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Multiscale modeling
Mathematical field

Multiscale modeling or multiscale mathematics is the field of solving problems that have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic acids as well as various physical and chemical phenomena (like adsorption, chemical reactions, diffusion).

An example of such problems involve the Navier–Stokes equations for incompressible fluid flow.

ρ 0 ( ∂ t u + ( u ⋅ ∇ ) u ) = ∇ ⋅ τ , ∇ ⋅ u = 0. {\displaystyle {\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}}}

In a wide variety of applications, the stress tensor τ {\displaystyle \tau } is given as a linear function of the gradient ∇ u {\displaystyle \nabla u} . Such a choice for τ {\displaystyle \tau } 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.

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History

Horstemeyer 2009,18 201219 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.20 Martin Karplus, Michael Levitt, and Arieh Warshel received the Nobel Prize in Chemistry in 2013 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.212223

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 models (information about electrons is included), level of molecular dynamics 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 since it allows the prediction of material properties or system behavior based on knowledge of the process-structure-property relationships.

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.

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 ~ 500 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 ~ 100 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 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:

See also

Further reading

  • Hosseini, SA; Shah, N (2009). "Multiscale modelling of hydrothermal biomass pretreatment for chip size optimization". Bioresource Technology. 100 (9): 2621–8. doi:10.1016/j.biortech.2008.11.030. PMID 19136256.
  • Tao, Wei-Kuo; Chern, Jiun-Dar; Atlas, Robert; Randall, David; Khairoutdinov, Marat; Li, Jui-Lin; Waliser, Duane E.; Hou, Arthur; et al. (2009). "A Multiscale Modeling System: Developments, Applications, and Critical Issues". Bulletin of the American Meteorological Society. 90 (4): 515–534. Bibcode:2009BAMS...90..515T. doi:10.1175/2008BAMS2542.1. hdl:2060/20080039624.

References

  1. Chen, Shiyi; Doolen, Gary D. (1998-01-01). "Lattice Boltzmann Method for Fluid Flows". Annual Review of Fluid Mechanics. 30 (1): 329–364. Bibcode:1998AnRFM..30..329C. doi:10.1146/annurev.fluid.30.1.329. /wiki/Bibcode_(identifier) ↩

  2. Steinhauser, M. O. (2017). Multiscale Modeling of Fluids and Solids - Theory and Applications. ISBN 978-3662532225. 978-3662532225 ↩

  3. Martins, Ernane de Freitas; da Silva, Gabriela Dias; Salvador, Michele Aparecida; Baptista, Alvaro David Torrez; de Almeida, James Moraes; Miranda, Caetano Rodrigues (2019-10-28). "Uncovering the Mechanisms of Low-Salinity Water Injection EOR Processes: A Molecular Simulation Viewpoint". OTC-29885-MS. OTC. doi:10.4043/29885-MS. https://onepetro.org/OTCBRASIL/proceedings/19OTCB/3-19OTCB/Rio%20de%20Janeiro,%20Brazil/180751 ↩

  4. Steinhauser, M. O. (2017). Multiscale Modeling of Fluids and Solids - Theory and Applications. ISBN 978-3662532225. 978-3662532225 ↩

  5. Oden, J. Tinsley; Vemaganti, Kumar; Moës, Nicolas (1999-04-16). "Hierarchical modeling of heterogeneous solids". Computer Methods in Applied Mechanics and Engineering. 172 (1): 3–25. Bibcode:1999CMAME.172....3O. doi:10.1016/S0045-7825(98)00224-2. /wiki/Bibcode_(identifier) ↩

  6. Zeng, Q. H.; Yu, A. B.; Lu, G. Q. (2008-02-01). "Multiscale modeling and simulation of polymer nanocomposites". Progress in Polymer Science. 33 (2): 191–269. doi:10.1016/j.progpolymsci.2007.09.002. /wiki/Doi_(identifier) ↩

  7. Baeurle, S. A. (2008). "Multiscale modeling of polymer materials using field-theoretic methodologies: A survey about recent developments". Journal of Mathematical Chemistry. 46 (2): 363–426. doi:10.1007/s10910-008-9467-3. S2CID 117867762. /wiki/Doi_(identifier) ↩

  8. Kmiecik, Sebastian; Gront, Dominik; Kolinski, Michal; Wieteska, Lukasz; Dawid, Aleksandra Elzbieta; Kolinski, Andrzej (2016-06-22). "Coarse-Grained Protein Models and Their Applications". Chemical Reviews. 116 (14): 7898–936. doi:10.1021/acs.chemrev.6b00163. ISSN 0009-2665. PMID 27333362. https://doi.org/10.1021%2Facs.chemrev.6b00163 ↩

  9. Levitt, Michael (2014-09-15). "Birth and Future of Multiscale Modeling for Macromolecular Systems (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 10006–10018. doi:10.1002/anie.201403691. ISSN 1521-3773. PMID 25100216. /wiki/Doi_(identifier) ↩

