A new, general hp-version axisymmetric finite element is derived for the boundary value problems of thin linearly elastic shells of revolution, applying a complementary strain energy-based three-field dual-mixed variational principle. For the interpolation of the mid-surface geometry, non-uniform rational B-splines—NURBS—is used. The independent field variables of the weak formulation are the a priori non-symmetric stress tensor, the displacement vector, and the infinitesimal skew-symmetric rotation tensor. The theoretical model of the shell formulation is based on a consistent dimensional reduction process and a systematic variable-number reduction procedure. The inverse of the unvaried three-dimensional constitutive equation is employed since neither the classical kinematical assumptions nor the stress hypotheses are built in the mathematical model; namely, both the through-the-thickness variation and the normal stress to the shell mid-surface are not excluded. The new hp axisymmetric shell finite element is tested by a representative model problem for extremely thin and moderately thick, singly and doubly curved shells of negative and positive Gaussian curvature. Following from the numerical experiments, the constructed hp-shell finite element gives locking-free results not only for the displacement but also for the stresses.

中文翻译：

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使用 NURBS 中面插值的双混合 hp 版轴对称壳有限元

应用基于互补应变能的三场双混合变分原理，推导出了一种新的通用 hp 版本轴对称有限元，用于求解薄线弹性旋转壳的边界值问题。对于中间曲面几何体的插值，使用非均匀有理 B 样条 - NURBS -。弱公式的独立场变量是先验非对称应力张量、位移矢量和无穷小偏对称旋转张量。壳公式的理论模型基于一致的降维过程和系统的变数降维程序。由于数学模型中既没有建立经典的运动学假设，也没有建立应力假设，因此采用不变的三维本构方程的逆；即，不排除整个厚度的变化和壳中表面的法向应力。新的 hp 轴对称壳有限元通过具有代表性的模型问题进行测试，用于极薄和中等厚的负高斯曲率和正高斯曲率的单曲和双曲壳。根据数值实验，构建的 hp-shell 有限元不仅对位移而且对应力给出了无锁定结果。新的 hp 轴对称壳有限元通过具有代表性的模型问题进行测试，用于极薄和中等厚的负高斯曲率和正高斯曲率的单曲和双曲壳。根据数值实验，构建的 hp-shell 有限元不仅对位移而且对应力给出了无锁定结果。新的 hp 轴对称壳有限元通过具有代表性的模型问题进行测试，用于极薄和中等厚的负高斯曲率和正高斯曲率的单曲和双曲壳。根据数值实验，构建的 hp-shell 有限元不仅对位移而且对应力给出了无锁定结果。