On G space theory

International Journal of Computational Methods

Volumn 6, Issue 2, 2009, Pages 257-289

  Liu, G R ^a,b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b SINGAPORE MIT ALLIANCE   (Singapore)

Author keywords

Compatibility; Finite element method; G space; Generalized smoothed Galerkin; Meshfree method; Numerical method; Point interpolation method; Solution bound; Variational principle; Weakened weakform

Indexed keywords

COMPATIBILITY; G SPACE; GENERALIZED SMOOTHED GALERKIN; MESHFREE METHOD; POINT INTERPOLATION METHOD; SOLUTION BOUND; VARIATIONAL PRINCIPLE; WEAKENED WEAKFORM;

APPROXIMATION THEORY; COMPUTATIONAL MECHANICS; FUNCTION EVALUATION; GALERKIN METHODS; INTERPOLATION; NUMBER THEORY; NUMERICAL METHODS; VARIATIONAL TECHNIQUES;

FINITE ELEMENT METHOD;

EID: 67549105641     PISSN: 02198762     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219876209001863     Document Type: Article

References (41)

1. Chen, J. S., Wu, C. T. and Belytschko, T. [2000] Regularization of material instabilities by meshfree approximations with intrinsic length scales, International Journal for Numerical Methods in Engineering 47, 1303-1322.
2. Chen, J. S., Wu, C. T., Yoon, S. and You, Y. [2001] A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50, 435-466.
3. Dai, K. Y., Liu, G. R. and Nguyen, T. T. [2007] An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Elements in Analysis and Design 43, 847-860.
4. Eringen, A. C. and Edelen, D. G. B. [1972] On nonlocal elasticity, Int. J. Eng. Sci. 10, 233-248.
5. Hughes, T. J. R. [1987] The Finite Element Method, Prentice-Hall, London.
6. Kee, B. B. T, Liu, G. R. and Lu, C. [2007] A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis, Computational Mechanics 40 837-853.
7. Liu, G. R. and Gu, Y. T. [2001] A point interpolation method for two-dimensional solids, International Journal for Numerical Methods in Engineering 50, 937-951.
8. Liu, G. R. and Liu, M. B. [2003] Smoothed Particle Hydrodynamics - A Meshfree Particle Method. (World Scientific: Singapore).
9. Liu, G. R. and Quek, S. S. [2003] Finite Element Method: A Practical Course (Butterworth-Heinemann, Burlington, MA).
10. Liu, G. R. and Gu, Y. T. [2005] An Introduction to MFree Methods and their Programming, (Springer).
11. Liu, G. R., Zhang, G. Y., Dai, K. Y., Wang, Y. Y., Zhong, Z. H., Li, G. Y. and Han, X. [2005] A linearly conforming point interpolation method (LC-PIM) for 2D mechanics problems, International Journal for Computational Methods 2, 645-665.
12. Liu, G. R. and Kee, B. B. T. [2006] A stabilized least-squares radial point collocation method (LS-RPCM) for adaptive analysis, Computer Method in Applied Mechanics and Engineering 195, 4843-4861.
13. Liu, G. R., Li, Y., Dai, K. Y., Luan, M. T. and Xue, W. [2006] A linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods 3, 401-428.
14. Liu, G. R., Dai, K. Y. and Nguyen, T. T. [2007] A smoothed finite element method for mechanics problems, Computational Mechanics 39, 859-877.
15. Liu, G. R., Nguyen, T. T., Dai, K. Y. and Lam, K. Y. [2007] Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71, 902-930.
16. Liu, G. R. [2008a] A generalized gradient smoothing technique and the smoothed bilinear form for galerkin formulation of a wide class of computational methods, International Journal of Computational Methods 5(2), 199-236.
17. Liu, G. R. [2008b] A G space and weakened weak (W2) form for a unified formulation of compatible and incompatible methods, part I-Theory and part II-Application to solid mechanics problems, International Journal for Numerical Methods in Engineering, accepted.
18. Liu, G. R. and Xu, G. X. [2008] A gradient smoothing method (GSM) for fluid dynamics problems, Int. J. Numer. Methods Fluids 56 10), 1101-1133.
