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Multiscale modeling or multiscale mathematics is the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids,
solids,
polymers,
proteins,
nucleic acids
Nucleic acids are biopolymers, macromolecules, essential to all known forms of life. They are composed of nucleotides, which are the monomers made of three components: a 5-carbon sugar, a phosphate group and a nitrogenous base. The two main clas ...
as well as various physical and chemical phenomena (like adsorption, chemical reactions,
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
).
An example of such problems involve the
Navier-Stokes equations for incompressible fluid flow.
In a wide-variety of applications, the stress tensor
is given as a linear function of the gradient
. Such a choice for
has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, it’s use for more complex fluids such as polymers is dubious. In such a case, it may be necessary to use multiscale modeling to accurately model the system such that the stress tensor can be extracted without requiring the computational cost of a full microscale simulation.
History
Horstemeyer 2009,
2012
presented a historical review of the different disciplines (mathematics, physics, and materials science) for solid materials related to multiscale materials modeling.
The aforementioned DOE multiscale modeling efforts were hierarchical in nature. The first concurrent multiscale model occurred when Michael Ortiz (Caltech) took the molecular dynamics code, Dynamo, (developed by
Mike Baskes at Sandia National Labs) and with his students embedded it into a finite element code for the first time.
Martin Karplus
Martin Karplus (born March 15, 1930) is an Austrian and American theoretical chemist. He is the Director of the Biophysical Chemistry Laboratory, a joint laboratory between the French National Center for Scientific Research and the University of St ...
,
Michael Levitt
Michael Levitt, ( he, מיכאל לויט; born 9 May 1947) is a South African-born biophysicist and a professor of structural biology at Stanford University, a position he has held since 1987. Levitt received the 2013 Nobel Prize in Chemistr ...
,
Arieh Warshel
Arieh Warshel ( he, אריה ורשל; born November 20, 1940) is an Israeli-American biochemist and biophysicist. He is a pioneer in computational studies on functional properties of biological molecules, Distinguished Professor of Chemistry ...
2013 were awarded a Nobel Prize in Chemistry for the development of a multiscale model method using both classical and quantum mechanical theory which were used to model large complex chemical systems and reactions.
Areas of research
In physics and chemistry, multiscale modeling is aimed at the calculation of material properties or system behavior on one level using information or models from different levels. On each level, particular approaches are used for the description of a system. The following levels are usually distinguished: level of
quantum mechanical model
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
s (information about electrons is included), level of
molecular dynamics
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
models (information about individual atoms is included),
coarse-grained models (information about atoms and/or groups of atoms is included), mesoscale or nano-level (information about large groups of atoms and/or molecule positions is included), level of continuum models, level of device models. Each level addresses a phenomenon over a specific window of length and time. Multiscale modeling is particularly important in
integrated computational materials engineering
Integrated Computational Materials Engineering (ICME) is an approach to design products, the materials that comprise them, and their associated materials processing methods by linking materials models at multiple length scales. Key words are "Inte ...
since it allows the prediction of material properties or system behavior based on knowledge of the process-structure-property relationships.
In
operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
, multiscale modeling addresses challenges for decision-makers that come from multiscale phenomena across organizational, temporal, and spatial scales. This theory fuses
decision theory
Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
and multiscale mathematics and is referred to as
multiscale decision-making Multiscale decision-making, also referred to as multiscale decision theory (MSDT), is an approach in operations research that combines game theory, multi-agent influence diagrams, in particular dependency graphs, and Markov decision processes to ...
. Multiscale decision-making draws upon the analogies between physical systems and complex man-made systems.
In meteorology, multiscale modeling is the modeling of the interaction between weather systems of different spatial and temporal scales that produces the weather that we experience. The most challenging task is to model the way through which the weather systems interact as models cannot see beyond the limit of the model grid size. In other words, to run an atmospheric model that is having a grid size (very small ~ ) which can see each possible cloud structure for the whole globe is computationally very expensive. On the other hand, a computationally feasible
Global climate model
A general circulation model (GCM) is a type of climate model. It employs a mathematical model of the general circulation of a planetary atmosphere or ocean. It uses the Navier–Stokes equations on a rotating sphere with thermodynamic terms f ...
(GCM), with grid size ~ , cannot see the smaller cloud systems. So we need to come to a balance point so that the model becomes computationally feasible and at the same time we do not lose much information, with the help of making some rational guesses, a process called Parametrization.
Besides the many specific applications, one area of research is methods for the accurate and efficient solution of multiscale modeling problems. The primary areas of mathematical and algorithmic development include:
*
Analytical modeling
*
Center manifold In the mathematics of evolving systems, the concept of a center manifold was originally developed to determine stability of degenerate equilibria. Subsequently, the concept of center manifolds was realised to be fundamental to mathematical modellin ...
and
slow manifold In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of ...
theory
*
Continuum modeling
*
Discrete modeling Discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, formulae are fit to discrete data—data that could potentially take on only a countable set of values, such as the integers, and which are not infinitely d ...
*
Network-based modeling
*
Statistical modeling
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...
See also
*
Computational mechanics
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science (also called scientific computing) as a "third w ...
*
Equation-free modeling Equation-free modeling is a method for multiscale computation and computer-aided analysis. It is designed for a class of complicated systems in which one observes evolution at a macroscopic, coarse scale of interest, while accurate models are only ...
*
Integrated computational materials engineering
Integrated Computational Materials Engineering (ICME) is an approach to design products, the materials that comprise them, and their associated materials processing methods by linking materials models at multiple length scales. Key words are "Inte ...
*
Multiphysics In computational modelling, multiphysics simulation (often shortened to simply "multiphysics") is defined as the simultaneous simulation of different aspects of a physical system or systems and the interactions among them. For example, simultaneous ...
*
Multiresolution analysis
A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introd ...
*
Space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The know ...
References
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Further reading
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External links
{{external links cleanup, date=September 2020
Mississippi State University ICME CyberinfrastructureMultiscale Modeling of Flow Flow* [https://web.archive.org/web/20080503073240/http://www-dick.chemie.uni-regensburg.de/group/stephan_baeurle/index.html Multiscale Modeling Group: Institute of Physical & Theoretical Chemistry, University of Regensburg, Regensburg, Germany]
Multiscale Materials Modeling: Fourth International Conference, Tallahassee, FL, USAMultiscale Modeling Tools for Protein Structure Prediction and Protein Folding Simulations, Warsaw, PolandMultiscale modeling for Materials Engineering: Set-up of quantitative micromechanical modelsMultiscale Material Modelling on High Performance Computer Architectures, MMM@HPC project''Modeling Materials: Continuum, Atomistic and Multiscale Techniques'' (E. B. Tadmor and R. E. Miller, Cambridge University Press, 2011)An Introduction to Computational Multiphysics II: Theoretical Background Part I Harvard University video seriesSIAM Journal of Multiscale Modeling and Simulation*
ttps://web.archive.org/web/20150123160454/http://multiscale.emsl.pnl.gov/ Department of Energy Summer School on Multiscale Mathematics and High Performance ComputingMultiscale Conceptual Model Figures for Biological and Environmental Science
Computational physics
Mathematical modeling