AI 中文总结
提出一种离散柯塞尔杆公式，将细长弹性杆表示为节点链，通过相对运动学评估应变度量，以柔顺形式表示本构行为。该公式源自混合彼得罗夫 - 伽辽金柯塞尔杆有限元公式，继承鲁棒性且无锁性，数值例子证实了方法的准确性、鲁棒性和收敛行为。
AI 中文摘要
在本通讯中，我们提出了一种离散柯塞尔杆公式，其中细长弹性杆表示为通过相邻节点对之间的柔顺弹性力和力矩耦合的刚体（节点）链。离散膨胀、剪切、扭转和曲率应变度量通过每个节点对的相对运动学来评估，而本构行为通过独立应力自由度以柔顺形式表示。我们表明，当内部虚功通过中点法则积分，外部和惯性贡献通过梯形法则积分时，所得模型严格源自线性运动插值阶次的混合彼得罗夫 - 伽辽金柯塞尔杆有限元公式（FEM）。所提出的公式继承了基础混合有限元的鲁棒性和无锁性，同时展现出一种双节点耦合结构，类似于计算机图形学界的离散杆模型。这与软机器人应用中常用的应变参数化降阶模型的密集耦合形成鲜明对比。三个数值例子，包括分段变化横截面、不同空间离散化下的腱驱动致动以及纵向 - 扭转耦合动力学，证实了所提出方法的准确性、鲁棒性和收敛行为。
英文摘要
In this communication we propose a discrete Cosserat rod formulation in which a slender elastic rod is represented as a chain of rigid bodies (nodes) coupled by compliant elastic forces and moments acting between adjacent node pairs. Discrete dilatation, shear, torsion and curvature strain measures are evaluated from the relative kinematics of each node pair, while the constitutive behavior is expressed in compliance form through independent stress degrees of freedom. We show that the resulting model arises rigorously from a mixed Petrov--Galerkin Cosserat rod finite element formulation (FEM) at linear kinematic interpolation order when the internal virtual work is integrated by the midpoint rule and the external and inertial contributions by the trapezoidal rule. The proposed formulation inherits the robustness and the absence of locking from the underlying mixed FEM while simultaneously exposing a two-node coupling structure that mirrors discrete rod models from the computer graphics community. This is in sharp contrast to the dense coupling of strain-parameterized reduced-order models often used in soft robotic applications. Three numerical examples involving piecewise-varying cross sections, tendon-driven actuation under different spatial discretizations, and coupled longitudinal-torsional dynamics confirm the accuracy, robustness, and convergence behavior of the presented approach.
