Table of Contents
ISRN Mechanical Engineering
Volume 2011, Article ID 967512, 11 pages
http://dx.doi.org/10.5402/2011/967512
Research Article

The Meshless Local Petrov-Galerkin Method for Large Deformation Analysis of Hyperelastic Materials

State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha 410082, China

Received 4 May 2011; Accepted 7 June 2011

Academic Editors: Ö. Civalek and D. Liao

Copyright © 2011 De'an Hu and Zhanhua Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. E. Green and W. Zerna, Theoretical Elasticity, Clarendon, Oxford, UK, 1966.
  2. D. S. Malkus and T. J. R. Hughes, “Mixed finite element methods reduced and selective integration techniques: a unification of concept,” Computer Methods in Applied Mechanics and Engineering, vol. 15, no. 1, pp. 63–81, 1978. View at Google Scholar · View at Scopus
  3. M. A. Crisfield, Non-Linear Finite Element Analysis of Solids and Structures, Thomson Press, London, UK, 1997.
  4. H. J. Al-Gahtani and N. J. Altiero, “Application of the boundary element method to rubber-like elasticity,” Applied Mathematical Modelling, vol. 20, no. 9, pp. 654–661, 1996. View at Publisher · View at Google Scholar · View at Scopus
  5. M. T. Bayliss and A. El-Zafrany, “Numerical modelling of elastomers using the boundary element method,” Communications in Numerical Methods in Engineering, vol. 20, no. 10, pp. 789–799, 2004. View at Publisher · View at Google Scholar · View at Scopus
  6. R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics: theory and application to non-spherical stars,” Monthly Notices of the Royal Astronomical Society, vol. 181, pp. 375–389, 1977. View at Google Scholar
  7. B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: diffuse approximation and diffuse elements,” Computational Mechanics, vol. 10, no. 5, pp. 307–318, 1994. View at Publisher · View at Google Scholar · View at Scopus
  8. T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229–256, 1994. View at Google Scholar · View at Scopus
  9. C. Duarte and J. T. Oden, “Hp-cloud a meshless method to solve boundary-value problems,” Computer Methods in Applied Mechanics Engineering, vol. 139, pp. 237–262, 1996. View at Google Scholar
  10. W. K. Liu, Y. Chen, R. A. Uras, and C. T. Chang, “Generalized multiple scale reproducing kernel particle methods,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 91–157, 1996. View at Google Scholar · View at Scopus
  11. N. Sukumar, B. Moran, and T. Belytschko, “The natural element method in solid mechanics,” International Journal for Numerical Methods in Engineering, vol. 43, no. 5, pp. 839–887, 1998. View at Google Scholar · View at Scopus
  12. H. Wendland, “Meshless Galerkin methods using radial basis functions,” Mathematics of Computation, vol. 68, no. 228, pp. 1521–1531, 1999. View at Google Scholar · View at Scopus
  13. S. N. Atluri and T. Zhu, “A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics,” Computational Mechanics, vol. 22, no. 2, pp. 117–127, 1998. View at Google Scholar · View at Scopus
  14. H. Lin and S. N. Atluri, “Meshless local Petrov-Galerkin (MLPG) method for convection-diffusion problems,” Computer Modeling in Engineering & Sciences, vol. 1, no. 2, pp. 45–60, 2000. View at Google Scholar · View at Scopus
  15. S. N. Atluri and T. Zhu, “A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computer modeling and simulation,” Computational Modeling and Simulation in Engineering, vol. 3, no. 3, pp. 187–196, 1998. View at Google Scholar
  16. H. G. Kim and S. N. Atluri, “Arbitrary placement of secondary nodes, and error control, in the meshless local Petrov-Galerkin (MLPG) method,” Computer Modeling in Engineering & Sciences, vol. 1, no. 3, pp. 11–32, 2000. View at Google Scholar · View at Scopus
