A meshless local Petrov-Galerkin (MLPG) approach based on the regular local boundary integral equation for linear elasticity

International Journal for Computational Methods in Engineering Science and Mechanics

Volumn 8, Issue 5, 2007, Pages 373-382

  Zhu, T L ^a   Zhang, J ^b   Wang, D ^b  

^a WIDENER UNIVERSITY   (United States)

^b XIAMEN UNIVERSITY   (China)

Author keywords

Local boundary integral equation; Meshless methods; MLPG; Moving least squares approximation

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ELASTICITY; GALERKIN METHODS; INTERPOLATION; LEAST SQUARES APPROXIMATIONS; PROBLEM SOLVING;

LOCAL BOUNDARY INTEGRAL EQUATION; MESHLESS METHODS; MOVING LEAST SQUARES APPROXIMATION;

INTEGRAL EQUATIONS;

EID: 34547525447     PISSN: 15502287     EISSN: 15502295     Source Type: Journal    
DOI: 10.1080/15502280701471434     Document Type: Article

References (37)

1. L.B. Lucy, A Numerical Approach to the Testing of the Fission Hypothesis, The Astronomical J., vol. 82, pp. 1013-1024, 1977.
2. B. Nayroles, G. Touzot, P. Villon, Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements, Comput. Mech., vol. 10, pp. 307-318, 1992.
3. T. Belytschko, Y.Y. Lu, L. Gu, Element-Free Galerkin Methods, Inter. J. Numer. Methods Engrg., vol. 37, pp. 229-256, 1994.
4. T. Belytschko, D. Organ, Y. Krongauz, A Coupled Finite Element-Element-Free Galerkin Method, Comput. Mech., vol. 17, pp. 186-195, 1995.
5. Y.Y. Lu, T. Belytschko, L. Gu, A New Implementation of the Element Free Galerkin Method, Comput. Methods Appl. Mech. Engrg., vol. 113, pp. 397-414, 1994.
6. T. Zhu, S.N. Atluri, A Modified Collocation & a Penalty Formulation for Enforcing the Essential Boundary Conditions in the Element Free Galerkin Method, Comput. Mech., vol. 21, pp. 211-222, 1998.
7. J.S. Chen, C.T. Wu, S. Yoon, Y You, A Stabilized Conforming Nodal Integration for Galerkin Meshfree Methods, Int. J. Numer. Methods Engrg., vol. 50, pp. 435-466, 2001.
8. S. Rahman, H. Xu, A Meshless Method for Computational Stochastic Mechanics, Int. J. Comput. Methods Engrg. Sci. Mech., vol. 6, pp. 41-58, 2005.
9. W.K. Liu, Y Chen, Wavelet and Multiple Scale Reproducing Kernel Methods, Inter. J. Numer. Methods Fluids, vol. 21, pp. 901-931, 1995.
10. W.K. Liu, Y Chen, C.T. Chang, T. Belytschko, Advances in Multiple Scale Kernel Particle Methods, Comput. Mech., vol. 18, pp. 73-111, 1996.
11. W.K. Liu, W. Hao, Y Chen, S. Jun, J. Gosz, Multiresolution Reproducing Kernel Particle Methods, Comput. Mech., vol. 20, pp. 295-309, 1997.
12. W.K. Liu, S. Jun, Multiple Scale Reproducing Kernel Particle Methods for Large Deformation Problems, Inter. J. Numer. Methods Engrg., vol. 41, pp. 1339-1362, 1998.
13. J.S. Chen, X. Zhang, T. Belytschko, An Implicit Gradient Model by a Reproducing Kernel Strain Regularization in Strain Localization Problems, Comput. Methods Appl. Mech. Engrg, vol. 193, pp. 2827-2844, 2004.
14. S.N. Atluri, T. Zhu, A New Meshless Local Petrov-Galerkin (MLPG) Approach in Computational Mechanics, Comput. Mech., vol. 22, pp. 117-127, 1998.
15. S.N. Atluri, T.L. Zhu, A New Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Nonlinear Problems in Computer Modeling and Simulation, Comput. Modeling Simul. Engrg., vol. 3, pp. 187-196, 1998.
