Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014 (2014), Article ID 247172, 13 pages
http://dx.doi.org/10.1155/2014/247172
Review Article

A Review on Recent Contribution of Meshfree Methods to Structure and Fracture Mechanics Applications

1Department of Mechanical Engineering, Babaria Institute of Technology, Vadodara, Gujarat 391240, India
2Department of Mechanical Engineering, M. S. University, Vadodara, Gujarat 390020, India

Received 17 August 2013; Accepted 2 October 2013; Published 2 January 2014

Academic Editors: D. Qian and X. Zhou

Copyright © 2014 S. D. Daxini and J. M. Prajapati. 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. G. R. Liu and M. B. Liu, Smoothed Particle Hydrodynamics: A Meshfree Particle Method, World Scientific Publishing, Singapore, 2003.
  2. L. B. Lucy, “A numerical approach to the testing of the fission hypothesis,” Astronomical Journal, vol. 82, pp. 1013–1024, 1977. View at Publisher · View at Google Scholar
  3. R. A. Gingold and J. J. Monaghan, “Smoothed particle hydrodynamics: theory and allocation to non-spherical stars,” Monthly Notices of the Royal Astronomical Society, vol. 181, pp. 375–389, 1977. View at Google Scholar
  4. J. J. Monaghan, “An introduction to SPH,” Computer Physics Communications, vol. 48, no. 1, pp. 89–96, 1988. View at Google Scholar · View at Scopus
  5. 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
  6. K. M. Liew, Y. Cheng, and S. Kitipornchai, “Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems,” International Journal for Numerical Methods in Engineering, vol. 65, no. 8, pp. 1310–1332, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. W. K. Liu, S. Jun, and Y. F. Zhang, “Reproducing kernel particle methods,” International Journal for Numerical Methods in Fluids, vol. 20, no. 8-9, pp. 1081–1106, 1995. View at Google Scholar · View at Scopus
  8. G. R. Liu, Meshfree Methods: Moving beyond the Finite Element Method, CRC Press, Taylor & Francic Group, 2nd edition, 2010.
  9. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: an overview and recent developments,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 3–47, 1996. View at Google Scholar · View at Scopus
  10. S. Li and W. K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Reviews, vol. 55, no. 1, pp. 1–34, 2002. View at Publisher · View at Google Scholar · View at Scopus
  11. V. P. Nguyen, T. Rabczuk, S. Bordas, and M. Duflot, “Meshless methods: a review and computer implementation aspects,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 763–813, 2008. View at Publisher · View at Google Scholar · View at Scopus
  12. K. M. Liew, X. Zhao, and A. J. M. Ferreira, “A review of meshless methods for laminated and functionally graded plates and shells,” Composite Structures, vol. 93, no. 8, pp. 2031–2041, 2011. View at Publisher · View at Google Scholar · View at Scopus
  13. S. N. Atluri and T. L. 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. Y. Y. Lu, T. Belytschko, and L. Gu, “A new implementation of the element free Galerkin method,” Computer Methods in Applied Mechanics and Engineering, vol. 113, no. 3-4, pp. 397–414, 1994. View at Google Scholar · View at Scopus
  15. Y. Krongauz and T. Belytschko, “Enforcement of essential boundary conditions in meshless approximations using finite elements,” Computer Methods in Applied Mechanics and Engineering, vol. 131, no. 1-2, pp. 133–145, 1996. View at Google Scholar · View at Scopus
  16. F. C. Günther and W. K. Liu, “Implementation of boundary conditions for meshless methods,” Computer Methods Appllied Mechanics and Engineering, vol. 163, no. 1–4, pp. 205–230, 1998. View at Publisher · View at Google Scholar
  17. S. Fernández-Méndez and A. Huerta, “Imposing essential boundary conditions in mesh-free methods,” Computer Methods Applied Mechanics and Engineering, vol. 193, no. 12–14, pp. 1257–1273, 2004. View at Publisher · View at Google Scholar
  18. J. S. Chen and H. P. Wang, “New boundary condition treatments in meshfree computation of contact problems,” Computer Methods in Applied Mechanics and Engineering, vol. 187, no. 3-4, pp. 441–468, 2000. View at Google Scholar · View at Scopus
  19. J. Ren and K. M. Liew, “Mesh-free method revisited: two new approaches for the treatment of essential boundary conditions,” International Journal of Computational Engineering Science, vol. 3, no. 2, pp. 219–233, 2002. View at Publisher · View at Google Scholar
