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اهداف و چشم انداز

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فهرست داوران همکار

مروری بر روش‌های چندمقیاسی همزمان در تحلیل مسائل ناپیوستگی در مقیاس ریز

نوع مقاله : مقاله مروری

نویسندگان

1 آزمایشگاه محاسبات سریع، دانشکده مهندسی عمران، دانشکدگان فنی، دانشگاه تهران

2 دانشکده مهندسی عمران، دانشگاه تهران

چکیده

مدلسازی مسائل ترک و ناپیوستگی اهمیت بسزایی در صنایع مختلف دارد و از دیرباز مورد توجه بوده است. مدلسازی رفتار ترک در مقیاس‌های مختلف بویژه مقیاس‌های اتمی می‌تواند باعث شناخته شدن بهتر رفتار ریز ساختار ماده و پیش بینی رفتار آن در مقیاس‌های بزرگتر گردد. از سوی دیگر مدلسازی مبتنی بر مقیاس‌های اتمی بدلیل بیشتر بودن تعداد درجات آزادی نیاز به صرف هزینه محاسباتی بسیار زیادی در مقایسه با سایر روش‌ها دارد. مجموعه روش‌های چندمقیاسی همزمان برای حل این مشکلات بوجود آمده‌اند و به عنوان تکنیک‌های قابل پذیرش برای مدلسازی ناپیوستگی در دهه‌های اخیر مورد استقبال محققین قرار گرفته است. مطالعات نشان می‌دهد روش‌های چندمقیاسی همزمان توانسته‌اند کلیه رفتارهای ریز مقیاس در مقابل تحریک‌های مکانیکی را شبیه‌سازی نمایند و ضمن دستیابی به تطابق مناسبی با مدل‌های آزمایشگاهی، ارتباط پیوسته با مقیاس‌های بزرگتر را برقرار کنند. در مطالعه حاضر، روش‌های چندمقیاسی همزمان همگن سازی و دامنه مجزا که در طول چند دهه گذشته در مدلسازی مسائل ناپیوستگی مورد استفاده قرار گرفته‌اند بررسی شده اند. جهت ایجاد زمینه مناسب برای مقایسه روش‌های توسعه داده شده در طول چد دهه گذشته، مسئله گسترش ترک لبه بازطراحی و مدلسازی شده است و نتایج حاصل از شبیه‌سازی و دقت محاسباتی مهم‌ترین روش‌ها با یکدیگر مقایسه شده اند.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

A review of concurrent multiscale methods for the analysis of fine scale discontinuity problems

نویسندگان [English]

  • Omid Alizadeh 1
  • S. Mohammadi 2

1 High Performance Computing Lab, School of Civil Engineering, College of Engineering, University of Tehran

چکیده [English]

Modeling of crack and discontinuity related problems has had a great influence on numerous industries for a long time. Simulation of discontinuity behavior in different scales, especially in atomistic scales, can lead to better insight of the crack/discontinuity initiation and propagation phenomena and prediction of its behavior in larger scales. On the other hand, modeling based on fully refined scales requires huge computational effort compared to other methods due to the higher number of degrees of freedom. Concurrent multiscale methods have been developed to overcome the high computational cost issues of refined models, while preserving sufficient accuracy. Studies have shown that concurrent multiscale methods are capable of simulating all atomic behaviors in order to establish a compatible solution with larger scales, and to accurately resemble the laboratory results. In the present review, concurrent multiscale methods, which could be categorized into homogenization and partitioned-domain methods, are briefly investigated and compared. These methods have been widely used for modelling of cracks, discontinuities and impurities in different types of problems in the past two decades. To create a suitable basis for comparing the main concurrent methods, the problem of edge crack propagation is redesigned and modeled, and the simulation results and their computational accuracy are compared.

