We present a fully explicit dynamic formulation for geometrically exact shear-deformable beams. The starting point of this work is an existing isogeometric collocation (IGA-C) formulation which is explicit in the strict sense of the time integration algorithm, but still requires a system matrix inversion due to the use of a consistent mass matrix. Moreover, in that work, the efficiency was also limited by an iterative solution scheme needed due to the presence of a nonlinear term in the time-discretized rotational balance equation. In the present paper, we address these limitations and propose a novel fully explicit formulation able to preserve high-order accuracy in space. This is done by extending a predictor–multicorrector approach, originally proposed for standard elastodynamics, to the case of the rotational dynamics of geometrically exact beams. The procedure relies on decoupling the Neumann boundary conditions and on a rearrangement and rescaling of the mass matrix. We demonstrate that an additional gain in terms of computational cost is obtained by properly removing the angular velocity-dependent nonlinear term in the rotational balance equation without any significant loss in terms of accuracy. The high-order spatial accuracy and the improved efficiency of the proposed formulation compared to the existing one are demonstrated through some numerical experiments covering different combinations of boundary conditions.

中文翻译：

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用于几何精确光束动力学的完全显式等几何配置公式


我们提出了一个几何精确的剪切变形梁的完全显式动力学公式。这项工作的起点是一个现有的等几何搭配 （IGA-C） 公式，它在严格意义上的时间积分算法中是明确的，但由于使用了一致的质量矩阵，它仍然需要系统矩阵反转。此外，在该工作中，由于时间离散旋转平衡方程中存在非线性项，效率也受到所需的迭代求解方案的限制。在本文中，我们解决了这些限制，并提出了一种新颖的完全显式公式，能够在空间中保持高阶精度。这是通过将最初为标准弹性动力学提出的预测器-多重校正器方法扩展到几何精确梁的旋转动力学来实现的。该过程依赖于 Neumann 边界条件的解耦以及质量矩阵的重排和重新缩放。我们证明，通过正确去除旋转平衡方程中与角速度相关的非线性项，可以在精度方面获得任何显著的损失，从而在计算成本方面获得额外的收益。通过一些涵盖不同边界条件组合的数值实验，证明了所提出的公式与现有公式相比的高阶空间精度和更高的效率。