Geometrically Nonlinear Analyses of Isotropic and Laminated Shells by a Hierarchical Quadrature Element Method

In this work, the Hierarchical Quadrature Element Method (HQEM) formulation of geometrically exact shells is proposed and applied for geometrically nonlinear analyses of both isotropic and laminated shells. The stress resultant formulation is developed within the HQEM framework, consequently significantly simplifying the computations of residual force and stiffness matrix. The present formulation inherently avoids shear and membrane locking, benefiting from its high-order approximation property. Furthermore, HQEM’s independent nodal distribution capability conveniently supports local p-refinement and flexibly facilitates mesh generation in various structural configurations through the combination of quadrilateral and triangular elements. Remarkably, in lateral buckling analysis, the HQEM outperforms the weak-form quadrilateral element (QEM) in accuracy with identical nodal degrees of freedom (three displacements and two rotations). Under high-load nonlinear response, the QEM exhibits a maximum relative deviation of approximately 9.5% from the reference, while the HQEM remains closely aligned with the benchmark results. In addition, for the cantilever beam under tip moment, HQEM produces virtually no out-of-plane deviation, compared to a slight deviation of 0.00001 with QEM, confirming its superior numerical reliability. In summary, the method demonstrates high accuracy, superior convergence, and robustness in handling large rotations and complex post-buckling behaviors across a series of benchmark problems.