Structural Engineering and Mechanics

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Strain-smoothed polygonal finite elements

  • Hoontae Jung (Department of Mechanical Engineering, Korean Advanced Institute for Science and Technology) ;
  • Chaemin Lee (Department of Mechanical Engineering, Korean Advanced Institute for Science and Technology) ;
  • Phill-Seung Lee (Department of Mechanical Engineering, Korean Advanced Institute for Science and Technology)
  • Received : 2023.01.19
  • Accepted : 2023.03.22
  • Published : 2023.05.10
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Abstract

Herein, we present effective polygonal finite elements to which the strain-smoothed element (SSE) method is applied. Recently, the SSE method has been developed for conventional triangular and quadrilateral finite elements; furthermore, it has been shown to improve the performance of finite elements. Polygonal elements enable various applications through flexible mesh handling; however, further development is still required to use them more effectively in engineering practice. In this study, piecewise linear shape functions are adopted, the SSE method is applied through the triangulation of polygonal elements, and a smoothed strain field is constructed within the element. The strain-smoothed polygonal elements pass basic tests and show improved convergence behaviors in various numerical problems.

Keywords

Acknowledgement

This work was supported by the Nuclear Safety Research Program through the Korea Foundation Of Nuclear Safety (KoFONS) using the financial resource granted by the Nuclear Safety and Security Commission (NSSC) of the Republic of Korea (No. 2106045).

