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Theoretical aspects of the smoothed finite element method (SFEM). (English) Zbl 1194.74432
Summary: This paper examines the theoretical bases for the smoothed finite element method (SFEM), which was formulated by incorporating cell-wise strain smoothing operation into standard compatible finite element method (FEM). The weak form of SFEM can be derived from the Hu-Washizu three-field variational principle. For elastic problems, it is proved that 1D linear element and 2D linear triangle element in SFEM are identical to their counterparts in FEM, while 2D bilinear quadrilateral elements in SFEM are different from that of FEM: when the number of smoothing cells (SCs) of the elements equals 1, the SFEM solution is proved to be ‘variationally consistent’ and has the same properties with those of FEM using reduced integration; when SC approaches infinity, the SFEM solution will approach the solution of the standard displacement compatible FEM model; when SC is a finite number larger than 1, the SFEM solutions are not ‘variationally consistent’ but ‘energy consistent’, and will change monotonously from the solution of SFEM (SC = 1) to that of SFEM (SC \(\to \infty \)). It is suggested that there exists an optimal number of SC such that the SFEM solution is closest to the exact solution. The properties of SFEM are confirmed by numerical examples.

74S05 Finite element methods applied to problems in solid mechanics
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[1] Displacement and equilibrium models in the finite element method. In Stress Analysis, (eds). Wiley: London, 1965.
[2] . Mixed and Hybrid Finite Element Methods. Springer: New York, 1991. · Zbl 0788.73002 · doi:10.1007/978-1-4612-3172-1
[3] Simo, Journal of Applied Mechanics 53 pp 51– (1986)
[4] . Computational Inelasticity. Springer: New York, 1998. · Zbl 0934.74003
[5] Chen, International Journal for Numerical Methods in Engineering 50 pp 435– (2000)
[6] Yoo, International Journal for Numerical Methods in Engineering 60 pp 861– (2004)
[7] Liu, International Journal of Computational Methods (2006)
[8] Finite Element Procedures. MIT Press/Prentice-Hall: Cambridge, MA, Englewood Cliffs, NJ, 1996.
[9] . The Finite Element Method: A Practical Course. Butterworth Heinemann: Oxford, 2003. · Zbl 1027.74001
[10] . The Finite Element Method (5th edn). Butterworth Heinemann: Oxford, 2000.
[11] Liu, Computational Mechanics (2006)
[12] . Hybrid and Incompatible Finite Element Methods. CRC Press: Boca Raton, FL, 2006. · Zbl 1110.65003
[13] Reduced integration to give equilibrium models for assessing the accuracy of finite element analysis. Proceedings of the Third International Conference in Australia on Finite Element Methods, July 1979.
[14] Kelly, International Journal for Numerical Methods in Engineering 15 pp 1489– (1980)
[15] . Theory of Elasticity (3rd edn). McGraw-Hill: New York, 1970.
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