Multiscale modeling has become an essential tool in understanding and designing materials and physical systems with characteristics at multiple length and time scales. Although modern computational techniques are able to track the material behaviors from the nano-scale atomic vibrations at femtoseconds to the macroscopic plastic deformations of metals at seconds, simulations of physical phenomena of engineering interest are often limited by overwhelming computation time. The objective of multiscale methods is to predict the important physical behaviors without resolving the full details of the system, through averaging/coarse-graining the structure in length and/or extracting the slow time-scale dynamics. This paper reviews the state-of-the-art multiscale methods with applications in material science and biological systems.
Multiple Length and Time-scale Approaches in Materials Modeling, Advances in Materials. Special Issue: Advances in Multiscale Modeling Approach.
Vol. 6, No. 1-1,
2017, pp. 1-9.
Assyr Abdulle, E Weinan, Bjorn Engquist, and Eric Vanden-Eijnden. The heterogeneous multiscale method. Acta Numerica, 21:1-87, 2012.
Jacob Aboudi. The response of shape memory alloy composites. Smart materials and structures, 6(1):1, 1997.
A. Acharya. Parametrized invariant manifolds: a recipe for multiscale modeling? Computer Methods in Applied Mechanics and Engineering, 194(27-29):3067-3089, 2005.
Z. Artstein. Singularly perturbed ordinary differential equations with non-autonomous fast dynamics. Journal of Dynamics and Differential Equations, 11(2):297-318, 1999.
Z. Artstein. On singularly perturbed ordinary differential equations with measure-valued limits. Mathematica Bohemica, 127(2):139-152, 2002.
Z. Artstein, C. W. Gear, I. G. Kevrekidis, M. Slemrod, and E.S. Titi. Analysis and Computation of a discrete KDV-Burgers type equation with fast dispersion and slow diffusion. Arxiv preprint arXiv:0908.2752, 2009.
Z. Artstein, I.G. Kevrekidis, M. Slemrod, and E.S. Titi. Slow observables of singularly perturbed differential equations. Nonlinearity, 20:2463, 2007.
Z. Artstein and M. Slemrod. On singularly perturbed retarded functional differential equations. Journal of Differential Equations, 171(1):88-109, 2001.
Z. Artstein and A. Vigodner. Singularly perturbed ordinary differential equations with dynamic limits. In Proceedings of the Royal Society of Edinburgh-A-Mathematics, volume 126, pages 541-570. Cambridge Univ Press, 1996.
K. Bhattacharya. Phase boundary propagation in a heterogeneous body. Proceedings: Mathematical, Physical and Engineering Sciences, pages 757-766, 1999.
Yannick J Bomble and David A Case. Multiscale modeling of nucleic acids: insights into dna flexibility. Biopolymers, 89(9):722-731, 2008.
James G Boyd and Dimitris C Lagoudas. Thermomechanical response of shape memory composites. Journal of intelligent material systems and structures, 5(3):333-346, 1994.
J. Carr. Applications of centre manifold theory, volume 35. Springer, 1981.
Jack Carr. Applications of centre manifold theory, volume 35. Springer Science & Business Media, 2012.
Pedro Ponte Castaneda. Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I-theory. Journal of the Mechanics and Physics of Solids, 50(4):737-757, 2002.
Z Chen. Geometric nonlinearity and mechanical anisotropy in strained helical nanoribbons. Nanoscale, 6(16):9443-9447, 2014.
Jiahao Cheng and Somnath Ghosh. Computational modeling of plastic deformation and shear banding in bulk metallic glasses. Computational Materials Science, 69:494-504, 2013.
Jiahao Cheng and Somnath Ghosh. A crystal plasticity fe model for deformation with twin nucleation in magnesium alloys. International Journal of Plasticity, 67:148-170, 2015.
FT Fisher, RD Bradshaw, and LC Brinson. Effects of nanotube waviness on the modulus of nanotube-reinforced polymers. Applied Physics Letters, 80(24):4647-4649, 2002.
Yasubumi Furuya, Atsushi Sasaki, and Minoru Taya. Enhanced mechanical properties of tini shape memory fiber/al matrix composite. Materials transactions, JIM, 34(3):224-227, 1993.
John Guckenheimer and Philip J Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, volume 42. Springer Science & Business Media, 2013.
Qiaohang Guo, Zi Chen, Wei Li, Pinqiang Dai, Kun Ren, Junjie Lin, Larry A Taber, and Wenzhe Chen. Mechanics of tunable helices and geometric frustration in biomimetic seashells. EPL (Europhysics Letters), 105(6):64005, 2014.
I Hariton, EA Socolsky, et al. Neo-hookean fiber-reinforced composites in finite elasticity. Journal of the Mechanics and Physics of Solids, 54(3):533-559, 2006.
J. Harlim. Numerical strategies for filtering partially observed stiff stochastic differential equations. Journal of Computational Physics, 230(3):744-762, 2011.
