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A three-dimensional hybrid smoothed finite element method (H-SFEM) for nonlinear solid mechanics problems

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Abstract

This paper presents a novel three-dimensional hybrid smoothed finite element method (H-SFEM) for solid mechanics problems. In 3D H-SFEM, the strain field is assumed to be the weighted average between compatible strains from the finite element method (FEM) and smoothed strains from the node-based smoothed FEM with a parameter α equipped into H-SFEM. By adjusting α, the upper and lower bound solutions in the strain energy norm and eigenfrequencies can always be obtained. The optimized α value in 3D H-SFEM using a tetrahedron mesh possesses a close-to-exact stiffness of the continuous system, and produces ultra-accurate solutions in terms of displacement, strain energy and eigenfrequencies in the linear and nonlinear problems. The novel domain-based selective scheme is proposed leading to a combined selective H-SFEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed 3D H-SFEM is an innovative and unique numerical method with its distinct features, which has great potential in the successful application for solid mechanics problems.

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References

  1. LeVeque R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, New York (2002)

    Book  MATH  Google Scholar 

  2. Li R.H., Chen Z.Y., Wu W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    Google Scholar 

  3. Wu T.W.: Boundary Element in Acoustics: Fundamentals and Computer Codes. WIT Press, Southampton (2000)

    Google Scholar 

  4. Liu Y.J.: Fast Multipole Boundary Element Method: Theory and Applications in Engineering. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  5. Zienkiewicz O.C., Taylor R.L.: The Finite Element Method, 5th edn. Butterworth-Heinemann, Oxford (2000)

    MATH  Google Scholar 

  6. Liu G.R., Quek S.S.: Finite Element Method: A Practical Course. Butterworth-Heinemann, Burlington (2003)

    Google Scholar 

  7. Liu G.R., Liu M.B.: Smoothed Particle Hydrodynamics: A Meshfree Practical Method. World Scientific, Singapore (2003)

    Book  Google Scholar 

  8. Belytschko T., Lu Y.Y., Gu L.: Element-free Galerkin methods. Int. J. Numer. Methods Eng. 37, 229–256 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Liu G.R.: Meshfree Methods: Moving Beyond the Finite Element Method, 2nd edn. CRC Press, Boca Raton (2009)

    Book  Google Scholar 

  10. Arnold D.N.: Mixed finite element methods for elliptic problems. Comput. Methods Appl. Mech. Eng. 82, 281–300 (1990)

    Article  MATH  Google Scholar 

  11. Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  MATH  Google Scholar 

  12. Liu G.R., Dai K.Y., Nguyen T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)

    Article  MATH  Google Scholar 

  13. Liu G.R., Nguyen T.T., Dai K.Y. et al.: Theoretical aspects of the smoothed finite element method (SFEM). Int. J. Numer. Methods Eng. 71, 902–930 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Malkus D.S., Hughes T.J.R.: Mixed finite element methods—reduced and selective integration techniques: a unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81 (1978)

    Article  MATH  Google Scholar 

  15. Pian T.H.H., Sumihara K.: Rational approach for assumed stress finite elements. Int. J. Numer. Methods Eng. 20, 1685–1695 (1984)

    Article  MATH  Google Scholar 

  16. Reese S., Wriggers P.: A stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Methods Eng. 48, 79–109 (2000)

    Article  MATH  Google Scholar 

  17. Rong T.Y., Lu A.Q.: Generalized mixed variational principles and solutions for ill-conditioned problems in computational mechanics—part I: volumetric locking. Comput. Meth. Appl. Mech. Eng. 191, 407–422 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Liu G.R., Gu Y.T.: An Introduction to Meshfree Methods and Their Programming. Springer, Dordrecht (2005)

    Google Scholar 

  19. Chen J.S., Wu C.T., Yoon S. et al.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Methods Eng. 50, 435–466 (2001)

    Article  MATH  Google Scholar 

  20. Liu G.R., Nguyen T.T., Lam K.Y.: Novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements. Comput. Meth. Appl. Mech. Eng. 197, 3883–3897 (2008)

    Article  MATH  Google Scholar 

  21. Zhang G.Y., Liu G.R., Wang Y.Y. et al.: A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems. Int. J. Numer. Methods Eng. 72(113), 1524–1543 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  22. Liu G.R., Nguyen T.T., Nguyen H.X. et al.: A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems. Comput. Struct. 87, 14–26 (2009)

    Article  Google Scholar 

  23. Liu G.R., Zhang G.Y.: Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM). Int. J. Numer. Methods Eng. 74, 1128–1161 (2008)

    Article  MATH  Google Scholar 

  24. Liu G.R., Nguyen T.T., Lam K.Y.: An edge-based smoothed finite element method (ES-FEM) for static and dynamic problems of solid mechanics. J. Sound Vib. 320, 1100–1130 (2009)

    Article  Google Scholar 

  25. He Z.C., Li G.Y., Zhong Z.H. et al.: An edge-based smoothed tetrahedron finite element method (ES-T-FEM) for 3D static and dynamic problems. Comput. Mech. 52, 221–236 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Liu G.R.: A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods. Int. J. Comput. Methods. 5, 199–236 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu G.R.: A weakened weak (W2) form for a unified formulation of compatible and incompatible methods, Part I: Theory and Part II: Applications to solid mechanics problems. Int. J. Numer. Methods Eng. 81, 1093–1156 (2010)

    MATH  Google Scholar 

  28. Xu X., Gu Y.T., Liu G.R.: A hybrid smoothed finite element method (H-SFEM) to solid mechanics problems. Int. J. Comput. Methods. 10, 1–17 (2013)

    Article  MathSciNet  Google Scholar 

  29. Bathe K.J.: Finite Element Procedures. MIT Press, Prentice-Hall: Cambridge (1996)

    Google Scholar 

  30. Reddy J.N.: An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, Oxford (2004)

    Book  MATH  Google Scholar 

  31. Timoshenko S.P., Goodier J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  32. Hughes T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, People’s Republic of China

    Eric Li & Z. C. He

  2. College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun, 130012, People’s Republic of China

    Xu Xu

  3. School of Aerospace Systems, University of Cincinnati, Cincinnati, OH, 45221-0070, USA

    G. R. Liu

  4. School of Engineering Systems, Queensland University of Technology, Brisbane, QLD, 4001, Australia

    Y. T. Gu

Authors
  1. Eric Li
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  2. Z. C. He
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  3. Xu Xu
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  4. G. R. Liu
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  5. Y. T. Gu
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Correspondence to Eric Li.

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Li, E., He, Z.C., Xu, X. et al. A three-dimensional hybrid smoothed finite element method (H-SFEM) for nonlinear solid mechanics problems. Acta Mech 226, 4223–4245 (2015). https://doi.org/10.1007/s00707-015-1456-6

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  • DOI: https://doi.org/10.1007/s00707-015-1456-6

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