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

Yingying Lan, Bo Liu^*
The Solid Mechanics Research Centre, School of Aeronautic Science and Engineering, Beihang University (BUAA), Beijing, China
* Corresponding Author: Bo Liu. Email: email
(This article belongs to the Special Issue: Structural Reliability and Computational Solid Mechanics: Modeling, Simulation, and Uncertainty Quantification)

Computer Modeling in Engineering & Sciences https://doi.org/10.32604/cmes.2026.075706

Received 06 November 2025; Accepted 30 December 2025; Published online 21 January 2026

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Abstract

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.

Keywords

Geometrically exact shell; hierarchical quadrature element method; geometrically nonlinear; laminated shells; local p-refinement; shear and membrane locking; post-buckling behaviors

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

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