Abstract
The 9-node quadrilateral shell element MITC9i is developed for the Reissner-Mindlin shell kinematics, the extended potential energy and Green strain. The following features of its formulation ensure an improved behavior: 1. The MITC technique is used to avoid locking, and we propose improved transformations for bending and transverse shear strains, which render that all patch tests are passed for the regular mesh, i.e. with straight element sides and middle positions of midside nodes and a central node. 2. To reduce shape distortion effects, the so-called corrected shape functions of Celia and Gray (Int J Numer Meth Eng 20:1447–1459, 1984) are extended to shells and used instead of the standard ones. In effect, all patch tests are passed additionally for shifts of the midside nodes along straight element sides and for arbitrary shifts of the central node. 3. Several extensions of the corrected shape functions are proposed to enable computations of non-flat shells. In particular, a criterion is put forward to determine the shift parameters associated with the central node for non-flat elements. Additionally, the method is presented to construct a parabolic side for a shifted midside node, which improves accuracy for symmetric curved edges. Drilling rotations are included by using the drilling Rotation Constraint equation, in a way consistent with the additive/multiplicative rotation update scheme for large rotations. We show that the corrected shape functions reduce the sensitivity of the solution to the regularization parameter gamma of the penalty method for this constraint. The MITC9i shell element is subjected to a range of linear and non-linear tests to show passing the patch tests, the absence of locking, very good accuracy and insensitivity to node shifts. It favorably compares to several other tested 9-node elements.
Highlights
It has been clear since the earliest implementations of a basic 9-node element, that it has several limitations, i.e. it is excessively stiff and its accuracy is diminished by shape distortions.To alleviate the problem of an excessive stiffness, five types of modifications to the standard formulation of the 9node element have been proposed: 1. Uniform Reduced Integration (URI) in [53], with the 2 × 2 instead of the 3 × 3 Gauss integration scheme
The standard isoparametric shape functions are obtained by the assumption that the midside nodes (5,6,7,8) are located at the middle positions between the respective corner nodes and the central node 9 is located at the element center
Ui, Vi, i are coefficients for displacement components and the drilling rotation depending on the nodal values uk, vk, ωk (k = 1, . . . , 9)
Summary
It has been clear since the earliest implementations of a basic 9-node element, that it has several limitations, i.e. it is excessively stiff (locks) and its accuracy is diminished by shape distortions. The 2 × 2-point scheme for the (12) strain components provided the correct rank of the tangent matrix and was the most accurate Another problem with the 9-node elements is the sensitivity of solutions to shifts of the midside nodes and the central node from the middle positions, which causes loss of accuracy for all the element formulations listed above. We have to verify whether the shell element MITC9i using the corrected shape functions passes patch tests for the midside nodes shifted along straight element sides and for the central node in an arbitrary position. The 9-node MITC9i shell element with drilling rotations is tested on a range of linear and non-linear numerical examples, which are performed to check passing the patch tests, an absence of locking, accuracy, an insensitivity to node shifts, and correctness of implementation of the drilling rotation; a selection of these tests is presented in Sect. Reference results obtained by our 4-node shell elements are provided; the corresponding ones for triangular elements can be found e.g., in [10]
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