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 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：

---

几何精确梁动力学的完全显式等几何配置公式


我们提出了几何精确的剪切变形梁的完全显式动力学公式。这项工作的起点是现有的等几何配置（IGA-C）公式，该公式在时间积分算法的严格意义上是明确的，但由于使用一致的质量矩阵，仍然需要系统矩阵求逆。此外，在这项工作中，由于时间离散旋转平衡方程中存在非线性项，效率还受到所需迭代求解方案的限制。在本文中，我们解决了这些局限性，并提出了一种能够保持空间高阶精度的新颖公式。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的情况来完成的。该过程依赖于诺伊曼边界条件的解耦以及质量矩阵的重新排列和重新缩放。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，可以获得计算成本方面的额外收益，而不会造成精度方面的任何重大损失。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具有高阶空间精度和更高的效率。