On the approximation in the smoothed finite element method (SFEM)

This letter aims at resolving the issues raised in the recent short communication (Int. J. Numer Meth. Engng 2008; 76(8):1285-1295. DOI: 10.1002/nme.2460) and answered by (Int. J. Numer Meth. Engng 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) (Comput. Mech. 2007; 39(6):859-877. DOI: 10.1007/s00466-0060075-4; Commun. Numer Meth. Engng 2009: 25(1):19-34. DOI: 10.1002/cnm.1098; Int. J. Numer Meth. Engng 2007, 71(8):902-930; Comput. Meth. Appl. Mech. Engng 2008, 198(2):165-177. DOI: 10.1016/j.cma.2008.05.029; Comput. Meth. Appl. Mech. Engng 2007; submitted; Int. J. Numer Meth. Engng 2008; 74(2):175-208. DOI: 10.1002/nme.2146; Comput. Meth. Appl. Mech. Engng 2008; 197 (13-16):1184-1203. DOI: 10.1016/j.cma.2007.10.008) and resolve the existence, linearity and positivity deficiencies pointed out in (Int. J. Numer Meth. Engng 2008; 76(8):1285-1295). We show that Wachspress interpolants (A Rational Basis for Function Approximation. Academic Press, Inc.: New York, 1975) 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 in (Comput. Mech. 2007; 39(6):859-877. DOI: 10.1007/s00466-006-0075-4). These results suggest that the proposed approximation scheme forms a strong and rigorous basis for the construction of SFEMs. Copyright (C) 2009 John Wiley & Sons, Ltd.