  20. J. Dolbow and T. Belytschko, “Numerical integration of the Galerkin weak form in meshfree methods,” Computational Mechanics, vol. 23, no. 3, pp. 219–230, 1999. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Beissel and T. Belytschko, “Nodal integration of the element-free Galerkin method,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 49–74, 1996. View at Google Scholar · View at Scopus
  22. J. S. Chen, C. T. Wu, S. Yoon, and Y. You, “A stabilized conforming nodal integration for Galerkin mesh-free methods,” International Journal for Numerical Methods Engineering, vol. 50, no. 2, pp. 435–466, 2001. View at Publisher · View at Google Scholar
  23. P. Krysl and T. Belytschko, “ESFLIB: a library to compute the element free Galerkin shape functions,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 15–17, pp. 2181–2205, 2001. View at Publisher · View at Google Scholar · View at Scopus
  24. J. Dolbow and T. Belytschko, “An introduction to programming the meshless element free Galerkin method,” Archives Computational Methods in Engineering, vol. 5, no. 3, pp. 207–241, 1998. View at Publisher · View at Google Scholar
  25. W. Barry and S. Saigal, “A three-dimensional element-free Galerkin elastic and elastoplastic formulation,” International Journal for Numerical Methods in Engineering, vol. 46, no. 5, pp. 671–693, 1999. View at Google Scholar · View at Scopus
  26. C. Tiago and V. M. A. Leitão, “Analysis of free vibration problems with the element-free Galerkin method,” in Proceedings of the 9th International Conference on Numerical Methods in Continuum Mechanics, Žilina, Slovakia, September 2003.
  27. Z. Zhang, P. Zhao, and K. M. Liew, “Improved element-free Galerkin method for two-dimensional potential problems,” Engineering Analysis with Boundary Elements, vol. 33, no. 4, pp. 547–554, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. Z. Zhang, P. Zhao, and K. M. Liew, “Analyzing three-dimensional potential problems with the improved element-free Galerkin method,” Computational Mechanics, vol. 44, no. 2, pp. 273–284, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. 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
  30. S. N. Atluri and T. L. Zhu, “Meshless local Petrov-Galerkin (MLPG) approach for solving problems in elasto-statics,” Computational Mechanics, vol. 25, no. 2, pp. 169–179, 2000. View at Google Scholar · View at Scopus
  31. S. N. Atluri and S. 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 and Sciences, vol. 3, no. 1, pp. 11–51, 2002. View at Google Scholar · View at Scopus
  32. S. Long and S. N. Atluri, “A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate,” Computer Modeling in Engineering and Sciences, vol. 3, no. 1, pp. 53–63, 2001. View at Google Scholar · View at Scopus
  33. I. S. Raju and D. R. Phillips, A Meshless Local Petrov-Galerkin Method for Euler-Bernoulli Beam Problems, NASA Langley Research Center, Hampton, Va, USA, 2001.
  34. I. S. Raju, D. R. Phillips, and T. Krishnamurthy, Meshless Local-Petrov Galerkin Euler-Bernoulli Beam Problems: A Radial Basis Function Approach, NASA Langley Research Center, Hampton, Va, USA, 2001.
  35. Q. Li, S. Shen, Z. D. Han, and S. N. Atluri, “Application of meshless local Petrov-Galerkin (MLPG) to problems with singularities, and material discontinuities, in 3-D elasticity,” Computer Modeling in Engineering and Sciences, vol. 4, no. 5, pp. 571–585, 2003. View at Google Scholar · View at Scopus
  36. Z. D. Han and S. N. Atluri, “A Meshless Local Petrov-Galerkin (MLPG) approach for 3-dimensional elasto-dynamics,” Computers, Materials and Continua, vol. 1, no. 2, pp. 129–140, 2004. View at Google Scholar · View at Scopus
  37. S. Y. Long, K. Y. Liu, and D. A. Hu, “A new meshless method based on MLPG for elastic dynamic problems,” Engineering Analysis with Boundary Elements, vol. 30, no. 1, pp. 43–48, 2006. View at Publisher · View at Google Scholar · View at Scopus
  38. P. Pudjisuryadi, “Adaptive meshless local Petrov-Galerkin method with variable domain of influence in 2D elastostatic problems,” Civil Engineering Dimension, vol. 10, no. 2, pp. 99–108, 2008. View at Google Scholar
  39. A. Abdollahifar, M. R. Nami, and A. R. Shafiei, “A new MLPG method for elastostatic problems,” Engineering Analysis with Boundary Elements, vol. 36, no. 3, pp. 451–457, 2012. View at Publisher · View at Google Scholar · View at Scopus
  40. P. Hein, “Diffuse element method applied to Kirchhoff plates,” Technical Report, Department of Civil Engineering, Northwestern University, Evanston, Ill, USA, 1993. View at Google Scholar
