Fundamentals of materials simulation using Monte Carlo method
DOI:
https://doi.org/10.18041/1794-4953/avances.1.306Keywords:
Computational simulation, Metropolis algorithm, Monte Carlo methodAbstract
Monte Carlo method is a tool that has been widely used in materials science and engineering since its beginning in the decade of the 1940, but is also applicable in subjects as diverse as economics, society and its behavior, biology and even medicine. The working of the Monte Carlo method is based on the use of random numbers and to get to describe the behavior of a system or explain a phenomenon difficult to understand and to treat analytically. It aims to provide an introduction to the basics of Monte Carlo method applied to materials science, as well as show some examples based on the algorithm proposed by N. Metropolis and that has helped to address interesting problems in the field. At the same time basic examples are included with their own algorithms developed in the programming language FORTRAN 95 and illustrate quite well the foundations of the Monte Carlo method.
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Shankar V., Rowell C., Hall W.F., Moham-madian A.H., Schuh M. y Taylor K. (1992). “Gigaflop (billion floating point operations per second) performance for computational electromagnetics”. En revista: Computing Systems in Engineering, Volumen 3, edición 1-4, Pp. 139-151.
D’Ambrosio D., Iovine G., Spataro W. y Miyamoto H. (2007). “A macroscopic collisional model for debris-flow simulation”. En revista: Envi-ronmental Modelling & Software, Volumen 22, Pp. 1417-1436.
Antoniou N.G., Diakonos F.K., Saridakis E.N. y Tsolias G.A. (2007). “The critical ensemble: An efficient simulation of a macroscopic system at the criti-cal point”. En revista: Physica A, Volumen 376, Pp. 308-318.
Levy M., Levy H. y Solomon S. (2000, junio). “Microscopic simulations in various fields”, Micro-scopic Simulations of Financial Markets, Editorial: Academic Press, Pp. 123-140.
Mizuseki, H., Hongo, K., Kawazoe, Y. y Wille, L.T. (2002). “Multiscale simulation of cluster growth and deposition processes by hybrid model based on direct simulation Monte Carlo method”. En revista: Computational Materials Science, Volumen 24, Edición 1-2, Pp. 88-92.
Pei, Q.X., Lu, C. y Lee, H.P. (2007).“Large scale molecular dynamics study of nanometric machining of cooper”. En revista: Computational Materials Science, Volumen 41, Edición 2, Pp. 177-185.
Karakasidis, T.E. y Charitidis, C.A. (2007).“Multiscale modeling in nanomaterials science”. En revista: Materials Science and Engineering: C, Volumen 27, Edición 5-8, Pp. 1082-1089.
Landau, D.P. y Binder, K. (2000). “Molecular Dynamics”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 363-371.
Landau, D.P. y Binder, K. (2000). “Quasi-classical spin dynamics”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 372-375.
Landau, D.P. y Binder, K. (2000). “Langevin equations and variations (cell dynamics)”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge Univer-sity Press, Pp. 375-376.
Landau, D.P. y Binder, K. (2000).“Dissipative particle dynamics (DDP)”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 377-378.
Landau, D.P. y Binder, K. (2000). “Lattice gas cellular automata”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 378-379.
Landau, D.P. y Binder, K. (2000). “Lattice Boltzmann equation”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 379.
Saito, Y. (1997). “The Monte Carlo simulation of microsctructural evolution in metals”. En revista: Materials Science and Engineerign, Volumen A233, Pp. 114-124.
Bruschi, P. (2000). “Three-dimensional Monte Carlo simulations of electromigration in polycrystalline thin films.” En revista: Computational Materials Science, Volumen 17, Edición 2-4 Pp. 299-304.
Gilmer, G. y Yip, S. (2005). “Handbook of mate-rials modeling: Basic Monte Carlo models: equilibrium and kinetics”. Editorial: Springer Science and Business Media, Pp. 613.
Metropolis, N. y Ulam, S. (1949). “The Monte Carlo Method”. En revista: Journal of the American Statistical Association, Volumen 44, Edición 247, Pp. 335-341.
Hastings, W.K. (1970). “Monte Carlo sampling methods using Markov chains and their applications”. En revista: Biometrika, Volumen 57, Edición 1, Pp. 97-109.
Landau, D.P. y Binder, K. (2000). “Introduction”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición. Editorial: Cambridge University Press, Pp. 1.
Metropolis, N., Rosenbluth, A.W., Rosen-bluth, M.N y Teller, A.H. (1953). “Equation of state calculations by fast computing machines”. En revista: The Journal of Chemical Physics, Volumen 21, Edición 6, Pp. 1087-1092.[
Stephan, J., Schrader, S. y Brehmer, L. (2000). “Monte Carlo simulations of charge transport in molecular solids: a modified Miller Abrahams type jump rate approach”. En revista: Synthetic Metals, Volumen 111-112, Pp. 353-357.
Riaño-Rojas, J.C., Restrepo-Parra, E., Orozco-Hernández, G., Urrea-Serna, J.A. y Restrepo, J. (2010). “Influence of the shapes on the magnetic and electrical transport properties of La2/3Ca1/3MnO3 nanoparticles”. En revista: Journal of Materials Science, Volumen 45, Pp. 6455-6460.
Touzik, A., Hermann, H. y Wetzig, K. (2003). “General-purpose distributed software for Monte Carlo simulations in materials design”. En revista: Computational Materials Science, Volumen 28, Pp. 134-154.
