An energy–momentum conserving temporal integration scheme is presented for a recently proposed geometrically exact shell formulation with drilling degrees of freedom. The scheme is based on a novel idea of defining mixed discrete derivatives for holonomic constraint functions with displacements and rotations. By defining general discrete derivative expressions with unknown terms, the mixed discrete derivatives with second-order accuracy are constructed according to deformation modes to satisfy directionality and orthogonality properties simultaneously, thus preserving conservation laws of total energy and momenta. The analysis of shell structures is conducted using the weak form quadrature elements to ensure exact incorporation of constraints and conservation of total energy after discretization, as well as circumvent shear and membrane locking phenomena. Benchmark numerical examples are presented to demonstrate the validity of the present scheme.

中文翻译：

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具有钻孔自由度的几何精确壳的能量-动量守恒方案

对于最近提出的具有钻孔自由度的几何精确壳公式，提出了能量-动量守恒时间积分方案。该方案基于为具有位移和旋转的完整约束函数定义混合离散导数的新想法。通过定义未知项的一般离散导数表达式，根据变形模式构造二阶精度的混合离散导数，同时满足方向性和正交性，从而保持总能量和动量守恒定律。壳结构的分析是使用弱形式正交元素进行的，以确保在离散化后精确结合约束和总能量守恒，以及避免剪切和膜锁定现象。给出了基准数值例子来证明本方案的有效性。