The basis of meshless domain discretization: The meshless local Petrov-Galerkin (MLPG) method

Advances in Computational Mathematics

Volumn 23, Issue 1-2, 2005, Pages 73-93

  Atluri, Satya N ^a   Shen, Shengping ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Local weak form; Meshless method; MLPG

Indexed keywords

EID: 16244377097     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-004-1813-9     Document Type: Article

References (69)

1. N.R. Aluru and G. Li, Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation, Internat. J. Numer. Methods Engrg. 50(10) (2001) 2373-2410.
2. S.N. Atluri, J.Y. Cho and H.G. Kim, Analysis of the beams, using the meshless local Petrov-Galerkin method, with generalized moving least squares interpolations, Comput. Mech. 24 (1999) 334-347.
3. 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, Comput. Mech. 24 (1999) 348-372.
4. S.N. Atluri and S. Shen, The Meshless Local Petrov-Galerkin (MLPG) Method (Tech Science Press, Los Angeles, CA, 2002).
5. S.N. Atluri and S. Shen, The meshless local Petrov-Galerkin (MLPG) method: A simple and lesscostly alternative to the finite element and boundary element methods, CMES: Comput. Modeling Engrg. Sci. 3(1) (2002) 11-52.
6. S.N. Atluri, J. Sladek, V. Sladek and T. Zhu, The local boundary integral equation (LBIE) and its meshless implementation for linear elasticity, Comput. Mech. 25 (2000) 180-198.
7. S.N. Atluri and T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach to nonlinear problems in computational modeling and simulation, Comput. Modeling Simulation Engrg. 3 (1998) 187-196.
8. S.N. Atluri and T. Zhu, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Comput. Mech. 22 (1998) 117-127.
9. S.N. Atluri and T. Zhu, The meshless local Petrov-Galerkin (MLPG) approach for solving problems in elastostatics, Comput. Mech. 25 (2000) 169-179.
10. S.N. Atluri and T. Zhu, New concepts in meshless methods, Internat. J. Numer.Mech. Engrg. 47 (2000) 537-556.
11. I. Babuska and J.M. Melenk, The partition of unity method, Internat. J. Numer. Methods Engrg. 40 (1997) 727-758.
12. W. Barry and V. Thulasi, A wachspress meshless local Petrov-Galerkin method, www.sce.ait.ac.th/people/faculty/wjbarry/ (2002).
13. R.C. Batra and H.K. Ching, Analysis of elastodynamic deformations near a crack/notch tip by the meshless local Petrov-Galerkin (MLPG) method, CMES: Comput. Modeling Engr. Sci. 3(6) (2002) 717-730.
14. S. Beissel and T. Belytschko, Nodal integration of element-free Galerkin method, Comput. Methods Appl. Mech. Engrg. 139 (1996) 49-74.
15. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods Appl. Mech. Engrg. 139 (1996) 3-47.
16. T. Belytschko, Y.Y. Lu and L. Gu, Element-free Galerkin methods, Internat. J. Numer. Methods Engrg. 37 (1994) 229-256.
17. M.K. Chati and S. Mukherjee, The boundary node method for three-dimensional problems in potential theory, Internat. J. Numer. Methods Engrg. 47(9) (2000) 1523-1547.
18. J.S. Chen, C.T. Wu, S. Yoon and Y. You, A stabilized conforming nodal integration for Galerkin mesh-free methods, Internat. J. Numer. Methods Engrg. 50 (2001) 435-466.
19. H.K. Ching and R.C. Batra, Determination of crack tip fields in linear elastostatics by the meshless local Petrov-Galerkin (MLPG) method, CMES: Comput. Modeling Engrg. Sci. 2(2) (2001) 273-290.
20. J.Y. Cho and S.N. Atluri, Analysis of shear flexible beams, using the meshless local Petrov-Galerkin method, based on a locking-free formulation, Engrg. Comput. 18(1/2) (2001) 215-240.
21. S. De and K.J. Bathe, The method of finite spheres, Comput. Mech. 25(4) (2000) 329-345.
22. C. Duarte and J.T. Oden, Hp-cloud - a meshless method to solve boundary-value problems, Comput. Methods Appl. Mech. Engrg. 139 (1996) 237-262.
