标题： 一种混合 Petrov--Galerkin Cosserat 杆有限元公式化 显示英文标题

标题： A mixed Petrov--Galerkin Cosserat rod finite element formulation

摘要： 本文提出了一种总拉格朗日混合佩特罗夫-加莱尔金有限元公式，该方法为分析柯西-罗德提供了一种计算效率高的方法，且不存在奇点和锁死现象。 为了实现无奇点的杆件方向参数化，节点运动学未知量定义为节点中心线位置和单位四元数。 我们对所有节点运动学坐标应用拉格朗日插值，并结合非单位四元数的投影，从而得到具有正交横截面固定基的插值。 为了消除剪切锁死等锁死效应，应用了变分海林格-赖斯纳原理，从而得到一种包含结果接触力和力矩的附加场的混合方法。 由于混合公式以符合形式包含本构定律，因此自然地包含了约束理论，如基尔霍夫-洛维理论。 本研究特别考察了附加内力场对数值性能的影响，包括锁死缓解和鲁棒性。 通过使用已建立的基准示例，该方法展示了增强的计算鲁棒性和效率，这由在应用标准牛顿-拉夫森方法时所需载荷步数和迭代次数的减少所证明。 显示英文摘要

摘要： This paper presents a total Lagrangian mixed Petrov--Galerkin finite element formulation that provides a computationally efficient approach for analyzing Cosserat rods that is free of singularities and locking. To achieve a singularity-free orientation parametrization of the rod, the nodal kinematical unknowns are defined as the nodal centerline positions and unit quaternions. We apply Lagrangian interpolation to all nodal kinematic coordinates, and in combination with a projection of non-unit quaternions, this leads to an interpolation with orthonormal cross-section-fixed bases. To eliminate locking effects such as shear locking, the variational Hellinger--Reissner principle is applied, resulting in a mixed approach with additional fields composed of resultant contact forces and moments. Since the mixed formulation contains the constitutive law in compliance form, it naturally incorporates constrained theories, such as the Kirchhoff--Love theory. This study specifically examines the influence of the additional internal force fields on the numerical performance, including locking mitigation and robustness. Using well-established benchmark examples, the method demonstrates enhanced computational robustness and efficiency, as evidenced by the reduction in required load steps and iterations when applying the standard Newton--Raphson method.