19. Liu, G. R. and Zhang, G. Y. [2008a] Edge-based smoothed point interpolation method (ES-PIM), International Journal of Computational Methods 5(4), 621-646.
20. Liu, G. R. and Zhang, G. Y. [2008b] Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM), Int. J. Numer. Mech. Engrg. 74 1128-1161.
21. Liu, G. R., Nguyen, T. T. and Lam, K. Y. [2008] An edge-based smoothed finite element method (E-SFEM) for static and dynamic problems of solid mechanics, J. Sound. Vib., (in press), Doi: 10. 1016/ j.jsv.2008.08.027.
22. Liu, G. R. [2009] Meshfree Methods: Moving Beyond the Finite Element Method, 2nd Edition (CRC Press, Boca Raton, USA).
23. 2) formulation of a cell-based smoothed point interpolation methods (CS-PIM), International Journal of Computational Methods 6 1), 147-179.
24. Liu, G. R., Nguyen, T. T., Nguyen, X. H. and Lam, K. Y. [2009] A node-based smoothed finite element method (N-SFEM) for upper bound solutions to solid mechanics problems, Computers and Structures 87, 14-26.
25. Liu, M. B., Liu, G. R. and Zong, Z. [2008] An overview on smoothed particle hydrodynamics, International Journal of Computational Methods 5(1), 135-188.
26. Lucy, L. B. [1977] Numerical approach to testing the fission hypothesis, Astronomical Journal 82, 1013-1024.
27. Monaghan, J. J. [1982] Why particle methods work (hydrodynamics), SIAM Journal on Scientific and Statistical Computing 3, 422-433.
28. Nguyen, T. T., Liu, G. R., Lam, K. Y. and Zhang, G. Y. [2008] A face-based smoothed finite element method (F-SFEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements, International Journal for Numerical Methods in Engineering, (in press) doi:10.1002/nme.2491.
29. Oliveira Eduardo, R. and De Arantes, E. [1968] Theoretical foundations of the finite element method. International Journal of Solids and Structures 4, 929-952.
30. Peraire, J. [1999] Lecture Notes on Finite Element Methods for Elliptic Problems, (MIT).
31. Pian, T. H. H. and Wu, C. C. [2006] Hybrid and Incompatible Finite Element Methods (CRC Press, Boca Raton).
32. Strang, G. and Fix, G. J. [1973] An Analysis of the Finite Element Method (Prentice-Hall).
33. Susanne, C., Brenner, S. C. and Scott, L. R. [2008] The Mathematical Theory of Finite Element Methods, 3rd Edition, Springer.
34. Wang, J. G. and Liu, G. R. [2002] A point interpolation meshless method based on the radial basis functions, International Journal for Numerical Methods in Engineering 54, 1623-1648.
35. Wikipedia-the Free Encyclopedia, http://en.wikipedia.org/wiki/ Integration_by_parts.
36. Wu, S. C., Liu, G. R., Zhang, H. 0. and Zhang, G. Y. [2008] A node-based smoothed point interpolation method (NS-PIM) for three-dimensional thermoelastic problems, Numerical Heat Transfer, Part A: Applications 54, 1121-1147.
37. Xu, G. X., Liu, G. R. and Tani, A. [2009] An adaptive gradient smoothing method (GSM) for fluid dynamics problems, Int. J. Numer. Methods Fluids, Published Online, DOI: 10.1002/fld.2032.
38. Zhang, G. Y., Liu, G. R., Wang, Y. Y., Huang, H. T., Zhong, Z. H., Li, G. Y. and Han, X. [2007] A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems, International Journal for Numerical Methods in Engineering 72, 1524-1543.
39. Zhang, G. Y., Liu, G. R., Nguyen, T. T., Song, C. X., Han, X., Zhong, Z. H. and Li, G. Y. [2007] The upper bound property for solid mechanics of the linearly conforming radial point interpolation method (LC-RPIM), International Journal of Computational Methods 4(3), 521-541.
40. Zhang, Y. Q., Liu, G. R. and Han, X. [2006] Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure, Physics Letters A 349, 370-376.
41. Zienkiewicz, 0. C. and Taylor, R. L. [2000] The Finite Element Method (5th edition), (Butterworth Heinemann, Oxford).