  17. H. Lin and S. N. Atluri, “The meshless local Petrov-Galerkin (MLPG) method for solving incompressible Navier-stokes equations,” Computer Modeling in Engineering & Sciences, vol. 2, no. 2, pp. 117–142, 2001. View at Google Scholar · View at Scopus
  18. S. Y. Long, “A local Petrov-Galerkin method for the elasticity problem,” Chinese Journal of Theoretical and Applied Mechanics, vol. 33, no. 4, pp. 508–518, 2001. View at Google Scholar
  19. S. Y. Long, “A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate,” Computer Modeling in Engineering & Sciences, vol. 3, no. 1, pp. 53–63, 2002. View at Google Scholar · View at Scopus
  20. Y. T. Gu and G. R. Liu, “A meshless local Petrov-Galerkin (MLPG) formulation for static and free vibration analysis of thin plates,” Computer Modeling in Engineering & Sciences, vol. 2, no. 4, pp. 463–476, 2001. View at Google Scholar · View at Scopus
  21. H. K. Ching and R. C. Batra, “Determination of crack tip fields in linear elastostatics by the Meshless Local Petrov-Galerkin (MLPG) method,” Computer Modeling in Engineering & Sciences, vol. 2, no. 2, pp. 273–290, 2001. View at Google Scholar · View at Scopus
  22. R. C. Batra and H. K. Ching, “Analysis of elastodynamic deformation near a Crack/Nothc tip by the meshless local Petrov-Galerkin (MLPG) method,” Computer Modeling in Engineering & Sciences, vol. 3, pp. 717–730, 2002. View at Google Scholar
  23. J. Sladek, V. Sladek, and S. N. Atluri, “Meshless local Petrov-Galerkin method in anisotropic elasticity,” Computer Modeling in Engineering & Sciences, vol. 6, no. 5, pp. 477–489, 2004. View at Google Scholar · View at Scopus
  24. J. Sladek, V. Sladek, J. Krivacek, and Ch. Zhang, “Meshless local Petrov-Galerkin method for stress and crack analysis in 3-D axisymmetric FGM bodies,” Computer Modeling in Engineering & Sciences, vol. 8, no. 3, pp. 259–270, 2005. View at Google Scholar · View at Scopus
  25. S. N. Atluri and S. P. Shen, “The Meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods,” Computer Modeling in Engineering & Sciences, vol. 3, no. 1, pp. 11–51, 2002. View at Google Scholar · View at Scopus
  26. G. R. Liu, Meshfree Methods: Moving Beyond the Finite Element Method, CRC Press, Singapore, 2003.
  27. J. R. Xiao and M. A. McCarthy, “A local heaviside weighted meshless method for two-dimensional solids using radial basis functions,” Computational Mechanics, vol. 31, no. 3-4, pp. 301–315, 2003. View at Google Scholar · View at Scopus
  28. J. S. Chen and C. H. Pan, “A pressure projection method for nearly incompressible rubber hyperelasticity, part I: theory,” Journal of Applied Mechanics, vol. 63, no. 4, pp. 862–868, 1996. View at Google Scholar · View at Scopus
  29. J. S. Chen, H. P. Wang, S. Yoon, and Y. You, “Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact,” Computational Mechanics, vol. 25, no. 2, pp. 137–156, 2000. View at Google Scholar · View at Scopus
  30. J. S. Chen, C. T. Wu, and C. H. Pan, “A pressure projection method for nearly incompressible rubber hyperelasticity, part II: applications,” Journal of Applied Mechanics, vol. 63, no. 4, pp. 869–876, 1996. View at Google Scholar · View at Scopus
  31. W. K. Liu, T. Belytschko, and J. S. Chen, “Nonlinear versions of flexurally superconvergent elements,” Computer Methods in Applied Mechanics and Engineering, vol. 71, no. 3, pp. 241–258, 1988. View at Google Scholar · View at Scopus
  32. T. Y. P. Chang, A. F. Saleeb, and G. Li, “Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle,” Computational Mechanics, vol. 8, no. 4, pp. 221–233, 1991. View at Publisher · View at Google Scholar · View at Scopus