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{{Short description|Mathematical field}}
{{Over-quotation|date=August 2019}}
[[File:Holec2016P40.svg|thumb|Modeling approaches and their scales]]
'''Multiscale modeling''' or '''multiscale mathematics''' is the [[Branches of science|field]] of solving problems
An example of such problems involve the [[Navier–Stokes equations]] for incompressible fluid flow.
▲'''Multiscale modeling''' or '''multiscale mathematics''' is the [[Branches of science|field]] of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids,<ref>{{Cite journal|last=Chen|first=Shiyi|last2=Doolen|first2=Gary D.|date=1998-01-01|title=Lattice Boltzmann Method for Fluid Flows|journal=Annual Review of Fluid Mechanics|volume=30|issue=1|pages=329–364|doi=10.1146/annurev.fluid.30.1.329|bibcode=1998AnRFM..30..329C}}</ref><ref name="Steinhauser 20082">{{cite book|title=Multiscale Modeling of Fluids and Solids - Theory and Applications|year=2017|isbn=978-3662532225|first1=M. O.|last1=Steinhauser}}</ref> solids,<ref name="Steinhauser 20082" /><ref>{{Cite journal|last=Oden|first=J. Tinsley|last2=Vemaganti|first2=Kumar|last3=Moës|first3=Nicolas|date=1999-04-16|title=Hierarchical modeling of heterogeneous solids|journal=Computer Methods in Applied Mechanics and Engineering|volume=172|issue=1|pages=3–25|doi=10.1016/S0045-7825(98)00224-2|bibcode=1999CMAME.172....3O}}</ref> polymers,<ref>{{Cite journal|last=Zeng|first=Q. H.|last2=Yu|first2=A. B.|last3=Lu|first3=G. Q.|date=2008-02-01|title=Multiscale modeling and simulation of polymer nanocomposites|journal=Progress in Polymer Science|volume=33|issue=2|pages=191–269|doi=10.1016/j.progpolymsci.2007.09.002}}</ref><ref name="Baeurle 20092">{{cite journal|year=2008|title=Multiscale modeling of polymer materials using field-theoretic methodologies: A survey about recent developments|journal=Journal of Mathematical Chemistry|volume=46|issue=2|pages=363–426|doi=10.1007/s10910-008-9467-3|last1=Baeurle|first1=S. A.}}</ref> proteins,<ref>{{Cite journal|last=Kmiecik|first=Sebastian|last2=Gront|first2=Dominik|last3=Kolinski|first3=Michal|last4=Wieteska|first4=Lukasz|last5=Dawid|first5=Aleksandra Elzbieta|last6=Kolinski|first6=Andrzej|date=2016-06-22|title=Coarse-Grained Protein Models and Their Applications|journal=Chemical Reviews|doi=10.1021/acs.chemrev.6b00163|issn=0009-2665|pmid=27333362|volume=116|issue=14|pages=7898–936|doi-access=free}}</ref><ref name=":0">{{Cite journal|last=Levitt|first=Michael|date=2014-09-15|title=Birth and Future of Multiscale Modeling for Macromolecular Systems (Nobel Lecture)|journal=Angewandte Chemie International Edition|language=en|volume=53|issue=38|pages=10006–10018|doi=10.1002/anie.201403691|issn=1521-3773|pmid=25100216}}</ref><ref name=":1" /><ref name=":2" /> [[nucleic acids]]<ref name="de Pablo 20112">{{cite journal|year=2011|title=Coarse-Grained Simulations of Macromolecules: From DNA to Nanocomposites|journal=Annual Review of Physical Chemistry|volume=62|pages=555–74|doi=10.1146/annurev-physchem-032210-103458|pmid=21219152|last1=De Pablo|first1=Juan J.|bibcode=2011ARPC...62..555D}}</ref> as well as various physical and chemical phenomena (like adsorption, chemical reactions, [[diffusion]]).<ref name=":1" /><ref name="Knizhnik2">{{cite journal|last2=Bagaturyants|first2=A.A.|last3=Belov|first3=I.V.|last4=Potapkin|first4=B.V.|last5=Korkin|first5=A.A.|year=2002|title=An integrated kinetic Monte Carlo molecular dynamics approach for film growth modeling and simulation: ZrO2 deposition on Si surface|journal=Computational Materials Science|volume=24|issue=1–2|pages=128–132|doi=10.1016/S0927-0256(02)00174-X|last1=Knizhnik|first1=A.A.}}</ref><ref name="Adams2">{{cite journal|last2=Astapenko|first2=V.|last3=Chernysheva|first3=I.|last4=Chorkov|first4=V.|last5=Deminsky|first5=M.|last6=Demchenko|first6=G.|last7=Demura|first7=A.|last8=Demyanov|first8=A.|last9=Dyatko|first9=N.|year=2007|title=Multiscale multiphysics nonempirical approach to calculation of light emission properties of chemically active nonequilibrium plasma: Application to Ar GaI3 system|journal=Journal of Physics D: Applied Physics|volume=40|issue=13|pages=3857–3881|bibcode=2007JPhD...40.3857A|doi=10.1088/0022-3727/40/13/S06|author1=Adamson|first1=S.|last10=Eletzkii|first10=A|last11=Knizhnik|first11=A|last12=Kochetov|first12=I|last13=Napartovich|first13=A|last14=Rykova|first14=E|last15=Sukhanov|first15=L|last16=Umanskii|first16=S|last17=Vetchinkin|first17=A|last18=Zaitsevskii|first18=A|last19=Potapkin|first19=B|display-authors=8}}</ref>
<math>\begin{array}{lcl} \rho_0(\partial_t\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u})=\nabla\cdot\tau,
\\ \nabla\cdot\mathbf{u}=0. \end{array}</math>
In a wide variety of applications, the stress tensor <math>\tau</math> is given as a linear function of the gradient <math>\nabla u</math>. Such a choice for <math>\tau</math> has been proven to be sufficient for describing the dynamics of a broad range of fluids. However, its 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.<ref>{{Cite book |last=E |first=Weinan |title=Principles of multiscale modeling |date=2011 |publisher=Cambridge University Press |isbn=978-1-107-09654-7 |___location=Cambridge |oclc=721888752}}</ref>
==History==
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{{Quote box
|quote = The recent surge of multiscale modeling from the smallest scale (atoms) to full system level (e.g., autos) related to solid mechanics that has now grown into an international multidisciplinary activity was birthed from an unlikely source. Since the US Department of Energy (DOE) national labs started to reduce nuclear underground tests in the mid-1980s, with the last one in 1992, the idea of simulation-based design and analysis concepts were birthed. Multiscale modeling was a key in garnering more precise and accurate predictive tools. In essence, the number of large
Essentially, the idea of filling the space of system
The advent of parallel computing also contributed to the development of multiscale modeling. Since more degrees of freedom could be resolved by parallel computing environments, more accurate and precise algorithmic formulations could be admitted. This thought also drove the political leaders to encourage the simulation-based design concepts.