کلیدواژه‌ها [English]

  • Concurrent multiscale methods
  • homogenization
  • discrete domain methods
  • discontinuity
مراجع
  1. Rashid, Y.R., “Analysis of Prestressed Concrete Reactor Vessels”, Nuclear Engineering and Design, 7, No. 4, pp. 334-344, 1968.
  2. Beissel, S., Johnson, G. and Popelar, C., “An Element-Failure Algorithm for Dynamic Crack Propagation in General Directions”, Engry Fracture Mechanics, 61, No. 3, pp. 407-425, 1998.
  3. Cindy, L., Rountree, R.K., and Kalia, E.L., “Atomistic Aspects of Crack Propagation :Multimillion Atom Molecular Dynamics Simulations”, Annual Review of Materials Research, 32, pp. 377-400, 2002.
  4. Tadmor, E.B. and Miller, R.E., “Modeling Materials; Continuum, Atomistic and Multiscale Techniques”, New York: Cambridge University Press, 2011.
  5. Sun, T., Mirzoev, A., Minhas, V., Korolev, N., Lyubartsev, A. P. and Nordenskiöld, L., “A Multiscale Analysis of DNA Phase Separation: from Atomistic to Mesoscale Level”, Nucleic Acids Research, 47, No. 11, pp. 5550-5562, 2019.
  6. Eftekhari, M., Mohammadi, S. and, M., “A Hierarchical Nano to Macro Multiscale Analysis of Monotonic Behavior of Concrete Columns Made of CNT-Reinforced Cement Composite”, Construction and Building Materials, 175, pp. 134-143, 2018.
  7. Shahi, S. and Mohammadi, S., “A Comparative Study of Transversely Isotropic Material Models for Prediction of Mechanical Behavior of the Aortic Valve Leaflet”, International Journal of Research in engineering and Technology, 2, No. 4, pp. 192-196, 2013.
  8. Shahi, S. and Mohammadi, S., “A Multiscale Finite Element Simulation of Human Aortic Heart Valve”, Applied Mechanics and Materials, 367, pp. 275-279, 2013.
  9. Talebi, H., Silani, M., Bordas, S. P. A., Kerfriden, P. and Rabczuk, T., “A Computational Library for Multiscale Modeling of Material Failure”, Computational Mechanics, 53, p. 1047–1071, 2014.
  10. Hassani, B. and Hinton, E., “A Review of Homogenization and Topology Optimization I—Homogenization Theory for Media with Periodic Structure”, Computers & Structures, 69, No. 6, pp. 707-717, 1998.
  11. Hassani, B. and Hinton, E., “A Review of Homogenization and Topology Opimization II—Analytical and Numerical Solution of Homogenization Equations”, Computers & Structures, 69, No. 6, pp. 719-738, 1998.
  12. Nemat-Nasser, S. and Hori, M., “Micromechanics: Overall Properties of Heterogeneous Materials”, New York: Elsevier, 1998.
  13. Larsson, R. and Diebels, S., “A Second‐Order Homogenization Procedure for Multi‐Scale Analysis Based on Micropolar Kinematics”, Numerical Methods in Engineering, 69, No. 12, pp. 2485-2512, 2006.
  14. Bayesteh, H. and Mohammadi, S., “Micro-Based Enriched Multiscale Homogenization Method for Analysis of Heterogeneous Materials”, International Journal of Solids and Structures, 125, pp. 22-42, 2017.
  15. Mohammadi, S., “Multiscale Biomechanics: Theory and Applications”, Wiely, 2023.
  16. Fish, J. and Fan, R., “Mathematical Homogenization of Nonperiodic Heterogeneous Media Subjected to Large Deformation Transient Loading”, International Journal for Numerical Methods in Engineering, 76, pp. 1044-1064, 2008.
  17. Markenscoff, X. and Dascalu, C., “Asymptotic Homogenization Analysis for Damage Amplification Due to Singular Interaction of Micro-Cracks”, Journal of the Mechanics and Physics of Solids, 60, No. 8, pp. 1478-1485, 2012.
  18. Yang, Y. , Ma, F., Lei, C., Liu, Y. and Li, J., “Nonlinear Asymptotic Homogenization and the Effective Behavior of Layered Thermoelectric Composites”, Journal of the Mechanics and Physics of Solids, 61, No. 8, pp. 1768-1783, 2013.
