The principal application considered is the rupture problem of solid mechanics. The equations of elasticity break down in the presence of cracks in a solid. In the absence of a generally valid theory of crack propagation, atomic level computations are used to follow crack propagation. Cracks arise due to local superposition of mean-field effects and thermal oscillations around crystalline equilibrium positions. The mean-field effects are given by local values of stress and strain given by solutions of the continuum elasticity equations. AMR is used to dynamically focus on regions of high stress. Once identified, these regions are treated using a hierarchical direct simulation Monte Carlo approach, essentially extending AMR ideas to microscopic computations. Grid levels now correspond to various levels of coarse graining. A number of algorithmic issues arise and are discussed. In continuum AMR finer grid values are typically obtained by interpolation from coarser grids. For microscopic computations this procedure is replaced by instantiation of specific statistical distributions of atomic velocities. Fixups from finer to coarser grid values must also be reinterpreted as an averaging procedure that eliminates thermal components of motion on fine grids to obtain quantities relevant to macroscopic variables. The microscopic simulation provides constitutive relations for the continuum simulation in regions where cracks are formed. The constitutive relations are updated dynamically during the course of computation.
The focus of the talk is on general guidelines for AMR codes that can be discerned from applications such as the crack propagation problem, both in terms of code structure and of numerical algorithms. A brief review of other physical problems in which the same type of issues arise will be presented.