J. Harlim and AJ Majda. Mathematical strategies for filtering complex systems: Regularly spaced sparse observations. Journal of Computational Physics, 227(10):5304-5341, 2008.
Zupan Hu, Wei Lu, and MD Thouless. Slip and wear at a corner with coulomb friction and an interfacial strength. Wear, 338:242-251, 2015.
Zupan Hu, Wei Lu, MD Thouless, and JR Barber. Simulation of wear evolution using fictitious eigenstrains. Tribology International, 82:191-194, 2015.
Zupan Hu, Wei Lu, MD Thouless, and JR Barber. Effect of plastic deformation on the evolution of wear and local stress fields in fretting. International Journal of Solids and Structures, 82:1-8, 2016.
Lu Huang, Chao Pu, Richard K Fisher, Deidra JH Mountain, Yanfei Gao, Peter K Liaw, Wei Zhang, and Wei He. A zr-based bulk metallic glass for future stent applications: Materials properties, finite element modeling, and in vitro human vascular cell response. Acta biomaterialia, 25:356-368, 2015.
Jay D Humphrey. Cardiovascular solid mechanics: cells, tissues, and organs. Springer Science & Business Media, 2013.
Ming Ji, Likun Tan, Linda Jen-Jacobson, and Sunil Saxena. Insights into copper coordination in the ecori-dna complex by esr spectroscopy. Molecular physics, 112(24):3173-3182, 2014.
E.L. Kang and J. Harlim. Filtering partially observed multiscale systems with heterogeneous multiscale methods-based reduced climate models. Monthly Weather Review, 140(3):860-873, 2012.
Dimitrios Karagiannis, Daniele Carnevale, and Alessandro Astolfi Invariant manifold based reduced-order observer design for nonlinear systems. IEEE Transactions on Automatic Control, 53(11):2602-2614, 2008.
Tomas Lindahl. Instability and decay of the primary structure of dna. nature, 362(6422):709-715, 1993.
Oscar Lopez-Pamies and Pedro Ponte Castaneda. Second-order estimates for the macroscopic response and loss of ellipticity in porous rubbers at large deformations. Journal of elasticity, 76(3):247-287, 2004.
FE Luborsky. Amorphous metallic alloys. Butterworth and Co (Publishers): London, UK, 1983.
P Kumar Mehta. Concrete. structure, properties and materials. 1986.
Graeme W Milton. The theory of composites. The Theory of Composites, by Graeme W. Milton, pp. 748. ISBN 0521781256. Cambridge, UK: Cambridge University Press, May 2002., page 748, 2002.
RG Muncaster. Invariant manifolds in mechanics I: The general construction of coarse theories from fine theories. Archive for Rational Mechanics and Analysis, 84(4):353-373, 1984.
F. Noe, I. Horenko, C. Schutte, and J.C. Smith. Hierarchical analysis of conformational dynamics in biomolecules: Transition networks of metastable states. The Journal of chemical physics, 126:155102, 2007.
F. Noe, C. Schutte, E. Vanden-Eijnden, L. Reich, and T.R. Weikl. Constructing the equilibrium ensemble of folding pathways from short off-equilibrium simulations. Proceedings of the National Academy of Sciences, 106(45):19011-19016, 2009.
Laurent Nottale. Scale relativity, fractal space-time and quantum mechanics. Chaos, Solitons & Fractals, 4(3):361- 388, 1994.
N Panasenko and NS Bakhvalov. Homogenization: Averaging processes in periodic media: Mathematical problems in the mechanics of composite materials, 1989.
G Papanicolau, A Bensoussan, and J-L Lions. Asymptotic analysis for periodic structures, volume 5. North Holland, 1978.
Kyoungsoo Park and Glaucio H Paulino. Cohesive zone models: a critical review of traction separation relationships across fracture surfaces. Applied Mechanics Reviews, 64(6):060802, 2011.
Treval Clifford Powers and Theodore Lucius Brownyard. Studies of the physical properties of hardened portland cement paste. In Journal Proceedings, volume 43, pages 101-132, 1946.
Chao Pu. Failure simulations at multiple length scales in high temperature structural alloys. PhD Thesis, 2015.
Chao Pu and Yanfei Gao. Crystal plasticity analysis of stress partitioning mechanisms and their microstructural dependence in advanced steels. Journal of Applied Mechanics, 82(3):031003, 2015.
Zhiwei Qian. Multiscale modeling of fracture processes in cementitious materials. TU Delft, Delft University of Technology, 2012.
W. Ren et al. Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. Journal of Computational Physics, 204(1):1-26, 2005.
Enrique Sanchez-Palencia. Non-homogeneous media and vibration theory. In Non-homogeneous media and vibration theory, volume 127, 1980.
Daniel V Santi, Anne Norment, and Charles E Garrett. Covalent bond formation between a dnacytosine methyltransferase and dna containing 5-azacytosine. Proceedings of the National Academy of Sciences, 81(22):6993-6997, 1984.