  19. Fatemi Dehaghani, P., Hatefi Ardakani, S., Bayesteh, H. and Mohammadi, S., “3D Hierarchical Multiscale Analysis of Heterogeneous SMA Based Materials”, International Journal of Solids and Structures, 118-119, pp. 24-40, 2017.
  20. Hashin, Z. and Shtrikman, S., “On Some Variational Principles in Anisotropic and Nonhomogeneous Elasticity”, Journal of the Mechanics and Physics of Solids, 10, No. 4, pp. 335-342, 1962.
  21. Hashin, Z. and Shtrikman, S., “A Variational Approach to the Theory of the Elastic Behaviour of Multiphase Materials”, Journal of the Mechanics and Physics of Solids, 11, No. 2, pp. 127-140, 1963.
  22. Hill, R., “Elastic Properties of Reinforced Solids: Some Theoretical Principles”, Journal of the Mechanics and Physics of Solids, 11, No. 5, pp. 357-372, 1963.
  23. Hill, R., “A Self-Consistent Mechanics of Composite Materials”, Journal of the Mechanics and Physics of Solids, 13, No. 4, pp. 213-222, 1965.
  24. Hill, R., “Continuum Micro-Mechanics of Elastoplastic Polycrystals”, Journal of the Mechanics and Physics of Solids, 13, No. 2, pp. 89-101, 1965.
  25. Budiansky, B., “On the Elastic Moduli of Some Heterogeneous Materials”, Journal of the Mechanics and Physics of Solids, 13, No. 4, pp. 223-227, 1965.
  26. Hashin, Z., “The Elastic Moduli of Heterogeneous Materials”, Appl. Mech., Vol. 29, No. 1, pp. 143-150, 1962.
  27. Eshelby, J. D., “The Determination of the Elastic Field of An Ellipsoidal Inclusion, and Related Problems”, Proceedings of the Royal Society A, 241, No. 1226, 1957.
  28. Feyel, F., “A Multilevel Finite Element Method (FE2) to Describe the Response of Highly Non-Linear Structures Using Generalized Continua”, Computer Methods in Applied Mechanics and Engineering, 192, No. 28, pp. 3233-3244, 2003.
  29. Shu, W., and Stanciulescu, I., “Computational Modeling and Multiscale Homogenization of Short Fiber Composites Considering Complex Microstructure and Imperfect Interfaces”, Composite Structures, 306, 116592, 2023.
  30. Özdemir, I., Brekelmans, W. and Geers, M., “FE2 Computational Homogenization for the Thermo-Mechanical Analysis of Heterogeneous Solids”, Computer Methods in Applied Mechanics and Engineering, 198, No. 3-4, pp. 602-613, 2008.
  31. Feyel, F., “Multiscale FE2 Elastoviscoplastic Analysis of Composite Structures”, Computational Materials Science, 16, No. 1-4, pp. 344-354, 1999.
  32. Otero, F., Oller, S. and Martinez, X., “Multiscale Computational Homogenization: Review and Proposal of a New Enhanced-First-Order Method”, Archives of Computational Methods in Engineering, 25, pp. 479–505, 2018.
  33. Petracca, M., Pelà, L. , Rossi, R., Oller, S., Camata, G. and Spacone, E., “Multiscale Computational First Order Homogenization of Thick Shells for the Analysis of Out-of-Plane Loaded Masonry Walls”, Computer Methods in Applied Mechanics and Engineering, 315, No. 1, pp. 273-301, 2017.
  34. Wang, C., Li, C., Ling, Y., and Wah, M. A., “Investigation on Fretting Fatigue Crack Initiation in Heterogenous Materials Using a Hybrid of Multiscale Homogenization and Direct Numerical Simulation”, Tribology International, 169, 107470, 2022.
  35. Geers, M., Kouznetsova, V. and Brekelmans, W., “Multi-Scale Computational Homogenization: Trends and Challenges”, Journal of Computational and Applied Mathematics, 234, pp. 2175–2182, 2010.
  36. Kouznetsova, V., Geers, M. G. D., and Brekelman, W. A. M., “Multi-Scale Constitutive Modelling of Heterogeneous Materials with A Gradient-Enhanced Computational Homogenization Scheme”, International Journal for Numerical Methods in Engineering, 54, No. 8, pp. 1235-1260, 2002.
