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Eliminating Shear-Locking in Meshless Methods: A Critical Overview and a New Framework for Structural Theories

  • Conference paper
Advances in Meshfree Techniques

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 5))

Abstract

Meshfree methods are recent additions to the family of numerical techniques for the solution of partial differential equations. Issues like shear locking are yet to be clarified. Earlier expectations that these methods would be free from locking problems, were not confirmed. In particular, it is shown that, for the case of moderately thick structural beam and plate theories, if the same approximation is used for both displacement and rotation(s), then shear locking occurs.

Considering that reduced integration is not an option in the present context, a review of the available methodologies is presented. It is proven that one of these methodologies, that of consistent fields, i.e., where the rotation(s) approximation(s) are obtained from the displacement approximation, always leads to linearly dependent approximations.

As an alternative, a shear deformable framework for structural theories is presented where, instead of the rotations, the shear strain(s) are approximated in addition to that of the displacement field. The particularization for beams and plates is presented and a comparison with the traditional thin (irreducible) and moderately thick (shear deformable) theories is made. One aspect to emphasize is the presence of second order differentials in the dual pair of equilibrium and compatibility operators.

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References

  1. S. N. Atluri, H.-G. Kim, and J. Y. Cho. A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods. Computational Mechanics, 24(5):348–372, 1999.

    Article  MATH  Google Scholar 

  2. S. Beissel and T. Belytschko. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 139(1–4):49–74, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  3. T. Belytschko, Y. Y. Lu, and L. Gu. Element-Free Galerkin Methods. International Journal for Numerical Methods in Engineering, 37(2):229–256, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  4. J. Y. Cho and S. N. Atluri. Analysis of shear flexible beams, using the meshless local Petrov-Galerking method, based on a locking-free formulation. Engineering Computations, 18(1–2):215–240, 2001.

    Article  MATH  Google Scholar 

  5. S. De and K.-J. Bathe. Displacement/pressure mixed interpolation in the method of finite spheres. International Journal for Numerical Methods in Engineering, 51(3):275–292, 2001.

    Article  MATH  Google Scholar 

  6. P. de T. R. Mendonça, C. S. de Barcellos, and A. Duarte. Investigations on the hp-cloud method by solving Timoshenko beam problems. Computational Mechanics, 25(2–3):286–295, 2000.

    MATH  Google Scholar 

  7. J. Dolbow and T. Belytschko. Volumetric locking in the element free Galerkin method. International Journal for Numerical Methods in Engineering, 46(6):925–942, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. M. Donning and W. K. Liu. Meshless methods for shear-deformable beams and plates. Computer Methods in Applied Mechanics and Engineering, 152(1–2):47–71, 1998.

    Article  MATH  Google Scholar 

  9. C. A. Duarte and J. Tinsley Oden. H-p clouds — An h-p meshless method. Numerical Methods for Partial Differential Equations, 12(6):673–705, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  10. O. Garcia, E. A. Fancello, C. S. de Barcellos, and C. A. Duarte. hp-Clouds in Mindlin’s thick plate model. International Journal for Numerical Methods in Engineering, 47(8):1381–1400, 2000.

    Article  MATH  Google Scholar 

  11. T. J. R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola, NY, 2000.

    MATH  Google Scholar 

  12. T. J. R. Hughes and T. E. Tezduyar. Finite elements based upon mindlin plate theory with particular reference to the four-node isoparametric element. Journal of Applied Mechanics, Transactions of the ASME, 48(3):587–596, 1981.

    Article  MATH  Google Scholar 

  13. W. Kanok-Nukulchai, W. Barry, K. S.-Yasoontorn, and P. H. Bouillard. On elimination of shear locking in the element-free Galerkin method. International Journal for Numerical Methods in Engineering, 52(7):705–725, 2001.

