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COUPLE MICROSCALE PERIODIC PATCHES TO SIMULATE MACROSCALE EMERGENT DYNAMICS

Part of: Mathematical economics Approximation methods and numerical treatment of dynamical systems Basic methods in fluid mechanics Numerical problems in dynamical systems

Published online by Cambridge University Press:  30 January 2018

HAMMAD ALOTAIBI Open the ORCID record for HAMMAD ALOTAIBI [Opens in a new window] ,
BARRY COX Open the ORCID record for BARRY COX [Opens in a new window]  and
A. J. ROBERTS Open the ORCID record for A. J. ROBERTS [Opens in a new window]
HAMMAD ALOTAIBI
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email [email protected], [email protected], [email protected]
BARRY COX
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email [email protected], [email protected], [email protected]
A. J. ROBERTS*
Affiliation:
School of Mathematical Sciences, University of Adelaide, South Australia, Australia email [email protected], [email protected], [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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Macroscale “continuum” level predictions are made by a new way to construct computationally efficient “wrappers” around fine-scale, microscopic, detailed descriptions of dynamical systems, such as molecular dynamics. It is often significantly easier to code a microscale simulator with periodicity: so the challenge addressed here is to develop a scheme that uses only a given periodic microscale simulator; specifically, one for atomistic dynamics. Numerical simulations show that applying a suitable proportional controller within “action regions” of a patch of atomistic simulation effectively predicts the macroscale transport of heat. Theoretical analysis establishes that such an approach will generally be effective and efficient, and also determines good values for the strength of the proportional controller. This work has the potential to empower systematic analysis and understanding at a macroscopic system level when only a given microscale simulator is available.

Keywords

macroscale dynamicsemergenceperiodic microscalepatch schemeequation-free analysis

MSC classification

Primary: 37M05: Simulation
Secondary: 65P20: Numerical chaos 76M28: Particle methods and lattice-gas methods 91B69: Heterogeneous agent models
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 3 , January 2018 , pp. 313 - 334
Copyright
© 2018 Australian Mathematical Society 

