Digest of [@E2011].


We don't need to know every thing at every scales to get what we need.

As a result, engineers are able to design structures and bridges without acquiring much understanding about the origins of the cohesion between the atoms in the material.

Making problems complex, or making your tools complex is not helpful. This comment also applies to engineering fields, with Shannon in mind.

Indeed the hallmark of deep physical insight has been the ability to describe complex phenomena using simple ideas.

But how do you know a system under inquiry is complex though?

Overall, empirical approaches have had limited success for complex systems or for small-scale systems in which the discrete or finite-size effect are important.

Two motivations for multiscale modeling

After all, there have been considerable efforts to try to understand the relations between microscopic and macroscopic models.

There have also been several classical success stories of combining physical models at different levels of detail for the efficient and accurate modeling of complex processes of interest.

Prerequisite of a successful multiscale modeling

Many fundamental issues have to be addressed before its [multiscale modeling] expected impact becomes a reality. These issues include: 1. a detailed understanding of the relation between the different level of physical models; 2. boundary conditions for atomistic models such as molecular dynamics; 3. systematic and accurate coarse-graining procedures.


1 Introduction

Examples of multiscale problems

  1. multiscale data
  2. differential equations with multiscale data
    • wave packets
    • composite materials
  3. differential equations with small parameters (singular perturbation)

The necessity of justification before adopting multiscale modeling:

Therefore, broadly speaking, it is not incorrect to say that multiscale modeling encompasses almost every aspect of modeling. However, adopting such a position would make it impossible to cary out serious discussion in any kind of depth.

Multi-physics problems

Scale-dependent phenomena: Some phenomena are scale-dependent, i.e. their behavior changes as the scales involved change, which is caused by the change of dominating physical effects.

Traditional approaches to modeling:

In traditional approaches to modeling we tend to focus on one particular scale: the effects of smaller scales are modeled through the constitutive relation; the effects of larger scales are neglected by assuming that the system is homogeneous at theses scales.

Constitutive relations

The essence of constitutive relations:

Central to any kind of coarse-grained model is a constitutive relation, which represents the effect of the microscopic processes at the macroscopic level.

Elements of constitutive relations:

In engineering applications the constitutive relations are often empirically based on very simple considerations such as: the second law of thermodynamics; symmetry and invariance properties; linearization and Taylor expansion.

For more information on constitutive relations, see section 4.1.

Multiscale modeling

Motivation for multiscale modeling: Some constitutive relations turns out to be simple and universal, but sometimes empirical models become inadequate. At other times, we are interested in some crucial micro-structural information.

Microscopic models may be more accurate, but too expensive for problems of real interest.

The philosophy of multiscale, multi-physics modeling: Information from and models of a finer scale can provide details for a coarse scale model; in the same way, micro-structural information at some crucial regions can also be obtained.

  1. Any system of interest can always be described by a hierarchy of models of different complexity. This allows us to think about more detailed models when a coarse-grained model is no longer adequate. It also gives us a basis for understanding coarse-grained models from more detailed ones. In particular, when empirical coarse-grained models are inadequate, one might still be able to capture the macro scale behavior of the system with the help of micro scale models.
  2. In many situations the system of interest can be described adequately by a coarse-grained model except in some small regions, where more detailed models are needed. These small regions may contain singularities, defects, chemical reactions or some other interesting events. In such cases, by coupling models of different complexity[^2] in different regions we may be able to develop modeling strategies[^3] that have an efficiency comparable with coarse-graining as well as an accuracy comparable with that of the more detailed models.

[^2]: In this case, complexity seems to be detailedness.

[^3]: Does multiscale modeling stop at modeling? By the way, does modeling here mean mathematical models, or merely computational simulation?


4 The hierarchy of physical models

Physical models at different scales, exemplified by gas dynamics

Physical model Governing equation Quantity of interest
continuum mechanics Euler and Navier-Stokes equations macroscopic density, velocity and temperature field of the gas
kinetic theory Boltzmann equation phase-space probability distribution of gas particles
molecular theory Newton's equation phase-space dynamics of gas particles
quantum mechanics Schrodinger equation collision process between gas particles

Continuum mechanics

Examples of constitutive relations:

  • the equation of state for gases
  • the stored energy density function for solids
  • the stress-strain relation for solids
  • the stress as a function of the rate of strain for solids

Construction of constitutive equations in continuum mechanics, with the stress-strain relation in elasticity in mind.

In fact, in many cases, constitutive modeling proceeds by {Groot and Mazur, Non-Equilibrium Thermodynamics, 1984}:

  1. writing down linear relations between the generalized forces and the generalized fluxes
  2. using symmetry properties to reduce the number of independent coefficients in the linear relations

Carefully thought, the Newton's equation is also a constitutive equation, with mass being the only experimental coefficient.

Stress and strain in solids

Variational principles in elasticity theory

Conservation laws