C. Schutte, F. Noe, J. Lu, M. Sarich, and E. Vanden-Eijnden. Markov state models based on milestoning. J. Chem. Phys., 134(20):204105, 2011.
Nuo Sheng, Mary C Boyce, David M Parks, GC Rutledge, JI Abes, and RE Cohen. Multiscale micromechanical modeling of polymer/clay nanocomposites and the effective clay particle. Polymer, 45(2):487-506, 2004.
Ellad B Tadmor, Michael Ortiz, and Rob Phillips. Quasicontinuum analysis of defects in solids. Philosophical magazine A, 73(6):1529-1563, 1996.
Li Kun Tan, Erik Schlangen, and Guang Ye. Simulation of failure in hydrating cement particles systems. In Key Engineering Materials, volume 348, pages 737-740. Trans Tech Publ, 2007.
Likun Tan, Amit Acharya, and Kaushik Dayal. Coarse variables of autonomous ode systems and their evolution. Computer Methods in Applied Mechanics and Engineering, 253:199-218, 2013.
Likun Tan, Amit Acharya, and Kaushik Dayal. Modeling of slow time-scale behavior of fast molecular dynamic systems. Journal of the Mechanics and Physics of Solids, 64:24-43, 2014.
A.N. Tikhonov. Systems of differential equations containing small parameters in the derivatives. Matematicheskii Sbornik, 73(3):575-586, 1952.
Ferdinand Verhulst and JA Sanders. Averaging methods in nonlinear dynamical systems. Applied Mathematical Sciences, 59, 1985.
E. Weinan, B. Engquist, and Z. Huang. Heterogeneous multiscale method: a general methodology for multiscale modeling. Physical Review B, 67(9):092101, 2003.
E Weinan, Bjorn Engquist, Xiantao Li, Weiqing Ren, and Eric Vanden-Eijnden. Heterogeneous multiscale methods: a review. Commun. Comput. Phys, 2(3):367-450, 2007.
Tao Xie. Tunable polymer multi-shape memory effect. Nature, 464(7286):267-270, 2010.
Zhongyu Yang, Michael R Kurpiewski, Ming Ji, Jacque E Townsend, Preeti Mehta, Linda Jen-Jacobson, and Sunil Saxena. Esr spectroscopy identifies inhibitory cu2+ sites in a dna-modifying enzyme to reveal determinants of catalytic specificity. Proceedings of the National Academy of Sciences, 109(17):E993-E1000, 2012.
Julien Yvonnet and Q-C He. The reduced model multiscale method (r3m) for the non-linear homogenization of hyperelastic media at finite strains. Journal of Computational Physics, 223(1):341-368, 2007.
J Zhang, J Johnston, and Aditi Chattopadhyay. Physics-based multiscale damage criterion for fatigue crack prediction in aluminium alloy. Fatigue & Fracture of Engineering Materials & Structures, 37(2):119-131, 2014.
Jinjun Zhang, Bonsung Koo, Yingtao Liu, Jin Zou, Aditi Chattopadhyay, and Lenore Dai. A novel statistical spring-bead based network model for self-sensing smart polymer materials. Smart Materials and Structures, 24(8):085022, 2015.
Jinjun Zhang, Bonsung Koo, Nithya Subramanian, Yingtao Liu, and Aditi Chattopadhyay. An optimized cross-linked network model to simulate the linear elastic material response of a smart polymer. Journal of Intelligent Material Systems and Structures, page 1045389X15595292, 2015.
Jinjun Zhang, Kuang Liu, Chuntao Luo, and Aditi Chattopadhyay. Crack initiation and fatigue life prediction on aluminum lug joints using statistical volume element-based multiscale modeling. Journal of Intelligent Material Systems and Structures, 24(17):2097-2109, 2013.
Ji, Ming, Sharon Ruthstein, and Sunil Saxena. "Paramagnetic metal ions in pulsed ESR distance distribution measurements." Accounts of chemical research 47.2 (2013): 688-695.
Yang, Zhongyu, Ming Ji, and Sunil Saxena. "Practical aspects of copper ion-based double electron electron resonance distance measurements." Applied Magnetic Resonance 39.4 (2010): 487-500.
Ruthstein, Sharon, Ming Ji, Preeti Mehta, Linda Jen-Jacobson, and Sunil Saxena. "Sensitive Cu2+–Cu2+ Distance Measurements in a Protein–DNA Complex by Double-Quantum Coherence ESR." The Journal of Physical Chemistry B 117, no. 20 (2013): 6227-6230.
Ruthstein, Sharon, Ming Ji, Byong-kyu Shin, and Sunil Saxena. "A simple double quantum coherence ESR sequence that minimizes nuclear modulations in Cu 2+-ion based distance measurements." Journal of Magnetic Resonance 257 (2015): 45-50.