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Mathematical Problems in Engineering
Volume 2014, Article ID 323945, 13 pages
http://dx.doi.org/10.1155/2014/323945
Research Article

An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions

1Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China
2Faculty of Science, Ningbo University of Technology, Ningbo 315016, China

Received 23 December 2013; Accepted 21 January 2014; Published 2 March 2014

Academic Editor: Miaojuan Peng

Copyright © 2014 F. X. Sun et al. 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. Z. Zhang, K. M. Liew, Y. Cheng, and Y. Y. Lee, “Analyzing 2D fracture problems with the improved element-free Galerkin method,” Engineering Analysis with Boundary Elements, vol. 32, no. 3, pp. 241–250, 2008. View at Publisher · View at Google Scholar · View at Scopus
  2. K. M. Liew, J. Ren, and J. N. Reddy, “Numerical simulation of thermomechanical behaviours of shape memory alloys via a non-linear mesh-free Galerkin formulation,” International Journal for Numerical Methods in Engineering, vol. 63, no. 7, pp. 1014–1040, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. D. Li, F. Bai, Y. Cheng, and K. M. Liew, “A novel complex variable element-free Galerkin method for two-dimensional large deformation problems,” Computer Methods in Applied Mechanics and Engineering, vol. 233–236, pp. 1–10, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. W. Ju-Feng, S. Feng-Xin, and C. Rong-Jun, “Element-free Galerkin method for a kind of KdV equation,” Chinese Physics B, vol. 19, no. 6, Article ID 060201, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. Z. Zhang, D.-M. Li, Y.-M. Cheng, and K. M. Liew, “The improved element-free Galerkin method for three-dimensional wave equation,” Acta Mechanica Sinica, vol. 28, no. 3, pp. 808–818, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. Z. Zhang, J. F. Wang, Y. M. Cheng, and K. M. Liew, “The improved element-free Galerkin method for three-dimensional transient heat conduction problems,” Science China Physics, Mechanics & Astronomy, vol. 56, no. 8, pp. 1568–1580, 2013. View at Google Scholar
  8. Z. Zhang, S. Y. Hao, K. M. Liew, and Y. M. Cheng, “The improved element-free Galerkin method for two-dimensional elastodynamics problems,” Engineering Analysis with Boundary Elements, vol. 37, no. 12, pp. 1576–1584, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Chen and Y. M. Cheng, “Reproducing kernel particle method with complex variables for elasticity,” Acta Physica Sinica, vol. 57, no. 1, pp. 1–10, 2008. View at Google Scholar · View at MathSciNet
  11. L. Chen and Y. M. Cheng, “Complex variable reproducing kernel particle method for transient heat conduction problems,” Acta Physica Sinica, vol. 57, no. 10, pp. 6047–6055, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. Chen and Y. M. Cheng, “The complex variable reproducing kernel particle method for elasto-plasticity problems,” Science China, vol. 53, no. 5, pp. 954–965, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. L. Chen and Y.-M. Cheng, “The complex variable reproducing kernel particle method for two-dimensional elastodynamics,” Chinese Physics B, vol. 19, no. 9, Article ID 090204, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. Y. M. Cheng, M. J. Peng, and J. H. Li, “The complex variable moving least-square approximation and its application,” Chinese Journal of Theoretical and Applied Mechanics, vol. 37, no. 6, pp. 719–723, 2005. View at Google Scholar · View at MathSciNet
  15. Y. M. Cheng and J. H. Li, “A meshless method with complex variables for elasticity,” Acta Physica Sinica, vol. 54, no. 10, pp. 4463–4471, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Cheng and J. Li, “Complex variable meshless method for fracture problems,” Science in China, Series G, vol. 49, no. 1, pp. 46–59, 2006. View at Publisher · View at Google Scholar · View at Scopus
  17. K. M. Liew, C. Feng, Y. Cheng, and S. Kitipornchai, “Complex variable moving least-squares method: a meshless approximation technique,” International Journal for Numerical Methods in Engineering, vol. 70, no. 1, pp. 46–70, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. Peng, P. Liu, and Y. Cheng, “The complex variable element-free Galerkin (CVEFG) method for two-dimensional elasticity problems,” International Journal of Applied Mechanics, vol. 1, no. 2, pp. 367–385, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. M. Peng, D. Li, and Y. Cheng, “The complex variable element-free Galerkin (CVEFG) method for elasto-plasticity problems,” Engineering Structures, vol. 33, no. 1, pp. 127–135, 2011. View at Publisher · View at Google Scholar · View at Scopus