  10. Karplus, Martin (2014-09-15). "Development of Multiscale Models for Complex Chemical Systems: From H+H2 to Biomolecules (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 9992–10005. doi:10.1002/anie.201403924. ISSN 1521-3773. PMID 25066036. /wiki/Doi_(identifier) ↩

  11. Warshel, Arieh (2014-09-15). "Multiscale Modeling of Biological Functions: From Enzymes to Molecular Machines (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 10020–10031. doi:10.1002/anie.201403689. ISSN 1521-3773. PMC 4948593. PMID 25060243. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4948593 ↩

  12. De Pablo, Juan J. (2011). "Coarse-Grained Simulations of Macromolecules: From DNA to Nanocomposites". Annual Review of Physical Chemistry. 62: 555–74. Bibcode:2011ARPC...62..555D. doi:10.1146/annurev-physchem-032210-103458. PMID 21219152. /wiki/Bibcode_(identifier) ↩

  13. Karplus, Martin (2014-09-15). "Development of Multiscale Models for Complex Chemical Systems: From H+H2 to Biomolecules (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 9992–10005. doi:10.1002/anie.201403924. ISSN 1521-3773. PMID 25066036. /wiki/Doi_(identifier) ↩

  14. Knizhnik, A.A.; Bagaturyants, A.A.; Belov, I.V.; Potapkin, B.V.; Korkin, A.A. (2002). "An integrated kinetic Monte Carlo molecular dynamics approach for film growth modeling and simulation: ZrO2 deposition on Si surface". Computational Materials Science. 24 (1–2): 128–132. doi:10.1016/S0927-0256(02)00174-X. /wiki/Doi_(identifier) ↩

  15. Adamson, S.; Astapenko, V.; Chernysheva, I.; Chorkov, V.; Deminsky, M.; Demchenko, G.; Demura, A.; Demyanov, A.; et al. (2007). "Multiscale multiphysics nonempirical approach to calculation of light emission properties of chemically active nonequilibrium plasma: Application to Ar GaI3 system". Journal of Physics D: Applied Physics. 40 (13): 3857–3881. Bibcode:2007JPhD...40.3857A. doi:10.1088/0022-3727/40/13/S06. S2CID 97819264. /wiki/Bibcode_(identifier) ↩

  16. da Silva, Gabriela Dias; de Freitas Martins, Ernane; Salvador, Michele Aparecida; Baptista, Alvaro David Torrez; de Almeida, James Moraes; Miranda, Caetano Rodrigues (2019). "From Atoms to Pre-salt Reservoirs: Multiscale Simulations of the Low-Salinity Enhanced Oil Recovery Mechanisms". Polytechnica. 2 (1–2): 30–50. doi:10.1007/s41050-019-00014-1. ISSN 2520-8497. http://link.springer.com/10.1007/s41050-019-00014-1 ↩

  17. E, Weinan (2011). Principles of multiscale modeling. Cambridge: Cambridge University Press. ISBN 978-1-107-09654-7. OCLC 721888752. 978-1-107-09654-7 ↩

  18. Horstemeyer, M. F. (2009). "Multiscale Modeling: A Review". In Leszczyński, Jerzy; Shukla, Manoj K. (eds.). Practical Aspects of Computational Chemistry: Methods, Concepts and Applications. pp. 87–135. ISBN 978-90-481-2687-3. 978-90-481-2687-3 ↩

  19. Horstemeyer, M. F. (2012). Integrated Computational Materials Engineering (ICME) for Metals. ISBN 978-1-118-02252-8. 978-1-118-02252-8 ↩

  20. Tadmore, E.B.; Ortiz, M.; Phillips, R. (1996-09-27). "Quasicontinuum Analysis of Defects in Solids". Philosophical Magazine A. 73 (6): 1529–1563. Bibcode:1996PMagA..73.1529T. doi:10.1080/01418619608243000. /wiki/Bibcode_(identifier) ↩

  21. Levitt, Michael (2014-09-15). "Birth and Future of Multiscale Modeling for Macromolecular Systems (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 10006–10018. doi:10.1002/anie.201403691. ISSN 1521-3773. PMID 25100216. /wiki/Doi_(identifier) ↩

  22. Karplus, Martin (2014-09-15). "Development of Multiscale Models for Complex Chemical Systems: From H+H2 to Biomolecules (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 9992–10005. doi:10.1002/anie.201403924. ISSN 1521-3773. PMID 25066036. /wiki/Doi_(identifier) ↩

  23. Warshel, Arieh (2014-09-15). "Multiscale Modeling of Biological Functions: From Enzymes to Molecular Machines (Nobel Lecture)". Angewandte Chemie International Edition. 53 (38): 10020–10031. doi:10.1002/anie.201403689. ISSN 1521-3773. PMC 4948593. PMID 25060243. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4948593 ↩