Giudici, P. (1998). “Markov Chain Monte Carlo methods for probabilistic network model determi-nation”. En revista: Journal of the Italian Statistical Society, Volumen 2, Pp. 171-183.
Dietrich, M. (1996).“Monte Carlo experiments and the defense of diffusion models in molecular population genetics”. En revista: Biology and Philosophy, Volumen 11, Edición 3, Pp. 339-356.
Zhen-min, Z. (2011).“Application of Monte Carlo method in recharge calculation of underground water resources”. En revista: Procedia Engineer-ing, Volumen 23, Pp. 316-319.
Landau, D.P. y Binder, K. (2000). “Ballistic deposition”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 343.
Miranda, R., Ramos, R. y Cadilhe, A. (2003). “Finite-size scaling study of the ballistic deposition model in (1+1)-dimensions”. En revista: Compu-tationals Materials Science, Volumen 27, Pp. 224-229.
Wu, M., Fjeld, A. y Ludwig, A. (2010). “Model-ling mixed columnar-equiaxed solidification with melt convection and grain sedimentation – Part I: Model description”. En revista: Computational Materi-als Science, Volumen 50, Pp. 32-42.
Landau, D.P. y Binder, K. (2000). “Sedimentation”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, Pp. 343-344.
Landau, D.P. y Binder, K. (2000).“Kinetic Monte Carlo and MBE growth”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición. Editorial: Cambridge University Press, Pp. 344-347.
Corbett, C. (2008). “The Kinetic Monte Carlo Method: Foundation, implementation and applica-tion”. En revista: Computer Methods in Applied Mechanics and Engineering, Volu-men 197, Pp. 3386-3398.
Donev, A., Bulatov, V., Oppelstrup, T., Gilmer, H., Sadigh, B. y Kalos, H. (2010).“A First-Passage Kinetic Monte Carlo algorithm for com-plex diffusion-reaction systems”. En revista: Journal of Computational Physics, Volumen 229, Pp. 3214-3236.
Wu, D., Zhao, D y Qian, R. (1986). “Monte Carlo simulation of a confined random-walk chain”.En revista: Polymer, Volumen 27, Edición 7, Pp. 1087-1090.
Landau, D.P. y Binder, K. (2000). “Generation of random walks”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición. Editorial: Cambridge University Press, Pp. 61-66.
Santoso, A.M., Phoon, K.K. y Quek S.T. (2011). “Modified Metropolis-Hastings algorithm with reduced chain correlation for efficient subset simu-lation”. En revista: Probabilistic Engineering Mechanics, Volumen 26, Pp. 331-341.
Holden, L. (1998). “Geometric Convergence of the Metropolis-Hastings simulation algorithm”. En revista: Statistic & Probability Letters, Volu-men 39, Pp. 371-377.
Landau, D.P. y Binder, K. (2000). “Simple Sam-plinge Monte Carlo Methods”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, P. 48.
Ruszczynski A. y Shapiro, A. (2003). “Monte Carlo Sampling Methods. Handbooks in Operations Research and Management Science”, Volumen 10, Pp. 353-425.
Ascasibar, Y. (2008). “FiEstAS sampling-a Monte Carlo algorithm for multidimensional numerical integration”. En revista: Computer Physics Communications, Volumen 179, Pp. 881-887.
Landau, D.P. y Binder, K. (2000). “Importance Sampling Monte Carlo Methods”.A Guide to Monte Carlo Simulations in Statistical Physics, Segunda Edición.Editorial: Cambridge University Press, P. 68.
Kuczera, G. y Parent, E. (1998). “Monte Carlo assessment of parameter uncertainty in conceptual catch-ment models: the Metropolis algorithm”. En revista: Journal of Hydrology, Volumen 211, Pp. 69-85.
Lu, Z. y Zhang, D. (2003). “On importance sam-pling Monte Carlo approach to uncertainty analysis for flow and transport in porous media”. En revista: Advances in water resources, Volumen 26, Pp. 1177-1188.
Yanhong, W., Xu, Y., Dong, Z., y Zhan, X. (2008) “Three-dimensional Monte Carlo simulation of discontinuous grain growth in HAz of stainless steel during GTAW process”. En revista: Journal of Materials Processing Technology, Volumen 209 Pp. 1466 -1470.
Wei, Y., Xu, Y., Dong, Z. y Zhan, X. (2009). “Three-dimensional Monte Carlo simulation of discon-tinuous grain growth in HAz of stainless steel during GTAW process”. En revista: Journal of Materi-als Processing Technology, Volumen 209, Pp. 1466-1470.
Mehnert, K. (1996). “Monte Carlo simulation of grain growth in textured metals using anisotropic grain boundary mobilities”. En revista: Computational Materials Science, Volumen 7, Pp. 103-108.
Fjeldberg, E. y Marthinsen, K. (2010). “A 3D Monte Carlo study of the effect of grain bound-ary anisotropy and particles on the size distribution of grains after recrystallisation and grain growth”. En revista: Computational Materials Science, Volumen 48, Pp. 267-281.
Restrepo-Parra, E., Orozco - Hernández, G., Urrea-Serna, J.A., Jurado, J. F., Vargas-Hernández, C., Riaño-Rojas, J. C. y Restrepo. J. (2010). “Interface roughness influence on exchange bias effect in La2/3Ca1/3MnO3/La1/3Ca2/3MnO3 bilayers”. En revista: Journal of Materials Sci-ence, Volumen 45, Edición 24, Pp. 6763-6768.