  37. Kouznetsova, V., Geers, M., and Brekelman, W. A. M., “Multi-Scale Second-Order Computational Homogenization of Multi-Phase Materials: A Nested Finite Element Solution Strategy”, Computer Methods in Applied Mechanics and Engineering, 193, No. 48, pp. 5525-5550, 2004.
  38. Kaczmarczyk, Ł., Pearce, C. J. and Bićanić, N., “Scale Transition and Enforcement of RVE Boundary Conditions in Second-Order Computational Homogenization”, International Journal for Numerical Methods in Engineering, 74, No. 3, pp. 506-522, 2008.
  39. Lesičar, T., Sorić, J. and Tonković, Z., “Large Strain, Two-Scale Computational Approach Using C1 Continuity Finite Element Employing A Second Gradient Theory”, Computer Methods in Applied Mechanics and Engineering, 298, pp. 303-324, 2016.
  40. Sánchez, P., Blanco, P. and Huespe, A., “Failure-Oriented Multi-scale Variational Formulation: Micro-Structures with Nucleation and Evolution of Softening Bands”, Computer Methods in Applied Mechanics and Engineering, 257, pp. 221-247, 2013.
  41. Miller, R. E., and Tadmor, E. B., “A Unified Framework and Performance Benchmark of Fourteen Multiscale Atomistic/Continuum Coupling Methods”, Modelling and Simulation in Materials Science and Engineering, 17, No. 5,pp. 1-51, 2009.
  42. Daw, M. S., and Baskes, M. I., “Embedded-Atom Method: Derivation and Application to Impurities, Surfaces, and Other Defects in Metals”, Physical Review B, 29, No. 12,pp. 6443-6453, 1984.
  43. Tadmor, E. B., Ortiz, M., and Phillips, R., “Quasicontinuum Analysis of Defects in Solids”, Philosophical Magazine A, 1529-1563, 1996.
  44. “QC Method”, [Online]. Available: qcmethod.org,2023.
  45. Qiu, R. Z., Lin, Y. C., Fang, T. H., and Tsai, L. R., “The Crack Growth and Expansion Characteristics of Fe and Ni Using Quasi-Continuum Method”, Materials Research Express, 4, No. 3,pp. 1-10, 2017.
  46. Xu, T., Fan, J., Stewart, R., Zeng, X., and Yao, A., “Quasicontinuum Simulation of Brittle Cracking in Single-Crystal Material”, Crystal Research and Technology, 52, No. 3,pp. 1-15, 2017.
  47. RingdalenVatne, I., Stby, E., Thaulow, C., and Farkas, D., “Quasicontinuum Simulation of Crack Propagation in Bcc-Fe”, Materials Science and Engineering A, 528, No. 15, pp. 5122–5134, 2011.
  48. Zhou, T., Yang, X., and C. Chen, “Quasicontinuum Simulation of Single Crystal Nano-Plate with A Mixed-Mode Crack”, International Journal of Solids and Structures, 46, No. 9, pp. 1975-1980, 2009.
  49. Huang, S., and Zhou, C., “Modeling and Simulation of Nanoindentation”, JOM, 69, No. 11, pp. 2256-2263, 2017.
  50. Alizadeh, O., Toloee Eshlaghi, G., and Mohammadi, S., “Nanoindentation Simulation of Coated Aluminum Thin Film Using Quasicontinuum Method”, Computational Materials Science, 111, No. 1, pp. 12-22, 2016.
  51. Moslemzadeh, H., Alizadeh, O., and Mohammadi, S., “Quasicontinuum Multiscale Modeling of the Effect of Rough Surface on Nanoindentation Behavior”, Meccanica, 54, pp. 411-427, 2019.
  52. Mikes, K., and Jirasek, M., “Quasicontinuum Method Extended to Irregular Lattices”, Computers & Structures, 192, pp. 50-70, 2017.
  53. Qiu, R. Z., Lin, Y. C., and Fang, T. H., “Fatigue Crack Growth Characteristics of Fe and Ni under Cyclic Loading Using A Quasi-Continuum Method”, Beilstein Journal of Nanotechnology, 9, pp. 1000-1014, 2018.
  54. Alizadeh, O., Moslemzadeh, H., and Mohammadi, S., “Concurrent Multiscale Modeling of Interlaminar Nano Crack Propagation under Cyclic Loading”, In 11th International Congress on Civil Engineering, Tehran, 2018.
  55. Chen, P. and Shen, Y., “Nanocontact Between BCC Tungsten and FCC Nickel Using the Quasicontinuum Method”, International Journal of Solids and Structures, 45, No. 24, pp. 6001-6017, 2008.