References

  1. Bathe, K.J. (1996), Finite Element Procedures, Prentice Hall, New 
  2. Beirao da Veiga, L., Brezzi, F. and Marini, L.D. (2013), "Virtual elements for linear elasticity problems", SIAM J. Numer. Anal., 51(2), 794-812. https://doi.org/10.1137/120874746. 
  3. Biabanaki, S.O.R. and Khoei, A.R. (2012), "A polygonal finite element method for modeling arbitrary interfaces in large deformation problems", Comput. Mech., 50, 19-33. https://doi.org/10.1007/s00466-011-0668-4. 
  4. Biabanaki, S.O.R., Khoei, A.R. and Wriggers, P. (2014), "Polygonal finite element methods for contact-impact problems on non-conformal meshes", Comput. Meth. Appl. Mech. Eng., 269, 198-221. https://doi.org/10.1016/j.cma.2013.10.025. 
  5. Chen, J.S., Wu, C.T., Yoon, S. and You, Y. (2001), "Stabilized conforming nodal integration for Galerkin mesh-free methods", Int. J. Numer. Meth. Eng., 50, 435-466. https://doi.org/10.1002/1097-0207(20010120)50:2<435::AIDNME32>3.0.CO;2-A. 
  6. Choi, H.G. and Lee, P.S. (2023), "Towards improving the 2D-MITC4 element for analysis of plane stress and strain problems", Comput. Struct., 275, 106933. https://doi.org/10.1016/j.compstruc.2022.106933. 
  7. Cook, R.D. (2007), Concepts and Applications of Finite Element Analysis, John Wiley & Sons, New York. 
  8. Dai, K.Y., Liu, G.R. and Nguyen, T.T. (2007), "An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics", Finite Elem. Anal. Des., 43, 847-860. https://doi.org/10.1016/j.finel.2007.05.009. 
  9. Floater, M.S. (2003), "Mean value coordinates", Comput. Aid. Geom. Des., 20, 19-27. https://doi.org/10.1016/S0167-8396(03)00002-5. 
  10. Ho-Nguyen-Tan, T. and Kim, H.G. (2018), "A new strategy for finite-element analysis of shell structures using trimmed quadrilateral shell meshes: A paving and cutting algorithm and a pentagonal shell element", Int. J. Numer. Meth. Eng., 114, 1-27. https://doi.org/10.1002/nme.5730. 
  11. Huang, X., Ding, W., Zhu, Y. and Yang, C. (2017), "Crack propagation simulation of polycrystalline cubic boron nitride abrasive materials based on cohesive element method", Comput. Mater. Sci., 138, 302-314. https://doi.org/10.1016/j.commatsci.2017.07.007. 
  12. Hughes, T.J.R. (2000), The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publications, Mineola, New York. 
  13. Jeon, H.M., Lee, P.S. and Bathe, K.J. (2014), "The MITC3 shell finite element enriched by interpolation covers", Comput. Struct., 134, 128-142. https://doi.org/10.1016/j.compstruc.2013.12.003. 
  14. Jun, H., Yoon, K., Lee, P.S. and Bathe, K.J. (2018), "The MITC3+ shell element enriched in membrane displacements by interpolation covers", Comput. Meth. Appl. Mech. Eng., 337, 458-480. https://doi.org/10.1016/j.cma.2018.04.007. 
  15. Jung, J., Jun, H. and Lee, P.S. (2022), "Self-updated four-node finite element using deep learning", Comput. Mech., 69, 23-44. https://doi.org/10.1007/s00466-021-02081-7. 
  16. Jung, J., Yoon, K. and Lee, P.S. (2020), "Deep learned finite elements", Comput. Meth. Appl. Mech. Eng., 372, 113401. https://doi.org/10.1016/j.cma.2020.113401. 
  17. Khoei, A.R., Yasbolaghi, R. and Biabanaki, S.O.R (2015a), "A polygonal finite element method for modeling crack propagation with minimum remeshing", Int. J. Fract., 194, 123-148. https://doi.org/10.1007/s10704-015-0044-z. 
  18. Khoei, A.R., Yasbolaghi, R. and Biabanaki, S.O.R (2015b), "A polygonal-FEM technique in modeling large sliding contact on non-conformal meshes: A study on polygonal shape functions", Eng. Comput., 32(5), 1391-1431. https://doi.org/10.1108/EC04-2014-0070. 
  19. Kim, S. and Lee, P.S. (2018), "A new enriched 4-node 2D solid finite element free from the linear dependence problem", Comput. Struct., 202, 25-43. https://doi.org/10.1016/j.compstruc.2018.03.001. 
  20. Kim, S. and Lee, P.S. (2019), "New enriched 3D solid finite elements: 8-node hexahedral, 6-node prismatic, and 5-node pyramidal elements", Comput. Struct., 216, 40-63. https://doi.org/10.1016/j.compstruc.2018.12.002. 
  21. Ko, Y. and Lee, P.S. (2017), "A 6-node triangular solid-shell element for linear and nonlinear analysis", Int. J. Numer. Meth. Eng., 111(13), 1203-1230. https://doi.org/10.1002/nme.5498. 
  22. Ko, Y., Lee, P.S. and Bathe K.J. (2016), "The MITC4+ shell element and its performance", Comput. Struct., 169, 57-68. https://doi.org/10.1016/j.compstruc.2016.03.002. 
  23. Ko, Y., Lee, P.S. and Bathe K.J. (2017a), "A new MITC4+ shell element", Comput. Struct., 182, 404-418. https://doi.org/10.1016/j.compstruc.2016.11.004. 
  24. Ko, Y., Lee, P.S. and Bathe K.J. (2017b), "A new 4-node MITC element for analysis of two-dimensional solids and its formulation in a shell element", Comput. Struct., 192, 34-49. https://doi.org/10.1016/j.compstruc.2017.07.003. 
  25. Lee, C. and Lee, P.S. (2018), "A new strain smoothing method for triangular and tetrahedral finite elements", Comput. Meth. Appl. Mech. Eng., 341, 939-955. https://doi.org/10.1016/j.cma.2018.07.022. 
  26. Lee, C. and Lee, P.S. (2019), "The strain-smoothed MITC3+ shell finite element", Comput. Struct., 223, 106096. https://doi.org/10.1016/j.compstruc.2019.07.005. 
