A moving IRBFN-based integration-free meshless method

CMES - Computer Modeling in Engineering and Sciences

Volumn 61, Issue 1, 2010, Pages 63-109

  Le, Phong B H ^a   Rabczuk, Timon ^b   Mai Duy, Nam ^a   Tran Cong, Thanh ^a  

^a UNIVERSITY OF SOUTHERN QUEENSLAND   (Australia)

^b BAUHAUS UNIVERSITY WEIMAR   (Germany)

Author keywords

Collocation method; Crack; Elasticity; First order system; Local IRBF; Locking; Meshless; Moving IRBF; RBF

Indexed keywords

COLLOCATION METHOD; FIRST ORDER SYSTEMS; LOCAL IRBF; LOCKING; MESHLESS; MOVING IRBF; RBF;

APPROXIMATION THEORY; ATTITUDE CONTROL; COMPUTATIONAL EFFICIENCY; CONVERGENCE OF NUMERICAL METHODS; CRACKS; ELASTICITY; INTERPOLATION;

RADIAL BASIS FUNCTION NETWORKS;

EID: 77957590327     PISSN: 15261492     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (46)

1. Atluri, S.; Shen, S. (2002): The meshless local Petrov-Galerkin (MLPG) method:A simple & less-costly alternative to the finite element and the boundary element methods.CMES: Computer Modeling in Engineering & Sciences, vol.3, pp. 11-51.
2. Atluri, S.N.; Liu, H.T.; Han, Z.D. (2006): Meshless local Petrov-Galerkin (MLPG) mixed collocation method for elasticity problems.CMES: Computer Modeling in Engineering & Sciences, vol.14, no.3, pp. 141-152.
3. Babuška, I.; Melenk, I. (1997): The partition of unity method.International. Journal for Numerical Methods in Engineering, vol.40, no.4, pp. 727-758.
4. Baxter, B. (2002): Preconditioned conjugate gradient, radial basis functions, and toeplitz matrices.Compu. Math.Appl., vol.43, pp. 305-318.
5. Belytschko, T.; Krongauz, Y.; Fleming, M.; Organ, D.; Liu, W. (1996): Smoothing and accelerated computations in the element-free Galerkin method.J. Comput.Appl. Math, vol.74, pp. 111-126. (Pubitemid 126381010)
6. Belytschko, T.; Lu, Y.Y.; Gu, L. (1994): Element-free Galerkin methods.International. Journal for Numerical Methods in Engineering, vol.37, pp. 229-256.
7. Bordas, S.; Duflot, M.; Le, P. (2008): A simple error estimator for extended finite elements.Communications. In Numerical Methods In Engineering, vol.24, pp. 961-971.
8. Brown, D.; Ling, L.; Kansa, E.; Levesley, J. (2005): On approximate cardinal preconditioning methods for solving pdes with radial basis functions.Engineering. Analysis with Boundary Elements, vol.29, pp. 343-353.
9. Chen, J.; Hu,W.; Hu, H. (2008): Reproducing kernel enhanced local radial basis collocation method.International. Journal for Numerical Methods in Engineering, vol.75, pp. 600-627.
10. Cheng, A.; Golberg, M.; Kansa, E.J.; Zamitto, G. (2003): Exponential convergnce and h-c multiquadric collocation method for partial differential equations.Numerical. Methods for Partial Differential Equations, vol.19, pp. 571-594.
11. Fasshauer, G. (1997): Solving partial differential equations by collocation with radial basis functions.In. Méhauté, A.L.; Rabut, C.; Schumaker, L.(Eds): Surface Fitting and Multiresolution Methods, pp. 131-138. Vanderbilt University Press.
12. Han, Z.D.; Atluri, S.N. (2003): Truly meshless local petrov-galerkin (MLPG) solutions of traction & displacement BIEs.CMES: Computer Modeling in Engineering & Sciences, vol.4, no.6, pp. 665-678.
13. Jung, J.H. (2007): A note on the Gibbs phenomenon with multiquadratic radial basis functions.Applied. Numerical Mathematics, vol.57, pp. 213-229.
14. Kansa, E.; Hon, Y. (2000): circumventing the ill-conditioning problem with multiquadric radial basis functions: application to elliptic partial differential equations.Compu. Math.Appl., vol.39, pp. 123-137.
15. Le, P.; Mai-Duy, N.; Tran-Cong, T.; Baker, G. (2007): A numerical study of strain localization in elasto-therno-viscoplastic materials using radial basis function networks.CMC: Computers, Materials & Continua, vol.5, pp. 129-150.
16. Le, P.; Mai-Duy, N.; Tran-Cong, T.; Baker, G. (2008): A meshless modeling of dynamic strain localization in quasi-britle materials using radial basis function networks.CMES: Computer Modeling in Engineering & Sciences, vol.25, no.1, pp. 43-66.
17. Le, P.; Mai-Duy, N.; Tran-Cong, T.; Baker, G. (2010): A Cartesian-grid Collocation Technique with Integrated Radial Basis Functions for mixed boundary value problems.International. Journal for Numerical Methods in Engineering, vol.82, pp. 435-463.
18. Li, Q.; Shen, S.; Han, Z.D.; Atluri, S.N. (2003): Application of meshless local Petrov-Galerkin (MLPG) to problems with singularities, and material discontinuities, in 3-D elasticity.CMES: Computer Modeling in Engineering & Sciences, vol.4, no.5, pp. 571-585.
19. Libre, N.A.; Emdadi, A.; Kansa, E.J.; Rahimian, M.; Shekarchi, M. (2008): A stabilized RBF collocation scheme for Neumann type boundary value problems.CMES: Computer Modeling in Engineering & Sciences, vol.24, pp. 63-82.
20. Ling, L.; Kansa, E. (2004): Preconditioning for radial basis functions with domain decomposition methods.Mathematical and Computer Modelling, vol.40, pp. 1413-1427.
