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On the approximation in the smoothed finite element method (SFEM). (English) Zbl 1183.74261
Summary: This letter aims at resolving the issues raised in the recent short communication [Int. J. Numer. Meth. Eng. 76, No. 8, 1285–1295 (2008; doi:10.1002/nme.2460)] and answered in [Int. J. Numer. Meth. Eng. (2009; doi:10.1002/nme.2587)] by proposing a systematic approximation scheme based on non-mapped shape functions, which both allows to fully exploit the unique advantages of the smoothed finite element method (SFEM) and resolve the existence, linearity and positivity deficiencies pointed out in [doi:10.1002/nme.2460]. We show that Wachspress interpolants [E. Wachspress, Lect. Notes Math. 228, 223-252 (1971; Zbl 0229.65019)] computed in the physical coordinate system are very well suited to the SFEM, especially when elements are heavily distorted (obtuse interior angles). The proposed approximation leads to results that are almost identical to those of the SFEM initially proposed by G. R. Liu, K. Y. Dai and T. T. Nguyen [Comput. Mech. 39, No. 6, 859–877 (2007; Zbl 1169.74047)]. These results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of SFEMs.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Software:
XFEM
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References:
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