New superconvergent points of the 8-node serendipity plane element for patch recovery

The Gaussian quadrature points, which are generally observed to be the same as the Barlow points for lower order elements, have so far been used as the sampling points for the superconvergent patch recovery (SPR). Recent developments on the best- t method to calculate the optimal sampling points suggest that, for higher order elements, Barlow points need not be the optimal sampling points and also need not be the same as the Gaussian quadrature points. In this paper the best- t method is extended to predict the optimal points of the 8-node serendipity rectangular element, and it is observed that best- t points do not exist. Next, a novel method is proposed, in which, the expressions for stress-error based on the best- t are used in the least-square t of the patch recovery, and thereby the superconvergent points are obtained more directly. Application of this method to the 8-node serendipity element reveals the existence of two sets of superconvergent points for patch recovery, one of which is the well-known Gaussian points, (±1= √ 3;±1= √ 3), and the other is the set of four points given by {(0;± 2=3); (± 2=3; 0)}, the existence of which has not been known before. A detailed numerical study on the patch recovery of stresses for two demonstrative problems reveals that there indeed exist two sets of superconvergent points as predicted by the proposed method. The comparative performance of the two sets of points is tested for typical demonstrative problems and the results are discussed