Discrete-to-continuum scale bridging

Abstract

The book chapter describes several information-passing and concurrent discrete-to-continuum scale bridging approaches. In the concurrent approach both, the discrete and continuum scales are simultaneously resolved, whereas in the information-passing schemes, the discrete scale is modelled and its gross response is infused into the continuum scale. Most of the information-passing approaches provide sublinear computational complexity, (i.e., scales sublinearly with the cost of solving a fine scale problem). Among the information-passing bridging techniques, we present the Generalized Mathematical Homogenization (GMH) theory, which constructs an equivalent continuum description directly from molecular dynamics (MD) equations; the Multiscale Enrichment based on Partition of Unity (MEPU) method, which gives rise to the enriched coarse grained formulation, the Heterogeneous Multiscale Method (HMM), which provides equivalent coarse scale integrands; the Variational Multiscale Method (VMS), which can be viewed as an equivalent coarse scale element builder; the Coarse-Grained Molecular Dynamics (CGMD), which derives effective Hamiltonian for the coarse-grained problem; the Discontinuous Galerkin Method, which constructs discontinuous enrichment; the Equation-Free Method (EFM), which makes no assumption on the response of the coarse scale problem; and the Kinetic Monte Carlo (KMC)-based methods, which bridge diverse time scales by calibrating certain KMC parameters from molecular dynamics or quantum mechanics calculations. The second part of the book chapter focuses on multiscale systems, whose response depends inherently on physics at multiple scales, such as turbulence, crack propagation, friction, and problems involving nano-like devices. For these types of problems, multiple scales have to be simultaneously resolved in different portions of the problem domain. Among the Concurrent Multiscale techniques, we describe Domain Bridging (DBCM), Local Enrichment (DBCM) and Multigrid (MGCM) based concurrent multiscale methods. A space-time variant of the MGCM for bridging discrete scales with either coarse grained discrete or continuum scales is presented. The method consists of the wave-form relaxation scheme aimed at capturing the high frequency response of the atomistic vibrations and the coarse scale space-time solution (explicit or implicit) intended to resolve the coarse scale features of the discrete medium.