×

zbMATH — the first resource for mathematics

A face-based smoothed finite element method (FS-FEM) for 3D linear and geometrically nonlinear solid mechanics problems using 4-node tetrahedral elements. (English) Zbl 1183.74299
Summary: This paper presents a novel face-based smoothed finite element method (FS-FEM) to improve the accuracy of the finite element method (FEM) for three-dimensional (3D) problems. The FS-FEM uses 4-node tetrahedral elements that can be generated automatically for complicated domains. In the FS-FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS-FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non-linear solid mechanics problems. In addition, a novel domain-based selective scheme is proposed leading to a combined FS/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS-FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS-FEM is found better than that of the FEM.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Dohrmann, A least squares approach for uniform strain triangular and tetrahedral finite elements, International Journal for Numerical Methods in Engineering 42 pp 1181– (1998) · Zbl 0912.73055
[2] 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
[3] Puso, A formulation and analysis of a stabilized nodally integrated tetrahedral, International Journal for Numerical Methods in Engineering 67 pp 841– (2006) · Zbl 1113.74075
[4] Liu, Meshfree Methods: Moving Beyond the Finite Element Method (2002)
[5] Gu, Meshfree methods and their comparisons, International Journal of Computational Methods 2 pp 477– (2005) · Zbl 1137.74302
[6] Chen, A stabilized conforming nodal integration for Galerkin meshfree method, International Journal for Numerical Methods in Engineering 50 pp 435– (2000)
[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 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 pp 199– (2008) · Zbl 1222.74044
[9] Liu, A 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 (2008)
[10] Liu, A linearly conforming point interpolation method (LC-PIM) for 2D solid mechanics problems, International Journal of Computational Methods 2 (4) pp 645– (2005) · Zbl 1137.74303
[11] Liu, 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
[12] Liu, A linearly conforming radial point interpolation method for solid mechanics problems, International Journal of Computational Methods 3 (4) pp 401– (2006) · Zbl 1198.74120
[13] Liu, A smoothed finite element method for mechanics problems, Computational Mechanics 39 pp 859– (2007) · Zbl 1169.74047
[14] Liu, Theoretical aspects of the smoothed finite element method (SFEM), International Journal for Numerical Methods in Engineering 71 pp 902– (2007) · Zbl 1194.74432
[15] Nguyen-Xuan, Smooth finite element methods: convergence, accuracy and properties, International Journal for Numerical Methods in Engineering 74 pp 175– (2008) · Zbl 1159.74435
[16] Liu, A node-based smoothed finite element method for upper bound solution to solid problems (NS-FEM), Computers and Structures (2008)
[17] Dai, An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Elements in Analysis and Design 43 pp 847– (2007)
[18] Nguyen-Xuan, A smoothed finite element method for plate analysis, Computer Methods in Applied Mechanics and Engineering 197 pp 1184– (2008) · Zbl 1159.74434
[19] Nguyen-Thanh, A smoothed finite element method for shell analysis, Computer Methods in Applied Mechanics and Engineering (2008) · Zbl 1194.74453
[20] Liu, Upper bound solution to elasticity problems: a unique property of the linearly conforming point interpolation method (LC-PIM), International Journal for Numerical Methods in Engineering 74 pp 1128– (2008) · Zbl 1158.74532
[21] 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
[22] Duarte, Arbitrarily smooth generalized finite element approximations, Computer Methods in Applied Mechanics and Engineering 196 pp 33– (2006) · Zbl 1120.74816
[23] Liu, An edge-based smoothed finite element method (ES-FEM) for static, free and forced vibration analyses of solids, Journal of Sound and Vibration (2008)
[24] Nguyen-Thoi, An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics, Finite Elements in Analysis and Design (2008)
[25] Bathe, Finite Element Procedures (1996)
[26] Liu, The Finite Element Method: A Practical Course (2003) · Zbl 1027.74001
[27] Zienkiewicz, The Finite Element Method (2000) · Zbl 0962.76056
[28] Pian, Hybrid and Incompatible Finite Element Methods (2006) · Zbl 1110.65003
[29] Simo, On the variational foundations of assumed strain methods, Journal of Applied Mechanics 53 pp 51– (1986) · Zbl 0592.73019
[30] Reddy, An Introduction to Nonlinear Finite Element Analysis (2004) · Zbl 1057.65087
[31] Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987) · Zbl 0634.73056
[32] Puso, Meshfree and finite element nodal integration methods, International Journal for Numerical Methods in Engineering 74 pp 416– (2008) · Zbl 1159.74456
[33] 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
[34] Rabczuk, Stable particle methods based on Lagrangian kernels, Computer Methods in Applied Mechanics and Engineering 193 pp 1035– (2004) · Zbl 1060.74672
[35] Nguyen, Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation (2008) · Zbl 1152.74055
[36] Timoshenko, Theory of Elasticity (1970)
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.