23. R.A. Gingold and J.J. Monaghan, Smoothed particle hydrodynamics: Theory and application to non-spherical stars, Mon. Not. Roy. Astronom. Soc. 181 (1977) 375-389.
24. Y.T. Gu and G.R. Liu, A meshless local Petrov-Galerkin (MLPG) formulation for static and free vibration analysis of thin plates, CMES: Comput. Modeling Engrg. Sci. 2(4) (2001) 463-476.
25. S. Hao, U.S. Park and W.K. Liu, Moving particle finite element method, Internat. J. Numer. Methods Engrg. 53(8) (2002) 1937-1958.
26. X. Jin, G. Li and N.R. Aluru, On the equivalence between least-squares and kernel approximations in meshless methods, CMES: Comput. Modeling Engrg. Sci. 2(4) (2001) 463-476.
27. H.G. Kim and S.N. Atluri, Arbitrary placement of secondary nodes and error control in the meshless local Petrov-Galerkin (MPLG) method, CMES: Comput. Modeling Engrg. Sci. 1(3) (2000) 11-32.
28. G. Li and N.R. Aluru, Boundary cloud method: A combined scattered point/boundary integral approach for boundary-only analysis, Comput. Methods Appl. Math. 191(21/22) (2002) 2337-2370.
29. 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, CMES: Comput. Modeling Engrg. Sci. 4(5) (2003) 567-581.
30. H. Lin and S.N. Atluri, Meshless local Petrov-Galerkin (MPLG) method for convection-diffusion problems, CMES: Comput. Modeling Engrg. Sci. 1(2) (2000) 45-60.
31. H. Lin and S.N. Atluri, The meshless local Petrov-Galerkin (MPLG) method for solving incompressible Navier-Stokes equations, CMES: Comput. Modeling Engrg. Sci. 2(2) (2001) 117-142.
32. W.K. Liu, Y. Chen, R.A. Uras and C.T. Chang, Generalized multiple scale reproducing kernel particle methods, Comput. Methods Appl. Mech. Engrg. 139 (1996) 91-157.
33. G.R. Liu and Y.T. Gu, Comparisons of two meshfree local point interpolation methods for structural analyses, CMES: Comput. Mech. 29(2) (2002) 107-121.
34. S.Y. Long and S.N. Atluri, A meshless local Petrov-Galerkin method for solving the bending problem of a thin plate, CMES: Comput. Modeling Engrg. Sci. 3(1) (2002) 53-63.
35. M. Luan, R. Tian and Q. Yang, A new numerical method - finite-cover based element-free method, in: Proc. of the 10th Internat. Conf. on Computer Methods and Advances in Geomechanics, University of Arizona, Tucson, AZ, USA, 2001.
36. J.J. Monaghan, Particle methods for hydrodynamics, Comput. Phys. Rep. 3 (1985) 71-124.
37. B. Nayroles, G. Touzot and P. Villon, Generalizing the finite element method: Diffuse approximation and diffuse elements, Comput. Mech. 10 (1992) 307-318.
38. A. Needleman, Material rate dependent and mesh sensitivity in localization problems, Comput. Methods Appl. Mech. Engrg. 67 (1988) 68-85.
39. E. Onate, S. Idelsohn, O.C. Zienkiewicz and R.L. Taylor, A finite point method in computational mechanics. Applications to convective transport and fluid flow, Internat. J. Numer. Methods Engrg. 39 (1996) 3839-3866.
40. E. Pardo, Meshless method for linear elastostatics based on a path integral formulation, Internat. J. Numer. Methods Engrg. 47(8) (2000) 1463-1480.
41. E. Pardo, Convergence and accuracy of the path integral approach for elastostatics, Comput. Methods Appl. Mech. 191(19/20) (2002) 2191-2219.
42. E. Pardo, Blurred derivatives and meshless methods, Internat. J. Numer. Methods Engrg. 56(2) (2003) 295-324.
43. S.H. Park and S.K. Young, The least-squares meshfree method, Internat. J. Numer. Methods Engrg. 52 (2001) 997-1012.
44. M.J.D. Powell, The theory of radial basis function approximation in 1990, in: Advances in Numerical Analysis, Vol. 2, ed. W. Light (Clarendon Press, Oxford, 1992) pp. 105-210.
45. L.F. Qian, R.C. Batra and L.M. Chen, Elastostatic deformations of a thick plate by using a higher-order shear and normal deformable plate theory and two meshless local Petrov-Galerkin (MLPG) method, CMES: Comput. Modeling Engrg. Sci. 4(1) (2003) 161-176.
