C1-Continuous stress recovery in finite element analysis

Abstract

A Penalized Discrete-Least-Squares (PDLS) variational principle is employed to recover C 1-continuous, smooth stress fields from finite element stresses. The error functional involves discrete least-squares with a penalty constraint to enforce C 1 continuity of the recovered stresses. The recovery procedure uses a smoothing element analysis (SEA), with element-based interpolation functions, to minimize the error functional. SEA/PDLS recovers a superconvergent stress field of higher accuracy and continuity than the underlying, ‘consistent’ finite element stress field. General and specialized formulations of the functional are given, and an appropriate discretization strategy for the SEA is discussed. Numerical results for both one- and two-dimensional stress fields are presented and compared with the corresponding solutions for the superconvergent patch recovery (SPR) method of Zienkiewicz and Zhu. The results demonstrate that, in addition to achieving a higher degree of continuity, SEA/PDLS produces generally more accurate stresses than SPR. Although an additional global analysis is required, results indicate that two ‘coarse’ mesh analyses (finite element and smoothing) produce solutions comparable to a single finite element analysis with a much greater degree of refinement and more computational effort. The procedure is unique in that the recovered stress field is essentially C 1 continuous. The results are shown to be insensitive to the value of the penalty parameter above a threshold, and hence uncertainty in the choice of an appropriate penalty value is eliminated.