References

Alotaibi, H., “Developing multiscale methodologies for computational fluid mechanics”, Ph. D. Thesis, School of Mathematical Sciences, University of Adelaide, 2017.Google Scholar
Aulbach, B. and Wanner, T., “The Hartman–Grobman theorem for Caratheodory-type differential equations in Banach spaces”, Nonlinear Anal. 40 (2000) 91–104; doi:10.1016/S0362-546X(00)85006-3.CrossRefGoogle Scholar
Bechhoefer, J., “Feedback for physicists: a tutorial essay on control”, Rev. Modern Phys. 77 (2005) 783–836; doi:10.1103/RevModPhys.77.783.CrossRefGoogle Scholar
Boxler, P., “A stochastic version of the centre manifold theorem”, Probab. Theory Related Fields 83 (1989) 509–545; doi:10.1007/BF01845701.CrossRefGoogle Scholar
Bunder, J. E. and Roberts, A. J., “Patch dynamics for macroscale modelling in one dimension”, in: Proc. 10th Biennial Engineering Mathematics and Applications Conf., EMAC-2011 (eds Nelson, M. et al. ), ANZIAM J. 53 (2012) C280–C295; doi:10.21914/anziamj.v53i0.5074.Google Scholar
Bunder, J. E. and Roberts, A. J., “Resolution of subgrid microscale interactions enhances the discretisation of nonautonomous partial differential equations”, Appl. Math. Comput. 304 (2017) 164–179; doi:10.1016/j.amc.2017.01.056.Google Scholar
Bunder, J. E., Roberts, A. J. and Kevrekidis, I. G., “Good coupling for the multiscale patch scheme on systems with microscale heterogeneity”, J. Comput. Phys. (2017) (in press); doi:10.1016/j.jcp.2017.02.004.CrossRefGoogle Scholar
Cao, M. and Roberts, A. J., “Multiscale modelling couples patches of wave-like simulations”, in: Proc. 16th Biennial Computational Techniques and Applications Conf., CTAC-2012 (eds McCue, S. et al. ), ANZIAM J. 54 (2013) C153–C170; doi:10.21914/anziamj.v54i0.6137.Google Scholar
Cao, M. and Roberts, A. J., “Multiscale modelling couples patches of nonlinear wave-like simulations”, IMA J. Appl. Math. 81 (2016) 228–254; doi:10.1093/imamat/hxv034.CrossRefGoogle Scholar
Carr, E. J., Perré, P. and Turner, I. W., “The extended distributed microstructure model for gradient-driven transport: a two-scale model for bypassing effective parameters”, J. Comput. Phys. 327 (2016) 810–829; doi:10.1016/j.jcp.2016.10.004.CrossRefGoogle Scholar
Cheng, H., Greengard, L. and Rokhlin, V., “A fast adaptive multipole algorithm in three dimensions”, J. Comput. Phys. 155 (1999) 468–498; doi:10.1006/jcph.1999.6355.CrossRefGoogle Scholar
Dove, M. T., “An introduction to atomistic simulation methods”, Sem. SEM 4 (2008) 7–37; http://www.ehu.eus/sem/seminario_pdf/SEM_SEM_4_7-37.pdf.Google Scholar
Evans, D. J. and Hoover, W. G., “Flows far from equilibrium via molecular dynamics”, Annu. Rev. Fluid Mech. 18 (1986) 243–264; doi:10.1146/annurev.fl.18.010186.001331.CrossRefGoogle Scholar
Frederix, Y. et al. , “Equation-free methods for molecular dynamics: a lifting procedure”, Proc. Appl. Meth. Mech. 7 (2007) 20100003–20100004; doi:10.1002/pamm.200700025.Google Scholar
Gear, C. W. and Kevrekidis, I. G., “Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum”, SIAM J. Sci. Comput. 24 (2003) 1091–1106; doi:10.1137/S1064827501388157.CrossRefGoogle Scholar
Givon, D., Kupferman, R. and Stuart, A., “Extracting macroscopic dynamics: model problems and algorithms”, Nonlinearity 17 (2004) R55–R127; doi:10.1088/0951-7715/17/6/R01.CrossRefGoogle Scholar
Hairer, E., Lubich, C. and Wanner, G., “Geometric numerical integration illustrated by the Stormer–Verlet method”, Acta Numer. 12 (2003) 399–450; doi:10.1017/S0962492902000144.CrossRefGoogle Scholar
Hassard, P., Turner, I., Farrell, T. and Lester, D., “Simulation of micro-scale porous flow using smoothed particle hydrodynamics”, in: Proc. 17th Biennial Computational Techniques and Applications Conf., CTAC-2014 (eds Sharples, J. and Bunder, J.), ANZIAM J. 56 (2016) C463–C480; doi:10.21914/anziamj.v56i0.9408.Google Scholar
Horstemeyer, M. F., “Multiscale modeling: a review”, in: Practical aspects of computational chemistry (eds Leszczynski, J. and Shukla, M. K.), (Springer, Dordrecht, 2009) Ch. 4, 87–135; doi:10.1007/978-90-481-2687-3_4.CrossRefGoogle Scholar
Kalweit, M. and Drikakis, D., “Multiscale simulation strategies and mesoscale modelling of gas and liquid flows”, IMA J. Appl. Math. 76 (2011) 661–671; doi:10.1093/imamat/hxr048.CrossRefGoogle Scholar
Kevrekidis, I. G. and Samaey, G., “Equation-free multiscale computation: algorithms and applications”, Annu. Rev. Phys. Chem. 60 (2009) 321–344; ;doi:10.1146/annurev.physchem.59.032607.093610.CrossRefGoogle ScholarPubMed
Koplik, J. and Banavar, J. R., “Continuum deductions from molecular hydrodynamics”, Annu. Rev. Fluid Mech. 27 (1995) 257–292; doi:10.1146/annurev.fl.27.010195.001353.CrossRefGoogle Scholar
Koumoutsakos, P., “Multiscale flow simulations using particles”, Annu. Rev. Fluid Mech. 37 (2005) 457–487; doi:10.1146/annurev.fluid.37.061903.175753.CrossRefGoogle Scholar
Liu, P., Samaey, G., Gear, C. W. and Kevrekidis, I. G., “On the acceleration of spatially distributed agent-based computations: a patch dynamics scheme”, Appl. Numer. Math. 92 (2015) 54–69; doi:10.1016/j.apnum.2014.12.007.CrossRefGoogle Scholar
Moller, J., Runborg, O., Kevrekidis, P. G., Lust, K. and Kevrekidis, I. G., “Equation-free, effective computation for discrete systems: a time stepper based approach”, Internat. J. Bifur. Chaos 15 (2005) 975–996; http://www.worldscinet.com/ijbc/15/1503/S0218127405012399.html.CrossRefGoogle Scholar
Plimpton, S. et al. , “Large-scale atomic/molecular massively parallel simulator”, Technical Report, Sandia National Laboratories, Department of Energy, USA, 2016; http://lammpssandiagov.Google Scholar
Roberts, A. J., “Low-dimensional modelling of dynamics via computer algebra”, Comput. Phys. Comm. 100 (1997) 215–230; doi:10.1016/S0010-4655(96)00162-2.CrossRefGoogle Scholar
Roberts, A. J., “Resolving the multitude of microscale interactions accurately models stochastic partial differential equations”, LMS J. Comput. Math. 9 (2006) 193–221; ;doi:10.1112/S146115700000125X.CrossRefGoogle Scholar
Roberts, A. J., Model emergent dynamics in complex systems (SIAM, Philadelphia, PA, 2015); http://bookstore.siam.org/mm20/.Google Scholar
Roberts, A. J. and Kevrekidis, I. G., “General tooth boundary conditions for equation free modelling”, SIAM J. Sci. Comput. 29 (2007) 1495–1510; doi:10.1137/060654554.CrossRefGoogle Scholar
Roberts, A. J., MacKenzie, T. and Bunder, J., “A dynamical systems approach to simulating macroscale spatial dynamics in multiple dimensions”, J. Engrg. Math. 86 (2014) 175–207; doi:10.1007/s10665-013-9653-6.CrossRefGoogle Scholar
Roose, D., Nies, E., Li, T., Vandekerckhove, C., Samaey, G. and Frederix, Y., “Lifting in equation-free methods for molecular dynamics simulations of dense fuids”, Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 855–874; doi:10.3934/dcdsb.2009.11.855.Google Scholar
Samaey, G., Kevrekidis, I. G. and Roose, D., “The gap-tooth scheme for homogenization problems”, Multiscale Model. Simul. 4 (2005) 278–306; doi:10.1137/030602046.CrossRefGoogle Scholar
Samaey, G., Roose, D. and Kevrekidis, I. G., “Patch dynamics with buffers for homogenization problems”, J. Comput. Phys. 213 (2006) 264–287; doi:10.1016/j.jcp.2005.08.010.CrossRefGoogle Scholar
Wagner, G. J. et al. , “Accelerated molecular dynamics and equation-free methods for simulating diffusion in solids”, Technical Report SAND2011-6659, Sandia National Laboratories, Department of Energy, USA, 2011; http://www.osti.gov/scitech/biblio/1030307.Google Scholar
Yoshida, H., “Recent progress in the theory and application of symplectic integrators”, Celestial Mech. Dynam. Astronom. 56 (1993) 27–43; doi:10.1007/BF00699717.CrossRefGoogle Scholar
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