  56. Peron-Luhrs, V., Sansoz, F., and Noels, L., “Quasicontinuum Study of the Shear Behavior of Defective Tilt Grain Boundaries in Cu”, Acta Materialia, 64, pp. 419-428, 2014.
  57. Yu, W. S., and Wang, Z. Q., “Interactions Between Edge Lattice Dislocations and Sigma 11 Symmetrical Tilt Grain Boundaries in Copper: A Quasi-Continuum Method Study”, Acta Materialia, 60, pp. 5010–5021, 2012.
  58. Yu, W., Wang, Z., and Shen, S., “Edge Dislocations Interacting with a Σ11 Symmetrical Grain Boundary in Copper Upon Mixed Loading: A Quasi Continuum Method Study”, Computational Materials Science, 137, pp. 162-170, 2017.
  59. Amelang, J. S., and Kochmann, D. M., “Surface Effects in Nanoscale Structures Investigated by a Fully-Nonlocal Energy-Based Quasicontinuum Method”, Mechanics of Materials, 90, pp. 166-184, 2015.
  60. Dupuy, L. M., Tadmor, E. B., Miller, R. E., and Phillips, R., “Finite-Temperature Quasicontinuum: Molecular Dynamics Without All the Atoms”, Physical Review Letters, 95,No. 6, pp. 1-4, 2005.
  61. Tadmor, E., Legoll, F., Kim, W., Dupuy, L., and Miller, R., “Finite-Temperature Quasi-Continuum”, Applied Mechanics Reviews, 65, No. 1,pp. 1-27, 2013.
  62. Wang, X. and Guo, X., “Numerical Simulation for Finite Deformation of Single-Walled Carbon Nanotubes at Finite Temperature Using Temperature-Related Higher Order Cauchy-Born Rule Based Quasi-Continuum Model”, Computational Materials Science, 55, No. 1, pp. 273–283, 2012.
  63. Beex, L., Rokoš, O., Zeman J., and Bordas, S., “Higher‐Order Quasicontinuum Methods for Elastic and Dissipative Lattice Models: Uniaxial Deformation and Pure Bending”, GAMM-Mitteilungen, 38, No. 2, pp. 344-368, 2015.
  64. Kochmann, D. M., and Venturini, G. N., “A Meshless Quasicontinuum Method Based on Local Maximum-Entropy Interpolation”, Modelling and Simulation in Materials Science and Engineering,Vol 22, No. 3, pp. 1-5,
  65. Amelang, J. S., Venturini, G. N., and Kochmann, D. M., “Summation Rules for A Fully-Nonlocal Energy-Based Quasicontinuum Method”, Journal of the Mechanics and Physics of Solids, 82, pp. 378–413, 2015.
  66. Beex, L., Peerlings, R., and Geers, M., “Central Summation in the Quasicontinuum Method”, Journal of the Mechanics and Physics of Solids, 70, pp. 242-261, 2014.
  67. Ortner, C., and Zhang, L., “Atomistic/Continuum Blending with Ghost Force Correction”, SIAM Journal of Scientific Computing, A346–A375, 2016.
  68. Dobson, M., and Luskin, M., “An Analysis of the Effect of Ghost Force Oscillation on Quasicontinuum Error”, Mathematical Modelling and Numerical Analysis, 43, pp. 591–604, 2009.
  69. Sorkin, V., Elliott, R. S., and Tadmor, E. B., “A Local Quasicontinuum Method for 3D Multilattice Crystalline Materials: Application to Shape-Memory Alloys”, Modelling and Simulations in Materials Science and Engineering, 22, No. 5, pp. 1-21, 2014.
  70. Dobson, M., Elliott, R., Luskin, M., and Tadmor, E., “A Multilattice Quasicontinuum for Phase Transforming Materials: Cascading Cauchy Born kinematics”, Journal of Computer-Aided Materials Design, 14, pp. 219-237, 2007.
  71. Smith, G. S., Tadmor, E. B., and Kaxiras, E., “Multiscale Simulation of Loading and Electrical Resistance in Silicon Nano Indentation”, Physics Review Letters, Vol. 84, No. 3,pp. 1260–1263
  72. Lu, H. B., Li, J. W., Ni, Y. S., Mei, J. F., and Wang, H. S., “Multiscale Analysis of Defect Initiation on the Atomistic Crack Tip in Body-Centered-Cubic Metal Ta”, Acta Physica Sinica, 60,No. 10, pp.1-7, 2011.
  73. Vatne, I. R., Ostby, E., Thaulow, C., and Farkas, D., “Quasicontinuum Simulation of Crack Propagation in bcc-Fe”, Materials Science and Engineering A, 528, pp. 5122–5134, 2011.
  74. Shao, Y. F. and Wang, S. Q., “Quasicontinuum Simulation of Crack Propagation in Nanocrystalline Ni”, Acta Physica Sinica, 59, pp. 7258–7265, 2010.
