Structural Engineering and Mechanics

  • 제92권4호
  • /
  • Pages.349-363
  • /
  • 2024
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

The enriched 2D solid finite elements in geometric nonlinear analysis

  • Hyung-Gyu Choi (Ground Technology Research Institute, Agency for Defense Development) ;
  • Chaemin Lee (Department of Safety Engineering, Chungbuk National Univesity) ;
  • San Kim (Department of Mechanical Convergence Engineering, Gyeongsang National University)
  • 투고 : 2024.09.20
  • 심사 : 2024.11.06
  • 발행 : 2024.11.25
⟨ 이전 논문 다음 논문 ⟩

초록

The primary strength of the enriched finite element method (enriched FEM) is its ability to enhance solution accuracy without mesh refinement. It also allows for the selective determination of cover function degrees based on desired accuracy. Furthermore, there is an adaptive enrichment strategy that applies enriched elements to targeted areas where accuracy may be lacking rather than across the entire domain, demonstrating its powerful use in engineering applications. However, its application to solid and structural problems encounters a linear dependence (LD) issue induced by using polynomial functions as cover functions. Recently, enriched finite elements that address the LD problem in linear analysis have been developed. In light of these advancements, this study is devoted to a robust extension of the enriched FEM to nonlinear analysis. We propose a nonlinear formulation of the enriched FEM, employing 3-node and 4-node 2D solid elements for demonstration. The formulation employs a total Lagrangian approach, allowing for large displacements and rotations. Numerical examples demonstrate that the enriched elements effectively improve solution accuracy and ensure stable convergence in nonlinear analysis. We also present results from adaptive enrichment to highlight its effectiveness.

키워드

과제정보

This research was funded by a National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT), grant number (NRF-2020R1G1A1006911). This work was supported by Chungbuk National University BK21 program (2023).

