Vol. 1 (2025)

Articles

One Element Per Grain: An Effcient Virtual Element Method for Heat Conduction in Polycrystalline Materials

Published 2025-12-31

  • Sundararajan Natarajan

Sundararajan Natarajan
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai - 600036, Tamil Nadu, India

Abstract

An efficient numerical framework based on the virtual element method is proposed to study the heat conduction in polycrystalline materials. Voronoi tessellation is used to generate a representative micro-structure, with grain boundaries explicitly modelled in two dimensions. The proposed framework treats each grain/grain boundary as one finite element and this feature drastically reduces the computational time. The numerical results from the proposed framework are compared with traditional finite element, where each grain/grain boundary is discretized with simplex elements. The numerical assessment demonstrates the effectiveness and robustness of the framework and yield accurate results at a fraction of computational effort without compromising accuracy. From the numerical study, it is observed that the proposed framework requires at least an order of magnitude fewer dofs when compared to the FEM.

Keywords

Effective thermal conductivity, Grain boundary, Grain core, Polycrystalline material, Polygon, Virtual element method

PDF

References

  1. [1] Torquato S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Interdisciplinary Applied Mathematics, Springer New York, 2013. URL https://books.google.co.in/books?id=UTfoBwAAQBAJ
  2. [2] Droste M, Jarvenpaa A, Jaskari M, Motylenko M, Weidner A, Karjalainen P, Biermann H. The role of grain size in static and cyclic deformation behaviour of a laser reversion annealed metastable austenitic steel. Fatigue & Fracture of Engineering Materials & Structures 2021; 44(1): 43-62.
  3. https://doi.org/10.1111/ffe.13326
  4. [3] Schuh CA, Lu K. Stability of nanocrystalline metals: The role of grain-boundary chemistry and structure. MRS Bulletin 2021; 1-11.
  5. https://doi.org/10.1557/s43577-021-00055-x
  6. [4] Symington AR, Molinari M, Dawson JA, Statham JM, Purton J, Canepa P, Parker SC. Elucidating the nature of grain boundary resistance in lithium lan thanum titanate. Journal of Materials Chemistry A 2021; 9(10): 6487-6498.
  7. https://doi.org/10.1039/D0TA11539H
  8. [5] Badry F, Ahmed K. A new model for the effective thermal conductivity of poly285 crystalline solids. AIP Advances 2020; 10(10): 105021.
  9. https://doi.org/10.1063/5.0022375
  10. [6] Gulizzi V, Milazzo A, Benedetti I. An enhanced grain-boundary framework for computational homogenization and micro-cracking simulations of polycrystalline materials. Computational Mechanics 2015; 56(4): 631-651.
  11. https://doi.org/10.1007/s00466-015-1192-8
  12. [7] Cascio ML, Gulizzi V, Milazzo A, Benedetti I. A model for pol-ycrystalline thermo-mechanical homogenisation and micro-cracking. Procedia Structural In tegrity 2024; 52: 618-624.
  13. https://doi.org/10.1016/j.prostr.2023.12.063
  14. [8] Manzini G, Russo A, Sukumar N. New perspectives on polygonal and polyhedral finite element methods. Mathematical Models and Methods in Applied Sciences 2014; 24(08): 1665-1699.
  15. https://doi.org/10.1142/S0218202514400065
  16. [9] Veiga LBD, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A. Basic principles of Virtual Element Methods. Math Models Methods Appl Sci 2013; 23: 199-214.
  17. https://doi.org/10.1142/S0218202512500492
  18. [10] Veiga LBD, Brezzi F, Marini LD, Russo A. The Hitchikkers Guide to the Virtual Element Method. Math Models Methods Appl Sci 2014; 24: 1541-1574.
  19. https://doi.org/10.1142/S021820251440003X
  20. [11] Veiga LBD, Brezzi F, Marini LD, Russo A. Virtual Element Method for general second-order elliptic problems. Math Models Methods Appl Sci 2016; 26(4): 729-750.
  21. https://doi.org/10.1142/S0218202516500160
  22. [12] Vacca G, Beirao Da Veiga L. Virtual element methods for parabolic problems on polygonal meshes. Numer Meth Part D E 2015; 31(6): 2110-2134.
  23. https://doi.org/10.1002/num.21982
  24. [13] Vacca G. Virtual element methods for hyperbolic problems on polygonal meshes. Comput Math Appl 2017; 74(5): 882-898.
  25. https://doi.org/10.1016/j.camwa.2016.04.029
  26. [14] A. Cangiani, G. Manzini, O. J. Sutton, Conforming and nonconforming virtual element methods for elliptic problems. IMA Journal of Numerical Analysis 2017; 37(3): 1317-1354.
