A new hierarchical partition of unity formulation of EFG meshless methods. (English) Zbl 1423.74906

Summary: Meshless methods have shown increased accuracy and better convergence rates compared to other well-known simulation methods in a variety of computational mechanics problems. Their computational cost is relatively higher especially in adaptivity analysis, when new nodes are inserted at each step, where the construction of updated shape functions requires a significant amount of computational effort. In this paper, two hierarchical formulations are proposed in the context of an \(h\)-type refinement scheme. The addition of new nodes and subsequently the re-calculation of the influenced moment matrices, that are necessary for obtaining the shape functions and their derivatives and subsequently for the construction of the stiffness matrix, are properly addressed. Both hierarchical schemes do not need the recalculation of the initial stiffness matrix but only the additional node contributions to each shape function field, while the second scheme produces purely hierarchically refined stiffness matrices leaving the initial stiffness matrix unmodified.


74S05 Finite element methods applied to problems in solid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs


Full Text: DOI


[1] Belytschko, T.; Krongauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: an overview and recent developments, Comput. Methods Appl. Mech. Engrg., 139, 3-47, (1996) · Zbl 0891.73075
[2] Li, S.; Liu, W. K., Meshfree particle methods, (2004), Springer · Zbl 1073.65002
[3] Li, S.; Liu, W. K., Meshfree and particle methods and their applications, Appl. Mech. Rev., 55, 1-34, (2002)
[4] Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Internat. J. Numer. Methods Fluids, 20, 1081-1106, (1995) · Zbl 0881.76072
[5] Le, C. V.; Askes, H.; Gilbert, M., Adaptive element-free Galerkin method applied to the limit analysis of plates, Comput. Methods Appl. Mech. Engrg., 199, 37-40, 2487-2496, (2010) · Zbl 1231.74485
[6] Liu, W. K.; Uras, R. A.; Chen, Y., Enrichment of the finite element method with the reproducing kernel particle method, J. Appl. Mech., 64, 4, 861-870, (1997) · Zbl 0920.73366
[7] Belytschko, T.; Organ, D.; Krongauz, Y., A coupled finite element-element-free Galerkin method, Comput. Mech., 17, 186-195, (1995) · Zbl 0840.73058
[8] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Internat. J. Numer. Methods Engrg., 37, 229-256, (1994) · Zbl 0796.73077
[9] Lu, Y. Y.; Belytschko, T.; Gu, L., A new implementation of the element free Galerkin method, Comput. Methods Appl. Mech. Engrg., 113, 3-4, 397-414, (1996) · Zbl 0847.73064
[10] Nguyen, V. P.; Rabczuk, T.; Bordas, S.; Duflot, M., Meshless methods: a review and computer implementation aspects, Math. Comput. Simul., 79, 763-813, (2008) · Zbl 1152.74055
[11] Duarte, C. A.; Oden, J. T., An h-p adaptive method using clouds, Comput. Methods Appl. Mech. Engrg., 139, 237-262, (1996) · Zbl 0918.73328
[12] Melenk, J. M.; Babuška, I., The partition of unity method: basic theory and applications, Internat. J. Numer. Methods Engrg., 40, 4, 727-758, (1997) · Zbl 0949.65117
[13] Yuan, W.-R.; Chen, P.; Liu, K.-X., High performance sparse solver for unsymmetrical linear equations with out-of-core strategies and its application on meshless methods, Appl. Math. Mech. (English Ed.), 27, 1339-1348, (2006) · Zbl 1167.65354
[14] Wu, S. C.; Zhang, H. O.; Zheng, C.; Zhang, J. H., A high performance large sparse symmetric solver for the meshfree Galerkin method, Int. J. Comput. Methods, 5, 533-550, (2008) · Zbl 1264.80032
[15] Danielson, K. T.; Hao, S.; Liu, W. K.; Uras, R. A.; Li, S., Parallel computation of meshless methods for explicit dynamic analysis, Internat. J. Numer. Methods Engrg., 47, 1323-1341, (2000) · Zbl 0981.74078
[16] Metsis, P.; Papadrakakis, M., Overlapping and non-overlapping domain decomposition methods for large-scale meshless EFG simulations, Comput. Methods Appl. Mech. Engrg., 229-232, 128-141, (2012) · Zbl 1253.74110
[17] Karatarakis, A.; Metsis, P.; Papadrakakis, M., GPU-acceleration of stiffness matrix calculation and efficient initialization of EFG meshless methods, Comput. Methods Appl. Mech. Engrg., 258, 63-80, (2013) · Zbl 1286.65162
[18] Barbieri, E.; Meo, M., A fast object-oriented MATLAB implementation of the reproducing kernel particle method, Comput. Mech., 49, 5, 581-602, (2011) · Zbl 1398.74450
[19] Wang, J. G.; Liu, G. R., A point interpolation meshless method based on radial basis functions, Internat. J. Numer. Methods Engrg., 54, 1623-1648, (2002) · Zbl 1098.74741
[20] Wenterodt, C.; von Estorff, O., Optimized meshfree methods for acoustics, Comput. Methods Appl. Mech. Engrg., 200, 25-28, 2223-2236, (2011) · Zbl 1230.76043
[21] Fernández-Méndez, S.; Huerta, A., Imposing essential boundary conditions in mesh-free methods, Comput. Methods Appl. Mech. Engrg., 193, 12-14, 1257-1275, (2004) · Zbl 1060.74665
[22] Miller, K. S., On the inverse of the sum of matrices, Math. Mag., 54, 2, 67-72, (1981) · Zbl 0462.15004
[23] Papadrakakis, M.; Babilis, G. P., Solution techniques for the \(p\)-version of the adaptive finite element method, Internat. J. Numer. Methods Engrg., 37, 1413-1431, (1994) · Zbl 0805.73064
[24] Lee, C. K.; Zhou, C. E., On error estimation and adaptive refinement for element free Galerkin method: part I: stress recovery and a posteriori error estimation, Comput. Struct., 82, 4-5, 413-428, (2004)
[25] Zienkiewicz, O. C.; Zhu, J. Z., The superconvergent patch recovery and a posteriori error estimates. part 2: error estimates and adaptivity, Internat. J. Numer. Methods Engrg., 33, 1365-1382, (1992) · Zbl 0769.73085
[26] Tabbara, M.; Blacker, T.; Belytschko, T., Finite element derivative recovery by moving least square interpolants, Comput. Methods Appl. Mech. Engrg., 117, 1-2, 211-223, (1994) · Zbl 0848.73072
[27] Chung, H.-J.; Belytschko, T., An error estimate in the EFG method, Comput. Mech., 21, 91-100, (1998) · Zbl 0910.73060
[28] Szabó, B.; Babuška, I., Finite element analysis, (1991), Wiley New York
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.