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[[File:Holec2016P40.svg|thumb|Modeling approaches and their scales]]
'''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>
An example of such problems involve the [[Navier–Stokes equations|Navier-Stokes equations]] for incompressible fluid flow.
<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, 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.<ref>{{Cite book |last=E |first=Weinan |url=https://www.worldcat.org/oclc/721888752 |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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