  41. Y. Y. Lu, L. Gu, and T. Belytschko, “Diffuse element method applied to Kirchhoff plates,” Internal report, 1996.
  42. P. Krysl and T. Belytschko, “Analysis of thin plates by the element-free Galerkin method,” Computational Mechanics, vol. 17, no. 1-2, pp. 26–35, 1995. View at Publisher · View at Google Scholar · View at Scopus
  43. P. Krysl and T. Belytschko, “Analysis of thin shells by the element-free Galerkin method,” International Journal of Solids and Structures, vol. 33, no. 20–22, pp. 3057–3080, 1996. View at Google Scholar · View at Scopus
  44. G. R. Liu and X. L. Chen, “A mesh-free method for static and free vibration analyses of thin plates of complicated shape,” Journal of Sound and Vibration, vol. 241, no. 5, pp. 839–855, 2001. View at Publisher · View at Google Scholar · View at Scopus
  45. X. L. Chen, G. R. Liu, and S. P. Lim, “An element free Galerkin method for the free vibration analysis of composite laminates of complicated shape,” Composite Structures, vol. 59, no. 2, pp. 279–289, 2003. View at Publisher · View at Google Scholar · View at Scopus
  46. K. Y. Dai, G. R. Liu, K. M. Lim, and X. L. Chen, “A mesh-free method for static and free vibration analysis of shear deformable laminated composite plates,” Journal of Sound and Vibration, vol. 269, no. 3–5, pp. 633–652, 2004. View at Publisher · View at Google Scholar · View at Scopus
  47. K. Y. Dai, G. R. Liu, X. Han, and K. M. Lim, “Thermomechanical analysis of functionally graded material (FGM) plates using element-free Galerkin method,” Computers and Structures, vol. 83, no. 17-18, pp. 1487–1502, 2005. View at Publisher · View at Google Scholar · View at Scopus
  48. L. X. Peng, S. Kitipornchai, and K. M. Liew, “Analysis of rectangular stiffened plates under uniform lateral load based on FSDT and element-free Galerkin method,” International Journal of Mechanical Sciences, vol. 47, no. 2, pp. 251–276, 2005. View at Publisher · View at Google Scholar · View at Scopus
  49. K. Liew, L. Peng, and S. Kitipornchai, “Buckling analysis of corrugated plates using a mesh-free Galerkin method based on the first-order shear deformation theory,” Computational Mechanics, vol. 38, no. 1, pp. 61–75, 2006. View at Publisher · View at Google Scholar · View at Scopus
  50. L. X. Peng, K. M. Liew, and S. Kitipornchai, “Analysis of stiffened corrugated plates based on the FSDT via the mesh-free method,” International Journal of Mechanical Sciences, vol. 49, no. 3, pp. 364–378, 2007. View at Publisher · View at Google Scholar · View at Scopus
  51. K. M. Liew, L. X. Peng, and S. Kitipornchai, “Nonlinear analysis of corrugated plates using a FSDT and a meshfree method,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 21–24, pp. 2358–2376, 2007. View at Publisher · View at Google Scholar · View at Scopus
  52. J. Belinha and L. M. J. S. Dinis, “Nonlinear analysis of plates and laminates using the element free Galerkin method,” Composite Structures, vol. 78, no. 3, pp. 337–350, 2007. View at Publisher · View at Google Scholar · View at Scopus
  53. R. Vaghefi, G. H. Baradaran, and H. Koohkan, “Three-dimensional static analysis of thick functionally graded plates by using meshless local Petrov-Galerkin (MLPG) method,” Engineering Analysis with Boundary Elements, vol. 34, no. 6, pp. 564–573, 2010. View at Publisher · View at Google Scholar · View at Scopus
  54. A. R. Mojdehi, A. Darvizeh, A. Basti, and H. Rajabi, “Three dimensional static and dynamic analysis of thick functionally graded plates by the meshless local Petrov-Galerkin (MLPG) method,” Engineering Analysis with Boundary Elements, vol. 35, no. 11, pp. 1168–1180, 2011. View at Publisher · View at Google Scholar · View at Scopus
  55. C. V. Le, H. Askes, and M. Gilbert, “Adaptive element-free Galerkin method applied to the limit analysis of plates,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 37–40, pp. 2487–2496, 2010. View at Publisher · View at Google Scholar · View at Scopus