  27. Lee, C. and Park, J. (2021), "A variational framework for the strain-smoothed element method", Comput. Math. Appl., 94, 76-93. https://doi.org/10.1016/j.camwa.2021.04.025. 
  28. Lee, C., Kim, S. and Lee, P.S. (2021), "The strain-smoothed 4-node quadrilateral finite element", Comput. Meth. Appl. Mech. Eng., 373, 113481. https://doi.org/10.1016/j.cma.2020.113481. 
  29. Lee, P.S. and Bathe, K.J. (2004), "Development of MITC isotropic triangular shell finite elements", Comput. Struct., 82(11-12), 945-962. https://doi.org/10.1016/j.compstruc.2004.02.004. 
  30. Lee, Y., Lee, P.S. and Bathe, K.J. (2014), "The MITC3+ shell element and its performance", Comput. Struct., 138, 12-23. https://doi.org/10.1016/j.compstruc.2014.02.005. 
  31. Lee, Y., Yoon, K. and Lee, P.S. (2012), "Improving the MITC3 shell finite element by using the Hellinger-Reissner principle", Comput. Struct., 110-111, 93-106. https://doi.org/10.1016/j.compstruc.2012.07.004. 
  32. Lien, S. and Kajiya, J.T. (1984), "A symbolic method for calculating the integral properties of arbitrary nonconvex polyhedral", IEEE Comput. Graph. Appl., 4(10), 35-42. https://doi.org/10.1109/MCG.1984.6429334. 
  33. Liu, F., Yu, C. and Yang, Y. (2018), "An edge-based smoothed numerical manifold method and its application to static, free and forced vibration analyses", Eng. Anal. Bound. Elem., 86, 19-30. https://doi.org/10.1016/j.enganabound.2017.10.006. 
  34. Liu, G.R., Dai, K.Y. and Nguyen, T.T. (2007), "A smoothed finite element method for mechanics problems", Comput. Mech., 39, 859-877. https://doi.org/10.1007/s00466-006-0075-4. 
  35. Liu, G.R., Nguyen-Thoi, T. and Lam, K.Y. (2009b), "An edgebased smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids", J. Sound Vib., 320, 1100-1130. https://doi.org/10.1016/j.jsv.2008.08.027. 
  36. Liu, G.R., Nguyen-Thoi, T., Nguyen-Xuan, H. and Lam, K.Y. (2009a), "A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems", Comput. Struct., 87, 14-26. https://doi.org/10.1016/j.compstruc.2008.09.003. 
  37. Natarajan, S., Bordas, S.P.A. and Ooi, E.T. (2015), "Virtual and smoothed finite elements: A connection and its application to polygonal/polyhedral finite element methods", Int. J. Numer. Meth. Eng., 104(13), 1173-1199. https://doi.org/10.1002/nme.4965. 
  38. Natarajan, S., Bordas, S.P.A. and Roy Mahapatra, D. (2009), "Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping", Int. J. Numer. Meth. Eng., 80, 103-134. https://doi.org/10.1002/nme.2589. 
  39. Nguyen, N.V., Lee, D., Nguyen-Xuan, H. and Lee, J. (2020), "A polygonal finite element approach for fatigue crack growth analysis of interfacial cracks", Theor. Appl. Fract. Mech., 108, 102576. https://doi.org/10.1016/j.tafmec.2020.102576. 
  40. Nguyen-Thanh, N., Rabczuk, T., Nguyen-Xuan, H. and Bordas, S.P.A. (2008), "A smoothed finite element method for shell analysis", Comput. Meth. Appl. Mech. Eng., 198, 165-177. https://doi.org/10.1016/j.cma.2008.05.029. 
  41. Nguyen-Thoi, T., Liu, G.R. and Nguyen-Xuan, H. (2011), "An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics", Int. J. Numer. Meth. Biomed. Eng., 27, 1446-1472. https://doi.org/10.1002/cnm.1375. 
  42. Nguyen-Thoi, T., Liu, G.R., Lam, K.Y. and Zhang, G.Y. (2009), "A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically non-linear solid mechanics problems using 4-node tetrahedral elements", Int. J. Numer. Meth. Eng., 78, 324-353. https://doi.org/10.1002/nme.2491. 
  43. Nguyen-Xuan, H. (2017), "A polygonal finite element method for plate analysis", Comput. Struct., 188, 45-62. https://doi.org/10.1016/j.compstruc.2017.04.002. 
  44. Nguyen-Xuan, H., Liu, G.R., Bordas, S.P.A., Natarajan, S. and Rabczuk, T. (2013), "An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order", Comput. Meth. Appl. Mech. Eng., 253, 252-273. https://doi.org/10.1016/j.cma.2012.07.017. 
  45. Sukumar, N. and Tabarraei, A. (2004), "Conforming polygonal finite elements", Int. J. Numer. Meth. Eng., 61, 2045-2066. https://doi.org/10.1002/nme.1141. 
  46. Tabarraei, A. and Sukumar, N. (2006), "Application of polygonal finite elements in linear elasticity", Int. J. Comput. Meth., 03, 503-520. https://doi.org/10.1142/S021987620600117X. 
  47. Talischi, C., Paulino, G.H., Pereira, A. and Menezes, I.F.M. (2012), "PolyMesher: A general-purpose mesh generator for polygonal elements written in MATLAB", Struct. Multidisc. Optim., 45, 309-328. https://doi.org/10.1007/s00158-011-0706-z. 
  48. Talischi, C., Pereira, A., Paulino, G.H., Menezes, I.F.M. and Carvalho, M.S. (2014), "Polygonal finite elements for incompressible fluid flow", Int. J. Numer. Meth. Fluid., 74, 134-151. https://doi.org/10.1002/fld.3843. 
  49. Thomes, R.L. and Menandro, F.C.M. (2020), "Polygonal finite element: A comparison of the stiffness matrix integration methods". Appl. Math. Comput., 375, 125089. https://doi.org/10.1016/j.amc.2020.125089. 
  50. Timoshenko, S. and Goodier, J.N. (1970), Theory of Elasticity, 3rd Edition, McGraw-Hill, New York. 
  51. Wachspress, E.L. (1975), A Rational Finite Element Basis, Academic Press, New York. 
  52. Yan, D.M., Wang, W., Levy, B. and Liu, Y. (2013), "Efficient computation of clipped Voronoi diagram for mesh generation", Comput. Aid. Des., 45, 843-852. https://doi.org/10.1016/j.cad.2011.09.004. York