On using degenerated solid shell elements in adaptive refinement analysis

Abstract

Three di!erent degenerated shell elements are studied in an adaptive re"nement procedure for the solution of shell problems. The stress recovery procedure expressed in a convective patch co-ordinate system is used for the construction of continuous smoothed stress "elds for the a posteriori error estimation. The performance of the stress recovery procedure, the error estimator and the adaptive re"nement strategy are tested by solving three benchmark shell problems. It is found that when adaptive re"nement is used, the adverse e!ects of boundary layers and stress singularities are eliminated and all the elements tested are able to achieve their optimal convergence rates. It is also found that the accuracy of the shell elements increases with the number of polynomial terms included in the stress and strain approximations. In addition, if complete Lagrangian polynomial terms are used, the element will be less sensitive to shape distortion than the one in which only complete polynomial terms are employed. Figure 11. Contour plots for Problem 1, last uniform mesh: (Left column: Quada9s element, Central column: SQ9S element, Right column: ANS6S element) (a) contour plots for principal membrane force N ; (b) contour plots for shear force Q WWFigure 12. Contour plots for Problem 2. (Left column: Quada9s element, Central column: SQ9S element, Right column: ANS6S element): (a) contour plots for principal membrane force N ; (b) contour plots for shear force Q WW DEGENERATED SOLID SHELL ELEMENTS IN ADAPTIVE REFINEMENT ANALYSIS 643 Figure 13. Results for Problem 1 by adaptive re"nement: (a) last adaptive meshes; (b) shear force Q WW ; (c) membrane force Nwith width comparable to the thickness of the shell (Figure 6(b)), the shear error norm will hardly be reduced unless the size of the element is of the order of the thickness of the shell. Hence, in the uniform re"nement, the convergence rate is reduced as the shear error gradually dominates the total error. In adaptive re"nement, as the mesh is re"ned, more elements (Figure 13(a)) are automatically generated along the soft-simply supported and the free edges as the shear error become relative large. As a result, the convergence rate is improved and the shear error was reduced for all the elements tested in adaptive re"nement as shown in Figure Figure 15. (a) Convergence of Problem 1, total relative error; (b) convergence of Problem 1, shear relative error a value of 0)6}0)7. The number of polynomial terms used also explains the relative accuracy of these three elements. In the Quada9s element, at least six terms are used in the interpolation of all the strain components while at most "ve terms and three terms are used in the SQ9S and the ANS6S elements, respectively. Such an e!ect can also be seen by comparing the contour plots shown in Figures 13 and 14. 648 Figure 16. (a) Convergence of Problem 2, total relative error; (b) convergence of Problem 2, shear relative error Results of Problem 3For this problem, four meshes are used in the uniform re"nement and their characteris...