  56. A. R. Z. Abidin and B. A. Izzuddin, “Meshless local buckling analysis of steel beams with irregular web openings,” Engineering Structures, vol. 50, pp. 197–206, 2013. View at Publisher · View at Google Scholar
  57. E. Jaberzadeh, M. Azhari, and B. Boroomand, “Inelastic buckling of skew and rhombic thin thickness-tapered plates with and without intermediate supports using the element-free Galerkin method,” Applied Mathematical Modelling, vol. 37, no. 10-11, pp. 6838–6854, 2013. View at Publisher · View at Google Scholar
  58. L. Liu, G. R. Liu, and V. B. C. Tan, “Element free method for static and free vibration analysis of spatial thin shell structures,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 51-52, pp. 5923–5942, 2002. View at Publisher · View at Google Scholar · View at Scopus
  59. M. Foroutan, R. Moradi-Dastjerdi, and R. Sotoodeh-Bahreini, “Static analysis of FGM cylinders by a mesh-free method,” Steel and Composite Structures, vol. 12, no. 1, pp. 1–11, 2012. View at Google Scholar · View at Scopus
  60. S. Li, D. Qian, W. K. Liu, and T. Belytschko, “A meshfree contact-detection algorithm,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 24-25, pp. 3271–3292, 2001. View at Publisher · View at Google Scholar · View at Scopus
  61. C. Tiago and P. M. Pimenta, “An EFG method for the nonlinear analysis of plates undergoing arbitrarily large deformations,” Engineering Analysis with Boundary Elements, vol. 32, no. 6, pp. 494–511, 2008. View at Publisher · View at Google Scholar · View at Scopus
  62. D. A. Hu, S. Y. Long, X. Han, and G. Y. Li, “A meshless local Petrov-Galerkin method for large deformation contact analysis of elastomers,” Engineering Analysis with Boundary Elements, vol. 31, no. 7, pp. 657–666, 2007. View at Publisher · View at Google Scholar · View at Scopus
  63. Q. Li and K. M. Lee, “An adaptive meshless method for analyzing large mechanical deformation and contacts,” Journal of Applied Mechanics, vol. 75, no. 4, Article ID 041014, 2008. View at Publisher · View at Google Scholar · View at Scopus
  64. D. Li, B. Bai, Y. Cheng, and K. M. Liew, “A novel complex variable element-free Galerkin method for two-dimensional large deformation problems,” Computer Methods Applied Mechanics and Engineering, vol. 233–236, pp. 1–10, 2012. View at Publisher · View at Google Scholar
  65. Z. Ullah and C. E. Augarde, “Finite deformation elasto-plastic modelling using an adaptive meshless method,” Computers and Structures, vol. 118, pp. 39–52, 2013. View at Publisher · View at Google Scholar
  66. F. Bobaru and S. Mukherjee, “Shape sensivity analysis and shape optimization in planar elasticity using th element-free Galerkin method,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 32-33, pp. 4319–4337, 2001. View at Publisher · View at Google Scholar · View at Scopus
  67. F. Bobaru and S. Mukherjee, “Meshless approach to shape optimization of linear thermoelastic solids,” International Journal for Numerical Methods in Engineering, vol. 53, no. 4, pp. 765–796, 2002. View at Publisher · View at Google Scholar · View at Scopus
  68. X. Zhang, M. Rayasam, and G. Subbarayan, “A meshless, compositional approach to shape optimal design,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 17–20, pp. 2130–2146, 2007. View at Publisher · View at Google Scholar · View at Scopus
  69. Z. Juan, L. Shuyao, and L. Guangyao, “The topology optimization design for continuum structures based on the element free Galerkin method,” Engineering Analysis with Boundary Elements, vol. 34, no. 7, pp. 666–672, 2010. View at Publisher · View at Google Scholar · View at Scopus
