Automatic adaptive refinement for shell analysis using nine-node assumed strain element

An automatic adaptive reÞnement procedure for the analysis of shell structures using the nine-node degenerated solid shell element is suggested. The basic adaptive reÞnement principle and the e¤ects of singularities and boundary layers on the convergence rate of the nine-node element used are discussed. A new stress recovery procedure based on the patch convective co-ordinate system concept is developed for the construction of a continuous smoothed stress Þeld over the shell domains. The stress recovery procedure is easy to implement, requires a modest computational e¤ort and needs only local patch information. It can be applied to shells with non-uniform thickness as well as to multi-layered shell structures. The smoothed recovered stress obtained is then used with the Zienkiewicz and Zhu error estimator for a posteriori error estimation during the adaptive reÞnement analysis. Numerical results which are in good agreement with theoretical predictions are obtained and they indicate that the current adaptive reÞnement procedure can eliminate the e¤ect of singularities inside the problem domains so that a near-optimal convergence rate is achieved in all the numerical examples. This also indicates that the stress recovery procedure can produce an accurate stress Þeld and as a result the error estimator can reßect the error distribution of the Þnite element solution. Even though in the current study only one type of element is used in the analysis, the whole adaptive reÞnement scheme can be readily applied to any other types of degenerated solid element