×

On error control in the element-free Galerkin method. (English) Zbl 1245.65161

Summary: We investigate discretisation error control in the element-free Galerkin method (EFGM) highlighting the differences from the finite element method (FEM). We demonstrate that the (now) conventional procedures for error analysis used in the finite element method require careful application in the EFGM, otherwise competing sources of error work against each other. Examples are provided of previous works in which adopting an FEM-based approach leads to dubious refinements. The discretisation error is here split into contributions arising from an inadequate number of degrees of freedom \(e_{h}\), and from an inadequate basis \(e_{p}\). Numerical studies given in this paper show that for the EFGM the error cannot be easily split into these component parts. Furthermore, we note that arbitrarily setting the size of nodal supports (as is commonly proposed in many papers) causes severe difficulties in terms of error control and solution accuracy. While no solutions to this problem are presented in this paper it is important to highlight these difficulties in applying an approach to errors from the FEM in the EFGM. While numerical tests are performed only for the EFGM, the conclusions are applicable to other meshless methods based on the concept of nodal support.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Software:

FEAPpv
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, 37, 229-256 (1994) · Zbl 0796.73077
[2] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 139, 3-47 (1996) · Zbl 0891.73075
[3] Fries TP, Matthies HG. Classification and overview of meshfree methods. Technical Report 2003-3, Technical University Braunschweig, Brunswick, Germany; July 2004.; Fries TP, Matthies HG. Classification and overview of meshfree methods. Technical Report 2003-3, Technical University Braunschweig, Brunswick, Germany; July 2004.
[4] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation, 79, 3, 763-813 (2008) · Zbl 1152.74055
[5] Zienkiewicz, O. C.; Morgan, K., Finite elements and approximations (2000), Dover Publications Inc.: Dover Publications Inc. Mineola, New York
[6] Ainsworth, M.; Oden, J. T., A posteriori error estimation in finite element analysis, Computer Methods in Applied Mechanics and Engineering, 142, 1-2, 1-88 (1997) · Zbl 0895.76040
[7] Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The finite element method: its basis and fundamentals (2005), Elsevier · Zbl 1307.74005
[8] Krysl, P.; Belytschko, T., Element-free Galerkin method: convergence of the continuous and discontinuous shape functions, Computer Methods in Applied Mechanics and Engineering, 148, 257-277 (1997) · Zbl 0918.73125
[9] Fleming, M.; Chu, Y. A.; Moran, B.; Belytschko, T., Enriched element-free Galerkin methods for crack tip fields, International Journal for Numerical Methods in Engineering, 40, 1483-1504 (1997)
[10] Belytschko, T.; Lu, Y. Y.; Gu, L.; Tabbara, M., Element-free Galerkin methods for static and dynamic fracture, International Journal for Numerical Methods in Engineering,, 32, 2547-2570 (1995) · Zbl 0918.73268
[11] Liu, G. R.; Tu, Z. H., An adaptive procedure based on background cells for meshless methods, Computer Methods in Applied Mechanics and Engineering, 191, 17-18, 1923-1943 (2002) · Zbl 1098.74738
[12] Rossi, R.; Alves, M. K., An h-adaptive modified element-free Galerkin method, European Journal of Mechanics: A: Solids, 24, 782-799 (2005) · Zbl 1125.74384
[13] Gavete, L.; Cuesta, J. L.; Ruiz, A., A numerical comparison of two different approximations of the error in a meshless method, European Journal of Mechanics: A: Solids, 21, 6, 1037-1054 (2002) · Zbl 1027.74074
[14] Dolbow, J.; Belytschko, T., Numerical integration of the Galerkin weak form in meshfree methods, Computational Mechanics, 23, 219-230 (1999) · Zbl 0963.74076
[15] Fries, T.-P.; Belytschko, T., The intrinsic partition of unity method, Computational Mechanics, 40, 803-814 (2007) · Zbl 1162.74049
[16] Askes, H.; de Borst, R.; Heeres, O., Conditions for locking-free elasto-plastic analyses in the element-Free Galerkin method, Computer Methods in Applied Mechanics and Engineering, 173, 1-2, 99-109 (1999) · Zbl 0962.74076
[17] Atluri, S. N.; Zhu, T., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, 22, 117-127 (1998) · Zbl 0932.76067
[18] Liu, G. R., Meshfree methods: moving beyond the finite element method (2003), CRC Press LLC: CRC Press LLC Florida
[19] Dolbow, J.; Belytschko, T., An introduction to programming the meshless element-free Galerkin method, Archives of Computational Methods in Engineering, 5, 207-241 (1998)
[20] Armentano, M. G., Error estimates in Sobolev spaces for moving least square approximations, SIAM Journal on Numerical Analysis, 39, 38-51 (2001) · Zbl 1001.65014
[21] Li, X.; Zhu, J., A Galerkin boundary node method and its convergence analysis, Journal of Computational and Applied Mathematics, 230, 1, 314-328 (2009) · Zbl 1189.65291
[22] Zuppa, C., Good quality point sets and error estimates for moving least square approximations, Applied Numerical Mathematics, 47, 575-585 (2003) · Zbl 1040.65034
[23] Washizu, K., Variational methods in elasticity and plasticity (1975), Pergamon: Pergamon New York · Zbl 0164.26001
[24] Timoshenko, S. P.; Goodier, J. N., Theory of elasticity (1970), McGraw-Hill: McGraw-Hill New York · Zbl 0266.73008
[25] Augarde, C. E.; Deeks, A. J., The use of Timoshenko’s exact solution for a cantilever beam in adaptive analysis, Finite Elements in Analysis and Design, 44, 595-601 (2008)
[26] Rabczuk, T.; Belytschko, T., Adaptivity for structured meshfree particle methods in 2D and 3D, International Journal for Numerical Methods in Engineering, 63, 1559-1582 (2005) · Zbl 1145.74041
[27] Le CV, Askes H, Gilbert M. A novel numerical procedure for limit analysis of plates: adaptive EFG method combined with SCP. In: Sansour C, editor. Proceedings of the 17th UK conference on computational mechanics, Nottingham, UK; 2009. p. 291-4.; Le CV, Askes H, Gilbert M. A novel numerical procedure for limit analysis of plates: adaptive EFG method combined with SCP. In: Sansour C, editor. Proceedings of the 17th UK conference on computational mechanics, Nottingham, UK; 2009. p. 291-4.
[28] Lu, Y.; Belytschko, T.; Gu, L., A new implementation of the element free Galerkin method, Computer Methods in Applied Mechanics and Engineering, 113, 397-414 (1994) · Zbl 0847.73064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.