  20. F.-N. Bai, D.-M. Li, J.-F. Wang, and Y.-M. Cheng, “An improved complex variable element-free Galerkin method for two-dimensional elasticity problems,” Chinese Physics B, vol. 21, no. 2, Article ID 020204, 2012. View at Publisher · View at Google Scholar · View at Scopus
  21. F.-N. Bai, D.-M. Li, J.-F. Wang, and Y.-M. Cheng, “An improved complex variable element-free Galerkin method for two-dimensional elasticity problems,” Chinese Physics B, vol. 21, no. 2, Article ID 020204, 2012. View at Publisher · View at Google Scholar · View at Scopus
  22. Y. M. Cheng, R. X. Li, and M. J. Peng, “Complex variable element-free Galerkin (CVEFG) method for viscoelasticity problems,” Chinese Physics B, vol. 21, no. 9, Article ID 090205, 2012. View at Google Scholar
  23. Y. M. Cheng, J. F. Wang, and R. X. Li, “The complex variable element-free Galerkin (CVEFG) method for two-dimensional elastodynamics problems,” International Journal of Applied Mechanics, vol. 4, no. 4, Article ID 1250042, 2012. View at Google Scholar
  24. S. C. Li and Y. M. Cheng, “Meshless numerical manifold method based on unit partition,” Acta Mechanica Sinica, vol. 36, no. 4, pp. 496–500, 2004. View at Google Scholar
  25. S. C. Li and Y. M. Cheng, “Numerical manifold method and its applications in rock mechanics,” Advances in Mechanics, vol. 34, no. 4, pp. 446–454, 2004. View at Google Scholar
  26. S. Li, Y. Cheng, and Y.-F. Wu, “Numerical manifold method based on the method of weighted residuals,” Computational Mechanics, vol. 35, no. 6, pp. 470–480, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. S. C. Li, S. C. Li, and Y. M. Cheng, “Enriched meshless manifold method for two-dimensional crack modeling,” Theoretical and Applied Fracture Mechanics, vol. 44, no. 3, pp. 234–248, 2005. View at Publisher · View at Google Scholar · View at Scopus
  28. S.-C. Li, Y.-M. Cheng, and S.-C. Li, “Meshless manifold method for dynamic fracture mechanics,” Acta Physica Sinica, vol. 55, no. 9, pp. 4760–4766, 2006. View at Google Scholar · View at Scopus
  29. H. F. Gao and Y. M. Cheng, “Complex variable numerical manifold method for elasticity,” Chinese Journal of Theoretical and Applied Mechanics. Lixue Xuebao, vol. 41, no. 4, pp. 480–488, 2009. View at Google Scholar · View at MathSciNet
  30. H. Gao and Y. Cheng, “A complex variable meshless manifold method for fracture problems,” International Journal of Computational Methods, vol. 7, no. 1, pp. 55–81, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. R. J. Cheng and K. M. Liew, “Analyzing two-dimensional sine-Gordon equation with the mesh-free reproducing kernel particle Ritz method,” Computer Methods in Applied Mechanics and Engineering, vol. 245-246, pp. 132–143, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. E. Oñate, S. Idelsohn, O. C. Zienkiewicz, and R. L. Taylor, “A finite point method in computational mechanics. Applications to convective transport and fluid flow,” International Journal for Numerical Methods in Engineering, vol. 39, no. 22, pp. 3839–3866, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  33. M. Dehghan and M. Tatari, “Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions,” Mathematical and Computer Modelling, vol. 44, no. 11-12, pp. 1160–1168, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. B. D. Dai and Y. M. Cheng, “Local boundary integral equation method based on radial basis functions for potential problems,” Acta Physica Sinica, vol. 56, no. 2, pp. 597–603, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. Y. M. Cheng and M. J. Chen, “A boundary element-free method for linear elasticity,” Acta Mechanica Sinica, vol. 35, no. 2, pp. 181–186, 2003. View at Google Scholar
  36. Y. Cheng and M. Peng, “Boundary element-free method for elastodynamics,” Science in China G, vol. 48, no. 6, pp. 641–657, 2005. View at Publisher · View at Google Scholar · View at Scopus
  37. K. M. Liew, Y. Cheng, and S. Kitipornchai, “Boundary element-free method (BEFM) for two-dimensional elastodynamic analysis using Laplace transform,” International Journal for Numerical Methods in Engineering, vol. 64, no. 12, pp. 1610–1627, 2005. View at Publisher · View at Google Scholar · View at Scopus
