one dimensional and two dimensional crystal lattices. The design of the method was motivated by appropriately analysing the error both at the atomistic and the continuum region. Its design is consistent and transferable. Next, we were concerned about the energy consistency and the variational consistency of the coupled methods. Consistency is a quantity that measures the extent to which an exact smooth solution does satisfy the numerical scheme. We proved that in one dimension the local contributions of the energy were of second order in the lattice spacing e, O(e²). The total energy error in one and two dimensions was second order. We analysed the error for rst variations both in one and two dimensions. Our analysis confirmed that the proposed methods were indeed ghost-force free and their variational consistency error was bounded by (e² + e²-1/p ) in the discrete W-1,p norm. We implemented the static atomistic problem and compared it to the static coupled method in one dimension. We considered energies from multi-body potentials. By using the symmetry properties of the potentials we derived energy consistency error bounds of order O(e²).