At LANL, LLNL, and ORNL, the multiscale modeling efforts were driven from the materials science and physics communities with a bottom-up approach. Each had different programs that tried to unify computational efforts, materials science information, and applied mechanics algorithms with different levels of success. Multiple scientific articles were written, and the multiscale activities took different lives of their own. At SNL, the multiscale modeling effort was an engineering top-down approach starting from continuum mechanics perspective, which was already rich with a computational paradigm. SNL tried to merge the materials science community into the continuum mechanics community to address the lower
Once this management infrastructure and associated funding was in place at the various DOE institutions, different academic research projects started, initiating various satellite networks of multiscale modeling research. Technological transfer also arose into other labs within the Department of Defense and industrial research communities.
The growth of multiscale modeling in the industrial sector was primarily due to financial motivations. From the DOE national labs perspective, the shift from large
|author = [[Mark Horstemeyer]]
|source = ''Integrated Computational Materials Engineering (ICME) for Metals'', Chapter 1, Section 1.3.
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}}
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
==Areas of research==
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In [[operations research]], multiscale modeling addresses challenges for decision-makers that come from multiscale phenomena across organizational, temporal, and spatial scales. This theory fuses [[decision theory]] and multiscale mathematics and is referred to as [[multiscale decision-making]]. Multiscale decision-making draws upon the analogies between physical systems and complex man-made systems.{{citation needed|date=August 2019}}
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 ~ {{val|500|u=m}}) 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]] (GCM), with grid size ~ {{val|100|u=km}}, 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
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:
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*[[Network theory|Network-based modeling]]
*[[Statistical mechanics|Statistical modeling]]
==See also==
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* [[Equation-free modeling]]
* [[Integrated computational materials engineering]]
* [[Multilevel model]]
* [[Multiphysics]]
* [[Multiresolution analysis]]
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<references>
<ref name="Horstemeyer 2009">
{{cite book |first1=M. F. |last1=Horstemeyer |year=2009 |chapter=Multiscale Modeling: A Review |chapter-url=https://books.google.com/books?id=esOANcsz5w8C&pg=PA87 |pages=87–135 |editor1-first=Jerzy |editor1-last=Leszczyński |editor2-first=Manoj K. |editor2-last=Shukla |title=Practical Aspects of Computational Chemistry: Methods, Concepts and Applications |publisher=Springer |isbn=978-90-481-2687-3}}
</ref>
<ref name="Horstemeyer 2012">
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==Further reading==
*{{cite journal |pmid=19136256 |year=2009 |last1=Hosseini |first1=SA |last2=Shah |first2=N |title=Multiscale modelling of hydrothermal biomass pretreatment for chip size optimization |volume=100 |issue=9 |pages=2621–8 |doi=10.1016/j.biortech.2008.11.030 |journal=Bioresource Technology|bibcode=2009BiTec.100.2621H }}
*{{cite journal |bibcode=2009BAMS...90..515T |title=A Multiscale Modeling System: Developments, Applications, and Critical Issues |
==External links==
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* [https://web.archive.org/web/20080220104618/http://www.mmm2008.org/bin/view.pl/Main/WebHome Multiscale Materials Modeling: Fourth International Conference, Tallahassee, FL, USA]
* [http://www.biocomp.chem.uw.edu.pl/multiscale_modeling.php Multiscale Modeling Tools for Protein Structure Prediction and Protein Folding Simulations, Warsaw, Poland]
* [http://www.e-xstream.com/applications/material-engineering/about-material-engineering Multiscale modeling for
* [https://web.archive.org/web/20190812193431/http://multiscale-modelling.eu/ Multiscale Material Modelling on High Performance Computer Architectures, MMM@HPC project]
* [http://www.modelingmaterials.org ''Modeling Materials: Continuum, Atomistic and Multiscale Techniques'' (E. B. Tadmor and R. E. Miller, Cambridge University Press, 2011)]
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