A theoretical study on the smoothed FEM (S-FEM) models: properties, accuracy and convergence rates.

*(English)*Zbl 1202.74180Summary: Incorporating the strain smoothing technique of meshfree methods into the standard finite element method (FEM), Liu et al. have recently proposed a series of smoothed finite element methods (S-FEM) for solid mechanics problems. In these S-FEM models, the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements, nodes, edges or faces, and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. We present in this paper a general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S-FEM models. First, an assumed strain field derived from the Hellinger – Reissner variational principle is shown to be identical to the smoothed strain field used in the S-FEM models. We then define a smoothing projection operator to modify the compatible strain field and show a set of properties. We next establish a general error bound of the S-FEM models. Some numerical examples are given to verify the theoretical properties established.

##### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

##### Keywords:

numerical methods; meshfree methods; displacement model; smoothed finite element method (S-FEM); equilibrium model; node-based smoothed finite elements (NS-FEM); edge-based smoothed finite elements (ES-FEM)##### Software:

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\textit{G. R. Liu} et al., Int. J. Numer. Methods Eng. 84, No. 10, 1222--1256 (2010; Zbl 1202.74180)

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##### References:

[1] | Liu, Mesh-Free Methods: Moving Beyond the Finite Element Method (2009) |

[2] | Nagashima, Node-by-node meshless approach and its applications to structural analyses, International Journal for Numerical Methods in Engineering 46 pp 341– (1999) · Zbl 0965.74079 |

[3] | Dohrmann, Node-based uniform strain elements for three-node triangular and four-node tetrahedral meshes, International Journal for Numerical Methods in Engineering 47 pp 1549– (2000) · Zbl 0989.74067 |

[4] | Puso, A stabilized nodally integrated tetrahedral, International Journal for Numerical Methods in Engineering 67 pp 841– (2006) · Zbl 1113.74075 |

[5] | Puso, Meshfree and finite element nodal integration methods, International Journal for Numerical Methods in Engineering 74 pp 416– (2008) · Zbl 1159.74456 |

[6] | Chen, A stabilized conforming nodal integration for Galerkin mesh-free methods, International Journal for Numerical Methods in Engineering 50 pp 435– (2001) · Zbl 1011.74081 |

[7] | Yoo, Stabilized conforming nodal integration in the natural-element method, International Journal for Numerical Methods in Engineering 60 pp 861– (2004) · Zbl 1060.74677 |

[8] | Liu, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, International Journal of Computational Methods 2 pp 645– (2006) |

[9] | Zhang, A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems, International Journal for Numerical Methods in Engineering 72 pp 1524– (2007) · Zbl 1194.74543 |

[10] | Liu, A linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods 3 pp 401– (2006) · Zbl 1198.74120 |

[11] | Liu, Edge-based smoothed point interpolation method (ES-PIM), International Journal of Computational Methods 5 (4) pp 621– (2008) |

[12] | Liu, 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) pp 199– (2008) · Zbl 1222.74044 |

[13] | Cescotto, A natural neighbour method for linear elastic problems based on Fraeijs de Veubeke variational principle, International Journal for Numerical Methods in Engineering 71 pp 1081– (2007) · Zbl 1194.74513 |

[14] | Liu, A smoothed finite element for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047 |

[15] | Liu, Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71 pp 902– (2007) · Zbl 1194.74432 |

[16] | Nguyen-Thoi, Selective smoothed finite element method, Tsinghua Science and Technology 12 (5) pp 497– (2007) |

[17] | Dai, An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Element Analysis and Design 43 pp 847– (2007) |

[18] | Nguyen-Xuan, Smooth finite element methods: convergence, accuracy and properties, International Journal for Numerical Methods in Engineering 74 pp 175– (2008) · Zbl 1159.74435 |

[19] | Nguyen-Xuan, Addressing volumetric locking and instabilities by selective integration in smoothed finite elements, Communications in Numerical Methods in Engineering 25 pp 19– (2009) · Zbl 1169.74044 |

[20] | Liu, On the essence and the evaluation of the shape functions for the smoothed finite element method (SFEM) (Letter to Editor), International Journal for Numerical Methods in Engineering 77 pp 1863– (2009) · Zbl 1181.74137 |

[21] | Nguyen-Xuan, A smoothed finite element method for plate analysis, Computer Methods in Applied Mechanics and Engineering 197 pp 1184– (2008) · Zbl 1159.74434 |