  75. Akhavan, S., Khodadad, M., Alizadeh, O., and Mohammadi, S., “Atomistic Modelling of Plastic Zone of Crack Tip in FCC Metals”, In 11th International Congress of Civil Engineering, Tehran, 2018.
  76. Wu, C. D., Fang, T. H., Su, W. C., and Fan, Y. C., “Effects of Constituting Material and Interfacial Crack on Mechanical Response of Nanoscale Metallic Bilayers - A Quasi-Continuum Study”, Molecular Simulation, 46, pp. 1155–1163, 2020.
  77. Wu, C. D., Fang, T. H., Su, W. C., and Fan, Y. C., “Fracture and Crack Propagation of Metallic Bilayers Using Quasi-Continuum Silumations”, Digest Journal of Nanomaterials & Biostructures (DJNB), 15, No. 2, pp. 319-327, 2020.
  78. Imaizumi, K., Ono, T., Kota, T., Okamoto, S., and Sa, S., “Transformation of Cubic Symmetry for Spherical Microdomains from Face-Centred to Body-Centred Cubic upon Uniaxial Elongation in an Elastomeric Triblock Copolymer”, Applied Crystallography, 36, pp. 976-981, 2003.
  79. Islam, S., [Online], Available: https://slideplayer.com/slide/14570656/.
  80. Shilkrot, L. E., Miller, R. E. and Curtin, W. A., “Coupled Atomistic and Discrete Dislocation Plasticity”, Physical Review Letters, 89, No. 2, pp. 025501, 2002.
  81. Shilkrot, L. E., Miller, R. E., and Curtin, W. A., “Multiscale Plasticity Modeling: Coupled Atomistic and Discrete Dislocation Mechanics”, Journal of the Mechanics and Physics of Solids, 52, No. 4, pp. 755-787, 2004.
  82. Anciaux, G., Junge, T., Hodapp, , Cho, J., Molinari, J. F., and Curtin, W., “The Coupled Atomistic/Discrete-Dislocation Method in 3d part I: Concept and Algorithms”, Journal of the Mechanics and Physics of Solids, Vol. 118, pp. 152-171, 2018.
  83. Hodapp, M., Anciaux, G., Molinari, J. F., and Curtin, W., “Coupled Atomistic/Discrete Dislocation Method in 3D Part II: Validation of the Method”, Journal of the Mechanics and Physics of Solids, 119, pp. 1-19, 2018.
  84. Cho, J., Molinari, J. F., Curtin, W. A., and Anciaux, G., “The Coupled Atomistic/Discrete-Dislocation Method in 3d. Part III: Dynamics of Hybrid Dislocations”, Journal of the Mechanics and Physics of Solids, 118, pp. 1-14, 2018.
  85. “A Study of Nano-Indentation Using Coupled Atomistic and Discrete Dislocation (CADD) Modeling”, Computational Fluid and Solid Mechanics 2003, 455-459, 2003.
  86. Miller, R. E., Shilkrot, L., and Curtin, A., “A Coupled Atomistics and Discrete Dislocation Plasticity Simulation of Nanoindentation into Single Crystal Thin Films”, Acta Materialia, Vol. 52, No. 2, pp. 271-284, 2004.
  87. Wagner, G. J., and Liu, W. K., “Coupling of Atomistic and Continuum Simulations Using A Bridging Scale Decomposition”, Journal of Computational Physics, 190, No. 1, pp. 249-274, 2003.
  88. Liu, W. K., Park, H. S., Qian, D., Karpov, E. G., Kadowaki, H., and Wagner, G. J., “Bridging Scale Methods for Nanomechanics and Materials”, Computer Methods in Appiled Mechanics and Engineering, 195, No. 13-16,pp. 1407-1421, 2006.
  89. Xiao, S., and Belytschko, T., “A Bridging Domain Method for Coupling Continua with Molecular Dynamics”, Computer Methods in Applied Mechanics and Engineering, 193, No. 17-20, pp. 1645-1669, 2004.
  90. Fish, Multiscale Methods: Bridging the Scales in Science and Engineering, Oxford Scholarship Online, 2009.
  91. Zhang, S., Khare, R., Lu, Q., and Belytschk, T., “A Bridging Domain and Strain Computation Method for Coupled Atomistic–Continuum Modelling of Solids”, International Journal for Numerical Methods in Engineering, 70, No. 8, pp. 913-933, 2006.
  92. Tabarraei, A., Wang, X., Sadeghirad, A. and Song, J. H., “An Enhanced Bridging Domain Method for Linking Atomistic and Continuum Domains”, Finite Elements in Analysis and Design , 92, pp. 36-49, 2014.
