The standard in rod finite element formulations is the Bubnov-Galerkin projection method, where the test functions arise from a consistent variation of the ansatz functions. This approach becomes increasingly complex when highly nonlinear ansatz functions are chosen to approximate the rod's centerline and cross-section orientations. Using a Petrov-Galerkin projection method, we propose a whole family of rod finite element formulations where the nodal generalized virtual displacements and generalized velocities are interpolated instead of using the consistent variations and time derivatives of the ansatz functions. This approach leads to a significant simplification of the expressions in the discrete virtual work functionals. In addition, independent strategies can be chosen for interpolating the nodal centerline points and cross-section orientations. We discuss three objective interpolation strategies and give an in-depth analysis concerning locking and convergence behavior for the whole family of rod finite element formulations.

翻译：杆有限元公式的标准方法是布勃诺夫-伽辽金投影法，其中测试函数由试函数的一致变分导出。当选择高度非线性的试函数来近似杆中心线和截面方向时，该方法变得日益复杂。采用佩特罗夫-伽辽金投影法，我们提出了一整族杆有限元公式，其中节点广义虚位移和广义速度被插值，而非使用试函数的一致变分和时间导数。该方法显著简化了离散虚功泛函中的表达式。此外，可以为插值节点中心点和截面方向选择独立的策略。我们讨论了三种客观插值策略，并对整族杆有限元公式的锁定和收敛行为进行了深入分析。