On the use of lagrange multiplier compatible modes for controlling accuracy and stability of mixed shell finite elements

Abstract

Classical hybrid-type formulations for shell finite elements can be developed from the Hellinger-Reissner virational principle combined with the use of stress basis functions which satisfy the homogeneous equilibrium equations a priori. In principle this technique can provide accurate stresses although implementation for general shell theories is challenging due to the complexity of the governing equations and the difficulty of satisfying coordinate invariance. An alternative approach is to provide independent approximations for all stress components in an extended form of the Hellinger-Reissner principle which includes additional variational constraints on the stresses to satisfy equilibrium through the use of Lagrange multipliers. The numerical stability and accuracy of the resulting formulations have a delicate dependency on the balance between the stress, displacement and Lagrange multiplier fields, but the introduction of the Lagrange multipliers provides a means of gaining control over this balance. It is shown in the context of Mindlin kinematics that the use of element-based compatible Lagrange multipliers with local bubble basis functions can lead to accurate stresses, including transverse shear stresses, and the complete elimination of transverse shear locking. Further, due to the use of uncoupled stress components, the element flexibility matrix can be inverted with minimal expense leading to an efficient procedure for the element stiffness matrix. A four-node shell element, which is proved to be numerically stable in the sense of the Babuška-Brezzi condition, is described in detail.