참고문헌

  1. Babuska, I. and Melenk, J.M. (1997), "The partition of unity method", Int. J. Numer. Meth. Eng., 40(4), 727-758. https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N.
  2. Bathe, K.J. (2016), Finite Element Procedures, 2nd Edition, 2014 and Higher Education Press, China.
  3. Bathe, K.J. and Dvorkin, E.N. (1986), "A formulation of general shell elements-the use of mixed interpolation of tensorial components", Int. J. Numer. Meth. Eng., 22(3), 697-722. https://doi.org/10.1002/nme.1620220312.
  4. Capuano, G. and Rimoli, J.J. (2019), "Smart finite elements: A novel machine learning application", Comput. Meth. Appl. Mech. Eng., 345, 363-381. https://doi.org/10.1016/j.cma.2018.10.046
  5. Chau-Dinh, T., Nguyen-Duy, Q. and Nguyen-Xuan, H. (2017), "Improvement on MITC3 plate finite element using edge-based strain smoothing enhancement for plate analysis", Acta Mechanica, 228(6), 2141-2163. https://doi.org/10.1007/s00707-017-1818-3. 
  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. Choi, H.G., Byun, Y.I., Song, C.K., Jun, M.B., Lee, C. and Kim, S. (2023), "A solution procedure to improve 3D solid finite element analysis with an enrichment scheme", Appl. Sci., 13(12), 7114. https://doi.org/10.3390/app13127114.
  8. Cook, R.D. (2007), Concepts and Applications of Finite Element Analysis, John Wiley & Sons.
  9. Daux, C., Moes, N., Dolbow, J., Sukumar, N. and Belytschko, T. (2000), "Arbitrary branched and intersecting cracks with the extended finite element method", Int. J. Numer. Meth. Eng., 48(12), 1741-1760. https://doi.org/10.1002/1097-0207(20000830)48:12<1741::AID-NME956>3.0.CO;2-L.
  10. Dolbow, J., Moes, N. and Belytschko, T. (2000), "Modeling fracture in Mindlin-Reissner plates with the extended finite element method", Int. J. Solid. Struct., 37(48-50), 7161-7183. https://doi.org/10.1016/S0020-7683(00)00194-3.
  11. Ham, S. and Bathe, K.J. (2012), "A finite element method enriched for wave propagation problems", Comput. Struct., 94, 1-12. https://doi.org/10.1016/j.compstruc.2012.01.001.
  12. Ibrahimbegovic, A. and Wilson, E.L. (1991), "A modified method of incompatible modes", Commun. Appl. Numer. Meth., 7(3), 187-194. https://doi.org/10.1002/cnm.1630070303.
  13. Jin, Y.F., Yuan, W.H., Yin, Z.Y. and Cheng, Y.M. (2020), "An edge-based strain smoothing particle finite element method for large deformation problems in geotechnical engineering", Int. J. Numer. Anal. Meth. Geomech., 44(7), 923-941. https://doi.org/10.1002/nag.3016.
  14. Jun, H. and Kim, S. (2018), "Benchmark tests of MITC triangular shell elements", Struct. Eng. Mech., 68(1), 17-38. https://doi.org/10.12989/sem.2018.68.1.017.
  15. Jung, J., Park, S. and Lee, C. (2022), "A posteriori error estimation via mode-based finite element formulation using deep learning", Struct. Eng. Mech., 83(2), 273-282. https://doi.org/10.12989/sem.2022.83.2.273.
  16. Kim, J. and Bathe, K.J. (2013), "The finite element method enriched by interpolation covers", Comput. Struct., 116, 35-49. https://doi.org/10.1016/j.compstruc.2012.10.001.
  17. 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.
  18. 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.
  19. Kwon, S.B., Bathe, K.J. and Noh, G. (2020), "An analysis of implicit time integration schemes for wave propagations", Comput. Struct., 230, 106188. https://doi.org/10.1016/j.compstruc.2019.106188.
  20. Lee, C. and Kim, S. (2020), "Towards improving finite element solutions automatically with enriched 2D solid elements", Struct. Eng. Mech., 76(3), 379-393. https://doi.org/10.12989/sem.2020.76.3.379.
  21. 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.
  22. 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.
  23. Lee, C. and Park, J. (2023), "Preconditioning for finite element methods with strain smoothing", Comput. Math. Appl., 130, 41-57. https://doi.org/10.1016/j.camwa.2022.11.018.
  24. 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.
  25. Lee, C., Lee, D.H. and Lee, P.S. (2022), "The strain-smoothed MITC3+ shell element in nonlinear analysis", Comput. Struct., 266, 106768. https://doi.org/10.1016/j.compstruc.2022.106768.
  26. Lee, C., Moon, M. and Park, J. (2022), "A gradient smoothing method and its multiscale variant for flows in heterogeneous porous media", Comput. Meth. Appl. Mech. Eng., 395, 115039. https://doi.org/10.1016/j.cma.2022.115039.
  27. Liu, G.R., Nguyen-Thoi, T. and Lam, K.Y. (2009), "An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids", J. Sound Vib., 320(4-5), 1100-1130. https://doi.org/10.1016/j.jsv.2008.08.027.
  28. Moes, N., Dolbow, J. and Belytschko, T. (1999), "A finite element method for crack growth without remeshing", Int. J. Numer. Meth. Eng., 46(1), 131-150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J.
  29. Simo, J.C. and Hughes, T. (1986), "On the variational foundations of assumed strain methods", J. Appl. Mech., 53(1), 51-54. https://doi.org/10.1115/1.3171737.
  30. Tian, R., Yagawa, G. and Terasaka, H. (2006), "Linear dependence problems of partition of unity-based generalized FEMs", Comput. Meth. Appl. Mech. Eng., 195(37-40), 4768-4782. https://doi.org/10.1016/j.cma.2005.06.030.
  31. Yu, M., Kim, S. and Noh, G. (2023), "Learned Gaussian quadrature for enriched solid finite elements", Comput. Meth. Appl. Mech. Eng., 414, 116188. https://doi.org/10.1016/j.cma.2023.116188.