  70. T. Belytschko, L. Gu, and Y. Y. Lu, “Fracture and crack growth by element free Galerkin methods,” Modelling and Simulation in Materials Science and Engineering, vol. 2, no. 3, article 007, pp. 519–534, 1994. View at Publisher · View at Google Scholar · View at Scopus
  71. T. Belytschko and M. Fleming, “Smoothing, enrichment and contact in the element-free Galerkin method,” Computers and Structures, vol. 71, no. 2, pp. 173–195, 1999. View at Publisher · View at Google Scholar · View at Scopus
  72. T. Belytschko and M. Tabbara, “Dynamic fracture using element-free Galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 39, no. 6, pp. 923–938, 1996. View at Google Scholar · View at Scopus
  73. P. Krysl and T. Belytschko, “The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks,” International Journal for Numerical Methods in Engineering, vol. 44, no. 6, pp. 767–800, 1999. View at Google Scholar · View at Scopus
  74. T. Belytschko, D. Organ, and C. Gerlach, “Element-free Galerkin methods for dynamic fracture in concrete,” Computer Methods in Applied Mechanics and Engineering, vol. 187, no. 3-4, pp. 385–399, 2000. View at Google Scholar · View at Scopus
  75. B. N. Rao and S. Rahman, “Efficient meshless method for fracture analysis of cracks,” Computational Mechanics, vol. 26, no. 4, pp. 398–408, 2000. View at Google Scholar · View at Scopus
  76. C. M. Tiago and V. M. A. Leitão, “Development of a EFG formulation for damage analysis of reinforced concrete beams,” Computers and Structures, vol. 82, no. 17–19, pp. 1503–1511, 2004. View at Publisher · View at Google Scholar · View at Scopus
  77. S. H. Lee and Y. C. Yoon, “Numerical prediction of crack propagation by an enhanced element-free Galerkin method,” Nuclear Engineering and Design, vol. 227, no. 3, pp. 257–271, 2004. View at Publisher · View at Google Scholar · View at Scopus
  78. T. Rabczuk and T. Belytschko, “Cracking particles: a simplified meshfree method for arbitrary evolving cracks,” International Journal for Numerical Methods in Engineering, vol. 61, no. 13, pp. 2316–2343, 2004. View at Publisher · View at Google Scholar · View at Scopus
  79. L. Kaiyuan, L. Shuyao, and L. Guangyao, “A simple and less-costly meshless local Petrov-Galerkin (MLPG) method for the dynamic fracture problem,” Engineering Analysis with Boundary Elements, vol. 30, no. 1, pp. 72–76, 2006. View at Publisher · View at Google Scholar · View at Scopus
  80. R. Brighenti, “Application of the element-free Galerkin meshless method to 3-D fracture mechanics problems,” Engineering Fracture Mechanics, vol. 72, no. 18, pp. 2808–2820, 2005. View at Publisher · View at Google Scholar · View at Scopus
  81. 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
  82. S. Parvanova, “Calculation of stress intensity factors based on force-displacement curve using element free Galerkin method,” Journal of Theoretical and Applied Mechanics, vol. 42, no. 1, pp. 23–40, 2012. View at Publisher · View at Google Scholar
  83. Y. Y. Zhang and L. Chen, “Impact simulation using simplified meshless method,” International Journal of Impact Engineering, vol. 36, no. 5, pp. 651–658, 2009. View at Publisher · View at Google Scholar · View at Scopus
  84. C. Gato, “Meshfree analysis of dynamic fracture in thin-walled structures,” Thin-Walled Structures, vol. 48, no. 3, pp. 215–222, 2010. View at Publisher · View at Google Scholar · View at Scopus
  85. C. Gato and Y. Shie, “Dynamic analysis of fracture in thin-walled pipes,” Journal of Pressure Vessel Technology, vol. 133, no. 6, Article ID 064501, 2011. View at Publisher · View at Google Scholar · View at Scopus
  86. M. Pant, I. V. Singh, and B. K. Mishra, “A novel enrichment criterion for modeling kinked cracks using element free Galerkin method,” International Journal of Mechanical Sciences, vol. 68, pp. 140–149, 2013. View at Publisher · View at Google Scholar
