Page 434 - Mechanics of Asphalt Microstructure and Micromechanics

426 C hapter T h ir te en

Currently, the simulated system size has increased significantly due to the fast increase
in computing power and reduction in computing costs. The linkage between classical
MD and ab initio QMD computations has become more clear and rigorous.
13.2.2.3 Multiscale Modeling
The explicit modeling of atomic/molecular structures through tracing the details of
atomic-scale processes, although having been enthusiastically pursued, has involved a
set of inherent spatial and temporal limitations. The spatial and temporal scale limita-
tions come from both small and large directions. Time scales of atoms are on the order
of femto-second and length scales on the order of angstroms. Typical atomistic simula-
tions therefore are limited to very small systems over very short times due to the very
small spatial and temporal scales. Although the computing power has grown signifi-
cantly, atomistic modeling methods are incapable of describing systems much larger
than billions of atoms (1μm geometric dimension) or longer than billions of femto-sec-
ond time steps (1 ms simulation period).
Realizing that continuum and atomistic modeling methods are complementary, the
idea of multiscale modeling came first to the mesoscopic modeling methods that are
developed to bridge this critical gap between the extremes of length scales, i.e., the
macroscale and atomic-scale. Continuum analyses tend to break down at the meso-
scale, at which atomistic methods start to reach their inherent spatial and temporal
limitations. There are two general solutions that have evolved so far to bridge the me-
soscopic gap. The first one is to study the dynamics of defect ensembles in the system,
instead of atomistically simulating the entire system. This approach has been success-
fully used in dynamic simulations of interacting cracks in brittle materials and disloca-
tions in crystalline materials. The second solution to the mesoscale problem is based on
statistical mechanics approaches, in which evolution equations for statistical averages
are solved for a complete description of the analyzed problem (Walgraef and Aifantis,
1985; H¨ahner et al., 1998; H¨ahner, 1996; Zaiser and H¨ahner, 1997; El-Azab, 2000).

13.2.2.4 Classification of Multiscale Problems
The very first step in multiscale modeling is to design an efficient multiscale method.
From this standpoint, multiscale problems can be divided into four general categories,
each sharing the special features that can be used when designing an efficient multi-
scale method for the category.
Type A: problems for which a macroscopic model is valid across a domain except for
localized region(s) in which the macroscopic model does not work spatially and/or
temporally and therefore a more accurate microscopic description is needed. Typical
type A problems include isolated defects or singularities such as cracks, dislocations,
shocks, and contact lines. For these problems, the microscopic model is necessary near
defects or singularities, whereas the continuum theory remains valid in the surround-
ing regions for establishing macroscopic model(s).
Type B: problems for which a closed macroscopic description is known to exist
across a domain, however, the macroscopic model may not be explicitly expressed or is
too expensive to obtain. Typical type B problems include transport of fluids and solutes
through heterogeneous porous media, complex fluids, and plasticity.
Type C: problems for which the macroscopic model is not explicitly known and does
not work correctly near defects. Type C problems have characteristics of both Type A
and Type B.