  38. K. M. Liew and Y. Cheng, “Complex variable boundary element-free method for two-dimensional elastodynamic problems,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 49–52, pp. 3925–3933, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. M. Peng and Y. Cheng, “A boundary element-free method (BEFM) for two-dimensional potential problems,” Engineering Analysis with Boundary Elements, vol. 33, no. 1, pp. 77–82, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. Y. Cheng, K. M. Liew, and S. Kitipornchair, “Reply to ‘Comments on Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems,” International Journal for Numerical Methods in Engineering, vol. 78, no. 10, pp. 1258–1260, 2009. View at Publisher · View at Google Scholar · View at Scopus
  41. J. Zhang, M. Tanaka, and T. Matsumoto, “Meshless analysis of potential problems in three dimensions with the hybrid boundary node method,” International Journal for Numerical Methods in Engineering, vol. 59, no. 9, pp. 1147–1166, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  42. S. N. Atluri, J. Sladek, V. Sladek, and T. Zhu, “Local boundary integral equation (LBIE) and it's meshless implementation for linear elasticity,” Computational Mechanics, vol. 25, no. 2, pp. 180–198, 2000. View at Google Scholar · View at Scopus
  43. B. Dai and Y. Cheng, “An improved local boundary integral equation method for two-dimensional potential problems,” International Journal of Applied Mechanics, vol. 2, no. 2, pp. 421–436, 2010. View at Publisher · View at Google Scholar · View at Scopus
  44. D. Shepard, “A two-dimensional interpolation function for irregularly spaced points,” in Proceeding of the 23rd ACM National Conference, pp. 517–524, 1968.
  45. P. Lancaster and K. Salkauskas, “Surfaces generated by moving least squares methods,” Mathematics of Computation, vol. 37, no. 155, pp. 141–158, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. T. Zhu and S. N. Atluri, “A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method,” Computational Mechanics, vol. 21, no. 3, pp. 211–222, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. T. Mostt and C. Bucher, “A Moving Least Squares weighting function for the Element-free Galerkin Method which almost fulfills essential boundary conditions,” Structural Engineering and Mechanics, vol. 21, no. 3, pp. 315–332, 2005. View at Google Scholar · View at Scopus
  48. T. Most and C. Bucher, “New concepts for moving least squares: an interpolating non-singular weighting function and weighted nodal least squares,” Engineering Analysis with Boundary Elements, vol. 32, no. 6, pp. 461–470, 2008. View at Publisher · View at Google Scholar · View at Scopus
  49. S. L. L. Verardi, J. M. Machado, and Y. Shiyou, “The application of interpolating MLS approximations to the analysis of MHD flows,” Finite Elements in Analysis and Design, vol. 39, no. 12, pp. 1173–1187, 2003. View at Publisher · View at Google Scholar · View at Scopus
  50. I. Kaljević and S. Saigal, “An improved element free Galerkin formulation,” International Journal for Numerical Methods in Engineering, vol. 40, no. 16, pp. 2953–2974, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. H.-P. Ren, Y.-M. Cheng, and W. Zhang, “An improved boundary element-free method (IBEFM) for two-dimensional potential problems,” Chinese Physics B, vol. 18, no. 10, pp. 4065–4073, 2009. View at Publisher · View at Google Scholar · View at Scopus
  52. R. Hongping, C. Yumin, and Z. Wu, “An interpolating boundary element-free method (IBEFM) for elasticity problems,” Science China, vol. 53, no. 4, pp. 758–766, 2010. View at Publisher · View at Google Scholar · View at Scopus
  53. H. Ren and Y. Cheng, “The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems,” Engineering Analysis with Boundary Elements, vol. 36, no. 5, pp. 873–880, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  54. H. Ren and Y. Cheng, “The interpolating element-free Galerkin (IEFG) method for two-dimensional potential problems,” Engineering Analysis with Boundary Elements, vol. 36, no. 5, pp. 873–880, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  55. H. Netuzhylov, “Enforcement of boundary conditions in meshfree methods using interpolating moving least squares,” Engineering Analysis with Boundary Elements, vol. 32, no. 6, pp. 512–516, 2008. View at Publisher · View at Google Scholar · View at Scopus
  56. J. F. Wang, F. X. Sun, and Y. M. Cheng, “An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems,” Chinese Physics B, vol. 21, no. 9, Article ID 090204, 2012. View at Google Scholar
  57. J. Wang, J. Wang, F. Sun, and Y. Cheng, “An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems,” International Journal of Computational Methods, vol. 10, no. 6, Article ID 1350043, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  58. A. P. S. Selvadurai, Partial Differential Equations in Mechanics, Springer, Berlin, Germany, 2000.