16. S.N. Atluri, T.L. Zhu, The Meshless Local Petrov-Galerkin (MLPG) Approach for Solving Problems in Elasto-Statics, Comput. Mech., vol. 25, pp. 169-179, 2000.
17. S.N. Atluri, T.L. Zhu, New Concepts in Meshless Methods, Int. J. Numer. Methods Engrg., vol. 47, pp. 537-556, 2000.
18. T. Zhun, J.D. Zhang, S.N. Atluri, A Local Boundary Integral Equation (LBIE) Method in Computational Mechanics, and a Meshless Discretization Approach, Comput. Mech., vol. 21, pp. 223-235, 1998.
19. T. Zhu, J.D. Zhang, S.N. Atluri, A Meshless Local Boundary Integral Equation (LBIE) Method for Solving Nonlinear Problems, Comput. Mech., vol. 22, pp. 174-186, 1998.
20. T. Zhu, J. Zhang, S.N. Atluri, A Meshless Numerical Method Based on the Local Boundary Integral Equation (LBIE) in Solving Linear and Nonlinear Boundary Value Problems, Engrg. Analysis Boundary Elements, vol. 23, pp. 375-389, 1999.
21. T. Zhu, A New Meshless Regular Local Boundary Integral Equation (MRL-BIE) Approach, Int. J. Numer. Methods Engrg., vol. 46, pp. 1237-1252, 1999.
22. S. De, K.J. Bathe, The Method of Fine Spheres, Comput. Mech., vol. 25, pp. 329-345, 2000.
23. S. De, K.J. Bathe, The Method of Fine Spheres with Improved Numerical Integration., Comput. Struct., vol. 79, pp. 2183-2196, 2001.
24. J.W. Hong, K.J. Bathe, On Analytical Transformations for Efficiency Improvements in the Method of Finite Spheres, in K.J. Bathe (eds.), Computational Fluid and Solid Mechanics, Elsevier, Oxford, 2003.
25. H. Li, J.Q. Cheng, T.Y. Ng, A New Variation of Point Collocation: The Meshless Finite Mixture (MFM) Method, Int. J. Comput. Methods Engrg. Sci. Mech., vol. 6, pp. 21-29, 2005.
26. C.T. Dyka, S. Saigal, On the Development of a Hybrid Particle Method Using the Differential Formulation, Int. J. Comput. Methods Engrg. Sci. Mech., vol. 6, pp. 65-75, 2005.
27. P. Lancaster, K. Salkauskas, Surface Generated by Moving Least Squares Methods, Math. Comput., vol. 37, pp. 141-158, 1981.
28. S.N. Atluri, S.P. Shen, The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple & Less-Costly Alternative to the Finite Element and Boundary Element Methods, Comput. Modeling Engrg. Sci., vol. 3, pp. 11-51, 2002.
29. P. Krysl, T. Belytschko, Analysis of Thin Plates by the Element-Free Galerkin Methods, Comput. Mech., vol. 17, pp. 26-35, 1995.
30. S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970.
31. T.J.R. Hughes, The Finite Element Method, Prentice-Hall, New Jersey, 1987.
32. G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, New Jersey, 1973.
33. R.B. Hetnarski, J. Ignaczak, Mathematical Theory of Elasticity, Taylor & Francis, New York, 2004.
34. F.Z. Li, C.F. Shih, A. Needleman, A Comparison of Methods for Calculating Energy Release Rates, Engrg. Fract. Mech., vol. 21, pp. 405-421, 1985.
35. B. Moran, C.F. Shih, Crack Tip and Associated Domain Integrals from Momentum and Energy Balance, Engrg. Fract. Mech., vol. 27, pp. 615-642, 1987.
36. B. Moran, C.F. Shih, A General Treatment of Crack Tip Contour Integrals, Int. J. Fracture, vol. 35, pp. 295-310, 1987.
37. H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, Del Research Corporation, Hellertown, Pennsylvania, 1973.