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A comparison of two formulations to blend finite elements and mesh-free methods.(English)Zbl 1059.65104

Summary: Mesh-free methods have since their early developments been blended to the finite element formulation in order to benefit from the advantages of both numerical techniques. In this paper, two recently proposed formulations to couple mesh-free and finite element methods are discussed and compared.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Cite

References:

[1]Belytschko, T.; Organ, D.; Krongauz, Y., A coupled finite element-element-free Galerkin method, Comput. Mech., 17, 3, 186-195 (1995) ·Zbl 0840.73058
[2]Hegen, D., Element free Galerkin methods in combination with finite element approaches, Comput. Methods Appl. Mech. Eng., 135, 1-2, 143-166 (1996) ·Zbl 0893.73063
[3]Huerta, A.; Fernández-Méndez, S., Enrichment and coupling of the finite element and meshless methods, Int. J. Numer. Methods Eng., 48, 11, 1615-1636 (2000) ·Zbl 0976.74067
[4]Fernández-Méndez, S.; Huerta, A., Coupling finite elements and particles for adaptivity: an application to consistently stabilized convection-diffusion, (Griebel, M.; Schweitzer, M. A., Meshfree Methods for Partial Differential Equations. Meshfree Methods for Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 26 (2002), Springer-Verlag: Springer-Verlag Berlin), 117-129, papers from the International workshop, Universität at Bonn, Germany, September 11-14, 2001 ·Zbl 1012.65120
[5]Huerta, A.; Fernández-Méndez, S.; Dı́ez, P., Enrichissement des interpolations d’ éléments finis en utilisant des méthodes de particules, ESAIM-Math. Model. Numer. Anal., 36, 6, 1027-1042 (2002) ·Zbl 1042.65085
[6]Fernández-Méndez, S.; Dı́ez, P.; Huerta, A., Convergence of finite elements enriched with meshless methods, Numer. Math., 96, 1, 43-59 (2003) ·Zbl 1049.65117
[7]Liu, W. K.; Uras, R. A.; Chen, Y., Enrichment of the finite element method with the reproducing kernel particle method, J. Appl. Mech. ASME, 64, 861-870 (1997) ·Zbl 0920.73366
[8]Wagner, G. J.; Liu, W. K., Hierarchical enrichment for bridging scales and mesh-free boundary conditions, Int. J. Numer. Methods Eng., 50, 3, 507-524 (2001) ·Zbl 1006.76073
[9]Han, W.; Wagner, G. J.; Liu, W. K., Convergence analysis of a hierarchical enrichment of Dirichlet boundary conditions in a mesh-free method, Int. J. Numer. Methods Eng., 53, 6, 1323-1336 (2002) ·Zbl 0995.65108
[10]Zhang, L.; Wagner, G.; Liu, W. K., A parallelized meshfree method with boundary enrichment for large-scale CFD, J. Comput. Phys., 176, 2, 483-506 (2002) ·Zbl 1060.76096
[11]Liu, W. K.; Jun, S.; Zhang, Y. F., Reproducing kernel particle methods, Int. J. Numer. Methods Fluids, 20, 8-9, 1081-1106 (1995) ·Zbl 0881.76072
[12]Li, S.; Liu, W. K., Meshfree and particle methods and their applications, Appl. Mech. Rev., 55, 4, 1-34 (2002)
[13]Liu, W. K.; Li, S.; Belytschko, T., Moving least square reproducing kernel methods Part I: Methodology and convergence, Comput. Methods Appl. Mech. Eng., 143, 1-2, 113-154 (1997) ·Zbl 0883.65088
[14]Chen, J.-S.; Han, W.; You, Y.; Meng, X., A reproducing kernel method with nodal interpolation property, Int. J. Numer. Methods Eng., 56, 7, 935-960 (2003) ·Zbl 1106.74424
[15]A. Huerta, S. Fernández-Méndez, Coupling element free Galerkin and finite element methods, in: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2000), 11-14 September, Barcelona, 2000, electronic publication ISBN: 84-89925-70-4; A. Huerta, S. Fernández-Méndez, Coupling element free Galerkin and finite element methods, in: Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2000), 11-14 September, Barcelona, 2000, electronic publication ISBN: 84-89925-70-4
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.
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