A rotation-based geometrically nonlinear spectral Reissner--Mindlin shell element
Nima Azizi, Wolfgang Dornisch
TL;DR
The paper tackles robust nonlinear shell analysis by introducing a rotation-based, geometrically nonlinear spectral element method for Reissner--Mindlin shells that uses an additive update of the director via Rodrigues rotations and a five-DOF-per-node formulation. It develops an isoparametric SEM framework (SEMI) and contrasts it with SEMN (CAD/NURBS-based geometry) and IGA variants with discrete and continuous rotation interpolations (IGA-RMD, IGA-RMC). Through four challenging nonlinear benchmarks (including Scordelis–Lo roof and curvature-rich free-form geometries), the work demonstrates that high-order SEM with a simple discrete rotation yields accuracy comparable to the most advanced IGA approaches while offering superior computational efficiency, especially in stiffness assembly. The results indicate that exact geometry representation is less critical than rotational formulation and interpolation characteristics, and they advocate using high-order elements on coarser SEM meshes for locking-limited nonlinear shell analyses. The study also outlines strategies for accelerating SEM computations and sketches future work on static condensation and CAD-based SEM extensions for complex patched geometries.
Abstract
In this paper, we propose a geometrically nonlinear spectral shell element based on Reissner--Mindlin kinematics using a rotation-based formulation with additive update of the discrete nodal rotation vector. The formulation is provided in matrix notation in detail. The use of a director vector, as opposed to multi-parameter shell models, significantly reduces the computational cost by minimizing the number of degrees of freedom. Additionally, we highlight the advantages of the spectral element method (SEM) in combination with Gauss-Lobatto-Legendre quadrature regarding the computational costs to generate the element stiffness matrix. To assess the performance of the new formulation for large deformation analysis, we compare it to three other numerical methods. One of these methods is a non-isoparametric SEM shell using the geometry definition of isogeometric analysis (IGA), while the other two are IGA shell formulations which differ in the rotation interpolation. All formulations base on Rodrigues' rotation tensor. Through the solution of various challenging numerical examples, it is demonstrated that although IGA benefits from an exact geometric representation, its influence on solution accuracy is less significant than that of shape function characteristics and rotational formulations. Furthermore, we show that the proposed SEM shell, despite its simpler rotational formulation, can produce results comparable to the most accurate and complex version of IGA. Finally, we discuss the optimal SEM strategy, emphasizing the effectiveness of employing coarser meshes with higher-order elements.