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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.

### MSC:

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

### Keywords:

finite element method (FEM); smoothed finite element method (SFEM); strain smoothing; reduced integration; compatible model; assumed strain method
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\textit{G. R. Liu} et al., Int. J. Numer. Methods Eng. 71, No. 8, 902--930 (2007; Zbl 1194.74432)

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