  93. Xu, M., Gracie, R., and Belytschko, T., “A Continuum‐to‐Atomistic Bridging Domain Method for Composite Lattices”, International Journal for Numerical Methods in Engineering, 81, No. 13, pp. 1635 - 1658, 2009.
  94. Guidault, P. and Belytschko, T., “Bridging Domain Methods for Coupled Atomistic–Continuum Models with L2 or H1 Couplings”, International Journal for Numerical Methods in Engineering, 77, No. 11, pp. 1566-1592, 2009.
  95. Talebi, H., Silani, M., and Rabczuk, T., “Concurrent Multiscale Modeling of Three Dimensional Crack and Dislocation Propagation”, Advances in Engineering Software, 80, pp. 82-92, 2015.
  96. Adelman, S. A., and Doll, J. D., “Generalized Langevin Equation Approach for Atom/Solid-Surface Scattering – Collinear Atom/Harmonic”, The Journal of Chemical Physics , 61, No. 10, pp. 4242–4245, 1974.
  97. Badia, S., Bochev, P., Lehoucq, R., Parks, M. L., Fish, J., Nuggehally, M., and Gunzburger, M., “A Force-Based Blending Model for Atomistic-to-Continuum Coupling”, International Journal for Multiscale Computational Engineering, 5, No. 5, pp. 387-406, 2007.
  98. Badia, S., Parks, M., Bochev, P., Gunzburger, M., and Lehoucq, R., “On Atomistic-to-Continuum Coupling by Blending”, Multiscale Modeling & Simulation , 7, No. 1, pp. 381-406, 2008.
  99. Fish, J., Nuggehally, M. A., Shephard, M. S., Picu, C. R., Badia, S., Parks, M. L., and Gunzburger, M., “Concurrent AtC Coupling Based on a Blend of the Continuum Stress and the Atomistic Force”, Computer Methods in Applied Mechanics and Engineering, 196, No. 45-48, pp. 4548-4560, 2007.
  100. Parks, M. L., Bochev, P. B., and Lehoucq, R. B., “Connecting Atomistic-to-Continuum Coupling and Domain Decomposition”, Multiscale Modeling & Simulation, 7, No. 1, pp. 362-380, 2008.
  101. Eidel, B., and Stukowski, A., “A Variational Formulation of the Quasicontinuum Method Based on Energy Sampling in Clusters”, Journal of the Mechanics and Physics of Solids, 57, No. 1, pp. 87-108, 2009.
  102. Alizadeh, O., “Enriched Multiscale Method”, Ph.D. Thesis, University of Tehran, Tehran, 2019.
  103. Zhu, H., “Graphene; Fabrication, Characterizations, Properties and Applications”,1st,272, Elsevier Inc, Amsterdam, 2018.
  104. Ericksen, J., “The Cauchy and Born Hypotheses for Crystals”, in Phase Transformations and Material Instabilities in Solids, New York, Academic Press, 1984, pp. 61-77.
  105. Marenić, E. and Ibrahimbegovic, A., “Multiscale Atomistic-to-Continuum Reduced Models for Micromechanical Systems”, in Computational Methods for Solids and Fluids, Springer, 2016, pp. 215-244.
  106. Atkins, P. and Paula, J. D., “Atkins' Physical Chemistry”, Oxford university press, 2010.
  107. Alizadeh, O. and Mohammadi, S., “The Variable Node Multiscale Approach: Coupling the Atomistic and Continuum Scales”, Computational Materials Science, 160, pp. 256-274, 2019.
  108. Moslemzadeh, H. and Mohammadi, S., “An Atomistic Entropy Based Finite Element Multiscale Method for Modeling Amorphous Materials”, International Journal of Solids and Structures, 256, 2022.
  109. Wang, S., Zhao, L., and Liu, Y., “A Concurrent Multiscale Method Based on Smoothed Molecular Dynamics for Large-Scale Parallel Computation at Finite Temperature”, Computer Methods in Applied Mechanics and Engineering, 406, 115898, 2023.
  110. Wang, S., Zhao, L., and Liu, Y., “An Improved Smoothed Molecular Dynamics Method with High-Order Shape Function”, International Journal for Numerical Methods in Engineering,Vol. 122, No. 13,pp. 3300-3322, 2021. 

 

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