In this study, an automated three-dimensional adaptive h-refinement procedure was presented in shell problems. By introducing the variable-node flat shell element with drilling degree-of-freedom, some drawbacks due to the displacement constraints in common adaptive procedure performed by quadrilateral elements are eliminated, which are imposed on the irregular nodes to preserve interelement compatibility. To establish this strategy, the following two major issues are intensively studied:
First, the variable-node flat shell element designated as CLS has been presented in this paper. The element has a variable number of mid-side nodes and each node has a drilling freedom. The element has been developed basically by combining a membrane element with drilling degree-of-freedom and a plate bending element. Thus the element possesses six degrees-of-freedom per node which, in addition to improvement of the element behavior, permits an easy connection to other six degrees-of-freedom per node elements. By introducing the variable-node elements which have physical midside nodes, some difficulties associated with connecting the different layer patterns in the common adaptive h-refinement on quadrilateral mesh, such as imposing displacement constraints on irregular nodes to enforce the inter-element compatibility can be easily overcome. It was verified from numerical tests that this element can be used for efficient analysis of shell structures, enabling to refine mesh locally and use this for adaptive mesh refinement.
Second, to obtain a better stress field for the error estimation, the superconvergent patch recovery which is proposed for one- and two-dimensional problems by Zienkiewicz and Zhu is extended to recover three-dimensional continuous nodal stresses in this study. The extension of the technique to three-dimensional problem is explained in detail and one typical example is presented to show that this can be effectively used for shell problems. The results show that the superconvergent patch recovery procedure gives better solution in comparison with others irrespective of mesh regularity and can provide good value at the domain boundary.
Using the simple error estimator derived from postprocessing of finite element stresses, some two- and three-dimensional numerical examples for the adaptive h-refinement using the variable-node transition element together with existing 4-node elements were presented. It is noted that the proposed h-refinement procedure is useful in practical engineering problems since the meshes are constructed based on the simple and efficient shell element.