  27. https://doi.org/10.1093/imanum/drw036
  28. [15] Benedetto MF, Berrone S, Borio A, Pieraccini S, Scialo S. Order preserving supg stabilization for the virtual element formulation of advection-di”usion problems. Comput Methods Appl Mech Eng 2016; 311: 18-40.
  29. https://doi.org/10.1016/j.cma.2016.07.043
  30. [16] Adak D, Natarajan E, Kumar S. Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes. Numer Meth Part D E 2019; 35(1): 222-245.
  31. https://doi.org/10.1002/num.22298
  32. [17] Adak D, Natarajan E, Kumar S. Virtual element method for semilinear hyper bolic problems on polygonal meshes. Int J Comput Math 2019; 96(5): 971-991.
  33. https://doi.org/10.1080/00207160.2018.1475651
  34. [18] Adak D, Natarajan S, Natarajan E. Virtual element method for semilinear elliptic problems on polygonal meshes. Appl Numer Math 2019; 145: 175-187.
  35. https://doi.org/10.1016/j.apnum.2019.05.021
  36. [19] Beirao Da Veiga L, Lovadina C, Mora D. A virtual element method for elastic and inelastic problems on polytope meshes, Comput Methods Appl Mech Eng 2015; 295: 327-346.
  37. https://doi.org/10.1016/j.cma.2015.07.013
  38. [20] Beirao Da Veiga L, Brezzi F, Marini LD, Virtual elements for linear elasticity problems. SIAM J Numer Anal 2013; 51: 794-812.
  39. https://doi.org/10.1137/120874746
  40. [21] Gain AL, Talischi C, Paulino GH. On the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes. Comput Methods Appl Mech Eng 2014; 282: 132-160.
  41. https://doi.org/10.1016/j.cma.2014.05.005
  42. [22] Antonietti PF, Beirao Da Veiga L, Scacchi S, Verani M. A c1 virtual element method for the cahn-hilliard equation with polygonal meshes. SIAM J Numer Anal 2016; 54(1): 34-56.
  43. https://doi.org/10.1137/15M1008117
  44. [23] Lo Cascio M, Milazzo A, Benedetti I. Virtual element method for computational homogenization of composite and heterogeneous materials. Composite Structures 2020; 232: 111523.
  45. https://doi.org/10.1016/j.compstruct.2019.111523
  46. [24] Hart KA, Rimoli JJ. Microstructpy: A statistical microstructure mesh gener ator in python. SoftwareX 2020; 12: 100595.
  47. https://doi.org/10.1016/j.softx.2020.100595
  48. [25] Bistri D, Di Leo CV. Modeling of chemo-mechanical multi-particle interactions in composite electrodes for liquid and solid-state liion batteries. Journal of The Electrochemical Society 2021; 168(3): 030515.
  49. https://doi.org/10.1149/1945-7111/abe8ea
  50. [26] Zuanetti B, Luscher DJ, Ramos K, Bolme C. Unraveling the implications of finite specimen size on the interpretation of dynamic experiments for polycrys talline aluminum through direct numerical simulations. International Journal of Plasticity 2021; 103080.
  51. https://doi.org/10.1016/j.ijplas.2021.103080
  52. [27] Quey R, Renversade L. Optimal polyhedral description of 3d polycrystals: Method and application to statistical and synchrotron x-ray di”raction data, Computer Methods in Applied Mechanics and Engineering 2018; 330: 308-333.
  53. https://doi.org/10.1016/j.cma.2017.10.029
  54. [28] Rimoli J, Ortiz M. A duality-based method for generating geometric representa tions of polycrystals. International Journal for Numerical Methods in Engineering 2011; 86(9): 1069-1081.
  55. https://doi.org/10.1002/nme.3090
  56. [29] Grilli N, Tarleton E, Cocks AC. Neper2cae and pycigen: Scripts to generate polycrystals and interface elements in abaqus. SoftwareX 2021; 13: 100651.
  57. https://doi.org/10.1016/j.softx.2020.100651
  58. [30] Geuzaine C, Remacle J-F. Gmsh 2009.
  59. [31] Ahmad B, Alsaedi A, Brezzi F, Marini D, Russo LA. Equivalent projectors for virtual element methods. Comput Math with Appl 2013; 66: 376-391.
  60. https://doi.org/10.1016/j.camwa.2013.05.015
  61. [32] Quey R, Dawson P, Barbe F. Large-scale 3D random polycrystals for the fi nite element method: Generation, meshing and remeshing. Computer Methods in Applied Mechanics and Engineering 2011; 200: 1729-1745.
  62. https://doi.org/10.1016/j.cma.2011.01.002
  63. [33] Saini A, Unnikrishnakurup S, Krishnamurthy C, Balasubramanian K, Sundararajan T. Numerical study using finite element method for heat conduction on heterogeneous materials with varying volume fraction, shape and size of fillers International Journal of Thermal Sciences 2021; 159: 106545.
  64. https://doi.org/10.1016/j.ijthermalsci.2020.106545