21. Liu, G.; Zhang, G.; Gu, Y. (2005): A meshfree radial point interpolation method (RPIM) for three-dimensional solids.Computational Mechanics, vol.36, no. 6, pp. 421-430. (Pubitemid 41361082)
22. Liu, G.; Zhang, J.; Li, H.; Lam, K.; Kee, B. (2006): Radial point interpolation based finite difference method for mechanics problems. International. Journal for Numerical Methods in Engineering, vol.68, pp. 728-754. (Pubitemid 44680299)
23. Liu, X.; Liu, G.; Tai, K. (2005): Radial point interpolation collocation method (RPICM) for partial differential equations.Computers. & Mathematics With Applications, vol.50, no.8-9, pp. 1425-1442. (Pubitemid 41770909)
24. Madych, W. (1992): Miscellaneous error bounds for multiquadric and related interpolators.Compu. Math.Appl., vol.24, pp. 121-138.
25. Mai-Duy, N.; Khennane, A.; Tran-Cong, T. (2007): Computation of laminated composite plates using indirect radial basis function networks.CMC: Computers, Materials & Continua, vol.5, pp. 63-77. (Pubitemid 46598761)
26. Mai-Duy, N.; Tran-Cong, T. (2001): Numerical solution of differential equations using multiquadric radial basic function networks.Neural Networks, vol.14, pp. 185-199. (Pubitemid 32157010)
27. Mai-Duy, N.; Tran-Cong, T. (2005): An efficient indirect RBFN-based method for numerical solution of PDEs.Numerical Methods for Partial Differetial Equations, vol.21, pp. 770-790.
28. Mai-Duy, N.; Tran-Cong, T. (2009): A Cartesian-Grid discretisation scheme based on local integrated IRBFNs for two-dimensional elliptic problems.CMES: Computer Modeling in Engineering & Sciences, vol.51, pp. 213-238.
29. Melenk, J.M.; Babuška, I. (1996): The partition of unity finite element method: Basic theory and applications.Computer Methods in Applied Mechanics and Engineering, vol.139, pp. 289-314.
30. Moës, N.; Dolbow, J.; Belytschko, T. (1999): A finite element method for crack growth without remeshing.International Journal for Numerical Methods in Engineering, vol.46, pp. 131-150.
31. Nguyen, V.; Rabczuk, T.; Bordas, S.; Duflot, M. (2008): Meshfree method: A review and computer implementation aspects.Mathematics and Computers in Simulation, vol.79, pp. 763-813.
32. Ohtake, Y.; Belyaev, A.; Seidel, H. (2006): Sparse surface reconstruction with adaptive partition of unity and radial basis functions.Graphical Models, vol.68, pp. 15-24.
33. Rabczuk, T.; Areias, P.; Belytschko, T. (2007): A meshfree thin shell method for non-linear dynamic fracture.International Journal for Numerical Methods in Engineering, vol.72, no.5, pp. 524-548.
34. Rabczuk, T.; Belytschko, T. (2004): Cracking particles: A simplified meshfree method for arbitrary evolving cracks.International Journal for Numerical Methods in Engineering, vol.61, no.13, pp. 2316-2343.
35. Schilling, R.J.; Caroll, J.J.; Al-Ajlouni, A.F. (2001): Approximation of nonlinear system with radial basis function neural networks.IEEE Transaction on Neural Networks, vol.12, pp. 1-15.
36. Shu, C.; Ding, H.; Yeo, K. (2003): Local radial basis functions based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stoke equations.Computer Methods in Applied Mechanics and Engineering, vol.192, pp. 941-954.
37. Shu, C.; Wu, Y. (2007): Integrated radial basis functions based differential quadrature method and its perfomance.International Journal for Numerical Methods in Fluids, vol.53, pp. 969-984.
38. Strouboulis, T.; Babuška, I.; Copps, K. (2000): The design and analysis of the generalized finite element method.Computer Methods in Applied Mechanics and Engineering, vol.181, pp. 43-71.
39. Strouboulis, T.; Copps, K.; Babuška, I. (2000): The generalized finite element method: an example of its implementation and illustration of its performance.International Journal for Numerical Methods in Engineering, vol.47, no.8, pp. 1401-1417.
40. Timoshenko, S.; Goodier, J. (1970): Theory of elasticity.McGraw-Hill, New York, USA, 3rd edition.
41. Tobor, I.; Reuter, P.; Schlick, C. (2004): Efficient reconstruction of large scattered geometric datasets using the partition of unity and radial basis functions.Journal of WSCG, vol.12, no.1-3. ISSN-1213-6972.
42. Tobor, I.; Reuter, P.; Schlick, C. (2006): Reconstructing multi-scale variational partition of unity implicit surfaces with attributes.Graphical Models, vol.68, pp. 25-41.
43. Šaler, B.; Vertnik, R. (2006): Meshfree local radial basis function collocation for diffusion problems.Compu. Math.Appl., vol.51, pp. 1269-1282.
44. Wendland, H. (1995): Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree.Advances in Computational Mathematics, vol.4, pp. 389-396.
45. Wendland, H. (2002): Fast evaluation of radial basis functions: Method based on partition of unity.In Chui, C.; Schumaker, L.; Stöckler, J.(Eds): Approximation theory X: Abstract and classical Analysis, pp. 473-483. Vanderbilt University Press.
46. Wright, G.; Fornberg, B. (2006): Scattered node compact finite difference-type formulas generated from radial basis functions.Journal of Computational Physics, vol.212, pp. 99-123.