Weinan E. Principles of Multiscale Modeling
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Chapter 7
Resolving Local Events or
Singularities
We now turn to type A problems. These are problems for which a m acroscopic model
is sufficiently accurate except in isolated regions where some important events take place
and a microscale model is needed to resolve the details of these local events. Examples
of such events include chemical reactions, interfaces, shocks, b oun dary layers, cracks,
contact lines, triple junctions, and dislocations. Conventional macroscale models fail
to resolve the important details of these events. Yet other than these local events, a
full microscopic description is unnecessary. Therefore, a coupled macro-micro modeling
approach is preferred in order to capture both the macroscale behavior of the system and
the important details of such local events or singularities.
One earliest and most well-known example of this type of approach is the coupled
quantum mechanics and molecular mechanics (QM-MM) method proposed by Warshel
and Levitt during the 1970’s for modeling chemical reactions of biomolecules [51] (see also
[50]). In this approach, quantum mechanics models are used in regions where the chemical
reaction takes places, and classical molecular mechanics models are used elsewhere. Such
a combined QM-MM approach has now become a standard tool in chemistry and biology.
This kind of philosophy has been applied to a variety of problems in different disciplines.
In the general setting, a locally coupled macro-micro mo d eling strategy involves the
following major components:
1. A coupling strategy. As we discuss below, there are several choices: The domain
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316 CHAPTER 7. RESOLVING LOCAL EVENTS OR SINGULARITIES
decomposition method, adaptive model refinement or adaptive model reduction,
and the heterogeneous multiscale method (HMM).
2. The data, usually in the form of boundary conditions, to be passed between the
macro and micro models.
3. The implementation of the boundary conditions for the macro and micro models.
The second and third components constitute the macro-micro interface conditions. This
is the main issue for type A problems, namely, how should we formulate the interface
conditions such that the overall macro-micro algorithm is stable and accurate? Of par-
ticular interest is the consistency b etween the different models at the interface where the
two models meet.
Since the nature of the different models can be quite different, linking them together
consistently can be a very non-trivial task. For example, quantum mechanics models
describe explicitly the charge density due to the electrons. This information is lost in
classical models. Consequently charge balance is a very serious issue in coupled QM-
MM formulations. As another example, temperature is treated as a continuous field in
continuum mechanics. However, in molecular dynamics, temperature is a measure of the
magnitude of the velocity fluctuations of the atoms. These are drastically different repre-
sentations of the same physical quantity. For this reason, coupling continuum mechanics
and molecular dynamics at finite temperature is a highly non-trivial task.
In what follows, we will first discuss general coupling strategies. We will then discuss
how to assess the performance of these coupling strategies in terms of stability and
consistency, two most important notions in classical numerical analysis. As expected,
these classical notions take on a new dimension in the current context, due to the different
nature of the physical models involved.
7.1 Domain decomposition method
This is a rather popular strategy. The QM-MM method, for example, uses such a
strategy. The basic idea is very similar to that of the standard domain decomposition
method (see Chapter 3), except that different mo d els are used on different domains.
The places wher e interesting local events occur are covered by domains on which the
microscale models are used. The size of the domain is chosen based on considerations of