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，其中测试函数由试探函数的一致变分导出。当选用高度非线性的试探函数来近似杆的形心线及横截面方向时，该方法变得日益复杂。我们采用彼得罗夫-伽辽金投影法，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通过插值获得，而非使用试探函数的一致变分及时间导数。这一方法显著简化了离散虚功泛函中的表达式。此外，可为节点形心线与横截面方向的插值选择独立的策略。我们讨论了三种客观插值策略，并对整族杆有限元公式的闭锁及收敛特性进行了深入分析。