Journal of the Architectural Institute of Korea Structure & Construction (대한건축학회논문집:구조계)

Architectural Institute of Korea (대한건축학회)
권호 표지
DOI QR코드

DOI QR Code

A Study on 3D Smoothed Finite Element Method for the Analysis of Nonlinear Nearly-incompressible Materials

비선형 비압축성 물질의 해석을 위한 3차원 Smoothed FEM

  • 이창계 (동아대 해양도시건설.방재연구소) ;
  • 이정재 (동아대 건축공학과)
  • Received : 2019.06.26
  • Accepted : 2019.08.28
  • Published : 2019.09.30
⟨ Previous Next ⟩

Abstract

This work presents the three-dimensional extended strain smoothing approach in the framework of finite element method, so-called smoothed finite element method (S-FEM) for quasi-incompressible hyperelastic materials undergoing the large deformations. The proposed method is known that the incompressible limits, such as over-estimation of stiffness and distorted mesh sensitivity, can be overcome in two dimensions. Therefore, in this paper, the idea of Cell-based, Edge-based and Node-based strain smoothing approaches is extended to three-dimensions. The construction of subcells and smoothing domains for each methods are explained. The smoothed strain-displacement matrix and the stiffness matrix are obtained on each smoothing domain in the same manner with two-dimensional S-FEM. Various numerical tests are studied to demonstrate the validity and accuracy of 3D-S-FEM. The obtained results are compared with analytical solutions to express the efficacy of the methods.

Keywords

Acknowledgement

Supported by : 한국연구재단

References

  1. Arnold, D., Brezzi, F., & Fortin, M. (1984). A stable finite element for the Stokes equations, Calcolo, 21(4), 337-344. https://doi.org/10.1007/BF02576171
  2. Belytschko, T., Moran B., & Liu, W.K. (1999). Nonlinear Finite Element for Continua and Structures, Wiley.
  3. Burtscher, S., Dorfmann, A., & Bergmeister, K. (1998). Mechanical aspects of high damping rubber, 2nd International PhD Symposium in Civil Engineering, Budapest.
  4. Cazes, F., & Meschke, G. (2013). An edge-based smoothed finite element method for 3D analysis of solid mechanics problems, International Journal for Numerical Methods in Engineering, 94(8), 715-739. https://doi.org/10.1002/nme.4472
  5. Chen, J.S., Wu, C.T., Yoon, S., & You, Y. (2001). A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering, 50(2), 435-466. https://doi.org/10.1002/1097-0207(20010120)50:2<435::AID-NME32>3.0.CO;2-A
  6. Feng, S.Z., Cui, X.Y., & Li, G.Y. (2013). Transient thermal mechanical analyses using a face-based smoothed finite element method (FS-FEM), International Journal of Thermal Sciences, 74, 95-103. https://doi.org/10.1016/j.ijthermalsci.2013.07.002
  7. Francis, A., Ortiz-Bernardin, A., Bordas, S.P.A., & Natarajan, S. (2017). Linear smoothed polyhedron and polyhedral finite elements, International Journal for Numerical Methods in Engineering, 109, 1263-1288. https://doi.org/10.1002/nme.5324
  8. He, Z.C., Liu, G.R., Zhong, Z.H., & Zhang, G.Y. (2010). Coupled analysis of 3D Structural-acoustic problems using the edge-based smoothed finite element method/finite element method, Finite Element in Analysis and Design, 46(12), 1114-1121. https://doi.org/10.1016/j.finel.2010.08.003
  9. Lee, C.K. (2015). Gradient Smoothing in Finite Elasticity: near-incompressibility, PhD thesis, Cardiff University.
  10. Lee, C.K., Angela Mihai, L., Hale, J.S., Kerfriden, P., & Bordas, S.P.A. (2017). Strain smoothing for compressible and nearly-incompressible finite elasticity, Computers & Structures, 2017, 182, 540-555. https://doi.org/10.1016/j.compstruc.2016.05.004
  11. Liu, G.R., Dai, K.Y., & Nguyen-Thoi, T. (2007). A smoothed finite element method for mechanics problems, Computational Mechanics, 39, 859-877. https://doi.org/10.1007/s00466-006-0075-4
  12. Liu, G.R., & Nguyen-Thoi, T. (2010). Smoothed Finite Element Methods, CRC Press.
  13. Liu, G.R., Nguyen-Thoi, T., Dai, K.Y., & Lam, K.Y. (2007). Theoretical aspects of the smoothed finite element method (SFEM). International Journal for Numerical Methods in Engineering, 78(8), 902-930.
  14. Liu, G.R., Nguyen-Thoi, T., & Lam, K.Y. (2009). An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, Journal of Sound and Vibration, 320, 1100-1130. https://doi.org/10.1016/j.jsv.2008.08.027
  15. Liu, G.R., Nguyen-Thoi, T., Nguyen-Xuan, H., & Lam, K.Y. (2009). A node based smoothed finite element method (NSFEM) for upper bound solution to solid mechanics problems, Computers & Structures, 87, 14-26. https://doi.org/10.1016/j.compstruc.2008.09.003
  16. Logg, A., Mardal, K.-A., & Wells, G.N. (2012). Automated Solution of Differential Equations by the Finite Element Method, Springer.
  17. Logg, A., & Wells, G.N. (2010). DOLFIN: Automated Finite Element Computing, ACM Transactions on Mathematical Software, 370.
  18. Nguyen-Thoi, T., Liu, G.R., Lam, K.Y., & Zhang, G.Y. (2009). A face-based smoothed finite element method (FS-FEM) for 3D linear and nonlinear solid mechanics using 4-node tetrahedral elements, International Journal for Numerical Methods in Engineering, 78, 324-353. https://doi.org/10.1002/nme.2491
  19. Nguyen-Thoi, T., Liu, G.R., Vu-Do, H.C., & Nguyen-Xuan, H. (2009). A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes, Computer Methods in Applied Mechanics and Engineering, 199(45-48), 3005-3027. https://doi.org/10.1016/j.cma.2010.06.017
  20. Oden, J.T. (2006). Finite Elements of Nonlinear Continua, Dover.
  21. Ong, T.H., Heaney, C.E., Lee, C.K., Nguyen-Xuan, H., & Liu, G.R. (2014). On stability, convergence and accuracy of bES-FEM and bFS-FEM for nearly incompressible elasticity, Computer Methods in Applied Mechanics and Engineering, 2014, 285, 315-345. https://doi.org/10.1016/j.cma.2014.10.022