  87. B. N. Rao and S. Rahman, “Mesh-free analysis of cracks in isotropic functionally graded materials,” Engineering Fracture Mechanics, vol. 70, no. 1, pp. 1–27, 2003. View at Publisher · View at Google Scholar · View at Scopus
  88. J. Sladek, V. Sladek, and Ch. Zhang, “Stress analysis in anisotropic functionally graded materials by the MLPG method,” Engineering Analysis with Boundary Elements, vol. 29, no. 6, pp. 597–609, 2005. View at Publisher · View at Google Scholar · View at Scopus
  89. I. Guiamatsia, B. G. Falzon, G. A. O. Davies, and L. Iannucci, “Element-Free Galerkin modelling of composite damage,” Composites Science and Technology, vol. 69, no. 15-16, pp. 2640–2648, 2009. View at Publisher · View at Google Scholar · View at Scopus
  90. S. S. Ghorashi, S. Mohammadi, and S. R. Sabbagh-Yazdi, “Orthotropic enriched element free Galerkin method for fracture analysis of composites,” Engineering Fracture Mechanics, vol. 78, no. 9, pp. 1906–1927, 2011. View at Publisher · View at Google Scholar · View at Scopus
  91. M. Duflot and H. Nguyen-Dang, “Fatigue crack growth analysis by an enriched meshless method,” Journal of Computational and Applied Mathematics, vol. 168, no. 1-2, pp. 155–164, 2004. View at Publisher · View at Google Scholar · View at Scopus
  92. U. Andreaus, R. C. Batra, and M. Porfiri, “Vibrations of cracked Euler-Bernoulli beams using meshless local Petrov-Galerkin (MLPG) method,” Computer Modeling in Engineering and Sciences, vol. 9, no. 2, pp. 111–131, 2005. View at Google Scholar · View at Scopus
  93. S. Wang and H. Liu, “Modeling brittle-ductile failure transition with meshfree method,” International Journal of Impact Engineering, vol. 37, no. 7, pp. 783–791, 2010. View at Publisher · View at Google Scholar · View at Scopus
  94. M. Pant, I. V. Singh, and B. K. Mishra, “Numerical simulation of thermo-elastic fracture problems using element free Galerkin method,” International Journal of Mechanical Sciences, vol. 52, no. 12, pp. 1745–1755, 2010. View at Publisher · View at Google Scholar · View at Scopus
  95. L. Bouhala, A. Makradi, and S. Belouettar, “Thermal and thermo-mechanical influence on crack propagation using an extended mesh free method,” Engineering Fracture Mechanics, vol. 88, pp. 35–48, 2012. View at Publisher · View at Google Scholar · View at Scopus
  96. T. Belytschko, Y. Krongauz, M. Fleming, D. Organ, and W. K. S. Liu, “Smoothing and accelerated computations in the element free Galerkin method,” Journal of Computational and Applied Mathematics, vol. 74, no. 1-2, pp. 111–126, 1996. View at Publisher · View at Google Scholar · View at Scopus
  97. S. Yoon, C. T. Wu, H. P. Wang, and J. S. Chen, “Efficient meshfree formulation for metal forming simulations,” Journal of Engineering Materials and Technology, vol. 123, no. 4, pp. 462–467, 2001. View at Google Scholar · View at Scopus
  98. D. Wang and J. S. Chen, “A locking-free meshfree curved beam formulation with the stabilized conforming nodal integration,” Computational Mechanics, vol. 39, no. 1, pp. 83–90, 2006. View at Publisher · View at Google Scholar · View at Scopus
  99. A. Khosravifard and M. R. Hematiyan, “A new method for meshless integration in 2D and 3D Galerkin meshfree methods,” Engineering Analysis with Boundary Elements, vol. 34, no. 1, pp. 30–40, 2010. View at Publisher · View at Google Scholar · View at Scopus
  100. H. J. Chung and T. Belytschko, “An error estimate in the EFG method,” Computational Mechanics, vol. 21, no. 2, pp. 91–100, 1998. View at Google Scholar · View at Scopus
  101. X. Zhuang, C. Heaney, and C. Augarde, “On error control in the element-free Galerkin method,” Engineering Analysis with Boundary Elements, vol. 36, no. 3, pp. 351–360, 2012. View at Publisher · View at Google Scholar · View at Scopus
  102. 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 and Sciences, vol. 1, no. 3, pp. 11–32, 2000. View at Publisher · View at Google Scholar · View at Scopus