[22] | Nguyen-Thanh, A smoothed finite element method for shell analysis, Computer Methods in Applied Mechanics and Engineering 198 pp 165– (2008) · Zbl 1194.74453 · doi:10.1016/j.cma.2008.05.029 |

[23] | Nguyen-Xuan, A stabilized smoothed finite element method for free vibration analysis of Mindlin-Reissner plates, Communications in Numerical Methods in Engineering 25 (8) pp 882– (2009) · Zbl 1172.74047 |

[24] | Liu, A node based smoothed finite element method (NS-FEM) for upper bound solution to solid mechanics problems, Computers and Structures 87 pp 14– (2009) |

[25] | Nguyen-Thoi, Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems, International Journal of Computational Methods 6 pp 633– (2009) · Zbl 1267.74115 |

[26] | Nguyen-Thoi, Adaptive analysis using the node-based smoothed finite element method (NS-FEM), Communications in Numerical Methods in Engineering (2009) · Zbl 1267.74115 |

[27] | Liu, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, Journal of Sound and Vibration 320 pp 1100– (2009) |

[28] | Nguyen-Thoi, An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics, Communications in Numerical Methods in Engineering (2009) |

[29] | Nguyen-Xuan, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, Smart Materials and Structures 18 pp 12– (2009) · doi:10.1088/0964-1726/18/6/065015 |

[30] | Nguyen-Xuan, An edge-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates, Computer Methods in Applied Mechanics and Engineering 199 (9-12) pp 471– (2010) |

[31] | Nguyen-Thoi, An edge-based smoothed finite element method (ES-FEM) for visco-elastoplastic analyses of 2D solids using triangular mesh, Computational Mechanics 45 pp 23– (2009) · Zbl 1398.74382 |

[32] | Tran, An edge-based smoothed finite element method for primal-dual shakedown analysis of structures, International Journal for Numerical Methods in Engineering 82 (7) pp 917– (2010) · Zbl 1188.74073 |

[33] | Nguyen-Thoi, A face-based Smoothed Finite Element Method (FS-FEM) for 3D linear and nonlinear solid mechanics problems using 4-node tetrahedral elements, International Journal for Numerical Methods in Engineering 78 pp 324– (2009) · Zbl 1183.74299 |

[34] | Nguyen-Thoi, A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh, Computer Methods in Applied Mechanics and Engineering 198 pp 3479– (2009) · Zbl 1230.74193 |

[35] | Liu, A novel Alpha Finite Element Method (\(\alpha\)FEM) for exact solution to mechanics problems using triangular and tetrahedral elements, Computer Methods in Applied Mechanics and Engineering 197 pp 3883– (2008) · Zbl 1194.74433 |

[36] | Fraeijs De Veubeke, Stress Analysis pp 145– (1965) · Zbl 0122.00205 |

[37] | Nguyen-Dang, Finite element equilibrium analysis of creep using the mean value of the equivalent shear modulus, Meccanica 15 pp 234– (1980) · Zbl 0454.73062 |

[38] | Nguyen-Dang H Sur la plasticité et le calcul des états limites par éléments finis 1985 |

[39] | Debongnie, Dual analysis with general boundary conditions, Computer Methods in Applied Mechanics and Engineering 122 pp 183– (1995) · Zbl 0851.73057 |

[40] | Debongnie, Proceedings of the Eighth International Conference on Computational Structures Technology (2006) |

[41] | Liu, 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 (2009) |

[42] | Bathe, Finite Element Procedures (1996) |

[43] | Hughes, The Finite Element Method (1987) · Zbl 0634.73056 |

[44] | Brezzi, Mixed and Hybrid Finite Element Methods (1991) · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1 |

[45] | Liu, A normed G space and weakened weak (W2) formulation of a cell-based Smoothed Point Interpolation Method, International Journal of Computational Methods 6 (1) pp 147– (2009) |

[46] | Liu, On a G space theory, International Journal of Computational Methods 6 (2) pp 257– (2009) |

[47] | Johnson, Some equilibrium finite element methods for two-dimensional elasticity problems, Numerische Mathematik 30 (1) pp 103– (1978) · Zbl 0427.73072 |

[48] | Felippa C 2009 http://www.colorado.edu/engineering/CAS/courses.d/IFEM.d/ |

[49] | Timoshenko, Theory of Elasticity (1987) |

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