46. I.S. Raju and D.R. Phillips, Further fevelopments in the MLPG method for beam problems, CMES: Comput. Modeling Engrg. Sci. 4(1) (2003) 141-160.
47. D. Shepard, A two-dimensional function for irregularly spaced points, in: Proc. of ACM National Conf., 1968, pp. 517-524.
48. J. Sladek and V. Sladek, A Trefftz function approximation in local boundary integral equations, Comput. Mech. 28(3/4) (2002) 212-219.
49. J. Sladek and V. Sladek, Application of local boundary integral equation method into micropolar elasticity, Engrg. Anal. Bound. Elem. 27(1) (2003) 81-90.
50. J. Sladek, V. Sladek and S.N. Atluri, Local boundary integral equation (LBIE) method for solving roblems of elasticity with nonhomogeneous material properties, Comput. Mech. 24(6) (2000) 456-462.
51. J. Sladek, V. Sladek and S.N. Atluri, A pure contour formulation for the meshless local boundary integral equation method in thermoelasticity, CMES: Comput. Modeling Engrg. Sci. 2(4) (2001) 423-434.
52. J. Sladek, V. Sladek and S.N. Atluri, Application of local boundary integral equation method to solve boundary value problems, Internat. Appl. Mech. 38(9) (2002) 3-27.
53. V. Sladek, J. Sladek, S.N. Atluri and R. Keer, Numerical integration of singularities in meshless implementation of the LBIE, Comput. Mech. 25 (2000) 394-403.
54. J. Sladek, V. Sladek and H.A. Mang, Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation, Comput. Methods Appl. Math. 191(51/52) (2002) 5943-5959.
55. J. Sladek, V. Sladek and R. Van Keer, Global and local Trefftz boundary integral formulations for sound vibration, Adv. Engrg. Software 33(7-10) (2002) 469-476.
56. N. Sukumar, B. Moran and T. Belytschko, The natural element method in solid mechanics, Internat. Numer. Methods Engrg. 43 (1998) 839-887.
57. Z. Tang, S. Shen and S.N. Atluri, Analysis of materials with strain gradient effects: A meshless local Petrov-Galerkin approach with nodal displacements only, CMES: Comput. Modeling Engrg. Sci. 4(1) (2003) 177-196.
58. J.G. Wang and G.R. Liu, A point interpolation meshless method based on radial basis functions, Internat. J. Numer. Methods Engrg. 54(11) (2002) 1623-1648.
59. H. Wendland, Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree, Adv. Comput. Math. 4 (1995) 389-396.
60. H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory 93 (1998) 258-272.
61. H. Wendland, Meshless Galerkin methods using radial basis function, Math. Comp. 68(228) (1999) 1521-1531.
62. Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl. 8 (1992) 1-10.
63. Z. Wu, Compactly supported positive definite radial functions, Adv. Comput. Math. 4 (1995) 283-292.
64. J.R. Xiao and M.A. McCarthy, Local variational inequality and meshless analysis of a beam involving unilateral contact conditions, in: Proc. of the ACMC-UK 10th Anniversary Conference, Swansea, 2002, pp. 223-226.
65. J.M. Zhang and Z.H. Yao, Meshless regular hybrid boundary node method, CMES: Comput. Modeling Engrg. Sci. 2(3) (2001) 307-318.
66. X. Zhang, K.Z. Song, M.W. Lu and X. Liu, Meshless methods based on collocation with radial basis functions, Comput. Mech. 26(4) (2000) 333-343.
67. T. Zhu and S.N. Atluri, A modified collocation and a penalty formulation for enforceing the essential boundary conditions in the element free Galerkin method, Comput. Mech. 21 (1998) 211-222.
68. T. Zhu, J.D. Zhang and S.N. Atluri, A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Comput. Mech. 21 (1998) 223-235.
69. T. Zhu, J.D. Zhang and S.N. Atluri, A meshless local boundary integral equation (LBIE) method for solving nonlinear problems, Comput. Mech. 22 (1998) 174-186.