    Article  MATH  Google Scholar 

  14. P. Lancaster and K. Šalkauskas. Surfaces genarated by moving least squares methods. Mathematics of Computation, 37(155):141–158, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  15. W. K. Liu, S. Jun, and Y. F. Zhang. Reproducing kernel particle methods. International Journal for Numerical Methods in Engineering, 20(8–9):1081–1106, 1995.

    MATH  MathSciNet  Google Scholar 

  16. W.-K. Liu, S. Li, and T. Belytschko. Moving least-square reproducing kernel methods (I): Methodology and convergence. Computer Methods in Applied Mechanics and Engineering, 143(1–2):113–154, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  17. Y. Y. Lu, T. Belytschko, and L. Gu. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 113(3–4):397–414, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. S. Malkus and T. J. R. Hughes. Mixed finite element methods-reduced and selective integration techniques: A unification of concepts. Computer Methods in Applied Mechanics and Engineering, 15(1):63–81, 1978.

    Article  MATH  Google Scholar 

  19. W. D. Pilkey and W. Wunderlich. Mechanics of Structures: Variational and Computational Methods, CRC Press, second edition, 2003.

    Google Scholar 

  20. G. Prathap. The Finite Element Method in Structural Mechanics: Principles and Practice of Design of Field-Consistent Elements for Structural and Solid Mechanics, Solid Mechanics and Its Applications, Vol. 24, Kluwer Academic Publishers, 1993.

    Google Scholar 

  21. J. C. Simo and M. S. Rifai. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering, 29(8):1595–1638, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  22. N. Sukumar, B. Moran, and T. Belytschko. The natural element method in solid mechanics. International Journal for Numerical Methods in Engineering, 43(5):839–887, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  23. C. Tiago and V. Leitão. On the procedures to eliminate shear locking in meshless methods. In V. Leitão, C. Alves, and C. Duarte (Eds), ECCOMAS Thematic Conference on Meshless Methods, Lisbon, Portugal, 2005, pp. A14.1–A14.8.

    Google Scholar 

  24. C. Tiago and P. Pimenta. Geometrically exact analysis of space frames by a meshless method. In V. Leitão, C. Alves, and C. Duarte (Eds), ECCOMAS Thematic Conference on Meshless Methods, Lisbon, Portugal, 2005, pp. C44.1–C44.8

    Google Scholar 

  25. C. Tiago and P. M. Pimenta. Geometrically exact analysis of shells by a meshless approach. In C. A. Mota Soares (Ed.), Third European Conference on Computational Mechanics — Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, June 2006.

    Google Scholar 

  26. D. Wang and J.-S. Chen. Locking-free stabilized conforming nodal integration for meshfree Mindlin-Reissner plate formulation. Computer Methods in Applied Mechanics and Engineering, 193(12–14):1065–1083, 2004.

    Article  MATH  Google Scholar 

  27. O. C. Zienkiewicz, R. L. Taylor, and J. M. Too. Reduced integration technique in general analysis of plates and shells. International Journal for Numerical Methods in Engineering, 3(2):275–290, 1971.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Instituto Superior Técnico, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Carlos Tiago & Vitor M. A. Leitão

Authors
  1. Carlos Tiago
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  2. Vitor M. A. Leitão
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Editor information

Editors and Affiliations

  1. Instituto Superior Técnico, Lisboa, Portugal

    V. M. A. Leitão  & C. J. S. Alves  & 

  2. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    C. Armando Duarte

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Tiago, C., Leitão, V.M.A. (2007). Eliminating Shear-Locking in Meshless Methods: A Critical Overview and a New Framework for Structural Theories. In: Leitão, V.M.A., Alves, C.J.S., Armando Duarte, C. (eds) Advances in Meshfree Techniques. Computational Methods in Applied Sciences, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6095-3_7

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  • DOI: https://doi.org/10.1007/978-1-4020-6095-3_7

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-6094-6

  • Online ISBN: 978-1-4020-6095-3

  • eBook Packages: EngineeringEngineering (R0)

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